poLCAParallel

Polytomous Variable Latent Class Analysis

With Bootstrap Likelihood Ratio Test

Sherman E. Lo, Queen Mary, University of London

A reimplementation of poLCA [CRAN, GitHub] in C++. It attempts to reproduce results and be as similar as possible to the original code, while running faster, especially with multiple repetitions, by utilising multiple threads.

About poLCAParallel

The package poLCAParallel reimplements the poLCA fitting, standard error calculations, goodness of fit tests and the bootstrap log-likelihood ratio test in C++. This was done using Rcpp and RcppArmadillo which allows R to run fast C++ code. Additional notes include:

• The API remains the same as the original poLCA with a few additions
• It tries to reproduce results from the original poLCA
• The code uses Armadillo for linear algebra
• Multiple repetitions are done in parallel using std::thread for multi-thread programming and std::mutex to prevent data races
• Direct inversion of matrices is avoided to improve numerical stability and performance
• Response probabilities are reordered to increase cache efficiency
• Use of std::map for the chi-squared calculations to improve performance

Further reading is available on the QMUL ITS Research Blog.

About poLCA

poLCA is a software package for the estimation of latent class models and latent class regression models for polytomous outcome variables, implemented in the R statistical computing environment.

Latent class analysis (also known as latent structure analysis) can be used to identify clusters of similar "types" of individuals or observations from multivariate categorical data, estimating the characteristics of these latent groups, and returning the probability that each observation belongs to each group. These models are also helpful in investigating sources of confounding and nonindependence among a set of categorical variables, as well as for density estimation in cross-classification tables. Typical applications include the analysis of opinion surveys; rater agreement; lifestyle and consumer choice; and other social and behavioral phenomena.

The basic latent class model is a finite mixture model in which the component distributions are assumed to be multi-way cross-classification tables with all variables mutually independent. The model stratifies the observed data by a theoretical latent categorical variable, attempting to eliminate any spurious relationships between the observed variables. The latent class regression model makes it possible for the researcher to further estimate the effects of covariates (or "concomitant" variables) on predicting latent class membership.

poLCA uses expectation-maximization and Newton-Raphson algorithms to find maximum likelihood estimates of the parameters of the latent class and latent class regression models.

Recommended Installation Instructions

The easiest way to install poLCAParallel is to use R with remotes.

Install From GitHub

Run the following in R to install the latest version

remotes::install_github("QMUL/poLCAParallel@package")

or for a previous version, for example,

remotes::install_github("QMUL/[email protected]")

Install From Releases

Download the .zip or .tar.gz file from the releases. Install it in R using

remotes::install_local(<PATH TO .zip OR .tar.gz FILE>)

User's Notes

Citation

Please consider citing the corresponding QMUL ITS Research Blog

• Lo, S.E. (2022). Speeding up and Parallelising R packages (using Rcpp and C++) | QMUL ITS Research Blog. [link]

and the publication below which this software was originally created for

• Eto F, Samuel M, Henkin R, Mahesh M, Ahmad T, et al. (2023). Ethnic differences in early onset multimorbidity and associations with health service use, long-term prescribing, years of life lost, and mortality: A cross-sectional study using clustering in the UK Clinical Practice Research Datalink. PLOS Medicine, 20(10): e1004300. https://doi.org/10.1371/journal.pmed.1004300

Tips

• When using model <- poLCAParallel::poLCA(), set the parameters calc.se=FALSE and calc.chisq=FALSE to avoid doing standard error and goodness of fit calculations respectively. This will save time if you do not require those results. You can always calculate them afterwards using model <- poLCAParallel::poLCAParallel.se(model) and model <- poLCAParallel::poLCAParallel.goodnessfit(model).
• Make use of multiple repetitions and threads. When using poLCAParallel::poLCA(), set nrep=1 to do a test run and gauge how long it takes. Afterwards, set nrep to a bigger number to try different initial values in parallel.
• When using poLCAParallel::poLCA(), set n.thread to set the number of threads to be used by the computer. By default, it uses all detectable threads.
• There is an experimental option to use Laplace smoothing on the response probabilities when doing standard error calculations. This provides better numerical stability and avoids very small standard errors. To use it, either
  • In poLCAParallel::poLCA(), set se.smooth=TRUE
  • Or in poLCAParallel::poLCAParallel.se(), set is_smooth=TRUE
• When using the regression model, it is encouraged to normalise your data frame to provide better numerical stability.
• Use set.seed() before using poLCAParallel::poLCA() to set the seed for random number generation. This ensures reproducibility when reporting what seed you have used.

Example Code

R scripts which compare poLCAParallel with poLCA are provided in exec/. An example use of a bootstrap likelihood ratio test is shown in exec/3_blrt.R.

Changes from the Original Code

• In poLCAParallel::poLCA(), the following arguments have been added:
  • n.thread is provided to specify the number of threads to use.
  • calc.chisq is provided to specify if you want to conduct goodness of fit tests or not.
  • se.smooth is provided if you wish to use Laplace smoothing on the response probabilities in the standard error calculations.
• The prior probabilities are a return value, accessible with $prior.
• The stopping condition of the EM algorithm has changed slightly. If the log-likelihood change after an iteration of EM is too small, the stopping condition is evaluated after the E step rather than the M step. This is so that the by-product of the E step is reused when calculating the log-likelihood.
• The Newton step uses a linear solver rather than directly inverting the Hessian matrix in the regression model.
• The output probs.start are the initial probabilities used to achieve the maximum log-likelihood from any repetition rather than from the first repetition.
• The output eflag is set to TRUE if any repetition has to be restarted, rather than the repetition which achieves maximum log-likelihood.
• The standard error is not calculated if calc.se is set to FALSE even in poLCA regression. Previously, the standard error is calculated regardless of calc.se in poLCA regression.
• In the standard error calculations, an SVD is done on the score matrix, rather than inverting the information matrix.
• Any errors in the input data will call stop() rather than return a NULL.
• No rounding in the return value predcell.

Developer's and Maintainer's Notes

Installing as a Developer

The following installation instructions are useful if you wish to develop the code and install a locally modified version of the package.

Requires the R packages for compiling and testing:

• Rcpp
• RcppArmadillo
• roxygen2
• testthat
• usethis

Requires the dependent R packages:

• MASS
• poLCA
• scatterplot3d

Git clone this repository

git clone https://github.com/QMUL/poLCAParallel.git

and change directory into it

cd poLCAParallel

From there, in the repository root, run the following to generate additional code and documentation so that the package can be compiled correctly

R -e "usethis::use_namespace()"
R -e "Rcpp::compileAttributes()"
R -e "roxygen2::roxygenize()"

Install the package using

R CMD INSTALL --preclean --no-multiarch .

Testing

The testing of the C++ code is done using Catch2 and the R code using testthat. All test codes are in tests/.

C++ with Catch2

The tests for the C++ code are done by compiling the test code, isolated from any R ecosystem, and running a compiled executable. It requires cmake, Catch2 and armadillo. To compile the code, from the repository root, make a new directory and use cmake inside it

mkdir build
cd build
cmake ..
cmake --build .

This will compile an executable called test_polca_parallel. Execute it to run the tests. Pass names or tags to run specific tests, see tests/*.cc.

R with testthat

To test the R code, run the following at the repository root

R -e "testthat::test_local()"

Apptainer

Apptainer definition files are provided, which can be used to install the package inside a container. These may be useful for further troubleshooting or development.

• The definition file poLCAParallel.def installs R and the package only. No version pinning
• The definition file poLCAParallel-dev.def installs the R package as well as generating documentation and running tests within the container. Versions of dependencies are pinned to the ones used during development or maintenance

To build the container, use the command (or similar)

apptainer build poLCAParallel-dev.sif poLCAParallel-dev.def

Within the container, the package is located in /usr/src/poLCAParallel. When using the definition file poLCAParallel-dev.def, the C++ doxygen documentation is located in /usr/src/poLCAParallel/html.

Git/GitHub Workflow Guide

All generated documents and codes, eg from

R -e "Rcpp::compileAttributes('poLCAParallel')"

and

R -e "roxygen2::roxygenize('poLCAParallel')"

shall not be included in the master branch. Instead, they shall be in the package branch so that this package can be installed using remotes::install_github("QMUL/poLCAParallel@package"). This is to avoid having duplicate documentation and generated code on the master branch.

Semantic versioning is used and tagged. Tags on the master branch shall have v prepended and -master appended, eg. v1.1.0-master. The corresponding tag on the package branch shall only have v prepended, eg. v1.1.0.

Development Notes

• The likelihood calculation is done by iteratively multiplying probabilities together. In the commit 85ee419, the multiplication starts from DOUBLE_XMAX to avoid underflows but was reverted. Consider investigating further in future releases.
• In the standard error calculations, the score matrix is typically ill-conditioned. Consider pre-conditioning the matrix.
• In the poLCA regression model, consider using multiple Newton steps instead of one single step in the EM algorithm.

Actions for the Next Minor Version(s)

• Add a feature where the likelihood calculation can be optionally done by summing the log probabilities rather than multiplying probabilities together. This should avoid underflows, especially when there are a large number of manifest variables (aka categories) or very small probabilities. Though it should be noted that working in log space is slower.

Actions for the Next Major Version

• The R package MASS is not required as a prerequisite.
• The default value for n.thread should be 1 instead of parallel::detectCores()

The following R functions, many of which are internal, are marked as deprecated and should be deleted

• poLCA.se() and poLCA.dLL2dBeta.C() - no longer needed because the standard error calculations are reimplemented in poLCAParallel.se()
• poLCA.probHat.C - no longer needed because the goodness of fit test is reimplemented in goodness_fit.cc
• poLCA.postClass.C() and poLCA.ylik.C() - no longer needed and reimplemented in polca_rcpp.cc
• poLCA.vectorize() and poLCA.unvectorize() - no longer needed and reimplemented in poLCAParallel.vectorize() and poLCAParallel.unvectorize() respectively

All C code in poLCA.C is deprecated because they are reimplemented in C++.

The R code should follow the Tidyverse style guide. In particular, variables, functions and parameters should be in snake case. This will result in

• Removing the poLCA. and poLCAParallel. prefix in function and file names
• Using a underscore instead of a dot in variable and parameter names, for example, na.rm should be called na_rm

C++ Style Guide

There was an attempt to use the Google C++ style guide.

C++ Source Code Documentation

The C++ code documentation can be created with Doxygen by running

doxygen

and viewed at html/index.html.

References

• Bandeen-roche, K., Miglioretti, D. L., Zeger, S. L., and Rathouz, P. J. (1997). Latent variable regression for multiple discrete outcomes. Journal of the American Statistical Association, 92(440):1375–1386. [link]
• Dziak, J. J., Lanza, S. T., & Tan, X. (2014). Effect size, statistical power, and sample size requirements for the bootstrap likelihood ratio test in latent class analysis. Structural Equation Modeling: A Multidisciplinary Journal, 21(4):534-552. [link]
• Linzer, D.A. & Lewis, J. (2013). poLCA: Polytomous Variable Latent Class Analysis. R package version 1.4. [link]
• Linzer, D.A. & Lewis, J.B. (2011). poLCA: An R package for polytomous variable latent class analysis. Journal of Statistical Software, 42(10): 1-29. [link]

License

The software is under the GNU GPL 2.0 license, as with the original poLCA code, stated in their documentation.