| Type: | Package |
| Title: | Fit Models to Two-Way Tables with Correlated Ordered Response Categories |
| Version: | 1.0.0.3 |
| Description: | Fit a variety of models to two-way tables with ordered categories. Most of the models are appropriate to apply to tables of that have correlated ordered response categories. There is a particular interest in rater data and models for rescore tables. Some utility functions (e.g., Cohen's kappa and weighted kappa) support more general work on rater agreement. Because the names of the models are very similar, the functions that implement them are organized by last name of the primary author of the article or book that suggested the model, with the name of the function beginning with that author's name and an underscore. This may make some models more difficult to locate if one doesn't have the original sources. The vignettes and tests can help to locate models of interest. For more dertaiils see the following references: Agresti, A. (1983) <doi:10.1016/0167-7152(83)90051-2> "A Simple Diagonals-Parameter Symmetry And Quasi-Symmetry Model", Agrestim A. (1983) <doi:10.2307/2531022> "Testing Marginal Homogeneity for Ordinal Categorical Variables", Agresti, A. (1988) <doi:10.2307/2531866> "A Model For Agreement Between Ratings On An Ordinal Scale", Agresti, A. (1989) <doi:10.1016/0167-7152(89)90104-1> "An Agreement Model With Kappa As Parameter", Agresti, A. (2010 ISBN:978-0470082898) "Analysis Of Ordinal Categorical Data", Bhapkar, V. P. (1966) <doi:10.1080/01621459.1966.10502021> "A Note On The Equivalence Of Two Test Criteria For Hypotheses In Categorical Data", Bhapkar, V. P. (1979) <doi:10.2307/2530344> "On Tests Of Marginal Symmetry And Quasi-Symmetry In Two And Three-Dimensional Contingency Tables", Bowker, A. H. (1948) <doi:10.2307/2280710> "A Test For Symmetry In Contingency Tables", Clayton, D. G. (1974) <doi:10.2307/2335638> "Some Odds Ratio Statistics For The Analysis Of Ordered Categorical Data", Cliff, N. (1993) <doi:10.1037/0033-2909.114.3.494> "Dominance Statistics: Ordinal Analyses To Answer Ordinal Questions", Cliff, N. (1996 ISBN:978-0805813333) "Ordinal Methods For Behavioral Data Analysis", Goodman, L. A. (1979) <doi:10.1080/01621459.1979.10481650> "Simple Models For The Analysis Of Association In Cross-Classifications Having Ordered Categories", Goodman, L. A. (1979) <doi:10.2307/2335159> "Multiplicative Models For Square Contingency Tables With Ordered Categories", Ireland, C. T., Ku, H. H., & Kullback, S. (1969) <doi:10.2307/2286071> "Symmetry And Marginal Homogeneity Of An r × r Contingency Table", Ishi-kuntz, M. (1994 ISBN:978-0803943766) "Ordinal Log-linear Models", McCullah, P. (1977) <doi:10.2307/2345320> "A Logistic Model For Paired Comparisons With Ordered Categorical Data", McCullagh, P. (1978) <doi:10.2307/2335224> A Class Of Parametric Models For The Analysis Of Square Contingency Tables With Ordered Categories", McCullagh, P. (1980) <doi:10.1111/j.2517-6161.1980.tb01109.x> "Regression Models For Ordinal Data", Penn State: Eberly College of Science (undated) https://online.stat.psu.edu/stat504/lesson/11 "Stat 504: Analysis of Discrete Data, 11. Advanced Topics I", Schuster, C. (2001) <doi:10.3102/10769986026003331> "Kappa As A Parameter Of A Symmetry Model For Rater Agreement", Shoukri, M. M. (2004 ISBN:978-1584883210). "Measures Of Interobserver Agreement", Stuart, A. (1953) <doi:10.2307/2333101> "The Estimation Of And Comparison Of Strengths Of Association In Contingency Tables", Stuart, A. (1955) <doi:10.2307/2333387> "A Test For Homogeneity Of The Marginal Distributions In A Two-Way Classification", von Eye, A., & Mun, E. Y. (2005 ISBN:978-0805849677) "Analyzing Rater Agreement: Manifest Variable Methods". |
| License: | MIT + file LICENSE |
| Encoding: | UTF-8 |
| LazyData: | true |
| Imports: | MASS |
| RoxygenNote: | 7.3.2 |
| Suggests: | knitr, rmarkdown, testthat |
| Config/testthat/edition: | 3 |
| Depends: | R (≥ 3.5) |
| VignetteBuilder: | knitr |
| NeedsCompilation: | no |
| Packaged: | 2025-09-12 23:00:33 UTC; johndonoghue |
| Author: | John R. Donoghue [aut, cre] |
| Maintainer: | John R. Donoghue <jdonoghue0823@gmail.com> |
| Repository: | CRAN |
| Date/Publication: | 2025-09-18 08:00:02 UTC |
Solves equation Agresti_f() = 0 for delta by method of bisection..
Description
Solves equation Agresti_f() = 0 for delta by method of bisection..
Usage
Agresti_bisection(p, pi_margin, x_low = 0, x_high = 1)
Arguments
p |
matrix of observed proportions |
pi_margin |
current value of (row and column) marginal proportion |
x_low |
lower bound for search. Default value is 0.0 |
x_high |
upper bound for search. Default value is 1.0 |
Value
value of kappa that makes the function 0.0
Computes value of lambda parameter
Description
Computes value of lambda parameter
Usage
Agresti_compute_lambda(p, pi)
Arguments
p |
matrix of observed proportions |
pi |
matrix of model-supplied proportions |
Value
value of the lambda parameter
Computes the matrix pi of model-based proportions
Description
Computes the matrix pi of model-based proportions
Usage
Agresti_compute_pi(pi_margin, kappa)
Arguments
pi_margin |
current value of (row and column) marginal proportion |
kappa |
current estimate of kappa coefficient |
Value
matrix of model-based proportions
Creates the design matrix for Agresti's simple diagonal quasi-symmetry model.
Description
This parameterization does not match equation (2.2) in the paper, but it yields results that are identical to those in the paper. Agresti, A. (1983), A simple diagonals-parameter symmetry and quasi-symmetry model. Statistics and Probability Letters I, 313-316.
Usage
Agresti_create_design_matrix(n_dim)
Arguments
n_dim |
the size of the date matrix |
Value
the design matrix for the model, that can bee used with ml_for_log_linear
First equation in section 3. Solved for kappa.
Description
First equation in section 3. Solved for kappa.
Usage
Agresti_equation_1(p, pi_margin, kappa)
Arguments
p |
matrix of observed proportions |
pi_margin |
current value of (row and column) marginal proportion |
kappa |
current value of coefficient kappa |
Second equation in section 3. Solved for pi_margin.
Description
Second equation in section 3. Solved for pi_margin.
Usage
Agresti_equation_2(p, pi_margin, lambda, kappa)
Arguments
p |
matrix of observed proportions |
pi_margin |
current value of (row and column) marginal proportion |
lambda |
value of quantity lambda defined in third equation |
kappa |
current value of coefficient kappa |
Third equation in section 3. Solved for lambda
Description
Third equation in section 3. Solved for lambda
Usage
Agresti_equation_3(p, pi_margin, kappa)
Arguments
p |
matrix of observed proportions |
pi_margin |
current value of (row and column) marginal proportion |
kappa |
current valye of coefficient kappa |
Extracts the quasi-symmetry information from the result provided.
Description
Extracts the quasi-symmetry information from the result provided.
Usage
Agresti_extract_delta(result)
Arguments
result |
result of call to log_linear_fit() |
Value
list consisting of beta: the beta coefficient se: the standard error of beta z: the ratio beta / se delta: the delta coefficient = exp(2.0 * beta)
Function value for first equation in section 3.
Description
Used by Agresti_bisection()
Usage
Agresti_f(p, pi_margin, kappa)
Arguments
p |
matrix of observed proportions |
pi_margin |
current value of (row and column) marginal proportion |
kappa |
current estimate of kappa coefficient |
Fits Agresti's agreement model that includes kappa as a parameter.
Description
Agresti, A. (1989). An agreement model with kappa as a parameter. Statistics and Probability Letters, 7, 271-273.
Usage
Agresti_kappa_agreement(n, verbose = FALSE)
Arguments
n |
matrix of observed counts |
verbose |
should cycle-by-cycle info be printed as messages? The default is FALSE. |
Value
a list containing kappa: value of kappa coefficient pi_margin: value of marginal p-values. They apply to rows and columns chisq: Pearson X^2 df: degrees of freedom expected: fitted frequencies
Agresti's simple diganal quasi-symmetry model.
Description
This parameterization does not match equation (2.2) in the paper, but it yields results that are identical to those in the paper. Agresti, A. (1983), A simple diagonals-parameter symmetry and quasi-symmetry model. Statistics and Probability Letters I, 313-316.
Usage
Agresti_simple_diagonals_parameter_quasi_symmetry(n)
Arguments
n |
the matrix of observed counts |
Value
a list containing expected: matrix of expected cell frequencies, chisq: Pearson X^2 g_squared: likelihood ratio G^2 df: degrees of freedom beta: the parameter estimated sigma_beta: standard error of beta z: z-score for beta delta: transformation of the the parameter into the model formulation
Examples
Agresti_simple_diagonals_parameter_quasi_symmetry(vision_data)
Computes staring values for marginal pi.
Description
Computes staring values for marginal pi.
Usage
Agresti_starting_values(p)
Arguments
p |
matrix of observed proportions |
Value
vector containing pi
Computes the weighted statistics listed in section 2.3.
Description
Computes weighted contrast of the two margins. Agresti, A. (1983). Testing marginal homogeneity for ordinal categorical variables. Biometrics, 39(2), 505-510.
Usage
Agresti_w_diff(w, n)
Arguments
w |
a vector of weights to be treated as scores |
n |
matrix of observed counts |
Value
a list containing diff: the weighted contrast computed using weights w sigma_diff: SE(diff) z_diff: z-score for diff
Examples
weights = c(-3.0, -1.0, 1.0, 3.0)
Agresti_w_diff(weights, vision_data)
Computes weighted tau from Section 2.1. Agresti, A. (1983). Testing marginal homogeneity for ordinal categorical variables. Biometrics, 39(2), 505-510.
Description
Computes weighted tau from Section 2.1. Agresti, A. (1983). Testing marginal homogeneity for ordinal categorical variables. Biometrics, 39(2), 505-510.
Usage
Agresti_weighted_tau(n)
Arguments
n |
matrix of observed counts |
Value
a list containing tau: value of tau-d coefficient sigma_tau: SE(tau) z_tau: z-score for tau
Bhapkar's (1979) test for marginal homogeneity
Description
Fits the marginal homogeneity model using WLS.
Usage
Bhapkar_marginal_homogeneity(n)
Arguments
n |
matrix containing the table to analyze |
Details
See: Bhapkar, V. P. (1966). A Note on the Equivalence of Two Test Criteria for Hypotheses in Categorical Data. Journal of the American Statistical Association, 61(313), pp.228-235.
Value
a list containing the chi-square statistic, the df and p-value.
Examples
Bhapkar_marginal_homogeneity(vision_data)
Bhapkar's 1979 test for quasi-symmetry.
Description
Fits the quasi-symmetry model using WLS. Bhapkar, V. P. (1979). On tests of marginal symmetry and quasi-symmetry in two and three-dimensional contingency tables. Biometrics 35(2), 417-426.
Usage
Bhapkar_quasi_symmetry(n)
Arguments
n |
the matrix to be analyzed |
Value
a list containing the chi-square and df.
Examples
Bhapkar_quasi_symmetry(vision_data)
Computes Bowker's test of symmetry.
Description
Computes the test of table symmetry in Bowker (1948). Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association 43, 572-574.
Usage
Bowker_symmetry(n)
Arguments
n |
the matrix to be tested for symmetry |
Value
a list containing the chi-square: Pearson X^2 g_square: likelihood ratio G^2 df: degrees of freedom p-value: p-value for Pearson X^2 expected: fitted values
Examples
Bowker_symmetry(vision_data)
Fits the tests comparing locations of the margins of a two-way table.
Description
The measure is based on the weighted cdfs. No "scores" are used, just the weighted (cumulative sums). Clayton, D. G. (1974) Odds ratio statistics for the analysis of ordered categorical data. Biometrika, 61(3), 525-531.
Usage
Clayton_marginal_location(wx, wy)
Arguments
wx |
vector containing frequencies for the first margin of the table |
wy |
vector containing frequencies for the second margin of the table |
Value
a list of results odds_ratios: odds ratios comparing cumulative frequencies of adjacent categories log_theta_hat: log of estimate of the common odds-ratio theta_hat: estimate of the common odds-ratio log_mh_theta_hat: log of the Mantel-Haenssel type odds-ratio mh_theta_hat: Mantel-Haenszel type odds-ratio var_log_theta_hat = variance of the log of the odds-ratios chisq_theta_hat: chi-square for odds-ratio chisq_mh_theta_hat: chi-square for Mantel-Haenszel odds-ratio df: degrees of freedom for chis-square = 1
Examples
Clayton_marginal_location(tonsils[1,], tonsils[2,])
Clayton's stratified version of the marginal location comparison.
Description
Compares marginal location conditional on a stratifying variable. Clayton, D. G. (1974) Odds ratio statistics for the analysis of ordered categorical data. Biometrika, 61(3), 525-531.
Usage
Clayton_stratified_marginal_location(mx, my)
Arguments
mx |
matrix with |
my |
matrix with |
Value
a list of results odds_ratios: odds ratios comparing cumulative frequencies of adjacent categories log_theta_hat: log of estimate of the common odds-ratio theta_hat: estimate of the common odds-ratio log_mh_theta_hat: log of the Mantel-Haenssel type odds-ratio mh_theta_hat: Mantel-Haenszel type odds-ratio var_log_theta_hat = variance of the log of the odds-ratios chisq_theta_hat: chi-square for odds-ratio chisq_mh_theta_hat: chi-square for Mantel-Haenszel odds-ratio df: degrees of freedom for chis-square = 1
See Also
[Clayton_marginal_location()]
Computes summary, cumulative proportions up to index provided
Description
Computes summary, cumulative proportions up to index provided
Usage
Clayton_summarize(weights, m)
Arguments
weights |
matrix of counts |
m |
index of summation, weights[1:m] |
Value
a list containing: n: the sum of the weights p: matrix of proportion values gamma: cumulative proportions 1:m
Analysis stratified by column variable j.
Description
Analysis stratified by column variable j.
Usage
Clayton_summarize_stratified(weight_matrix, m)
Arguments
weight_matrix |
matrix of cell weights from the table |
m |
the column index to stratify on |
Value
a list containing: n: the number of strata p: matrix of proportion values gamma: cumulative proportions
See Also
[Clayton_summarize()]
Clayton's stratified measure of association
Description
Quantifies association between two ordinal variables. Clayton, D. G. (1974) Odds ratio statistics for the analysis of oordered categorical data. Biometrika, 61(3), 525-531.
Usage
Clayton_two_way_association(f)
Arguments
f |
matrix of frequencies |
Value
a list of results log_theta_hat: log odds-ratio measure of association theta_hat: odds-ratio measure of association log_mh_theta_hat: log of Mantel-Haenszel odds-ratio measure of association mh_theta_hat: Mantel-Haenszel odds-ratio measure of association var_log_theta_hat: variance of the log odds-ration measures chisq_theta_hat: chi-square for measure of association chisq_mh_theta_hat: chi-square for Mantel-Haenszel measure of association df: degress of freedom = 1, corr_theta_hat: theta-hat association converted to correlation metric corr_mh_theta_hat: Mantel-Haenszel theta-hat converted to correlation metric
Converts two vectors containing scores and integer frequencies (cell counts) into a d-matrix
Description
Converts two vectors containing scores and integer frequencies (cell counts) into a d-matrix
Usage
Cliff_as_d_matrix(scores, cells, nrow = NULL)
Arguments
scores |
vector of scores, typically 1:r |
cells |
vector of integer weights, i.e. cell frequencies |
nrow |
number of score categories in table. Default is NULL. If NULL, takes 1:length(scores) |
Value
d-matrix of results
Computes between groups dominance matrix "d".
Description
Computes between groups dominance matrix "d".
Usage
Cliff_compute_d(x, y)
Arguments
x |
first vector of scores |
y |
second vector of scores |
Value
N X N dominance matrix
Generates counts from table frequencies for 2 category items
Description
Generates counts from table frequencies for 2 category items
Usage
Cliff_counts_2(mij)
Arguments
mij |
Matrix of counts. |
Value
a list containing wm1m1: for -1, -1 wm10: for -1, 0 wm11: for -1, 1 w00: for 0, 0 w01: for 0, 1 w11: for 1, 1
Generates counts from table frequencies for 3 category items
Description
Generates counts from table frequencies for 3 category items
Usage
Cliff_counts_3(mij)
Arguments
mij |
Matrix of counts. |
Value
a list containing wm1m1: for -1, -1 wm10: for -1, 0 wm11: for -1, 1 w00: for 0, 0 w01: for 0, 1 w11: for 1, 1
Generates counts from table frequencies for 4 category items
Description
Generates counts from table frequencies for 4 category items
Usage
Cliff_counts_4(mij)
Arguments
mij |
Matrix of counts. |
Value
a list containing wm1m1: for -1, -1 wm10: for -1, 0 wm11: for -1, 1 w00: for 0, 0 w01: for 0, 1 w11: for 1, 1
Generates counts from table frequencies for 5 category items
Description
Generates counts from table frequencies for 5 category items
Usage
Cliff_counts_5(mij)
Arguments
mij |
Matrix of counts. |
Value
a list containing wm1m1: for -1, -1 wm10: for -1, 0 wm11: for -1, 1 w00: for 0, 0 w01: for 0, 1 w11: for 1, 1
Generates counts from table frequencies for 6 category items
Description
Generates counts from table frequencies for 6 category items
Usage
Cliff_counts_6(mij)
Arguments
mij |
Matrix of counts. |
Value
a list containing wm1m1: for -1, -1 wm10: for -1, 0 wm11: for -1, 1 w00: for 0, 0 w01: for 0, 1 w11: for 1, 1
Computes Cliff's dependent d-statistics based on a dominance matrix.
Description
Takes the dominance matrix provided and computes the d-statistics: dw - within-subjects d-statistic db - between-subjects d-statistic db_dw - sum of dw and db, omnibus test of whether one group is higher than the other Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological Bulletin, 114(3), 494-509. Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mawhaw NJ: Lawerence Erlbaum.
Usage
Cliff_dependent(d_matrix)
Arguments
d_matrix |
N x N within-subjects dominance matrix |
Value
a list containing dw: within-subjects d-statistic sigma_dw: SE of dw z_dw: z-score for dw db: between-subjects d-statistic sigma_db: SE of db z_db: z-score for db db_dw: sum db + dw, omnibus measure sigma_db_dw: SE of db + dw z_db_dw: z-score of db _ dw cov_db_dw: covariance between db and dw
Examples
Cliff_dependent(interference_control_1)
Computes sum term in covariance db-dw for weighted dominance matrix.
Description
Computes sum term in covariance db-dw for weighted dominance matrix.
Usage
Cliff_dependent_compute_cov(wd)
Arguments
wd |
weighted dominance matrix |
Compute the sum in the covariance of db+dw
Description
Compute the sum in the covariance of db+dw
Usage
Cliff_dependent_compute_cov_from_d(d_matrix)
Arguments
d_matrix |
d-matrix of dominances |
Value
the sum for the covariance term
Computes Cliff's dependent d-statistics based on a dominance matrix.
Description
Takes the dominance matrix provided and computes the d-statistics: dw - within-subjects d-statistic db - between-subjects d-statistic db_dw - sum of db and dw, omnibus test of whether one group is higher than the other Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological Bulletin, 114(3), 494-509. Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mawhaw NJ: Lawerence-Erlbaum.
Usage
Cliff_dependent_compute_from_matrix(d_matrix)
Arguments
d_matrix |
N x N within-subjects dominance matrix |
Value
a list containing dw: within-subjects d-statistic sigma_dw: SE of dw z_dw: z-score for dw db: between-susbjects d-statistic sigma_db: SE of db z_db: z-score for db db_dw: sum db + dw, omnibus measure sigma_db_dw: SE of db + dw z_db_dw: z-score of db _ dw cov_db_dw: covariance between db and dw
Examples
Cliff_dependent_compute_from_matrix(interference_control_1)
Computes Cliff's dependent d-statistics based on a table of frequency counts.
Description
Takes the r X r table and returns: dw - within-subjects d-statistic db - between-subjects d-statistic db_dw - sum of dw and db, omnibus test of whether one group is higher than the other No intermediate dominance matrix is computed, so this is much faster than Cliff_dependent_compute_from_matrix(). Large number of terms are needed to compute intermediate d_ij_ji. These are contained in separate functions for r <= 6. Results for r [7, 10] are available, but the files are so large that they cause an error if included in the library.
Usage
Cliff_dependent_compute_from_table(mij)
Arguments
mij |
an r x r table of paired observations |
Details
See: Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological Bulletin, 114(3), 494-509. Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mawhaw NJ: Lawerence-Erlbaum.
Value
a list containing dw: within-subjects d-statistic sigma_dw: SE of dw z_dw: z-score for dw db: between-susbjects d-statistic sigma_db: SE of db z_db: z-score for db db_dw: sum db + dw, omnibus measure sigma_db_dw: SE of db + dw z_db_dw: z-score of db _ dw cov_db_dw: covariance between db and dw
See Also
[Cliff_dependent_compute_paired_d()]
Examples
Cliff_dependent_compute_from_table(movies)
Computes Cliff's dependent d-statistics based on cell frequencies.
Description
Computes d-matrix and then analyzes it. This can be time consuming. Try Cliff_dependent_from_table() instead. The current function is provided mainly for comparison & validation. For an example, compare running this function on vision_data to running Cliff_dependent_from_table(vision_data).
Usage
Cliff_dependent_compute_paired_d(cells)
Arguments
cells |
r x r matrix of frequencies |
Details
dw - within-subjects d-statistic db - between-subjects d-statistic db_dw - sum of dw and db, omnibus test of whether one group is higher than the other Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological Bulletin, 114(3), 494-509. Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mawhaw NJ: Lawerence-Erlbaum.
Value
a list containing dw: within-subjects d-statistic sigma_dw: SE of dw z_dw: z-score for dw db: between-subjects d-statistic sigma_db: SE of db z_db: z-score for db db_dw: sum db + dw, omnibus measure sigma_db_dw: SE of db + dw z_db_dw: z-score of db _ dw cov_db_dw: covariance between db and dw
See Also
[Cliff_dependent_compute_from_table()]
Examples
Cliff_dependent_compute_paired_d(movies)
Computes the independent groups d-statistic comparing the two vectors provided.
Description
Computes the independent groups d-statistic comparing the two vectors provided.
Usage
Cliff_independent(x, y)
Arguments
x |
vector of scores for first group |
y |
vector of scores for second group |
Value
list containing d, SE(d) and z(d)
Computes d-statistic from dominance matrix provided.
Description
Computes d-statistic from dominance matrix provided.
Usage
Cliff_independent_from_matrix(d)
Arguments
d |
N X M dominance matrix |
Value
list containing d, SE(d) and z(d)
Computes independent group's d-statistic from the matrix of frequencies provided.
Description
Computes intermediate d-matrix, so can be slow for large N
Usage
Cliff_independent_from_table(n)
Arguments
n |
matrix of counts |
Value
list containing d, SE(d) and z(d)
Computes d-statistic based on scores and integer weights(frequencies) for each group.
Description
Computes d-statistic based on scores and integer weights(frequencies) for each group.
Usage
Cliff_independent_weighted(x, w_x, y, w_y)
Arguments
x |
first vector of scores |
w_x |
weights associated with first vector of scores |
y |
second vector of scores |
w_y |
weights associated with second vector of scores |
Value
list containing d, SE(d) and z(d)
Computes weighted version of dominance matrix "d"
Description
Arguments are scores and associated weights. Not useful for tables. Use Cliff_compute_d_matrix instead.
Usage
Cliff_weighted_d_matrix(x, y, w.x = rep(1, length(x)), w.y = rep(1, length(y)))
Arguments
x |
first vector of scores |
y |
second vector of scores |
w.x |
first vector of weights, to apply to x. Defaults to vector of 1.0 |
w.y |
second vector of weights, to apply to y. Defaults to vector of 1.0 |
Value
an n X m d-matrix, where n is length(x) and m is length(y)
Fits the model where some of the delta parameters are constrained to be equal to one another.
Description
Fits the model where some of the delta parameters are constrained to be equal to one another.
Usage
Goodman_constrained_diagonals_parameter_symmetry(n, equality)
Arguments
n |
the matrix of observed counts |
equality |
logical vector indicating whether corresponding delta the parameter is part of the equality set. |
Value
a list containing pooled_chisq: Pearson chi-square for the pooled delta values pooled_df: degrees of freedom for pooled chisq omnibus_chisq: Pearson chi-square for overall model fit, subject to equality constraints omnibus_df; degrees of freedom for omnibus_chisq equality_chisq: Pearson chi-square for test that remaining deltas are all equal equality_df: degrees of freedom for equality_chisq delta_pooled: estimate of pooled delta
Examples
equality = c(TRUE, TRUE, FALSE)
Goodman_diagonals_parameter_symmetry(vision_data)
Fit's Goodman's diagonals parameter symmetry model.
Description
Goodman, L. A. (1979). Multiplicative models for square contingency tables with ordered categories. Biometrika, 66(3), 413-316.
Usage
Goodman_diagonals_parameter_symmetry(n)
Arguments
n |
the matrix of obsever counts |
Value
a list containing individual_chisq: chi-square value for each diagonal individual_df: degrees of freedom for individual_chisq omnibus_chisq: overall chi-square for the model omnibus_df: degrees for freedom for omnibus_chisq equality_chisq: chi-square for test that all delta values are equal equality_df: degrees of freedom from equality_chisq delta: the vector of estimated delta values (without any equality constraints)
Examples
Goodman_diagonals_parameter_symmetry(vision_data)
Fits the model with given parameters fixed to specific values.
Description
The model has simple closed form solutions when fitting either the unconstrained version of the version that species equality of delta parameters. However, I could not see how to adapt that to the case where specific parameters were constrained to have a specific value. This routine is to fit that model. It will also fit the unconstrained model, but Goodman gives the estimator for that case.
Usage
Goodman_fixed_parameter(
n,
delta,
fixed,
convergence = 1e-04,
max_iter = 50,
verbose = FALSE
)
Arguments
n |
the r X r matrix of observed counts |
delta |
the vector of asymmetry r - 1 parameters |
fixed |
r - 1 logical vector that specifies whether a delta parameter is fixed (TRUE) or allowed to be estimated (FALSE). |
convergence |
maximum change in a parameter across iterations. Default is 1.0e-4 |
max_iter |
maximum number of iterations, Default is 50. |
verbose |
should progress information be printed to the console. Default is FALSE, do not print. |
Value
list containing phi, delta, max_change largest change in parameter for last the iteration, chisq: Pearson chi-square g_squared: likelihood ratio G^2 df: degrees of freedom
See Also
[Goodman_diagonals_parameter_symmetry()]
[Goodman_ml()]
Examples
fixed <- c(FALSE, TRUE, FALSE)
delta <- c(1.0, 1.0, 1.0)
phi <- matrix(0.0, nrow=4, ncol=4)
diag(phi) = rep(1.0, 4)
Goodman_fixed_parameter(vision_data, delta, fixed)
Performs ML estimation of the model.
Description
The model has simple closed form solutions when fitting either the unconstrained version of the version that species equality of delta parameters. However, I could not see how to adapt that to the case where specific parameters were constrained to have a specific value. This routine is to fit that model. It will also fit the unconstrained model, but Goodman gives the estimator for that case.
Usage
Goodman_ml(n, phi, delta, fixed)
Arguments
n |
the r X r matrix of observed counts |
phi |
the symmetric matrix parameter |
delta |
the vector of asymmetry r - 1 parameters |
fixed |
r - 1 logical vector that specifies whether a delta parameter is fixed (TRUE) or allowed to be estimated (FALSE). |
Value
list containing new estimates of phi amd delta
See Also
[Goodman_diagonals_parameter_symmetry()]
Examples
fixed <- c(FALSE, TRUE, FALSE)
delta <- c(1.0, 1.0, 1.0)
phi <- matrix(0.0, nrow=4, ncol=4)
for (i in 1:4) {
phi[i, i] = 1.0
}
Goodman_ml(vision_data, phi, delta, fixed)
Fits Goodman's (1979) Model I
Description
Fits Goodman's (1979) Model I
Usage
Goodman_model_i(
n,
row_effects = TRUE,
column_effects = TRUE,
max_iter = 25,
verbose = FALSE,
exclude_diagonal = FALSE
)
Arguments
n |
matrix of observed counts |
row_effects |
should row effects be included in the model? Default is TRUE |
column_effects |
should column effects be included in the model? Default is TRUE |
max_iter |
maximum number of iterations. Default is 10 |
verbose |
logical. Should cycle-by-cycle output be printed? Default is no |
exclude_diagonal |
logical. For square tables, should the cells on the diagonal be excluded? Default is FALSE, include all cells |
Value
a list containing alpha: row effects beta: column effects gamma: row location weights delta: column location weights log_likelihood: log(likelihood) g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom
Fits Goodman's (1979) Model I*
Description
Fits Goodman's (1979) Model I*
Usage
Goodman_model_i_star(
n,
max_iter = 25,
verbose = FALSE,
exclude_diagonal = FALSE
)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations |
verbose |
should cycle-by-cycle information be printed out? Default is FALSE, do not print |
exclude_diagonal |
should the cells along the main diagonal be excluded? Default is FALSE, include all cells |
Value
a list containing alpha: vector of row parameters beta: vector of column parameters theta: vector of common row/column estimates log_likelihood: log(likelihood) at completion g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom
Fits Goodman's (1979) Model II
Description
Fits Goodman's (1979) Model II
Usage
Goodman_model_ii(
n,
rho = 1:nrow(n) - (nrow(n) + 1)/2,
sigma = 1:ncol(n) - (ncol(n) + 1)/2,
update_rows = TRUE,
update_columns = TRUE,
max_iter = 25,
verbose = FALSE,
exclude_diagonal = FALSE
)
Arguments
n |
matrix of observed counts |
rho |
values of row locations. Default is 1:nrow(n) - (nrow(n) + 1) / 2 |
sigma |
values of column locations. Default is 1:ncol(n) - (ncol(n) + 1) / 2 |
update_rows |
should values of row locations be updated? Default is TRUE, update |
update_columns |
should value of column locations be updated? Default is TRUE, update |
max_iter |
maximum number of iterations to perform. Default is 10 |
verbose |
should cycle-by-cycle output be produced? Default is FALSE |
exclude_diagonal |
logical. Should the diagonal be excluded from the computation. Default is FALSE. |
Value
a list containing alpha: row effects beta: column effects rho: centered row locations mu: row locations sigma: centered column locations nu: column locations log_likelihood: log(likelihood) g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom
Fits Goodman's (1979) model II*, where row and column effects are equal.
Description
Fits Goodman's (1979) model II*, where row and column effects are equal.
Usage
Goodman_model_ii_star(
n,
exclude_diagonal = FALSE,
max_iter = 25,
verbose = FALSE
)
Arguments
n |
matrix of observed counts |
exclude_diagonal |
should the cells of the main diagonal be excluded? Default is FALSE, include all cells |
max_iter |
maximum number of iterations |
verbose |
should cycle-by-cycle information be printed out? Default is FALSE, do not print |
Value
a list containing alpha: vector of alpha (row) parameters beta: vector of beta (column) parameters phi: vector of common row/column effects log_likelihood: value of the log(likelihood) function at completion g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom
Fits Goodman's L. A. (1979) Simple Models for the Analysis of Association in Cross-Classifications Having Ordered Categories
Description
null association model
Usage
Goodman_null_association(
n,
max_iter = 25,
verbose = FALSE,
exclude_diagonal = FALSE
)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations. Default is 10 |
verbose |
should cycle-by-cycle info be printed? Default is FALSE |
exclude_diagonal |
logical, Should the diagonal be excluded from the computations. Default is FALSE |
Value
a list containing alpha: row effects beta: column effects log_likelihood: log(likelihood) g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom
Computes the model-based probability for cell i, j
Description
Computes the model-based probability for cell i, j
Usage
Goodman_pi(phi, delta, i, j)
Arguments
phi |
symmetry matrix |
delta |
vector of asymmetry parameters |
i |
row index |
j |
column index |
Value
pi for that cell
Computes the full matrix of model-based cell probabilities.
Description
Computes the full matrix of model-based cell probabilities.
Usage
Goodman_pi_matrix(phi, delta)
Arguments
phi |
the symmetric matrix |
delta |
the vector of asymmetry parameters |
Value
matrix of model-based probabilities
Fits the symmetric association model from Goodman (1979). Note the model is a reparameterized version of the quasi-symmetry model, so the quasi-symmetry model has the same fit indices.
Description
Fits the symmetric association model from Goodman (1979). Note the model is a reparameterized version of the quasi-symmetry model, so the quasi-symmetry model has the same fit indices.
Usage
Goodman_symmetric_association_model(n)
Arguments
n |
matrix of observed counts |
Value
a list containing x: design matrix used for the glm() regression beta: parameter estimates se: standard errors of beta g_squared: G^2 measure of fit chisq: X^2 measure of fit df: degrees of freedom expected: model-based expected cell counts
Fits Goodman's (1979) uniform association model
Description
Fits Goodman's (1979) uniform association model
Usage
Goodman_uniform_association(
n,
max_iter = 25,
verbose = FALSE,
exclude_diagonal = FALSE
)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations. Default is 10. |
verbose |
should cycle-by-cycle info be printed out? Default is FALSE |
exclude_diagonal |
logical. Should the cells of the main diagonal be excluded from the computations? Default is FALSE, include all cells. |
Value
a list containing alpha: row effects beta: column effects theta: uniform association parameter log_likelihood: log(likelihood) g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom
Fits marginal homogeneity model
Description
Fits the marginal homogeneity model according to the minimum discriminant information. Ireland, C. T., Ku, H. H., & Kullback, S. (1969). Symmetry and marginal homogeneity of an r × r contingency table. Journal of the American Statistical Association, 64(328), 1323-1341.
Usage
Ireland_marginal_homogeneity(
n,
truncated = FALSE,
max_iter = 15,
verbose = FALSE
)
Arguments
n |
matrix of observed counts |
truncated |
should the diagonal be excluded. Default is FALSE, include the diagonal. |
max_iter |
maximum number of iterations to perform |
verbose |
should cycle-by-cycle information be printed out. Default is FALSE. |
Value
a list containing mdis: value of the minimum discriminant information statistic (appox chi-squared) df: dgrees of freedom x_star: matrix of model-based counts p_star: matrix of model-based p-values
Examples
Ireland_marginal_homogeneity(vision_data)
Computes the MDIS between the two matrices provided.
Description
Computes the MDIS between the two matrices provided.
Usage
Ireland_mdis(n, x_star, truncated = FALSE)
Arguments
n |
first matrix (usually observed counts) |
x_star |
second matrix (usually model-based) |
truncated |
should the diagonal be ignored. Default is FALSE, include the diagonal elements. |
Value
value of the MDIS criterion
Renormalize counts to account for truncation of diagonal
Description
Renormalize counts to account for truncation of diagonal
Usage
Ireland_normalize_for_truncation(n)
Arguments
n |
matrix of observed counts |
Value
matrix n with diagonal set to 0.0
Fit for quasi-symmetry model. Obtained by subtraction, so no model-based probabilities.
Description
Fit for quasi-symmetry model. Obtained by subtraction, so no model-based probabilities.
Usage
Ireland_quasi_symmetry(n, truncated = FALSE)
Arguments
n |
matrix of observed counts |
truncated |
should the diagonal be excluded, Default is FALSE, include the diagonal. |
Value
a list with mdis = MDIS value and df = degrees of freedom for quasi-symmetry model
See Also
[Ireland_quasi_symmetry_model()]
Examples
Ireland_quasi_symmetry(vision_data)
Fitss the quasi-symmetry model.
Description
Fits the model according to the MDIS criterion.
Usage
Ireland_quasi_symmetry_model(
n,
truncated = FALSE,
max_iter = 5,
verbose = FALSE
)
Arguments
n |
matrix of observed counts |
truncated |
should the diagonal be excluded. Default is FALSE, include diagonal cells. |
max_iter |
maximum number of iterations in minimizing the criterion. Default is 4 |
verbose |
logical variable, should cycle-by-cycle info be printed. Defaullt is FALSE. |
Value
a list containing mdis: value of the MDIS at termination df: degrees of freedom x_star: matrix of model-reproduced counts p_star: matrix of model-reproduced p-values
See Also
[Ireland_quasi_symmetry()]
Examples
Ireland_quasi_symmetry_model(vision_data)
Fits symmetry model.
Description
Ireland, C. T., Ku, H. H., & Kullback, S. (1969). Symmetry and marginal homogeneity of an r × r contingency table. Journal of the American Statistical Association, 64(328), 1323-1341.
Usage
Ireland_symmetry(n, truncated = FALSE)
Arguments
n |
matrix of observed counts |
truncated |
should the diagonal be excluded. Default is FALSE, include the diagonal. |
Value
a list containing mdis: value of the minimum discriminant information statistic (appox chi-squared) df: dgrees of freedom x_star: matrix of model-based counts p_star: matrix of model-based p-values
Examples
Ireland_symmetry(vision_data)
Compute the observed sums Nij
Description
Compute the observed sums Nij
Usage
McCullagh_compute_Nij(n)
Arguments
n |
the matrix of observed counts |
Value
a list containing Pij and Qij
Computes sums c+ used in maximizing the log(likelihod)
Description
Computes sums c+ used in maximizing the log(likelihod)
Usage
McCullagh_compute_c_plus(phi, alpha)
Arguments
phi |
matrix of symmetry parameters |
alpha |
vector of asymmetry parameters |
Value
list of c_i_plus and c_plus_i
Compute the linear constraint on psi elements for identifiablity.
Description
Compute the linear constraint on psi elements for identifiablity.
Usage
McCullagh_compute_condition(psi)
Arguments
psi |
symmetry matrix |
Value
value of the constraint
Computes cumulative sums for rows,
Description
Computes cumulative sums for rows,
Usage
McCullagh_compute_cumulative_sums(n)
Arguments
n |
matrix of observed counts |
Value
R where R[i, ] contains cumulative sum of n[i,]
Computes the model-based cumulative probability matrices pij and qij
Description
Computes the model-based cumulative probability matrices pij and qij
Usage
McCullagh_compute_cumulatives(psi, delta, alpha, c = 1)
Arguments
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
list containing matrices pij and qij
Computes the degrees of freedom for the model
Description
Computes the degrees of freedom for the model
Usage
McCullagh_compute_df(M, generalized = FALSE)
Arguments
M |
the size of the M X M observed matrix |
generalized |
is the generalized model being fit? Default is FALSE, regular model |
Computes gamma from x and beta
Description
Computes gamma from x and beta
Usage
McCullagh_compute_gamma(x, beta, s, c)
Arguments
x |
predictor variables |
beta |
vector of regression coefficients |
s |
number of rows in the table |
c |
number of score levels in table |
Value
vector of model-based gamma coefficients
Computes value of gamma from phi. Inverse of usual computation.
Description
Computes value of gamma from phi. Inverse of usual computation.
Usage
McCullagh_compute_gamma_from_phi(phi, j, gamma)
Arguments
phi |
value to compute from |
j |
index to use in computation |
gamma |
vector of gamma values (model-based cumulative logits) |
Value
gamma[j] given phi and gamma[j + 1]
Computes value of gamma[j + 1] from phi.
Description
Computes value of gamma[j + 1] from phi.
Usage
McCullagh_compute_gamma_plus_1_from_phi(phi, j, gamma)
Arguments
phi |
value used in computation |
j |
index to use in computation |
gamma |
vector of gamma values (model-based cumulative logits) |
Value
gamma[j + 1] given phi and gamma[j]
Coompute the model-based cumulative probabilities pij and qij.
Description
Coompute the model-based cumulative probabilities pij and qij.
Usage
McCullagh_compute_generalized_cumulatives(psi, delta_vec, alpha, c = 1)
Arguments
psi |
symmetry matrix |
delta_vec |
vector of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
normalizing constant so pis sum to 1. Defaults to 1.0 |
Value
matrices of model-based cumulative probabilities pij and qij
Cpompute matrix pi under generalized model.
Description
Cpompute matrix pi under generalized model.
Usage
McCullagh_compute_generalized_pi(psi, delta_vec, alpha, c = 1)
Arguments
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
the matrix pi
Computes lambda, log of cumulative odds.
Description
Computes lambda, log of cumulative odds.
Usage
McCullagh_compute_lambda(n, use_half = TRUE)
Arguments
n |
matrix of observed counts |
use_half |
logical whether of not to add half to the cell count before taking the logit. Default value is TRUE. |
Computes the log(likelihood) for the general nonlinear model.
Description
Computes the log(likelihood) for the general nonlinear model.
Usage
McCullagh_compute_log_l(n, phi)
Arguments
n |
matrix of observed counts |
phi |
vector of model-based parameters |
Value
log(likelihood)
Compute the value of the Lagrange multiplier for the constraint on psi.
Description
Compute the value of the Lagrange multiplier for the constraint on psi.
Usage
McCullagh_compute_omega(n, pi)
Arguments
n |
matrix of observed counts |
pi |
matrix of model-based probabilities pi. |
Value
the value of the Lagrange multiplier.
Computes phi based on gamma
Description
Computes phi based on gamma
Usage
McCullagh_compute_phi(gamma, j)
Arguments
gamma |
vector of gamma parameters |
j |
index of phi to compute |
Value
phi[j]
Compute matrix of model-based logits
Description
Compute matrix of model-based logits
Usage
McCullagh_compute_phi_matrix(gamma)
Arguments
gamma |
matrix of model-based cumulative odds |
Value
matrix of model-based logits
Compute the regular (non-cumulative) model-based pi values
Description
Compute the regular (non-cumulative) model-based pi values
Usage
McCullagh_compute_pi(psi, delta, alpha, c)
Arguments
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
the matrix pi
Computes matrix of p-values pi based on x and current value of beta.
Description
Computes matrix of p-values pi based on x and current value of beta.
Usage
McCullagh_compute_pi_from_beta(n, x, beta)
Arguments
n |
matrix of observed counts |
x |
design matrix |
beta |
current values of location model regression parameters |
Value
matrix of model-based pi values
Compute the cell probabilities pi from gamma.
Description
Compute the cell probabilities pi from gamma.
Usage
McCullagh_compute_pi_from_gamma(gamma)
Arguments
gamma |
matrix of gamma values |
Value
c X c matrix of p-values pi
Computes regression weights w; R_dot_j * (N - R_dot_j[j]) * (n_do_j[j] a= na_dot_j[j+ 1] )
Description
Computes regression weights w; R_dot_j * (N - R_dot_j[j]) * (n_do_j[j] a= na_dot_j[j+ 1] )
Usage
McCullagh_compute_regression_weights(n)
Arguments
n |
matrix of observed counts |
Value
list of w, and sum(w)
Compute sums too use in maximizing log(likelihood)
Description
Compute sums too use in maximizing log(likelihood)
Usage
McCullagh_compute_s_plus(n)
Arguments
n |
matrix of observed counts |
Value
list of s_i_plus and s_plus_i
Compute the Newton-Raphson update.
Description
Compute the Newton-Raphson update.
Usage
McCullagh_compute_update(gradient, hessian)
Arguments
gradient |
gradient vector of log(likelihood) wrt parameters |
hessian |
hessian of log(likelihood) wrt parameters |
Value
vector with update values for each of the parameters
Computes Z, where z is w * lambda.
Description
Computes Z, where z is w * lambda.
Usage
McCullagh_compute_z(lambda, w)
Arguments
lambda |
cumulative logits |
w |
weights to apply to the logits |
Value
z, sum pf product of lambda
Fits the McCullagh (1978) conditional-symmetry model.
Description
McCullagh, P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65(2) 413-418.
Usage
McCullagh_conditional_symmetry(n, max_iter = 5, verbose = FALSE)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations to maximize the log(likelihood) |
verbose |
should cycle-by-cycle info be printed. Default is FALSE. |
Value
a list containing theta: the asymmetry parameter chisq: chi-square g_squared: likelihood ratio G^2 df: degrees of freedom
Examples
McCullagh_conditional_symmetry(vision_data)
Computes sums used in maximizing theta.
Description
Computes sums used in maximizing theta.
Usage
McCullagh_conditional_symmetry_compute_s(n)
Arguments
n |
matrix of observed counts |
Value
list with s_i_plus and s_plus-i
Initializes symmetry matrix phi
Description
Initializes symmetry matrix phi
Usage
McCullagh_conditional_symmetry_initialize_phi(M)
Arguments
M |
the number of rows/columns in phi |
Value
the phi matrix
Maximizes log(likelihood) wrt phi.
Description
Maximizes log(likelihood) wrt phi.
Usage
McCullagh_conditional_symmetry_maximize_phi(n)
Arguments
n |
matrix of observed counts |
Value
phi matrix
Maximizes the log(likelihood) wrt theta.
Description
Maximizes the log(likelihood) wrt theta.
Usage
McCullagh_conditional_symmetry_maximize_theta(n)
Arguments
n |
matrix of observed counts |
Value
value of asymmetry parameter theta
Computes model-based proportions.
Description
Computes model-based proportions.
Usage
McCullagh_conditional_symmetry_pi(phi, theta)
Arguments
phi |
the symmetric matrix |
theta |
the asymmetry parameter |
Value
matrix of model-based p-values
Derivative of the condition wrt psi[i, j].
Description
Derivative of the condition wrt psi[i, j].
Usage
McCullagh_derivative_condition_wrt_psi(i, j)
Arguments
i |
first index of psi |
j |
second index of psi |
Value
derivative
Derivative of gamma j + 1 wrt phi.
Description
Derivative of gamma j + 1 wrt phi.
Usage
McCullagh_derivative_gamma_plus_1_wrt_phi(gamma, j, phi)
Arguments
gamma |
vector |
j |
index of gamma to take derivative of |
phi |
scalar phi taking derivative wrt |
Value
derivative
Derivative of gamma wrt phi.
Description
Version given in McCullagh isn't right.
Usage
McCullagh_derivative_gamma_wrt_phi(gamma, j, phi)
Arguments
gamma |
vector of cumulative logits |
j |
index of derivative sought |
phi |
scalar phi taking derivative wrt |
Value
derivative
Derivative of y wrt gamma.
Description
Assumes a logit link is being used.
Usage
McCullagh_derivative_gamma_wrt_y(gamma, i, j)
Arguments
gamma |
matrix of gamma values |
i |
row index of gamma |
j |
column index of gamma |
Value
derivative
Derivative of Lagrange multiplier wrt scalar delta.
Description
Derivative of Lagrange multiplier wrt scalar delta.
Usage
McCullagh_derivative_lagrangian_wrt_delta(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
symmetry matrix |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing coefficient so that sum o pi = 1. Default value is 1.0 |
Value
value of the derivative
Derivative of Lagrangian wrt delta_vec.
Description
Derivative of Lagrangian wrt delta_vec.
Usage
McCullagh_derivative_lagrangian_wrt_delta_vec(
n,
k,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec to compute derivative wrt |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of Lagrangian wrt psi[i1, j1].
Description
Derivative of Lagrangian wrt psi[i1, j1].
Usage
McCullagh_derivative_lagrangian_wrt_psi(n, i1, j1, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
i1 |
first index of psi |
j1 |
first index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of log(likelihood) wrt alpha[index].
Description
Derivative of log(likelihood) wrt alpha[index].
Usage
McCullagh_derivative_log_l_wrt_alpha(n, index, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of log(likelihood) wrt beta, as given in appendix of McCullagh.
Description
McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Stastical Society, Series B, 42(2), 109-142. With assist from appendix of Agresti, (1984). Agresti, A. (1984). Analysis of ordinal categorical data. New York, Wiley, p. 244-246.
Usage
McCullagh_derivative_log_l_wrt_beta(n, x, gamma)
Arguments
n |
matrix of observed counts |
x |
design matrix for location |
gamma |
matrix of model-based cumulative logits |
Value
derivative
Derivative of log(likelihood) wrt c.
Description
Derivative of log(likelihood) wrt c.
Usage
McCullagh_derivative_log_l_wrt_c(n, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of log(likelihood) wrt delta (scalar or vector0.
Description
Derivative of log(likelihood) wrt delta (scalar or vector0.
Usage
McCullagh_derivative_log_l_wrt_delta(n, psi, delta, alpha, c = 1, k = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
k |
index into delta_vac. Defaults to 1. |
Value
derivative
Derivative of log(likelihood) wrt delta_vec[k].
Description
Derivative of log(likelihood) wrt delta_vec[k].
Usage
McCullagh_derivative_log_l_wrt_delta_vec(n, k, psi, delta_vec, alpha, c = 1)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of log(likelihood) wrt parameters.
Description
Derivative of log(likelihood) wrt parameters.
Usage
McCullagh_derivative_log_l_wrt_params(n, x, beta)
Arguments
n |
matrix of observed counts |
x |
design matrix for location model |
beta |
vector of regression parameters for location model |
Value
gradient vector
Derivative of log(likelihood) wrt phi[i, j]
Description
Derivative of log(likelihood) wrt phi[i, j]
Usage
McCullagh_derivative_log_l_wrt_phi(n, phi, i, j)
Arguments
n |
matrix of observed counts |
phi |
matrix of phi-values |
i |
row index of phi |
j |
column index of phi |
Value
derivative
Derivative of log(likelihood) wrt psi.
Description
Derivative of log(likelihood) wrt psi.
Usage
McCullagh_derivative_log_l_wrt_psi(n, i1, j1, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of Lagrange multiplier omega wrt alpha[index].
Description
Derivative of Lagrange multiplier omega wrt alpha[index].
Usage
McCullagh_derivative_omega_wrt_alpha(n, index, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of Lagrange multiplier omega wrt c.
Description
Derivative of Lagrange multiplier omega wrt c.
Usage
McCullagh_derivative_omega_wrt_c(n, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of Lagrange multiplier omega wrt scalar delta.
Description
Derivative of Lagrange multiplier omega wrt scalar delta.
Usage
McCullagh_derivative_omega_wrt_delta(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of Lagrange multiplier omega wrt vector delta[k].
Description
Derivative of Lagrange multiplier omega wrt vector delta[k].
Usage
McCullagh_derivative_omega_wrt_delta_vec(n, k, psi, delta_vec, alpha, c = 1)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Derivative of Lagrange multiplier omega wrt psi[i, j].
Description
Derivative of Lagrange multiplier omega wrt psi[i, j].
Usage
McCullagh_derivative_omega_wrt_psi(n, i, j, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
i |
first index of psi |
j |
second index of psi |
psi |
symmetry matrix |
delta |
scalar or vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Defaults to 1.0 |
Derivative of phi wrt gamma.
Description
Derivative of phi wrt gamma.
Usage
McCullagh_derivative_phi_wrt_gamma(gamma, j)
Arguments
gamma |
vector of gamma values |
j |
index of gamma for which to compute the derivative |
Value
derivative
Derivative of pi[i, j] wrt alpha[index].
Description
Derivative of pi[i, j] wrt alpha[index].
Usage
McCullagh_derivative_pi_wrt_alpha(i, j, index, psi, delta, alpha, c = 1)
Arguments
i |
row index of pi |
j |
column index of pi |
index |
index of alpha |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Derivative pi[i, j] wrt c.
Description
Derivative pi[i, j] wrt c.
Usage
McCullagh_derivative_pi_wrt_c(i, j, psi, delta, alpha, c)
Arguments
i |
row index of pi |
j |
column index of pi |
psi |
the matrix of symmetry parameters |
delta |
the scalar or vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 |
Value
derivative
Derivative of pi[i, j] wrt delta.
Description
Derivative of pi[i, j] wrt delta.
Usage
McCullagh_derivative_pi_wrt_delta(i, j, psi, delta, alpha, c = 1)
Arguments
i |
row index of pi |
j |
column index of pi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Derivative pi[i, j] wrt delta[k].
Description
Derivative pi[i, j] wrt delta[k].
Usage
McCullagh_derivative_pi_wrt_delta_vec(i, j, k, psi, delta_vec, alpha, c = 1)
Arguments
i |
row index of pi |
j |
column index of pi |
k |
index of delta_vec |
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Derivative of pi[i, j] wrt psi[i1, j1].
Description
Derivative of pi[i, j] wrt psi[i1, j1].
Usage
McCullagh_derivative_pi_wrt_psi(i, j, i1, j1, psi, delta, alpha, c = 1)
Arguments
i |
row index of pi |
j |
column index of pi |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Derivative of pij[i, j] wrt alpha[index]
Description
Derivative of pij[i, j] wrt alpha[index]
Usage
McCullagh_derivative_pij_wrt_alpha(i, j, index, psi, delta, alpha, c = 1)
Arguments
i |
row index of pij |
j |
column index of pij |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar or vector of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
normalizing constant to make pi sum to 1.0. Default ot 1.0 |
Value
derivative
Derivative pij[i, j] wrt c.
Description
Derivative pij[i, j] wrt c.
Usage
McCullagh_derivative_pij_wrt_c(i, j, psi, delta, alpha, c)
Arguments
i |
row index of pij |
j |
column index of pij |
psi |
matrix of symmetry parameters |
delta |
scalar or vector of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
normalizing constant to make pi sum to 1.0 |
Value
derivative
Derivative of pij[i, j] wrt scalar delta.
Description
Derivative of pij[i, j] wrt scalar delta.
Usage
McCullagh_derivative_pij_wrt_delta(i, j, psi, delta, alpha, c = 1)
Arguments
i |
row index of pij |
j |
column index of pij |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing constant so that pi sum to 1.0. Default value is 1.0 |
Value
derivative
Derivative pij[i,j] wrt vector delta[k].
Description
Derivative pij[i,j] wrt vector delta[k].
Usage
McCullagh_derivative_pij_wrt_delta_vec(i, j, k, psi, delta_vec, alpha, c = 1)
Arguments
i |
row index of pij |
j |
column index of pij |
k |
index of delta |
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
list containing matrices pij and qij
Derivative of pij[a, b] wrt psi[h, k]
Description
Derivative of pij[a, b] wrt psi[h, k]
Usage
McCullagh_derivative_pij_wrt_psi(a, b, h, k, delta, alpha, c = 1)
Arguments
a |
row index of pi |
b |
column index of pi |
h |
row index of phi |
k |
column index of phi |
delta |
scalar or vector version of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
normalizing constant for to make pi sum to 1. Defaults to 1.0 |
Value
derivative
Extracts the weights to convert cumulative model-based probabilities to regular probabilities.
Description
Extracts the weights to convert cumulative model-based probabilities to regular probabilities.
Usage
McCullagh_extract_weights(i, j, M)
Arguments
i |
row index sought |
j |
column index sought |
M |
the number of rows/columns in observed matrix |
Value
a list containing w_psi for when i == j w_pij for when i < j w_qij for when j < i weight populated with correct entry based on actual i and j
Fit location model
Description
Fit location model
Usage
McCullagh_fit_location_regression_model(n, x, max_iter = 5, verbose = FALSE)
Arguments
n |
matrix of observed counts |
x |
design matrix for regression model |
max_iter |
maximum number of Fisher scoring iterations |
verbose |
logical: should cycle-by-cycle info be printed out? Default value is FALSE, do not print |
Value
a list containing beta: regression parameter estimates se: matrix of estimated standard errors cov: covariance matrix of parameter estimates g_squared: G^2 likelihood ratio chi-square for model chisq: Pearson chi-square for model df: degrees of freedom
Generalized version of palindromic symmetry model
Description
delta now is a vector, varying by index McCullagh, P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65(2). 413-416.
Usage
McCullagh_generalized_palindromic_symmetry(
n,
max_iter = 15,
verbose = FALSE,
start_values = FALSE
)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations to maximize log(likelihood) |
verbose |
should cycle-by-cycle information be printed out? Default is FALSE, do not print |
start_values |
logical should the regular palindomic symmetry model be fit first to get good starting values. Default is FALSE. |
Value
a list containing
a list containing delta: the vector of asymmetry parameter delta sigma_delta: vector of SE(delta) logL: value of log(likelihood) for final estimates chisq: Pearson chi-square for solution df: degrees of freedom for solution chisq psi: matrix of symmetry parameters alpha: c: constraint, sum of pi - values condition: constraint on psi to make model identified, Lagrange multiplier SE: vector of standard errors for all parameters
Examples
McCullagh_generalized_palindromic_symmetry(vision_data)
Computes culuative model probabilities for the generalized model using vector delta.
Description
Computes culuative model probabilities for the generalized model using vector delta.
Usage
McCullagh_generalized_pij_qij(i, j, psi, delta_vec, alpha, c1 = 1)
Arguments
i |
row index |
j |
column index |
psi |
symmetry matrix |
delta_vec |
vector of delta values |
alpha |
vector of asymmetry values |
c1 |
normalizing value for pi. Defaults to 1.0 |
Value
model-based cumulative probability pi_ij
Generates names to label the parameters.
Description
Generates names to label the parameters.
Usage
McCullagh_generate_names(psi, delta, alpha, c)
Arguments
psi |
matrix of symmetry parameters |
delta |
scalar of matrix of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
scling factor to ensure sup of pi is 1.0 |
Value
character vector of labels for the SE values
Computes summary statistics needed to compute estimate of delta.
Description
Computes summary statistics needed to compute estimate of delta.
Usage
McCullagh_get_statistics(m)
Arguments
m |
matrix of observed counts |
Value
a list containing: N: matrix of sums above and below the diagonal n: vector, size of binomial r: vector, observed sums, number of successes for binomail
Gradient vector of log(likelihood)
Description
Gradient vector of log(likelihood)
Usage
McCullagh_gradient_log_l(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar or vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
gradient vector of first-order partials wrt log(likelihood0)
Hessian matrix of log(likelihood)
Description
Hessian matrix of log(likelihood)
Usage
McCullagh_hessian_log_l(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar or vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
hessian matrix of second-order partials wrt log(likelihood0)
Initializes the beta vector.
Description
Initializes the beta vector.
Usage
McCullagh_initialize_beta(n, c, v)
Arguments
n |
matrix of observed counts |
c |
number of score levels in table |
v |
number of levels of beta beyond c |
Value
initialized beta vector
Compute initial values for scalar delta
Description
Compute initial values for scalar delta
Usage
McCullagh_initialize_delta(n)
Arguments
n |
matrix of observed counts |
Value
value of delta
Initialize vector delta
Description
Initialize vector delta
Usage
McCullagh_initialize_delta_vec(n)
Arguments
n |
matrix of observed counts |
Value
vector of delta values
Initialize the symmetry matrix psi
Description
Initialize the symmetry matrix psi
Usage
McCullagh_initialize_psi(n, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
delta |
scalar delta value |
alpha |
vector of asymmetry parameters |
c |
normalizing value of pi. Default is 1.0 |
Value
matrix psi
Initialize design matrix for location model.
Description
This is the simplest possible implementation, that fits thresholds and a single group contrast. More complex problems will implement the matrix X themselves.
Usage
McCullagh_initialize_x(s, c, v)
Arguments
s |
number of levels of stratification variable |
c |
number of score levels |
v |
number of predictors above thresholds |
Value
design matrix X
Logical test of whether a specific psi will be in the constraint set.
Description
Logical test of whether a specific psi will be in the constraint set.
Usage
McCullagh_is_in_constraint_set(i, j)
Arguments
i |
first index of psi |
j |
second index of psi |
Value
TRUE if it falls within the set, FALSE otherwise.
Test whether pi matrix is valid, i.e., 0 < all values.
Description
Test whether pi matrix is valid, i.e., 0 < all values.
Usage
McCullagh_is_pi_invalid(pi)
Arguments
pi |
matrix of pi values to be tested. |
Value
TRUE if all pi > 0, FALSE otherwise.
Computes the log(likelihood).
Description
Computes the log(likelihood).
Usage
McCullagh_log_L(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar or vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
MCCullagh's logistic model.
Description
McCullah, P. (1977). A logistic model for paired comparisons with ordered categorical data. Biometrika, 64(3), 449-453.
Usage
McCullagh_logistic_model(m)
Arguments
m |
matrix of observed counts |
Value
a list containing w_tilde: vector of model weights for sum of normally distributed components delta_tilde: delta parameter computed using w_tilde w_star: vector of weights for Mantel-Haenszel type numerator and denominator delta_star: delta parameter computed using w_star var: variance of delta estimate
Examples
McCullagh_logistic_model(coal_g)
Computed cumulative logits.
Description
Computed cumulative logits.
Usage
McCullagh_logits(cumulative, use_half = TRUE)
Arguments
cumulative |
vector of cumulative counts |
use_half |
logical indicting whether or not to add 0.5 to numerator and denominator counts before computing logits, Default value is TRUE, add 0.5. |
Maximize the log(likelihood) wrt parameters phi and alpha
Description
Maximize the log(likelihood) wrt parameters phi and alpha
Usage
McCullagh_maximize_q_symmetry(n, phi, alpha)
Arguments
n |
matrix of observed counts |
phi |
matrix of symmetry parameters |
alpha |
vector of asymmetry parameters |
Value
list with new values of phi and alpha
Newton-Raphson update.
Description
Using gradient and hessian, it finds the update direction. Then it tries increassingly smaller step sizes until the step*update yields a valid pi matrix.
Usage
McCullagh_newton_raphson_update(
n,
gradient,
hessian,
psi,
delta,
alpha,
c = 1,
max_iter = 50,
verbose = FALSE
)
Arguments
n |
matrix of observed counts |
gradient |
gradient vector |
hessian |
hessian matrix |
psi |
matrix of symmetry parameters |
delta |
scalar or vector of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
scaling factor to ensure pi sums to 1.0. Default is 1.0 |
max_iter |
maximum number of iterations. Default is 50. |
verbose |
should cycle-by-cycle into be printed out. Default is FALSE, do not print. |
Value
list containing new parameters psi: matrix of symmetry parameters delta; scalar or vector of asymmetry parameters alpha: vector of asymmetry parameters c: scaling coefficient to ensure pi sums to 1.0
McCullagh's palindromic symmetry model
Description
McCullagh, P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65(2). 413-416.
Usage
McCullagh_palindromic_symmetry(n, max_iter = 15, verbose = FALSE)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations to maximize the log(likelihood) |
verbose |
should cycle-by-cycle info be printed out? Default is FALSE, don't print. |
Value
a list containing delta: the value of the asymmetry parameter delta sigma_delta: SE(delta) logL: value of log(likelihood) for final estimates chisq: Pearson chi-square for solution df: degrees of freedom for solution chisq psi: matrix of symmetry parameters alpha: c: constraint, sum of pi - values condition: constraint on psi to make model identified, Lagrange multiplier SE: vector of standard errors for all parameters
Examples
McCullagh_palindromic_symmetry(vision_data)
Computes the penalized value of a derivative by adding the derivative of the penalty to it.
Description
Computes the penalized value of a derivative by adding the derivative of the penalty to it.
Usage
McCullagh_penalized(derivative, i1, j1, n, psi, delta, alpha, c = 1)
Arguments
derivative |
the base derivative |
i1 |
first index of psi |
j1 |
second index of psi |
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Compute model-based cumulative probabilities
Description
Compute model-based cumulative probabilities
Usage
McCullagh_pij_qij(i, j, psi, delta, alpha, c = 1)
Arguments
i |
row index |
j |
column index |
psi |
the symmetry matrix |
delta |
the asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for pi. Default is 1.0 |
Value
the model-based cumulative probability pi_ij
Computes the proportional hazards.
Description
Computes the proportional hazards.
Usage
McCullagh_proportional_hazards(n)
Arguments
n |
matrix of observed counts |
Value
loga(-log(survival))
Initializes the asymmetry vector alpha
Description
Initializes the asymmetry vector alpha
Usage
McCullagh_q_symmetry_initialize_alpha(M)
Arguments
M |
size of alpha vector to create = nrow(matrix to analyze) |
Value
vector of asymmetry parameters alpha
Initializes the phi matrix
Description
Initializes the phi matrix
Usage
McCullagh_q_symmetry_initialize_phi(M)
Arguments
M |
size of the psi matrix to create |
Value
the symmetry matrix phi
Computes the model-based p-values
Description
Computes the model-based p-values
Usage
McCullagh_q_symmetry_pi(phi, alpha)
Arguments
phi |
the matrix of symmetry parameters |
alpha |
the vector of asymmetry parameters |
Value
matrix pi of model-based p-values
Fits McCullagh's (1978) quasi-symmetry model.
Description
McCullagh, P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65(2) 413-418.
Usage
McCullagh_quasi_symmetry(n, max_iter = 15, verbose = FALSE)
Arguments
n |
matrix of observed counts |
max_iter |
maximum number of iterations in maximizing log(likelihood), Default is 15. |
verbose |
should cycle-by-cycle information be printed out? Default is FALSE, do not print |
Value
a list containing phi: symmetry matrix alpha: vector of asymmetry parameters chisq: Pearson chi-square value df; degrees of freedom
Examples
McCullagh_quasi_symmetry(vision_data)
Second derivative of Lagrangian wrt psi^2.
Description
Second derivative of Lagrangian wrt psi^2.
Usage
McCullagh_second_order_lagrangian_wrt_psi_2(
n,
i1,
j1,
i2,
j2,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
first row index of psi |
j1 |
first column index of psi |
i2 |
second row index of psi |
j2 |
second column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrangian wrt psi[i1, j1] and alpha[index].
Description
Second derivative of Lagrangian wrt psi[i1, j1] and alpha[index].
Usage
McCullagh_second_order_lagrangian_wrt_psi_alpha(
n,
i1,
j1,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
index |
second row index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrangian wrt psi[i1, j1] and delta.
Description
Second derivative of Lagrangian wrt psi[i1, j1] and delta.
Usage
McCullagh_second_order_lagrangian_wrt_psi_delta(
n,
i1,
j1,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrangian wrt psi[i1, j1] and delta_vec[k[.
Description
Second derivative of Lagrangian wrt psi[i1, j1] and delta_vec[k[.
Usage
McCullagh_second_order_lagrangian_wrt_psi_delta_vec(
n,
i1,
j1,
k,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
k |
index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt alpha^2.
Description
Second derivative of log(likelihood) wrt alpha^2.
Usage
McCullagh_second_order_log_l_wrt_alpha_2(
n,
index_a,
index_b,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
index_a |
first index of alpha |
index_b |
second column index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt alpha[index] and c.
Description
Second derivative of log(likelihood) wrt alpha[index] and c.
Usage
McCullagh_second_order_log_l_wrt_alpha_c(n, index, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. |
Value
derivative
Expected values of second order derivatives of log(likelihood) wrt beta.
Description
Appendix of McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42(2), 109-142. and appendix B3 of Agresti, A. (1984). Analysis of ordinal categorical data, New York, Wiley, p. 242-244.
Usage
McCullagh_second_order_log_l_wrt_beta_2(n, x, gamma)
Arguments
n |
matrix of observed counts |
x |
design matrix for location model |
gamma |
current value of model-based cumulative logits. |
Value
matrix of second order partial derivatives
Second derivative of log(likelihood) wrt c^2.
Description
Second derivative of log(likelihood) wrt c^2.
Usage
McCullagh_second_order_log_l_wrt_c_2(n, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt delta^2.
Description
Second derivative of log(likelihood) wrt delta^2.
Usage
McCullagh_second_order_log_l_wrt_delta_2(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt delta and alpha[index].
Description
Second derivative of log(likelihood) wrt delta and alpha[index].
Usage
McCullagh_second_order_log_l_wrt_delta_alpha(
n,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt scalar delta and c.
Description
Second derivative of log(likelihood) wrt scalar delta and c.
Usage
McCullagh_second_order_log_l_wrt_delta_c(n, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0.. |
Value
derivative
Second derivative of log(likelihood) wrt delta_vec^2.
Description
Second derivative of log(likelihood) wrt delta_vec^2.
Usage
McCullagh_second_order_log_l_wrt_delta_vec_2(
n,
k1,
k2,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
k1 |
first index of delta_vec |
k2 |
second index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt delta[k] and alpha[index].
Description
Second derivative of log(likelihood) wrt delta[k] and alpha[index].
Usage
McCullagh_second_order_log_l_wrt_delta_vec_alpha(
n,
k,
index,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likeloihood) wrt delta_vec[k] and c.
Description
Second derivative of log(likeloihood) wrt delta_vec[k] and c.
Usage
McCullagh_second_order_log_l_wrt_delta_vec_c(n, k, psi, delta_vec, alpha, c)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0 |
Value
derivative
Expected second order derivatives of log(likelihood)
Description
Expected second order derivatives of log(likelihood)
Usage
McCullagh_second_order_log_l_wrt_parms(n, x, beta)
Arguments
n |
matrix of observed counts |
x |
design matrix for location model |
beta |
vector of regression parameters for location model |
Value
matrix of expected second derivatives
Second derivative of log(likelihoood) wrt psi^2.
Description
Second derivative of log(likelihoood) wrt psi^2.
Usage
McCullagh_second_order_log_l_wrt_psi_2(
n,
i1,
j1,
i2,
j2,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
first row index of psi |
j1 |
first column index of psi |
i2 |
second row index of psi |
j2 |
second column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihoood) wrt ps[i1, j1] and alpha[index].
Description
Second derivative of log(likelihoood) wrt ps[i1, j1] and alpha[index].
Usage
McCullagh_second_order_log_l_wrt_psi_alpha(
n,
i1,
j1,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt psi[i1, j1] and c.
Description
Second derivative of log(likelihood) wrt psi[i1, j1] and c.
Usage
McCullagh_second_order_log_l_wrt_psi_c(n, i1, j1, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt psi[i1, j1] and scalar delta..
Description
Second derivative of log(likelihood) wrt psi[i1, j1] and scalar delta..
Usage
McCullagh_second_order_log_l_wrt_psi_delta(n, i1, j1, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of log(likelihood) wrt psi[i1, j1] and delta_vec[k].
Description
Second derivative of log(likelihood) wrt psi[i1, j1] and delta_vec[k].
Usage
McCullagh_second_order_log_l_wrt_psi_delta_vec(
n,
i1,
j1,
k,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
k |
second row index of delta |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt alpha^2.
Description
Second derivative of Lagrange multiplier omega wrt alpha^2.
Usage
McCullagh_second_order_omega_wrt_alpha_2(n, k1, k2, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
k1 |
first index of alpha |
k2 |
second index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt alpha[index] and c.
Description
Second derivative of Lagrange multiplier omega wrt alpha[index] and c.
Usage
McCullagh_second_order_omega_wrt_alpha_c(n, index, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
index |
row index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt c^2.
Description
Second derivative of Lagrange multiplier omega wrt c^2.
Usage
McCullagh_second_order_omega_wrt_c_2(n, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt scalae delta^2.
Description
Second derivative of Lagrange multiplier omega wrt scalae delta^2.
Usage
McCullagh_second_order_omega_wrt_delta_2(n, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt delta and alpha[index].
Description
Second derivative of Lagrange multiplier omega wrt delta and alpha[index].
Usage
McCullagh_second_order_omega_wrt_delta_alpha(
n,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt scalar delta and c.
Description
Second derivative of Lagrange multiplier omega wrt scalar delta and c.
Usage
McCullagh_second_order_omega_wrt_delta_c(n, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt delta_vec^2.
Description
Second derivative of Lagrange multiplier omega wrt delta_vec^2.
Usage
McCullagh_second_order_omega_wrt_delta_vec_2(
n,
k1,
k2,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
k1 |
first index of delta_vec |
k2 |
second index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt delta_vec[k] and alpha[index].
Description
Second derivative of Lagrange multiplier omega wrt delta_vec[k] and alpha[index].
Usage
McCullagh_second_order_omega_wrt_delta_vec_alpha(
n,
k,
index,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt delta_vec[k] and c.
Description
Second derivative of Lagrange multiplier omega wrt delta_vec[k] and c.
Usage
McCullagh_second_order_omega_wrt_delta_vec_c(n, k, psi, delta_vec, alpha, c)
Arguments
n |
matrix of observed counts |
k |
index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector of asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt psi^2.
Description
Second derivative of Lagrange multiplier omega wrt psi^2.
Usage
McCullagh_second_order_omega_wrt_psi_2(
n,
i1,
j1,
i2,
j2,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
first row index of psi |
j1 |
first column index of psi |
i2 |
second row index of psi |
j2 |
second column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt psi[i1, j1] and alpha[index].
Description
Second derivative of Lagrange multiplier omega wrt psi[i1, j1] and alpha[index].
Usage
McCullagh_second_order_omega_wrt_psi_alpha(
n,
i1,
j1,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
index |
index of alpha |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt psi[i1, j1] and c.
Description
Second derivative of Lagrange multiplier omega wrt psi[i1, j1] and c.
Usage
McCullagh_second_order_omega_wrt_psi_c(n, i1, j1, psi, delta, alpha, c)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt psi and scalar delta.
Description
Second derivative of Lagrange multiplier omega wrt psi and scalar delta.
Usage
McCullagh_second_order_omega_wrt_psi_delta(n, i1, j1, psi, delta, alpha, c = 1)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
matrix of symmetry parameters |
delta |
scalar asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of Lagrange multiplier omega wrt psi[i1, j1] and delta_vec[k].
Description
Second derivative of Lagrange multiplier omega wrt psi[i1, j1] and delta_vec[k].
Usage
McCullagh_second_order_omega_wrt_psi_delta_vec(
n,
i1,
j1,
k,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
n |
matrix of observed counts |
i1 |
row index of psi |
j1 |
column index of psi |
k |
index of delta_vec |
psi |
matrix of symmetry parameters |
delta_vec |
vector asymmetry parameter |
alpha |
vector of asymmetry parameters |
c |
normalizing factor to make pi sum to 1.0. Default is 1.0. |
Value
derivative
Second derivative of pi[i, j] wrt alpha^2.
Description
Second derivative of pi[i, j] wrt alpha^2.
Usage
McCullagh_second_order_pi_wrt_alpha_2(
i,
j,
index1,
index2,
psi,
delta,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
index1 |
index of first alpha |
index2 |
index of second aloha |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second derivaitve of pi[i, j] wrt alpha[index] and c.
Description
Second derivaitve of pi[i, j] wrt alpha[index] and c.
Usage
McCullagh_second_order_pi_wrt_alpha_c(i, j, index, psi, delta, alpha, c)
Arguments
i |
row index of pi |
j |
column index of pi |
index |
index of alpha |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 |
Value
derivative
Second order derivative of pi[i, j] wrt c^2.
Description
Second order derivative of pi[i, j] wrt c^2.
Usage
McCullagh_second_order_pi_wrt_c_2(i, j, psi, delta, alpha, c)
Arguments
i |
row index of pi |
j |
column index of pi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order derivative of pi[i, j] wrt scalar delta.
Description
Second order derivative of pi[i, j] wrt scalar delta.
Usage
McCullagh_second_order_pi_wrt_delta_2(i, j, psi, delta, alpha, c = 1)
Arguments
i |
row index of pi |
j |
column index of pi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order deriviative of pi[i, j] wrt scalar delta and alpha[index]
Description
Second order deriviative of pi[i, j] wrt scalar delta and alpha[index]
Usage
McCullagh_second_order_pi_wrt_delta_alpha(
i,
j,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
index |
index of alpha |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order derivative of pi[i, j] wrt scalae delta and c.
Description
Second order derivative of pi[i, j] wrt scalae delta and c.
Usage
McCullagh_second_order_pi_wrt_delta_c(i, j, psi, delta, alpha, c)
Arguments
i |
row index of pi |
j |
column index of pi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 |
Value
derivative
Derivative of pi[i, j] wrt delta^2.
Description
Derivative of pi[i, j] wrt delta^2.
Usage
McCullagh_second_order_pi_wrt_delta_vec_2(
i,
j,
k1,
k2,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
k1 |
first index of delta |
k2 |
second index of delta |
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order dertivative of pi[i, j] wrtt delta[k] alpha[index].
Description
Second order dertivative of pi[i, j] wrtt delta[k] alpha[index].
Usage
McCullagh_second_order_pi_wrt_delta_vec_alpha(
i,
j,
k,
index,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
k |
index of delta |
index |
index of alpha |
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second derivative of pi[i, j] wrt delta[k] and c.
Description
Second derivative of pi[i, j] wrt delta[k] and c.
Usage
McCullagh_second_order_pi_wrt_delta_vec_c(i, j, k, psi, delta_vec, alpha, c)
Arguments
i |
row index of pi |
j |
column index of pi |
k |
index of delta |
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order derivative wrt psi^2.
Description
Second order derivative wrt psi^2.
Usage
McCullagh_second_order_pi_wrt_psi_2(
i,
j,
i1,
j1,
i2,
j2,
psi,
delta,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
i1 |
first row index of psi |
j1 |
first column index of psi |
i2 |
second row index of psi |
j2 |
second column index of pis |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order derivative of pi[i, j] wrt psi[i1, j1] and alpha[index].
Description
Second order derivative of pi[i, j] wrt psi[i1, j1] and alpha[index].
Usage
McCullagh_second_order_pi_wrt_psi_alpha(
i,
j,
i1,
j1,
index,
psi,
delta,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
i1 |
row index of psi |
j1 |
column index of psi |
index |
index of alpha |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order derivative of pi[i, j] wrt psi[i1, j1] and c.
Description
Second order derivative of pi[i, j] wrt psi[i1, j1] and c.
Usage
McCullagh_second_order_pi_wrt_psi_c(i, j, i1, j1, psi, delta, alpha, c)
Arguments
i |
row index of pi |
j |
column index of pi |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 |
Value
derivative
Second order derivaitve of pi wrt pshi and scalar delta.
Description
Second order derivaitve of pi wrt pshi and scalar delta.
Usage
McCullagh_second_order_pi_wrt_psi_delta(i, j, i1, j1, psi, delta, alpha, c = 1)
Arguments
i |
row index of pi |
j |
column index of pi |
i1 |
row index of psi |
j1 |
column index of psi |
psi |
the matrix of symmetry parameters |
delta |
the scalar asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Second order derivaitve of pi[i, j] wrt psi[i1, j1] and kelta[k].
Description
Second order derivaitve of pi[i, j] wrt psi[i1, j1] and kelta[k].
Usage
McCullagh_second_order_pi_wrt_psi_delta_vec(
i,
j,
i1,
j1,
k,
psi,
delta_vec,
alpha,
c = 1
)
Arguments
i |
row index of pi |
j |
column index of pi |
i1 |
row index of psi |
j1 |
column index of psi |
k |
index of delta |
psi |
the matrix of symmetry parameters |
delta_vec |
the vector asymmetry parameter |
alpha |
the vector of asymmetry parameters |
c |
the normalizing constant for the pis to sum to 1.0 Default value is 1.0 |
Value
derivative
Update the parameters based on Newton-Raphson step.
Description
Update the parameters based on Newton-Raphson step.
Usage
McCullagh_update_parameters(update, step, psi, delta, alpha, c = 1)
Arguments
update |
vector of update values |
step |
size of candidate step along direction of update |
psi |
vector of symmetry parameters |
delta |
scalar or vector of asymmetry parameters |
alpha |
vector of asymmetry parameters |
c |
normalization factor to make sum pf pi = 1.0. Default value is 1.0. |
Value
list containing new parameters psi: matrix of symmetry parameters delta; scalar or vector of asymmetry parameters alpha: vector of asymmetry parameters c: scaling coefficient to ensure pi sums to 1.0
Compute v_inverse (from appendix).
Description
Compute v_inverse (from appendix).
Usage
McCullagh_v_inverse(gamma, i, j)
Arguments
gamma |
matrix of cumulative logits |
i |
row index |
j |
column index |
Value
V^(-1) : d phi / d gamma[i, j]
Computes the degrees of freedom for the model.
Description
Computes the degrees of freedom for the model.
Usage
Schuster_compute_df(pi_margin)
Arguments
pi_margin |
expected proportions for each of the categories |
Value
the df for the model
Compute matrix of model-based proportions pi.
Description
Compute matrix of model-based proportions pi.
Usage
Schuster_compute_pi(marginal_pi, kappa, v, validate = TRUE)
Arguments
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
validate |
logical. should the cells be validated within this function? Defaults to TRUE |
Value
matrix of model-based cell proportions
Computes starting values for the model.
Description
Patterned after example in code in appendix to article
Usage
Schuster_compute_starting_values(n)
Arguments
n |
matrix of observed counts |
Value
a list containing marginal_pi: vector of expected proportions for each category kappa: kappa coefficient of agreement v: matrix of symmetry parameters
Derivative of log(likelihood) wrt kappa.
Description
Derivative of log(likelihood) wrt kappa.
Usage
Schuster_derivative_log_l_wrt_kappa(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each category |
kappa |
current value of kappa coefficient |
v |
symmetry matrix |
Value
derivative of log(L) wrt kappa
Derivative of log(likelihood) wrt marginal_pi[k]
Description
Derivative of log(likelihood) wrt marginal_pi[k]
Usage
Schuster_derivative_log_l_wrt_marginal_pi(n, k, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
k |
index into marginal_pi |
marginal_pi |
expected proportions of each of the categories |
kappa |
current value of the kappa coefficient |
v |
symmetry matrix |
Value
derivative of log(L) wrt marginal_pi[k]
Derivative of log(likelihood) wrt v[i1, j1]
Description
Derivative of log(likelihood) wrt v[i1, j1]
Usage
Schuster_derivative_log_l_wrt_v(n, i1, j1, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
i1 |
first index into v |
j1 |
second index into v |
marginal_pi |
expected marginal proportions |
kappa |
current value of kappa coefficient |
v |
symmetry matrix |
Value
derivative of log(L) wrt v[i1, j1]
Derivative of pi[i, j] wrt kappa coefficient.
Description
Derivative of pi[i, j] wrt kappa coefficient.
Usage
Schuster_derivative_pi_wrt_kappa(i, j, marginal_pi, kappa, v)
Arguments
i |
first index into pi |
j |
second index into pi |
marginal_pi |
expected proportions in each category |
kappa |
current value of kappa coefficient |
v |
symmetry matrix |
Value
the derivative of pi[i, j] wrt kappa
Derivative of pi[i, j] wrt marginal_pi[k].
Description
Derivative of pi[i, j] wrt marginal_pi[k].
Usage
Schuster_derivative_pi_wrt_marginal_pi(i, j, k, marginal_pi, kappa, v)
Arguments
i |
first index into pi |
j |
second index into pi |
k |
index into marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
derivative of pi[i, j] wrt marginal_pi[k]
Computes derivative of pi[i, j] wrt v[i1, j1]
Description
Computes derivative of pi[i, j] wrt v[i1, j1]
Usage
Schuster_derivative_pi_wrt_v(i, j, i1, j1, marginal_pi, kappa, v)
Arguments
i |
first index into pi |
j |
second index into pi |
i1 |
first index into v |
j1 |
second index into v |
marginal_pi |
expected marginal proportions |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
value of derivative of specified pi wrt specified element of v
Computes derivative of v[i1, j1] wrt v[i2, j2]
Description
Needed because of computed v terms in column r
Usage
Schuster_derivative_v_wrt_v(i1, j1, i2, j2, marginal_pi, kappa, v)
Arguments
i1 |
first index into target v |
j1 |
second index into target v |
i2 |
first index into |
j2 |
second index into |
marginal_pi |
expected marginal proportions |
kappa |
current estimate of kappa coefficient |
v |
matrix of symmetry parameters |
Value
derivative of v[i1, j1] wrt v[i2, j2]
Compute v matrix subject to constraints on rows 1..r-1.
Description
Compute v matrix subject to constraints on rows 1..r-1.
Usage
Schuster_enforce_constraints_on_v(marginal_pi, kappa, v)
Arguments
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
new v matrix with last row/column set to agree with constraints. Element v[r, r] is set to v-tilde
Gradient vector log(L) wrt parameters.
Description
Work is delegated to functions that compute partial derivatives. This function is responsible for laying them out in correct positions in the vector.
Usage
Schuster_gradient(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
gradient vector
Computes the hessian matrix of second-order partial derivatives of log(L).
Description
Work is delegated to functions that compute second-order partial derivatives. This function is responsible for laying them out in correct positions in the matrix.
Usage
Schuster_hessian(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
Value
hessian matrix
Determines whether the candidate pi matrix is valid.
Description
All elements must lie in (0, 1)
Usage
Schuster_is_pi_valid(pi)
Arguments
pi |
matrix of model-based proportions |
Value
logical value indicating whether or not the matrix is valid.
Performs Newton-Raphson step.
Description
The step size is determined to be the largest that yields valid results for all quantities marginal_pi and v. Both must be positive, and the elements of marginal_pi must be valid proportions that sum to 1.0.
Usage
Schuster_newton_raphson(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
Value
a list containing updated versions of model quantities marginal_pi kappa v
Second order partial log(L) wrt kappa^2.
Description
Second order partial log(L) wrt kappa^2.
Usage
Schuster_second_deriv_log_l_wrt_kappa_2(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix second derivative of log(L) wrt kappa^2 |
Second order partial log(L) wrt kappa and v.
Description
Second order partial log(L) wrt kappa and v.
Usage
Schuster_second_deriv_log_l_wrt_kappa_v(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix second derivative of log(L) wrt kappa and v |
Second order partial log(L) wrt marginal_pi^2.
Description
Second order partial log(L) wrt marginal_pi^2.
Usage
Schuster_second_deriv_log_l_wrt_marginal_pi_2(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix second derivative of log(L) wrt marginal_pi^2 |
Second order partial log(L) wrt marginal_pi and kappa.
Description
Second order partial log(L) wrt marginal_pi and kappa.
Usage
Schuster_second_deriv_log_l_wrt_marginal_pi_kappa(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix second derivative of log(L) wrt marginal_pi and kappa |
Second order partial log(L) wrt marginal_pi and v.
Description
Second order partial log(L) wrt marginal_pi and v.
Usage
Schuster_second_deriv_log_l_wrt_marginal_pi_v(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix second derivative of log(L) wrt marginal_pi and v |
Second order partial log(L) wrt v^2.
Description
Second order partial log(L) wrt v^2.
Usage
Schuster_second_deriv_log_l_wrt_v_2(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each response category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix second derivative of log(L) wrt v^2 |
Second order partial wrt kappa, kappa
Description
Derivative is uniformly 0
Usage
Schuster_second_deriv_pi_wrt_kappa_2(i, j, marginal_pi, kappa, v)
Arguments
i |
first index of pi |
j |
second index of pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
Value
second order partial derivative
Second order partial wrt kappa, v
Description
Derivative is uniformly 0
Usage
Schuster_second_deriv_pi_wrt_kappa_v(i, j, i1, j1, marginal_pi, kappa, v)
Arguments
i |
first index of pi |
j |
second index of pi |
i1 |
first index of v |
j1 |
second index of v |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
Value
second order partial derivative
Second derivative of pi[i, j] wrt marginal_pi[k]^2
Description
Second derivative of pi[i, j] wrt marginal_pi[k]^2
Usage
Schuster_second_deriv_pi_wrt_marginal_pi_2(i, j, k, k2, marginal_pi, kappa, v)
Arguments
i |
first index into pi |
j |
second index into pi |
k |
index into marginal_pi |
k2 |
second index into marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
second derivative of pi[i, j] wrt marginal_pi^2
Second order partial wrt kappa, marginal_pi
Description
Derivative is uniformly 0
Usage
Schuster_second_deriv_pi_wrt_marginal_pi_kappa(i, j, k, marginal_pi, kappa, v)
Arguments
i |
first index of pi |
j |
second index of pi |
k |
index of marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
Value
second order partial derivative
Second order partial pi wrt marginal_pi and v
Description
Second order partial pi wrt marginal_pi and v
Usage
Schuster_second_deriv_pi_wrt_marginal_pi_v(
i,
j,
k,
i1,
j1,
marginal_pi,
kappa,
v
)
Arguments
i |
first index of pi |
j |
second index of pi |
k |
index of marginal_pi |
i1 |
first index of v |
j1 |
second index of v |
marginal_pi |
expected proportions of each of the categories |
kappa |
current value of kappa coefficient |
v |
symmetry matrix |
Value
derivative
Second order partial wrt v^2
Description
Derivative is uniformly 0
Usage
Schuster_second_deriv_pi_wrt_v_2(i, j, i1, j1, i2, j2, marginal_pi, kappa, v)
Arguments
i |
first index of pi |
j |
second index of pi |
i1 |
first index of first v |
j1 |
second index of first v |
i2 |
first index of second v |
j2 |
second index of second |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of the kappa coefficient |
v |
symmetry matrix |
Value
second order partial derivative
Solves for the last row and diagonal of symmetry matrix v (v-tilde) using constraint equations
Description
Solves for the last row and diagonal of symmetry matrix v (v-tilde) using constraint equations
Usage
Schuster_solve_for_v(marginal_pi, kappa, v)
Arguments
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
revised version of v matrix with last row and diagonal modified
Solves for the last row and diagonal of symmetry matrix v (parameteer v-tilde) using linear algebra formulation from paper.
Description
Solves for the last row and diagonal of symmetry matrix v (parameteer v-tilde) using linear algebra formulation from paper.
Usage
Schuster_solve_for_v1(marginal_pi, kappa, v)
Arguments
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
revised version of v matrix with last row and diagonal modified
Computes the model that has kappa as a coefficient and symmetry.
Description
Schuster, C. (2001). Kappa as a parameter of a symmetry model for rater agreement. Journal of Educational and Behavioral Statistics, 26(3), 331-342.
Usage
Schuster_symmetric_rater_agreement_model(
n,
verbose = FALSE,
max_iter = 10000,
criterion = 1e-07,
min_iter = 1000
)
Arguments
n |
the matrix of observed counts |
verbose |
logical. should cycle-by-cycle information be printed out |
max_iter |
integer. maximum number of iterations to perform |
criterion |
number. maximum change in log(likelihood) to decide convergence |
min_iter |
integer. minimum number of iterations to perform |
Value
a list containing marginal_pi: vector of expected proportions for each category kappa numeric: kappa coefficient v: matrix of symmetry parameters chisq: Pearson X^2 g_squared: likelihood ratio G^2 df: degrees of freedom
Computes the Newton-Raphson update
Description
Computes both gradient and hessian, and then solves the system of equations
Usage
Schuster_update(n, marginal_pi, kappa, v)
Arguments
n |
matrix of observed counts |
marginal_pi |
expected proportions for each category |
kappa |
current value of kappa coefficient |
v |
symmetry matrix |
Value
the vector of updates
Computes the common diagonal term v-tilde.
Description
Computes the common diagonal term v-tilde.
Usage
Schuster_v_tilde(marginal_pi, kappa, validate = TRUE)
Arguments
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
validate |
logical. should the value of pi[r,r] be checked for validity? Default is TRUE |
Value
v-tilde
Computes Stuart's Q test of marginal homogeneity.
Description
Stuart, A. (1955). A test for homogeneity of the marginal distributions in a two-way classification. Biometrika, 42(3/4), 412-416.
Usage
Stuart_marginal_homogeneity(n)
Arguments
n |
matrix of observed counts |
Value
a list containing q: value of q test-statistic df: degrees of freedom p: upper tail p-value of q
Examples
Stuart_marginal_homogeneity(vision_data)
Participation in household budgeting by psychiatric patients. Rows are ratings by patient, columns are ratings by relative. 1 - not at all 2 - doing some 3 - doing regularly
Description
Participation in household budgeting by psychiatric patients. Rows are ratings by patient, columns are ratings by relative. 1 - not at all 2 - doing some 3 - doing regularly
Usage
budget_actual
Format
## 'budget_actual' A matrix with 3 rows and 3 columns
Source
Schuster, C, (2001). Kappa as a parameter of a symmetry model for rater agreement. Journal of Educational and Behavioral Statistics, 26(3), 331-342.
Ratings of expected participation in household budgeting by psychiatric patients. Rows are ratings by patient, columns are ratings by relative. 1 - not at all 2 - doing some 3 - doing regularly
Description
Ratings of expected participation in household budgeting by psychiatric patients. Rows are ratings by patient, columns are ratings by relative. 1 - not at all 2 - doing some 3 - doing regularly
Usage
budget_expected
Format
## 'budget_expected' a matrix with 3 rows and 3 columns.
Source
Schuster, C, (2001). Kappa as a parameter of a symmetry model for rater agreement. Journal of Educational and Behavioral Statistics, 26(3), 331-342.
Degree of disease measured at two points in time for mine workers.
Description
Based on radiological measurements, the matrix contains the degree of pneumoconiosis in coal workers. 1 = least severe disease and 4 = most severe.
Usage
coal_g
Format
## 'coal_g' A matrix with 4 rows and 4 columns.
Source
McCullagh, P. (1977). A logistic model for paired comparisons with ordered categorical data. Biometrika, 64(3), 449-453.
Computes the constant of integration of a multinomial sample.
Description
N! / product(n[i]!)
Usage
constant_of_integration(n, exclude_diagonal = FALSE)
Arguments
n |
Matrix of observed counts |
exclude_diagonal |
logical. Should the diagonal cells of a square matrix be excluded from the computation. Default is FALSE, |
Value
value of constant of integration for observed matrix provided
Ratings of severity of patient's depression by two therapists.
Description
1 = slight 2 = moderate 3 = severe
Usage
depression
Format
## 'depression' A matrix with 3 rows and 3 columns.
Source
von Eye, A. & Mun, E. Y. (2005, p.41). Analyzing rater agreement: Manifest variable methods. Mahwah, NJ: Lawrence Erlbaum.
Dehydration in dogs data set.
Description
An interrater agreement data set from Shourki, M. M. (2005, p.80). It is agreement study of two clinicians evaluating whether dogs were dehydrated. The lowest score indicates normal, and the highest score indicates dehydrated (above 10 The "g" in the name indicates that this is taken from mine "G" in the original study.
Usage
dogs
Format
## 'dogs' A matrix with 4 rows and 4 columns.
Source
Shoukri, M. M. (2005). The measurement of interobserver agreement. New York: Chapman & Hall.
Severity of disturbing dreams in adolescent boys, measured at two ages..
Description
Severity of disturbing dreams in adolescent boys, measured at two ages..
Usage
dreams
Format
## 'dreams' A matrix with 4 rows and 4 columns.
Source
McCullagh, P. (1980, p.117). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42(2), 109-142.
Occurrence of side effects after gastro-intestinal surgery.
Description
Columns 1 = None 2 = Slight 3 = Moderate
Usage
dumping
Format
## 'dumping' A matrix with 4 rows and 3 columns
Details
Rows Hospital A Hospital B Hospital C Hospital D
Source
Agresti, A. (1984, p. 63). Analysis of ordinal categorical data. Naew York: Wiley.
Ratings of number of hot drinks consumed by cases with cancer of the esophagus, compared with control subjects.
Description
Ratings of number of hot drinks consumed by cases with cancer of the esophagus, compared with control subjects.
Usage
esophageal_cancer
Format
## 'esophageal_cancer' A matrix with 4 rows and 4 columns.
Source
Agresti, A. (1984, p. 217). Analysis of ordinal categorical data. New York, Wiley.
Converts weighted (x, w) pairs into unweighted data by replicating x[i] w[i] times
Description
Takes a set of (value, weight) pairs and converts into unweighted vector (w[i]) for each i Weights are assumed to be integers
Usage
expand(x, w)
Arguments
x |
Numeric vector of scores. |
w |
Numeric vector of weights. These are assumed to be integers |
Value
new unweighted vector of scores
Computes the "expit" function – inverse of logit.
Description
Computes the "expit" function – inverse of logit.
Usage
expit(z)
Arguments
z |
Numeric. Real valued argument to expit() function. |
Value
exp(z) / (1.0 + exp(z))
Family income for two years from US census.
Description
Family income for two years from US census.
Usage
family_income
Format
## 'family_income' A matrix with 2 rows and 7 columns. Rows are years 1960 and 1970. Columns are income range.
Source
McCullagh, P. (1980, p.114). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42(2), 109-142.
Ratings of visual acuity for men and women employed at the Royal Ordinance factories, 1943-1946.
Description
1 = best visual acuity 4 = worst visual acuity
Usage
gender_vision
Format
## 'gender_vision' A matrix with 2 rows for the genders and 4 columns for visual acuity.
Source
McCullagh, P. (1980, p. 119). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42(2), 109-142.
Case where j == r, i == k == k2
Description
Case where j == r, i == k == k2
Usage
handle_max_i_i(i, marginal_pi, kappa, v)
Arguments
i |
index into marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
second-order derivative
Case where j == r, i != k, i == k2
Description
Case where j == r, i != k, i == k2
Usage
handle_max_i_k(i, k, marginal_pi, kappa, v)
Arguments
i |
index into pi |
k |
index into v (other is i) |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
second-order derivative
Case where j == r, i != k && i != k2
Description
Case where j == r, i != k && i != k2
Usage
handle_max_k_k2(i, k, k2, marginal_pi, kappa, v)
Arguments
i |
index into pi |
k |
first index into marginal_pi |
k2 |
second index into marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
second-order derivative
Case where pi[i, r] with k and k2
Description
Case where pi[i, r] with k and k2
Usage
handle_one_maximum(i, j, k, k2, marginal_pi, kappa, v)
Arguments
i |
first index of pi |
j |
second index of pi |
k |
first index into marginal_pi |
k2 |
second index into marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
second order derivative
Case where i == j, i < r, j < r
Description
Case where i == j, i < r, j < r
Usage
handle_tied_below_maximum(j, k, k2, marginal_pi, kappa, v)
Arguments
j |
index of pi |
k |
first index into marginal_pi |
k2 |
second index into marginal_pi |
marginal_pi |
expected proportions for each of the categories |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
derivative
Case where pi[r, r] with k and k2
Description
Case where pi[r, r] with k and k2
Usage
handle_tied_maximum(k, k2, marginal_pi, kappa, v)
Arguments
k |
first index into marginal_pi |
k2 |
second index into marginal_pi |
marginal_pi |
expected proportions for each category |
kappa |
current estimate of kappa coefficient |
v |
symmetry matrix |
Value
second order derivative
Case where i != j, i < r && j < r
Description
Case where i != j, i < r && j < r
Usage
handle_untied_below_maximum(i, j, k, k2, marginal_pi, kappa, v)
Arguments
i |
first index of pi |
j |
second index of pi |
k |
first index of marginal_pi |
k2 |
second index of marginal_pi |
marginal_pi |
expected proportions of each of the categories |
kappa |
current value of kappa coefficient |
v |
symmetry matrix |
Data about charges of homicide in the state of Florida.
Description
Counts of cases charged with homicide. The rows and columns indicate whether there was an additional charge of a felony occurring in addition to the homicide. The data is actually 3-dimensional. It is stored as 4 related matrices, each with the leading word "homicide_" The rest of the name gives the race of the defendant and the race of the victim, separated by an underscore
Usage
homicide_black_black
Format
## 'homicide_black_black' Each is a matrix with 3 rows and 3 columns. Rows are classification by police and columns are classification by the court/prosecutor. 1 = No felony 2 = Possible felony 2 = Felony
Source
Agresti, A. (1984, p. 211). Analysis of ordinal categorical data. New York: Wiley.
Data about charges of homicide in the state of Florida.
Description
Counts of cases charged with homicide. The rows and columns indicate whether there was an additional charge of a felony occurring in addition to the homicide. The data is actually 3-dimensional. It is stored as 4 related matrices, each with the leading word "homicide_" The rest of the name gives the race of the defendant and the race of the victim, separated by an underscore.
Usage
homicide_black_white
Format
## 'homicide_black_white' Each is a matrix with 3 rows and 3 columns. Rows are classification by police and columns are classification by the court/prosecutor. 1 = No felony 2 = Possible felony 2 = Felony
Source
Agresti, A. (1984, p. 211). Analysis of ordinal categorical data. New York: Wiley.
Data about charges of homicide in the state of Florida.
Description
Counts of cases charged with homicide. The rows and columns indicate whether there was an additional charge of a felony occurring in addition to the homicide. The data is actually 3-dimensional. It is stored as 4 related matrices, each with the leading word "homicide_" The rest of the name gives the race of the defendant and the race of the victim, separated by an underscore
Usage
homicide_white_black
Format
## 'homicide_white_black' Each is a matrix with 3 rows and 3 columns. Rows are classification by police and columns are classification by the court/prosecutor. 1 = No felony 2 = Possible felony 2 = Felony
Source
Agresti, A. (1984, p. 211). Analysis of ordinal categorical data. New York: Wiley.
Data about charges of homicide in the state of Florida.
Description
Counts of cases charged with homicide. The rows and columns indicate whether there was an additional charge of a felony occurring in addition to the homicide. The data is actually 3-dimensional. It is stored as 4 related matrices, each with the leading word "homicide_" The rest of the name gives the race of the defendant and the race of the victim, separated by an underscore
Usage
homicide_white_white
Format
## 'homicide_white_white' Each is a matrix with 3 rows and 3 columns. Rows are classification by police and columns are classification by the court/prosecutor. 1 = No felony 2 = Possible felony 2 = Felony
Source
Agresti, A. (1984, p. 211). Analysis of ordinal categorical data. New York: Wiley.
Measures of men's hypothalamus taken from cadavers. First data set.
Description
Measures of men's hypothalamus taken from cadavers. First data set.
Usage
hypothalamus_1
Format
# 'hypothalamus_1' Each set is a dominance matrix (see e.g., Cliff 1996).
Source
Cliff, N. (1996), Ordinal methods for behavioral data analysis. Mahwah NJ: Lawrence Erlbaum.
Measures of men's hypothalamus taken from cadavers. Second data set.
Description
Measures of men's hypothalamus taken from cadavers. Second data set.
Usage
hypothalamus_2
Format
# 'hypothalamus_2' Each set is a dominance matrix (see e.g., Cliff 1996).
Source
Cliff, N. (1996), Ordinal methods for behavioral data analysis. Mahwah NJ: Lawrence Erlbaum.
Measures of interference in memory recall study.
Description
Measures are within subjects, comparing a control condition to two conditions with interference. Interference condition 1 v. interference condition 2
Usage
interference_12
Format
## 'interference_control_1', 'interference_control_2', 'interference_12' Within-persons dominance matrices.
Source
Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mahwah NJ: Lawrence Erlba
Measures of interference in memory recall study.
Description
Measures are within subjects, comparing a control condition to two conditions with interference. Control v. interference condition 1
Usage
interference_control_1
Format
## 'interference_control_1', 'interference_control_2', 'interference_12' Within-persons dominance matrices.
Source
Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mahwah NJ: Lawrence Erlbaum.
Measures of interference in memory recall study.
Description
Measures are within subjects, comparing a control condition to two conditions with interference. Control v. interference condition 2
Usage
interference_control_2
Format
## 'interference_control_1', 'interference_control_2', 'interference_12' Within-persons dominance matrices.
Source
Cliff, N. (1996). Ordinal methods for behavioral data analysis. Mahwah NJ: Lawrence Erlba
Tests whether a square matrix is invertible (non singular)
Description
from stackoverflow: https://stackoverflow.com/questions/24961983/how-to-check-if-a-matrix-has-an-inverse-in-the-r-language
Usage
is_invertible(X)
Arguments
X |
Matrix to be tested. It is assumed X is square |
Value
logical: TRUE if inversion succeeds, FALSE otherwise
Determines if its argument is not a valid number.
Description
Determines if its argument is not a valid number.
Usage
is_missing_or_infinite(x)
Arguments
x |
Numeric. Number of be evaluated |
Value
TRUE if is.na(), is.nan(), or is.infinite() returns TRUE. FALSE otherwise.
Computes Cohen's 1960 kappa coefficient
Description
Computes Cohen's 1960 kappa coefficient
Usage
kappa(n)
Arguments
n |
matrix of observed counts |
Value
kappa coefficient
Computes the likelihood ratio G^2 measure of fit.
Description
Computes the likelihood ratio G^2 measure of fit.
Usage
likelihood_ratio_chisq(n, pi, exclude_diagonal = FALSE)
Arguments
n |
Matrix of observed counts |
pi |
Matrix of same dimensions as n. Model-based matrix of predicted proportions |
exclude_diagonal |
logical. Should the diagonal cells of a square matrix be excluded from the computation. Default is FALSE. The effect of setting it to TRUE for non-square matrices may be unintuitive and should he avoided. |
Value
G^2
Function to load a data set written out using save().
Description
The first (should be the only) element read from the RData file is returned From: https://stackoverflow.com/questions/5577221/how-can-i-load-an-object-into-a-variable-name-that-i-specify-from-an-r-data-file
Usage
loadRData(file_name)
Arguments
file_name |
Character. Name of the file containing the RData |
Details
usage x <- loadRData(file_name="")
Value
the first object from the restored RData
Computes the logs of the cell frequencies.
Description
In the case of an observed 0, epsilon is inserted into the cell before the log is taken.
Usage
log_Linear_create_log_n(n, epsilon = 1e-06, all_cells = FALSE)
Arguments
n |
matrix of cell counts |
epsilon |
amount to be inserted into cell with observed 0. |
all_cells |
add epsilon to all cells or just those with 0 observed frequencies |
Value
a list containing: log_n – a vector of log frequencies and dat – modified version of the cell counts data
Computes the multinomial log(likelihood).
Description
Computes the multinomial log(likelihood).
Usage
log_likelihood(n, pi, exclude_diagonal = FALSE)
Arguments
n |
Matrix of observed counts |
pi |
Matrix of same dimensions as n. Model-based matrix of predicted proportions |
exclude_diagonal |
logical. Should diagonal cells of square matrix be excluded from the computation? Default is FALSE. The effect of setting it to TRUE for non-square matrices may be unintuitive and should he avoided. |
Value
log(likelihood)
Adds indicator variables for the diagonal cells in table n.
Description
Adds indicator variables for the diagonal cells in table n.
Usage
log_linear_add_all_diagonals(n, x)
Arguments
n |
the matrix of observed counts |
x |
the design matrix to be augmented |
Value
new design matrix with nrow(n) columns added. The columns are all 0 unless the row corresponds to a diagonal cell in n, in which case the entry is 1
Examples
x <- log_linear_main_effect_design(vision_data)
x_prime <- log_linear_add_all_diagonals(vision_data, x)
Appends a column to an existing design matrix.
Description
Takes the design matrix provided and appends the new column
Usage
log_linear_append_column(x, x_new, position = ncol(x) + 1)
Arguments
x |
the original design matrix |
x_new |
the column to be appended |
position |
column index within the new matrix for the new column. Defaults to last position = appending the column |
Value
the new design matrix
Examples
x <- log_linear_main_effect_design(vision_data)
new_column <- c(1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1)
x_prime <- log_linear_append_column(x, new_column)
Creates missing column names
Description
Creates missing column names
Usage
log_linear_create_coefficient_names(x, n, effect_names = NULL)
Arguments
x |
the design matrix being modified |
n |
the matrix of observed counts |
effect_names |
user specified names to be applied to effects after the intercept and main effects. Default is NULL |
Value
vector of names to apply to x
Creates a vector containing the linear-by-linear vector.
Description
Uses the ordinal ranks (1, 2, ..., nrow(n)) as data.
Usage
log_linear_create_linear_by_linear(n, centered = FALSE)
Arguments
n |
the matrix of observed cell counts |
centered |
should the variables be centered before the product is computed |
Value
a vector containing the new variable
Examples
linear <- log_linear_create_linear_by_linear(vision_data)
x <- log_linear_equal_weight_agreement_design(vision_data)
x_prime <- log_linear_append_column(x, linear)
Creates design matrix for model with main effects and a single agreement parameter delta.
Description
The model has main effects for rows and for columns, plus an additional parameter for the agreement (diagonal) cells.
Usage
log_linear_equal_weight_agreement_design(n, n_raters = 2)
Arguments
n |
the matrix of cell counts |
n_raters |
number of raters. Currently only 2 (the default) are supported. This is an extension point for future work. |
Value
design matrix for the model
Examples
x <- log_linear_equal_weight_agreement_design(vision_data)
Fits a log-linear model to the data provided, using the design matrix provided. Names for the effects will be "rows1", "cols1" etc. If there are remaining entries, they can be specified as the "effect_names" character vector. This function is a wrapper around a call to glm() that handles some of the details of the call and packages the output in a more convenient form.
Description
Fits a log-linear model to the data provided, using the design matrix provided. Names for the effects will be "rows1", "cols1" etc. If there are remaining entries, they can be specified as the "effect_names" character vector. This function is a wrapper around a call to glm() that handles some of the details of the call and packages the output in a more convenient form.
Usage
log_linear_fit(n, x, effect_names = NULL)
Arguments
n |
matrix of observed counts to be fit |
x |
design matrix for predictor variables |
effect_names |
character vector of additional names to apply to the columns of x The default is NULL, in which case the columns will be labeled "model1" etc. |
Value
a list containing x: the design matrix beta: the regression parameters se: the vector of standard errors g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Design matrix for baseline independence model with main effects for rows and columns.
Description
It is intended as a straw-man model as it assumes no agreement beyond chance.
Usage
log_linear_main_effect_design(n, n_raters = 2)
Arguments
n |
the matrix of cell counts |
n_raters |
number of raters. Currently only 2 (the default) are supported. This is an extension point for future work. |
Value
the design matrix for the model
Examples
x <- log_linear_main_effect_design(vision_data)
Converts a matrix of data into a vector suitable for use in analysis with the design matrices created. Unlike simply calling vector() on the matrix the resulting vector is organized by rows, then columns. This order corresponds to the order in the design matrix.
Description
Converts a matrix of data into a vector suitable for use in analysis with the design matrices created. Unlike simply calling vector() on the matrix the resulting vector is organized by rows, then columns. This order corresponds to the order in the design matrix.
Usage
log_linear_matrix_to_vector(dat)
Arguments
dat |
the matrix to be converted a vector |
Value
a vector suitable to use as dependent variable, e.g. in a call to glm()
Creates the design matrix for a quasi-symmetry design
Description
Creates the design matrix for a quasi-symmetry design
Usage
log_linear_quasi_symmetry_model_design(n)
Arguments
n |
matrix of observed counts |
Value
design matrix for quasi-symmetry design
Removes a column from an existing design matrix.
Description
Takes the design matrix provided and removes the column in the position specified
Usage
log_linear_remove_column(x, position = ncol(x))
Arguments
x |
the original design matrix |
position |
column index within the new matrix for the new column. Defaults to last position |
Value
the new design matrix
Examples
x <- log_linear_main_effect_design(vision_data)
linear <- log_linear_create_linear_by_linear(vision_data)
x_prime <- log_linear_append_column(x, linear)
x_again <- log_linear_remove_column(x_prime, ncol(x_prime))
Creates design matrix for symmetry model.
Description
Creates design matrix for symmetry model.
Usage
log_linear_symmetry_design(n)
Arguments
n |
matrix of observed counts |
Value
design matrix for the model
Computes the log-odds (logit) for the value provided
Description
Computes the log-odds (logit) for the value provided
Usage
logit(p)
Arguments
p |
Numeric. Assumed to lie in interval(0, 1) |
Value
log(p / (1.0 - p))
Relationship between child's mental health and parents' socioeconomic status.
Description
Rows are child's mental health (ranging from 1 = well to 4 = impaired), and columns are parents' socioeconomic status, A - F.
Usage
mental_health
Format
## 'mental_health' A matrix with 4 rows and 6 columns
Source
Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories.
Computes the column association values theta-hat
Description
Computes the column association values theta-hat
Usage
model_i_column_theta(fHat)
Arguments
fHat |
matrix of model-based expected counts |
Value
thetaHat vector of association parameters
Gets the overall effects for Model I.
Description
Gets the overall effects for Model I.
Usage
model_i_effects(result)
Arguments
result |
a Model I result object |
Value
a list containing theta: the overall association zeta_i_dot: row effects for association zeta_dot_j: column effects for association
Computes model-based expected cell counts for Model I
Description
Computes model-based expected cell counts for Model I
Usage
model_i_fHat(alpha, beta, gamma, delta)
Arguments
alpha |
row effects |
beta |
column effects |
gamma |
row location weights |
delta |
column location weights |
Value
matrix of model-based expected counts
Normalizes pi(fHat) to sum to 1.0. If exclude_diagonal is TRUE, the sum of the off-diagonal terms sums to 1.0.
Description
Normalizes pi(fHat) to sum to 1.0. If exclude_diagonal is TRUE, the sum of the off-diagonal terms sums to 1.0.
Usage
model_i_normalize_fHat(fHat, exclude_diagonal = FALSE)
Arguments
fHat |
matrix of model-based cell frequencies |
exclude_diagonal |
logical. Should the cells on the main diagonal be excluded? Default is FALSE, include all cells |
Value
matrix of model-based proportions pi
Computes the table of adjacent odds-ratios theta-hat.
Description
Computes the table of adjacent odds-ratios theta-hat.
Usage
model_i_row_column_odds_ratios(fHat)
Arguments
fHat |
matrix of model-based expected counts |
Value
thetaHat matrix of adjacent odds-ratios
Computes the row association values theta-hat
Description
Computes the row association values theta-hat
Usage
model_i_row_theta(fHat)
Arguments
fHat |
matrix of model-based expected counts |
Value
thetaHat vector of association parameters
Gets the Model I* effects.
Description
Gets the Model I* effects.
Usage
model_i_star_effects(result)
Arguments
result |
a Model I* effect object |
Value
a list containing theta: the overall association zeta: the row/column effect
Computes expected frequencies for Model I*
Description
Computes expected frequencies for Model I*
Usage
model_i_star_fHat(alpha, beta, theta)
Arguments
alpha |
row effect parameters |
beta |
column effect parameters |
theta |
row/column parameters |
Value
matrix of model-based expected cell counts
Updates the row/column parameters for Model I*.
Description
Updates the row/column parameters for Model I*.
Usage
model_i_star_update_theta(theta, n, fHat, exclude_diagonal = FALSE)
Arguments
theta |
vector of estimated row/column effects |
n |
matrix of observed counts |
fHat |
matrix of model-based expected frequencies |
exclude_diagonal |
should the cells of the main diagonal be excluded? Default is FALSE, include all cells |
Value
new value of theta vector
Computes crude starting values for Model I.
Description
Computes crude starting values for Model I.
Usage
model_i_starting_values(n)
Arguments
n |
matrix of observed counts |
Value
a list containing alpha: vector of row parameters beta: vector of column parameters gamma: vector of row locations delta: vector of column locations
Updates the estimate of the alpha vector for Model I
Description
Updates the estimate of the alpha vector for Model I
Usage
model_i_update_alpha(alpha, n, fHat, exclude_diagonal = FALSE)
Arguments
alpha |
current estimate of beta |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical. Should the diagonal be excluded from the computation? Default is FALSE, use all cells. |
Value
updated estimate of alpha vector
Updates the estimate of the beta vector for Model I
Description
Updates the estimate of the beta vector for Model I
Usage
model_i_update_beta(beta, n, fHat, exclude_diagonal = FALSE)
Arguments
beta |
current estimate of alpha |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical. Should the diagonal be excluded from the computation? Default is FALSE, use all cells |
Value
updated estimate of beta vector
Updates the estimate of the delta vector for Model I
Description
Updates the estimate of the delta vector for Model I
Usage
model_i_update_delta(delta, n, fHat, exclude_diagonal = FALSE)
Arguments
delta |
current estimate of delta |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical. Should the diagonal be excluded from the computation? Default is FALSE, use all cells |
Value
updated estimate of delta vector
Updates the estimate of the gamma vector for Model I
Description
Updates the estimate of the gamma vector for Model I
Usage
model_i_update_gamma(gamma, n, fHat, exclude_diagonal = FALSE)
Arguments
gamma |
current estimate of gamma |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical. Should the diagonal be excluded from the computation? Default is FALSE, use all cells |
Value
updated estimate of gamma vector
Computes the overall association theta and the row and column effects zeta
Description
Computes the overall association theta and the row and column effects zeta
Usage
model_i_zeta(odds)
Arguments
odds |
matrix of adjacent odds-ratios |
Value
a list containing theta: the overall association zeta_i_dot: row effects for association zeta_dot_j: column effects for association
Gets the effects phi, ksi_i_dot and ksi_dot_j for Model II results.
Description
Gets the effects phi, ksi_i_dot and ksi_dot_j for Model II results.
Usage
model_ii_effects(result)
Arguments
result |
a result object from Model II |
Value
a list containing: phi: the overall effect ksi_i_dot: the row effects ksi_dot_j: the column effects
Computes expected counts for Model II
Description
Computes expected counts for Model II
Usage
model_ii_fHat(alpha, beta, rho, sigma)
Arguments
alpha |
row effects |
beta |
column effects |
rho |
row locations |
sigma |
column locations |
Value
matrix of model-based expected counts
Gets the effects phi, ksi_i_dot and ksi_dot_j for Model II matrix of odds-ratios.
Description
Gets the effects phi, ksi_i_dot and ksi_dot_j for Model II matrix of odds-ratios.
Usage
model_ii_ksi(odds)
Arguments
odds |
matrix of adjacent odds-ratios |
Value
a list containing: phi: the overall effect in log metric ksi_i_dot: the row effects ksi_dot_j: the column effects
Gets the effects for Model II*
Description
Gets the effects for Model II*
Usage
model_ii_star_effects(result)
Arguments
result |
a Model II* result object |
Value
a list containing phi: common effect in log metric ksi: vector of ksi parameters
Computes expected counts for Model II*
Description
Computes expected counts for Model II*
Usage
model_ii_star_fHat(alpha, beta, phi)
Arguments
alpha |
row effects |
beta |
column effects |
phi |
row/column locations |
Value
matrix of model-based expected counts
Updates estimate of phi vector
Description
Updates estimate of phi vector
Usage
model_ii_star_update_phi(n, fHat, mu, phi, exclude_diagonal = FALSE)
Arguments
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
mu |
alternative row coefficients |
phi |
vector of column location parameters |
exclude_diagonal |
logical, Should the cells on the main diagonal be excluded? Default is FALSE, use all cells |
Value
list containing: phi: updated estimate of the phi vector mu: updated estimate of vector mu
Computes crude starting values for Model II
Description
Computes crude starting values for Model II
Usage
model_ii_starting_values(n)
Arguments
n |
matrix of observed counts |
Value
a list containing alpha: vector of row parameters beta: vector of column parameters rho: row coefficients sigma: column coefficients mu: alternative row coefficients nu: alternative column coefficients
Updates the estimate of the alpha vector for Model II
Description
Updates the estimate of the alpha vector for Model II
Usage
model_ii_update_alpha(alpha, n, fHat, exclude_diagonal = FALSE)
Arguments
alpha |
current estimate of alpha |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical, Should the cells on the main diagonal be excluded? Default is FALSE, use all cells |
Value
updated estimate of alpha vector
Updates the estimate of the beta vector for Model II
Description
Updates the estimate of the beta vector for Model II
Usage
model_ii_update_beta(beta, n, fHat, exclude_diagonal = FALSE)
Arguments
beta |
current estimate of beta |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical, Should the cells on the main diagonal be excluded? Default is FALSE, use all cells |
Value
updated estimate of beta vector
Updates the estimate of the rho vector for Model II
Description
Updates the estimate of the rho vector for Model II
Usage
model_ii_update_rho(n, fHat, mu, sigma, exclude_diagonal = FALSE)
Arguments
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
mu |
alternative row coefficients |
sigma |
vector of column location parameters |
exclude_diagonal |
logical, Should the cells on the main diagonal be excluded? Default is FALSE, use all cells |
Value
updated estimate of alpha vector
Updates the estimate of the sigma vector for Model II
Description
Updates the estimate of the sigma vector for Model II
Usage
model_ii_update_sigma(n, fHat, nu, rho, exclude_diagonal = FALSE)
Arguments
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
nu |
vector of column coefficients |
rho |
vector of row location parameters |
exclude_diagonal |
logical, Should the cells on the main diagonal be excluded? Default is FALSE, use all cells |
Value
updated estimate of sigma vector
Movie ratings by two film critics, Siskel and Ebert.
Description
Movie ratings by two film critics, Siskel and Ebert.
Usage
movies
Format
## 'movies' A matrix with 3 rows and 3 columns 1 is con 2 is mixed 3 is pro
Source
https://online.stat.psu.edu/stat504/lesson/11/11.3
Agreement between two clinicians on presence of multiple sclerosis based on file.
Description
See companion winnipeg_data.
Usage
new_orleans_data
Format
## 'new_orleans_data' A matrix with 4 rows and 4 columns Ratings range from definite presence of disease to definite absence.
Source
???
Computes expected counts for null association model
Description
Computes expected counts for null association model
Usage
null_association_fHat(alpha, beta)
Arguments
alpha |
row effects |
beta |
column effects |
Value
matrix of model-based expected counts
Cross tabulation of father's employment status with son's employment status.
Description
Higher numbers correspond to higher status occupation
Usage
occupational_status
Format
## 'occupational_status' A matrix with 6 rows and 6 columns
Source
???
Interrater agreement of two psychologists' ratings of paranoia.
Description
Severity corresponds to level 1 low 3 high
Usage
paranoia
Format
## 'paranoia' A matrix with 3 rows and 3 columns.
Source
von Eye, A. & Mun, E. Y. (2005, p. 70). Analyzing rater agreement: Manifest variable methods. Mahwah, NJ: Lawrence Erlbaum.
Computes the Pearson X^2 statistic.
Description
Computes the Pearson X^2 statistic.
Usage
pearson_chisq(n, pi, exclude_diagonal = FALSE)
Arguments
n |
Matrix of observed counts |
pi |
Matrix with same dimensions as n. Model-based matrix of predicted proportions |
exclude_diagonal |
logical. Should diagonal cells of square matrix be excluded from the computation? Default is FALSE. The effect of setting it to TRUE for non-square matrices may be unintuitive and should he avoided. |
Value
X^2
Interrater agreement of two radiologists diagnosis of severity of carcinoma.
Description
The data contains a comparison vector of (simulated) covariate data.
Usage
radiology
Format
## 'radiology' 'covariate' A matrix with 4 rows and 4 columns, and a vector of 16 elements.
Source
von Eye, A. & Mun, E. Y. (2005, p. 60). Analyzing rater agreement: Manifest variable methods. Mahwah, NJ: Lawrence Erlbaum.
Social mobility data with father's occupational social status and son's occupational social status.
Description
Social mobility data with father's occupational social status and son's occupational social status.
Usage
social_status
Format
## 'social_status' A matrix with 7 rows and 7 columns
Source
Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74(367), 537-552.
Social mobility data with father's occupational social status and son's occupational social status. * categories instead of 7 in social status..
Description
Social mobility data with father's occupational social status and son's occupational social status. * categories instead of 7 in social status..
Usage
social_status2
Format
## 'social_status2' A matrix with 8 rows and 8 columns
Source
Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74(367), 537-552.
Taste ratings
Description
Taste ratings
Usage
taste
Format
## 'taste' A matrix with 5 rows and 5 columns.
Source
McCullagh, P. (1980, p. 119). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42(2), 109-142.
Teachers ratings of their students intelligence.
Description
Interrater agreement data for two teachers asked to rate the intelligence of their students.
Usage
teachers
Format
## 'teachers' A matrix with 4 rows and 4 columns. Higher scores correspond to higher estimated intelligence.
Source
von Eye, A. & Mun, E. Y. (2005, p. 36). Analyzing rater agreement: Manifest variable methods. Mahwah, NJ: Lawrence Erlbaum.
Style of teachers rated by supervisors
Description
Ratings of style of teaching by supervisors. 1 indicates Authoritarian, 2 indicates Democratic, 3 indicates Permissive.
Usage
teaching_style
Format
An object of class matrix (inherits from array) with 3 rows and 3 columns.
Details
@format ## 'teaching_style' A matrix with 3 rows and 3 columns.
@source Agresti, A. (1989). An agreement model with kappa as parameter. Statistics & Probability Letters, 7, 271-273.
Relationship between size of child's tonsils and their status as a carrier of a disease.
Description
Relationship between size of child's tonsils and their status as a carrier of a disease.
Usage
tonsils
Format
## 'tonsils' A matrix with 2 rows and 3 columns. Rows are disease status and columns are ratings of tonsil size.
Source
McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42(2), 109-142.
Interrater agreement of two journalists' evaluation of proposed TV programs.
Description
Ratings go from low to high probability of the show's success.
Usage
tv
Format
## 'tv' A matrix of 6 rows and 6 columns.
Source
von Eye, A. & Mun, E. Y. (2005, p. 56). Analyzing rater agreement: Manifest variable methods. Mahwah, NJ: Lawrence Erlbaum.
Computes expected counts for uniform association model
Description
Computes expected counts for uniform association model
Usage
uniform_association_fHat(alpha, beta, theta)
Arguments
alpha |
row effects |
beta |
column effects |
theta |
association parameter |
Value
matrix of model-based expected counts
Updates estimate of theta value of the uniform association model
Description
Updates estimate of theta value of the uniform association model
Usage
uniform_association_update_theta(theta, n, fHat, exclude_diagonal = FALSE)
Arguments
theta |
current estimate of theta |
n |
matrix of observed counts |
fHat |
current model-based counts for each cell |
exclude_diagonal |
logical. Should the cells of the main diagonal be excluded from the computations? Defualt is FALSE, include all cells. |
Value
updated estimate of theta parameter
Computes the sampling variance of kappa.
Description
Formulas are from the paper by Fleiss,J. L., Cohen, J., & Everitt, B. S. (1969). Large sample standard errors of kappa and weighted kappa. Two results are returned in a list. var_kappa0 is the null case and would be used for testing the hypothesis that kappa = 0. The second is var_kappa and is for the non-null case, such as constructing CI for estimated kappa. Not that both are in the variance metric. Take the square root to get the standard error.
Usage
var_kappa(n)
Arguments
n |
matrix of observe counts |
Value
a list containing; var_kappa0: variance for the null case var_kappa: variance for the non-null case.
Computes the sampling variance of weighted kappa.
Description
Formulas are from the paper by Fleiss,J. L., Cohen, J., & Everitt, B. S. (1969). Large sample standard errors of kappa and weighted kappa. Two results are returned in a list. var_kappa0 is the null case and would be used for testing the hypothesis that kappa = 0. The second is var_kappa and is for the non-null case, such as constructing CI for estimated kappa. Not that both are in the variance metric. Take the square root to get the standard error.
Usage
var_weighted_kappa(n, w)
Arguments
n |
matrix of observe counts |
w |
matrix of penalty weights |
Value
a list containing; var_kappa0: variance for the null case var_kappa: variance for the non-null case.
Visual acuity of women factory workers.
Description
Measurements of unaided visual acuity for women working at the Royal Ordinance factories 1943-1946. Rows are right eye, columns are left eye. 1 indicates best vision, 4 is poorest.
Usage
vision_data
Format
## 'visual_data' A matrix with 4 rows and 4 columns.
Source
Stuart, A. (1953). The estimation and comparison of strengths of association in contingency tables. Biometrika, 40(1/2), 105-110.
Visual acuity of men factory workers.
Description
Measurements of unaided visual acuity for men working at the Royal Ordinance factories 1943-1946. Rows are right eye, columns are left eye. 1 indicates best vision, 4 is poorest.
Usage
vision_data_men
Format
## 'visual_data_men' A matrix with 4 rows and 4 columns.
Source
Stuart, A. (1953). The estimation and comparison of strengths of association in contingency tables. Biometrika, 40(1/2), 105-110.
Fits the diagonal effects model, where each category has its own parameter delta[k].
Description
Fits the diagonal effects model, where each category has its own parameter delta[k].
Usage
von_Eye_diagonal(n)
Arguments
n |
the matrix of observed counts |
Value
a list containing beta: the regression parameters. delta parameters are the final elements of beta g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Fits the diagonal effects model, where each category has its own parameter delta[k], while also incorporating a linear-by-linear term.
Description
Fits the diagonal effects model, where each category has its own parameter delta[k], while also incorporating a linear-by-linear term.
Usage
von_Eye_diagonal_linear_by_linear(n, center = TRUE)
Arguments
n |
the matrix of observed counts |
center |
should the linear-by-linear components be centered to have mean 0? Default is TRUE |
Value
a list containing beta: the regression parameters. delta parameters come after rows and columns and finally the linear-by-linear term g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Fits the diagonal effects model, where there is a single delta parameter for all categories, while also incorporating a linear-by-linear term.
Description
Fits the diagonal effects model, where there is a single delta parameter for all categories, while also incorporating a linear-by-linear term.
Usage
von_Eye_equal_weight_diagonal_linear(n, center = TRUE)
Arguments
n |
the matrix of observed counts |
center |
should the linear-by-linear components be centered to have mean 0? Default is TRUE |
Value
a list containing beta: the regression parameters. delta parameters come after rows and columns and finally the linear-by-linear term g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Fits the equal weighted diagonal model, where the diagonals all have an additional parameter delta, with the constraint that delta is equal across all categories.
Description
Fits the equal weighted diagonal model, where the diagonals all have an additional parameter delta, with the constraint that delta is equal across all categories.
Usage
von_Eye_equal_weighted_diagonal(n)
Arguments
n |
the matrix of observed counts |
Value
a list containing beta: the regression parameters g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Fits the basic independent rows and columns model incorporating a linear-by-linear term.
Description
Fits the basic independent rows and columns model incorporating a linear-by-linear term.
Usage
von_Eye_linear_by_linear(n, center = TRUE)
Arguments
n |
matrix of observed counts |
center |
should the linear-by-linear components be centered to have mean 0? Default is TRUE |
Value
a list containing beta: the regression parameters. The linear-by-linear parameter is last g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Fits the base model with only independent row and column effects.
Description
Fits the base model with only independent row and column effects.
Usage
von_Eye_main_effect(n)
Arguments
n |
the matrix of observed counts |
Value
a list containing beta: the regression parameters g_squared: G^2 fit measure chisq: X^2 fit measure df: degrees of freedom expected: matrix of expected frequencies
Creates design matrix for weight be response category model.
Description
The model specifies main effects for row and column, and a parameter for the agreement (diagonal) cells. This takes a design matrix for that model and applies domain-specific weights to the agreement parameters.
Usage
von_Eye_weight_by_response_category_design(n, x, w, n_raters = 2)
Arguments
n |
the matrix of cell counts |
x |
the original design matrix. |
w |
the vector of weights to apply to the agreement cells. Should have same number of entries as the number of diagonal elements (number of rows & of columns) |
n_raters |
number of raters. Currently only 2 (the default) are supported. This is an extension point for future work. |
Value
new design matrix with weights applied to the agreement cells.
Computes the weighted covariance
Description
Computes covariance between x and y using case weights in w
Usage
weighted_cov(x, y, w, use_df = TRUE)
Arguments
x |
Numeric vector. First variable |
y |
Numeric vector. Second variable |
w |
Numeric vector. case weights |
use_df |
Logical. should the divisor be sum of weights - 1 (TRUE) or N - 1 (FALSE) |
Value
the weighted covariance between x and y
Computes Cohen's 1968 weighted kappa coefficient
Description
Computes Cohen's 1968 weighted kappa coefficient
Usage
weighted_kappa(n, w = diag(rep(1, nrow(n))), quadratic = FALSE)
Arguments
n |
matrix of observed counts |
w |
matrix of weights. Defaults to identity matrix |
quadratic |
logical. Should quadratic weights be used? Default is FALSE. If TRUE, quadratic weights are used. These override the values in w. If FALSE, weights in w are used |
Value
value of weighted kappa
Computes the weighted variance
Description
Computes variance between x and y using case weights in w
Usage
weighted_var(x, w, use_df = TRUE)
Arguments
x |
Numeric vector. First variable |
w |
Numeric vector. Case weights |
use_df |
Logical. Should the divisor be sum of weights - 1 (TRUE) or N - 1 (FALSE) |
Value
the weighted covariance between x and y
Agreement between two clinicians on presence of multiple sclerosis based on file.
Description
See companion new_orleans_data.
Usage
winnipeg_data
Format
## 'winnipeg_data' A matrix with 4 rows and 4 columns Ratings range from definite presence of disease to definite absence.
Source
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