Bayesian Conditional Transformation Models

Abstract

Recent developments in statistical regression methodology shift away from pure mean regression towards distributional regression models. One important strand thereof is that of conditional transformation models (CTMs). CTMs infer the entire conditional distribution directly by applying a transformation function to the response condi- tionally on a set of covariates towards a simple log-concave reference distribution. Thereby, CTMs allow not only variance, kurtosis or skewness but the complete con- ditional distribution to depend on the explanatory variables. We propose a Bayesian notion of conditional transformation models (BCTMs) focusing on exactly observed continuous responses, but also incorporating extensions to randomly censored and discrete responses. Rather than relying on Bernstein polynomials that have been considered in likelihood-based CTMs, we implement a spline-based parametrization for monotonic effects that are supplemented with smoothness priors. Furthermore, we are able to benefit from the Bayesian paradigm via easily obtainable credible in- tervals and other quantities without relying on large sample approximations. A simu- lation study demonstrates the competitiveness of our approach against its likelihood- based counterpart but also Bayesian additive models of location, scale and shape and Bayesian quantile regression. Two applications illustrate the versatility of BCTMs in problems involving real world data, again including the comparison with various types of competitors.

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Data

All used data is freely available through packages:

• Framingham illustration: data("Cholesterol", package="qrLMM")
• Leukemia survival illustration: data("LeukSurv", package="spBayesSurv")
• Veteran lung cancer illustration: data(cancer, package="survival")

and via csv in the folder ./data.

Code

• folder ./code/ contains helper functions used by BCTM in setting up the model
• folder code/nuts/ contains the NUTS code used for posterior inference. Note that, currently, updates of the smoothing variances and results of the prior elicitation process for $\theta$ are hardcoded in many cases.
• folder code/rcpp/ contains rcpp source files for posteriors (up to a constant) and gradients
• folder code/sims/ contains R code to reproduce all simulations and plots from the paper and the supplement
  • plot_mads.R produces Figure 1
  • plot_quantile_deviations.R produces Figure 2
  • plot_quantile_scores.R produces Figure 3
  • plot_crps_decomp.R produces Figure 4
  • plot_coverages.R produces Figure 5 and Figure C.2
  • plot_simulated_effects.R produces Figures C.3 to 10 (generate_sims_lin.R and generate_sims_nonlin.R generates RData files)
• folder code/framingham/ contains R code to reproduce all models and plots for the Framingham heart study illustration
  • plot_fram.R produces Figure 6 and 7
  • folder ./models/ contains code for running the models that, together with fram_te_log_scores.R and fram_log_scores_load_sum.R are reported in Table B.2
  • fram_ess_rhat.R produces Table B.3 and B.4
• folder code/leukemia/ contains R code to reproduce all models and plots for the Leukemia survival illustration
  • leuk_comparison.R produces Table 1 and 2
  • plot_leuk.R produces Figure 8 and Table B.5
• folder code/veteran/ contains R code to reproduce all models and plots for the Veteran illustration
  • plot_veteran.R produces Figure B.1
• folder code/mlt_sims/ contains R code from Hothorn et al. (2018) to simulate data (obtained from Hothorn, T. (2017b).MLT: Most likely transformations: The mlt package, R package vignette version 0.1-5.Available at: https://CRAN.R-project.org/package=mlt.docreg.)
• file BCTM.rar is intended for review and contains only the blinded code and data

All scripts include a helper function that installs and loads used libraries. A majority of (but not all) libraries are captured in the renv.lock file (after installing the renvpackage use renv::init()).

Usage

All code in sims/framingham/leukemia/veteran is self-contained and can be used by simply running the scripts. Code can be used to reproduce the results in the paper and supplement, but is difficult to adapt for different tasks in its current state. The different files in code/nuts/ differ only in the updates of the smoothing variances and prior elicitation schemes (nuts.R is the default used in most of the applications). Naming:

• _sd refers to scale-dependent prior
• _te refers to tensor spline
• _re refers to random effect
• _vcm refers to varying coefficient model
• _spat refers to spatial effect

Dependencies

• splines (4.21)
• dplyr (1.0.10)
• tidyverse (1.3.2)
• mlt (1.4-2)
• tram (0.7-2)
• tramME (1.0.3)
• scam (1.2-13)
• mgcv (1.8-41)
• bamlss (1.1-8)
• Rcpp (1.0.9) RcppArmadillo (0.11.2.3.1) RcppEigen (0.3.3.92)
• qrLMM (2.1)
• viridis (0.6.2)
• MASS (7.3-58.1)
• mvtnorm (1.1-3)
• cowplot (1.1.1)
• gamboostLSS (2.0-6)
• lattice (0.20-45)
• microbenchmark (1.4.9)
• progress (1.2.2)
• devtools (2.4.4)
• doParallel (1.0.17)
• survival (3.4-0)
• spBayesSurv (1.1.6)
• Matrix (1.5-1)
• MCMCpack (1.6-3)
• profvis (0.3.7)
• sf (1.0-8)
• rgeos (0.5-9)
• RhpcBLASctl (0.21-247.1)
• Loo (2.5.1)
• BayesX (0.3-1.1)
• caret (6.0-93)
• sdPrior (1.0-0)
• scoringutils (1.0.1)

Processed Files

Folder ./processed_data/ contains all results used in the paper except for files that are larger than GitHub's size limits (due to inclusion of all samples).