Fitting Nonlinear Mixed-Effects Regression Models
Launch the FRApp application with the command
The application automatically opens in the browser. While the application is running, it is not possible to have access to the console. To close the application and get access to the R console again, you have to close the application by clicking on the red stop button located in the top right corner of the frame of the R console.
The application allows for:
Loading the data
The FRApp application accepts the csv files format. By default, it uses the semicolon as the field separator and the period as the decimal separator, but you can select different separators from the drop-down menu.
The first line of the file must contain the variable names.
The Browse… button allows you to load your own data file to be analyzed. Check that your import is correct by looking on the right at the summary description of the variables and number of observations imported.
Please, double-check variable types:
the response variable must be numeric (num)
the explanatory variable, e.g. time, can be either integer (int) or numeric (num).
For response and explanatory variables, factor variable type is not allowed.
Alternatively, to showcase the functionality and usability of FRApp, the Load example data button allows you to load the dataset analyzed in the paper and automatically selects the specification of the final selected model.
In the case of the example data, the response variable y, the fluorescence intensity, and the explanatory variable, time, are numeric.
Estimating and comparing exponential mixed-effects models
The following list highlights the main characteristics of an exponential mixed-effects model that can be implemented in FRApp and within our FRAP example.
Model fitting is perfomed with the nlme
function.
Fixed effects
For each unit, the response
variable \(y\) is modeled through an
asymptotic regression function on time \(t\), that is, \[y_i =
Asym+(R0−Asym)exp(−exp(lrc)t_i)+\epsilon_i,\quad \text{for } i =
1,...,n\] This asymptotic function is characterized by three
parameters: \(R0\), the response \(y\) at time \(t =
0\), the horizontal asymptote \(Asym\) (reached for \(t \rightarrow \infty\)) and the log rate
constant \(lrc\).
FRApp: Begin by selecting the response variable and the explanatory variable from the drop-down menu.
Example data: In the context of our example analysis, the fluorescence intensity \(y\) was the response variable, and \(time\) was the explanatory variable. The model parameters had a biological interpretation in describing the actin cytoskeleton dynamics in spines. Specifically, \(R0\) represented the fluorescence intensity after photobleaching, \(Asym\) denoted the fluorescence intensity recovered in time, and \(lrc\) measured how fast the fluorescence intensity recovered.
Random effects
Mixed-effects models combine
fixed effects describing the population mean and random effects
accounting for variation among groups. When the data show a nested
structure, the hierarchy can be easily handled specifying random effects
for each level of the hierarchy. In the FRApp, random effects are
assumed to be independent and normally distributed.
FRApp: Choose the hierarchical structure levels, starting with the highest level. You can specify up to three hierarchical levels. For each specified hierarchical level, you must select at least one random effect. There is no a priori rule for constructing the random effect structure. You can begin by defining a structure that best captures the expected variability between groups. Alternatively, you can fit the initial model with all possible random effects and adjust it by removing unnecessary random effects, i.e. those with variability almost close to zero.
Example data: In our case, the parameters \(Asym\), \(lrc\), and \(R0\) were expected to vary across cultures, neurons, and spines, which were organized in a hierarchical structure, i.e., spines (level 3) within neurons (level 2) within cultures (level 1). Therefore, we could specify a random effect for each level-specific parameter to capture the variability among entities belonging to the same group. For instance, the random effect on \(Asym\) at the culture level described the variability among asymptotes of different cultures.
Variance and temporal dependence
When model
residuals, which are the difference between fitted and observed values,
show a non constant behavior, several variance functions can be
evaluated to account for their heteroscedasticity. The temporal
dependence among data can be also modeled by introducing an
autoregressive (AR) additional step.
FRApp: Select the variance function from the drop-down menu. The variance section allows you to specify different variance functions to model the heteroscedasticity of the error term. The default function is Constant, which represents the assumption of constant error variance. The temporal dependence section allows you to select the order of the AR model on the error term. The default value is 0, which corresponds to the uncorrelated errors structure.
Example data: Through visual inspection of the residuals, we detected heteroscedasticity, which we modeled with a non-constant variance function. Our example data were recorded over time, so we anticipated the presence of temporal dependence among the data.
Interactions
Interactions with an experimental
condition can be included with different model components: random
effects, fixed effects, and the variance structure. Interactions allow
to estimate specific effects for each experimental condition.
FRApp: Select the variable representing the experimental condition from the drop-down menu and choose the interaction to be included in the model.
Example data: The experimental condition, here indicated as genetic.id, was the main effect of interest in the data analyzed in the paper.
It’s advisable to move from simplest structures towards greater model complexity by exploring model diagnostics including residual plots, random effects normality, and serial correlation.
You can save models for model comparison. Saved models must have different names. When multiple models are saved, the table at the bottom of the page displays comparisons among saved models as based on AIC, BIC, and likelihood ratio tests.
The Reset model list button allows you to delete the saved models from the list.
Printing a model report and exporting the data and the model
From the drop-down menu in the Download section, you can select a saved model from the list and download a report and some model’s objects.
You can download the model report using the Download report button. It includes the output shown in the app after model fitting, i.e. model summary, diagnostic plots (scatterplot of residuals vs. estimated values and quantile-quantile plot of the residuals), and the approximate 95% confidence intervals.
The Download RData button allows you to export an RData file containing six objects:
data: the dataset in the format used for the analysis
fit: the output of the estimated model
pred: the values estimated by the model
CI: the approximate 95% confidence intervals of the parameters of interest
resid: the residuals of the model
raneff: the random effects available at the different hierarchical levels.
You can open the Rdata file with RStudio, and use the included objects to create further plots including curves estimated at the different hierarchical levels (using the pred object), plots of residuals (with resid) or random effects (with raneff).
