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 tries to reproduce results and be as similar as possible to the original code but runs faster, especially with multiple repetitions by using multiple threads.

About poLCAParallel

The package poLCAParallel reimplements 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 a 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 Prerequisites

The following prerequisites are recommended to be installed:

• R packages for installing and compiling:
  • devtools
  • Rcpp
  • RcppArmadillo
  • roxygen2
• Dependent R packages:
  • MASS
  • parallel
  • poLCA
  • scatterplot3d

Recommended Installation Instructions

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

Install From GitHub

Run the following in R

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

Install From Releases

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

devtools::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 probabilites is 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 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. The instructions do not require the R package devtools.

Requires the R packages for compiling:

• Rcpp
• RcppArmadillo
• roxygen2

Requires the dependent R packages:

• MASS
• parallel
• poLCA
• scatterplot3d

Git clone this repository

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

Run the following to generate additional code and documentation so that the package can be compiled correctly

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

Install the package using

R CMD INSTALL --preclean --no-multiarch poLCAParallel

Troubleshooting

If the installation instructions fail or there are other problems, please check the possible following prerequisites are installed:

• A C++ compiler like gcc or clang
• Armadillo (see the installation manual for further details)
• LAPACK
• OpenBLAS

An Apptainer definition file poLCAParallel.def is provided which installs R, prerequisites and poLCAParallel in a container. This may be useful for further troubleshooting.

Development Notes

• When calculating the likelihood, probabilities are iteratively multiplied, this is much faster than taking the sum of log probabilities. However, to avoid underflow errors, the calculation of the likelihood uses the sum of log probabilities when an underflow is detected. See PosteriorUnnormalize() in src/em_algorithm.* for the implementation.
• 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 Major Version

The following should be actioned in the next major version:

• The R package MASS is not required as a prerequisite.

The following R functions (and their corresponding C functions if available) are marked as deprecated and should be deleted in the next major version

• poLCA.se() - no longer needed, reimplemented in poLCAParallel.se()
• poLCA.dLL2dBeta.C() - no longer needed, reimplemented in em_algorithm_regress.cc
• poLCA.probHat.C - no longer needed, the goodness of fit test is reimplemented in goodness_fit.cc

Code Style

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 devtools::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.

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

Armadillo objects are used sparingly, preferring the use of double* when passing vectors and matrices.

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.