# TreeBUGS: Introduction to Hierarchical MPT Modeling

#### Daniel W. Heck, Nina R. Arnold, & Denis Arnold

#### 2022-09-06

## General Procedure of Using TreeBUGS

In the most simple scenario, the following steps are required:

1. Define path to existing MPT model file in .eqn format (cf. multiTree; Moshagen, 2010)
2. Define path to data set with individual frequencies (.csv file: comma separated, rows=persons, columns=labeled categories)
3. Call one of the fitting functions `betaMPT` or `traitMPT` (examples below)
4. Check convergence of MCMC chains
5. Summarize and plot results

In the following, these steps are explained in more detail. Note that TreeBUGS requires a recent version of the software JAGS (https://mcmc-jags.sourceforge.io/).

### 1. Step: MPT model file in EQN syntax

The model needs to be passed in the standard .eqn file format (e.g., as in multiTree; Moshagen, 2010). As an example, consider the most simple two-high-threshold model (2HTM), each line defines a single processing path containing tree label, category label, and model equations:

    #####Title: 2HTM
    Target  Hit    Do
    Target    Hit    (1-Do)*g
    Target    Miss   (1-Do)*(1-g)
    Lure      FA     (1-Dn)*g
    Lure      CR     (1-Dn)*(1-g)
    Lure      CR     Dn

Note that category labels (e.g., hit, miss,…) must start with a letter (different to multiTree or HMMTree) and match the column names of . The model equations require the multiplication sign `*` and parameters should not be summarized, e.g., by `a^2*(1-a)`. As an input for TreeBUGS, the model file (e.g., `"2htm.txt"`) needs to be saved in the current working directory. Otherwise, the relative or absolute path to the file must be specified (e.g., `models/2htm.txt`). To check how TreeBUGS interprets a given .eqn-file, use:

    readEQN(file = "pathToFile.eqn",       # relative or absolute path
            restrictions = list("Dn=Do"),  # equality constraints
            paramOrder = TRUE)             # show parameter order

Equality restrictions on the MPT parameters can either be provided in a list:

    restrictions = list("Dn=Do", "g=0.5")

or by a path to a text file on the hard drive (e.g., `restrictions="pathToFile.txt"`) that contains the equality constraints, one per row:

    Dn=Do
    g=0.5

### 2. Step: Data Set with Individual Frequencies

Data can be loaded from a comma-separated text file (.csv) in the following format:

    Hit,   Miss,     FA,    CR
    20,      10,      5,    25
    13,       7,      9,    21
    15,       5,      6,    14
    .....

Note that the first line contains the category labels, which must match the category labels from the .eqn-file. The remaining rows contain individual frequencies. Similarly as for the .eqn file, the path to the data file can either be specified as `data_ind.csv` if it is in the current working directory, or as an relative or absolute path (e.g., `"C:/models/data_ind.csv"`).

When using TreeBUGS within R, a data.frame or matrix with appropriate column names that match the category labels can be provided.

### 3. Step: Fit Hierarchical MPT Model

An hierarchical Beta-MPT model is fitted with the following code:

    # load the package:
    library(TreeBUGS)

    # fit the model:
    fitHierarchicalMPT <- betaMPT(
          eqnfile="2htm.txt",         # .eqn file
          data="data_ind.csv",        # individual data
          restrictions=list("Dn=Do"), # parameter restrictions (or path to file)

          ### optional MCMC input:
          n.iter = 20000,             # number of iterations
          n.burnin = 5000,            # number of burnin samples that are removed
          n.thin = 5,                 # thinning rate of removing samples
          n.chains = 3                # number of MCMC chains (run in parallel)
    )

A latent-trait model is fitted similarly by replacing `betaMPT` by `traitMPT`.

### 4. Step: Check convergence of MCMC chains

The functions `betaMPT` and `traitMPT` return a list that includes the original samples from the MCMC sampler for convergence checks. The MCMC samples are stored in `fittedModel$runjags$mcmc` as an `mcmc.list` object (see the package for an overview of convergence diagniostics). TreeBUGS provides a handy wrapper to access the most important plotting functions:

    # Default: Traceplot and density
    plot(fitHierarchicalMPT,    # fitted model
         parameter ="mean"      # which parameter to plot
    )
    # further arguments are passed to ?plot.mcmc.list

    # Auto-correlation plots:
    plot(fitHierarchicalMPT,  parameter = "mean", type = "acf")

    # Gelman-Rubin plots:
    plot(fitHierarchicalMPT,  parameter = "mean", type = "gelman")

See and for further convergence statistics and plots. Note that inferences from the model can be invalid if the Markov-Chain Monte-Carlo (MCMC) sampler did not converge!

### 5. Step: Summarize and Plot Results

TreeBUGS produces an MPT-tailored summary of parameter estimates and convergence statistics:

- Information about the parameter posterior distribution: Mean, SD, Median, 2.5% and 97.5% quantiles
- Convergence: \(\hat R\)-Statistic (should be close to 1.00 for all parameters, e.g., \(\hat R < 1.05\)) and number of effective samples when accounting for auto-correlation (should be large)

To obtain the summary after fitting the model, simply use:

    summary(fitHierarchicalMPT)

The following functions allow to plot parameter estimates, distributions, goodness of fit, and raw frequencies:

    plotParam(fitHierarchicalMPT,     # estimated parameters
              includeIndividual=TRUE  # whether to plot individual estimates
    )
    plotDistribution(fitHierarchicalMPT)      # estimated hierarchical parameter distribution
    plotFit(fitHierarchicalMPT)               # observed vs. predicted mean frequencies
    plotFit(fitHierarchicalMPT, stat = "cov") # observed vs. predicted covariance
    plotFreq(fitHierarchicalMPT)              # individual and mean raw frequencies per tree
    plotPriorPost(fitHierarchicalMPT)         # comparison of prior/posterior (group level parameters)

Parameter estimates (posterior mean, median, SD) can be extracted and saved to a file by using:

    # matrix for further use within R:
    tt <- getParam(fitHierarchicalMPT, parameter="theta",
                   stat="mean")
    tt

    # save complete summary of individual estimates to file:
    getParam(fitHierarchicalMPT, parameter = "theta",
             stat = "summary", file = "parameter.csv")

## References

- Erdfelder, E., Auer, T.-S., Hilbig, B. E., Assfalg, A., Moshagen, M., & Nadarevic, L. (2009). Multinomial processing tree models: A review of the literature. Journal of Psychology, 217, 108–124. https://doi.org/10.1027/0044-3409.217.3.108

- Heck, D. W., & Wagenmakers, E. J. (2016). Adjusted priors for Bayes factors involving reparameterized order constraints. Journal of Mathematical Psychology, 73, 110–116. https://doi.org/10.1016/j.jmp.2016.05.004

- Klauer, K. C. (2010). Hierarchical multinomial processing tree models: A latent-trait approach. Psychometrika, 75, 70–98. https://doi.org/10.1007/s11336-009-9141-0

- Matzke, D., Dolan, C. V., Batchelder, W. H., & Wagenmakers, E.-J. (2015). Bayesian estimation of multinomial processing tree models with heterogeneity in participants and items. Psychometrika, 80, 205–235. https://doi.org/10.1007/s11336-013-9374-9

- Moshagen, M. (2010). multiTree: A computer program for the analysis of multinomial processing tree models. Behavior Research Methods, 42, 42–54. https://doi.org/10.3758/BRM.42.1.42

- Smith, J. B., & Batchelder, W. H. (2010). Beta-MPT: Multinomial processing tree models for addressing individual differences. Journal of Mathematical Psychology, 54, 167–183. https://doi.org/10.1016/j.jmp.2009.06.007