# SBMTrees: Introduction and Usage

# Introduction

The R package SBMTrees (Sequential Imputation with Bayesian Trees Mixed-Effects Models) provides a powerful Bayesian non-parametric framework for imputing missing covariates and outcomes in longitudinal data under the Missing at Random (MAR) assumption. The package leverages centralized Dirichlet Process (CDP) Normal Mixture priors to model non-normal random effects and errors, offering robust handling of model misspecification and capturing complex relationships in longitudinal data.

This vignette introduces the key functionalities of the package, including:

- Prediction: Using `BMTrees_prediction` to predict longitudinal outcomes.
- Imputation: Using `sequential_imputation` for imputing missing values.

# Install and load the package

    library(SBMTrees)
    library(mitml)
    #> *** This is beta software. Please report any bugs!
    #> *** See the NEWS file for recent changes.
    library(lme4)
    #> Loading required package: Matrix

# Prediction

The `BMTrees_prediction` function is used to predict longitudinal outcomes based on Bayesian Mixed-Effects Models. Below is an example of how to generate data, split it into training and testing datasets, and run predictions.

    # Simulate data
    data <- simulation_prediction(
      n_subject = 100,
      seed = 1234,
      nonlinear = TRUE,
      nonrandeff = TRUE,
      nonresidual = TRUE
    )
    #> Warning: No records are assigned to patterns 16, 18, 19, 40, 41. These patterns
    #> will not be generated. Consider reducing the number of patterns or increasing
    #> the dataset size.

    # Extract training and testing datasets
    X_train <- data$X_train
    Y_train <- data$Y_train
    Z_train <- data$Z_train
    subject_id_train <- data$subject_id_train

    X_test <- data$X_test
    Z_test <- data$Z_test
    subject_id_test <- data$subject_id_test

    Y_test_true <- data$Y_test_true

We then run the prediction model `BMTrees`, with 3 burn-in iterations and 4 posterior samples. The number of burn-in and posterior iterations should be increase to 3000 and 4000, respectively. Here we only use the small numbers to simply debug.

    # Fit the predictive model
    model <- BMTrees_prediction(
      X_train, Y_train, Z_train, subject_id_train,
      X_test, Z_test, subject_id_test,
      model = "BMTrees",
      binary = FALSE,
      nburn = 3L,
      npost = 4L,
      skip = 1L,
      verbose = FALSE,
      seed = 1234
    )

    # Posterior expectation for the testing dataset
    posterior_predictions <- model$post_predictive_y_test
    head(colMeans(posterior_predictions))
    #> [1]   58.09237  -49.36526  -71.31737 -109.58288  -79.30009  -32.51103

To evaluate the model’s predictive performance, we compute the Mean Absolute Error (MAE), and the Mean Square Error (MSE). We also calculate the 95% posterior predictive intervals to check coverage, and visualize the results using scatterplots of true versus predicted values.

    point_predictions = colMeans(posterior_predictions)

    # Compute MAE
    mae <- mean(abs(point_predictions - Y_test_true))
    cat("Mean Absolute Error (MAE):", mae, "\n")
    #> Mean Absolute Error (MAE): 15.15302

    # Compute MSE
    mse <- mean((point_predictions - Y_test_true)^2)
    cat("Mean Squared Error (MSE):", mse, "\n")
    #> Mean Squared Error (MSE): 366.9497

    # Compute 95% credible intervals
    lower_bounds <- apply(posterior_predictions, 2, quantile, probs = 0.025)
    upper_bounds <- apply(posterior_predictions, 2, quantile, probs = 0.975)

    # Check if true values fall within the intervals
    coverage <- mean(Y_test_true >= lower_bounds & Y_test_true <= upper_bounds)
    cat("95% Posterior Predictive Interval Coverage:", coverage * 100, "%\n")
    #> 95% Posterior Predictive Interval Coverage: 11.73913 %

    plot(Y_test_true, point_predictions,
         xlab = "True Values",
         ylab = "Predicted Values",
         main = "True vs Predicted Values")
    abline(0, 1, col = "red") # Add a 45-degree reference line

# Multiple Imputation

The `sequential_imputation` function is used to impute missing covariates and outcomes in longitudinal data. Below is an example of how to generate longitudinal data with MAR missing, and run imputations.

    # Simulate data with missing values
    data <- simulation_imputation(
      n_subject = 100,
      seed = 1234,
      nonrandeff = TRUE,
      nonresidual = TRUE,
      alligned = FALSE
    )

    # Extract components of the dataset
    X_mis <- data$X_mis   # Covariates with missing values
    Y_mis <- data$Y_mis   # Outcomes with missing values
    Z <- data$Z           # Random predictors
    subject_id <- data$subject_id

We then run the sequential imputation with `BMTrees`, with 3 burn-in iterations, 40 posterior iterations, and sample one posterior sample for every 2 posterior iterations, ensuring 20 multiply-imputed sets are generated. The number of burn-in and posterior iterations should be increase to 3000 and 4000, respectively. Here we only use the small numbers to simply debug.

    # Run the sequential imputation
    imputed_model <- sequential_imputation(
      X_mis, Y_mis, Z, subject_id,
      type = rep(0, ncol(X_mis)),
      binary_outcome = FALSE,
      model = "BMTrees",
      nburn = 3L,
      npost = 40L,
      skip = 2L,
      verbose = FALSE,
      seed = 1234
    )
    #> reordering: new covariates order is X1  X2  X3  X4  X5  X6  X8  X7  X9
    #> Start to initialize imputed missing data by LOCF and NOCB.
    #> Completed.
    #> Start to impute using Longitudinal Sequential Imputation with:
    #> BMTrees
    #> Outcome variable has missing values
    #> Start initializing models
    #>
    #>
    #> Complete initialization
    #>
    #> Finish imputation with 20 imputed sets

    # Extract imputed data
    imputed_data <- imputed_model$imputed_data
    dim(imputed_data) # Dimensions: posterior samples x observations x variables
    #> [1]  20 600  10

To evaluate the model’s imputation performance, we apply Rubin`s rule to estimate linear mixed-effects model on the multiply-imputed sets.

    # create structure which can be used in mitml
    MI_data = list()
    for (i in 1:dim(imputed_data)[1]) {
      MI_data[[i]] = cbind(as.data.frame(imputed_data[i,,]), Z, subject_id)
      colnames(MI_data[[i]]) = c(colnames(X_mis), "Y", "Z1", "Z2", "Z3", "subject_id")
    }
    MI_data <- as.mitml.list(MI_data)  # Replace with actual datasets
    # Fit the model on each imputed dataset
    lmm_results <- with(MI_data, lmer(Y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + (0 + Z1 + Z2 + Z3 | subject_id)))
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.0129595 (tol = 0.002, component 1)
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.00492905 (tol = 0.002, component 1)
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.0103055 (tol = 0.002, component 1)
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.00845373 (tol = 0.002, component 1)
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.0215275 (tol = 0.002, component 1)
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.00259377 (tol = 0.002, component 1)
    #> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
    #> Model failed to converge with max|grad| = 0.00296298 (tol = 0.002, component 1)
    #> boundary (singular) fit: see help('isSingular')
    #> boundary (singular) fit: see help('isSingular')
    #> boundary (singular) fit: see help('isSingular')

    # Pool fixed effects using Rubin's Rules
    pooled_results <- testEstimates(lmm_results)

    # Print pooled results
    print(pooled_results)
    #>
    #> Call:
    #>
    #> testEstimates(model = lmm_results)
    #>
    #> Final parameter estimates and inferences obtained from 20 imputed data sets.
    #>
    #>              Estimate Std.Error   t.value        df   P(>|t|)       RIV       FMI
    #> (Intercept)    -6.955     1.201    -5.791    93.387     0.000     0.822     0.462
    #> X1             -0.191     0.360    -0.531  1145.635     0.596     0.148     0.130
    #> X2             -0.306     0.446    -0.685    81.012     0.495     0.939     0.497
    #> X3              1.134     0.403     2.813   124.918     0.006     0.639     0.400
    #> X4              2.389     0.326     7.332    90.084     0.000     0.849     0.471
    #> X5             -0.588     0.445    -1.320    40.993     0.194     2.133     0.695
    #> X6              1.636     0.306     5.347   160.456     0.000     0.525     0.352
    #> X7             -0.691     0.122    -5.678   127.059     0.000     0.631     0.396
    #> X8              0.436     0.093     4.666    45.944     0.000     1.802     0.658
    #> X9             -0.664     0.082    -8.077    31.892     0.000     3.383     0.785
    #>
    #> Unadjusted hypothesis test as appropriate in larger samples.

# Summary

The SBMTrees package provides flexible tools for handling missing values and making predictions in longitudinal data. By leveraging Bayesian non-parametric methods, it effectively addresses challenges associated with model misspecification, non-normal random effects, and non-normal errors.

For further details, please refer to the package documentation and the paper: Nonparametric Bayesian Additive Regression Trees for Predicting and Handling Missing Covariates and Outcomes in Longitudinal Data.

# License

This vignette is part of the SBMTrees R package and is distributed under the terms of the GNU General Public License (GPL-2).