Bayesian Approaches for HTE Analysis

Description

This package contains the functions for running Bayesian models implemented in STAN for HTE analysis.

Notation

Consider a randomized two-arm clinical trial. Let Y denote the response and Z denote treatment arm assignment. For subgroup analysis, assume there are P baseline covariates, X_1,\ldots,X_P, of interest. The covariates can be binary, ordinal with numerical values, or nominal variables. Let \Omega = \{(X_1,\ldots,X_P)\} denote the collection of subgroups defined by the covariates. Let \theta_g denote the treatment effect in subgroup G=g, and let \widehat{\theta}_g be the estimated \theta in subgroup G=g with \widehat{\sigma}^2_g the estimated variance associated with \widehat{\theta}_g.

Models

We approximate the distribution of \widehat{\theta}_g by

\widehat{\theta}_g | \theta_g, \sigma^2_g \sim N(\theta_g, \sigma^2_g)

and assign an informative prior to \sigma_g.

We consider two options in the software: log-normal or uniform prior. The uniform prior is specified as:

\log \sigma_g | \widehat{\sigma}_g, \Delta \sim Unif( \log \widehat{\sigma}_g - \Delta, \log\widehat{\sigma}_g + \Delta)

and the log-normal prior is specified as:

\log \sigma_g | \widehat{\sigma}_g, \Delta \sim N( \log \widehat{\sigma}_g, \Delta)

where \Delta is a parameter specified by the users.

We consider a set of models together with the priors for \theta_g:

No subgroup effect model

This model assumes that patients in all the subgroups are exchangeable. That is, all the subgroups are statistically identical with regard to the treatment effect and there is no subgroup effect. Information about treatment effects can be directly combined from all subgroups for inference. The model is specified as follows:

\begin{array}{rcl} \theta_g &=& \mu\\ \mu &\sim& N(MU, B), \end{array}

where MU should be set to 0 in most cases, and B is large in relation to the magnitude of the treatment effect size so that the prior for \mu is essentially non-informative.

Full stratification model

The subgroups are fully distinguished from each other with regard to the treatment effect. There is no information about treatment effects shared between any subgroups. The model is specified as follows:

\begin{array}{rcl} \theta_g &=& \mu_g \\ \mu_g &\sim& N(MU, B). \end{array}

Simple regression model

The model introduces a first-order, linear regression structure. This model takes into account the information that the subgroups are formulated based on the set of baseline covariates. The coefficients are assumed to be exchangeable among subgroups. Information about treatment effects are shared between subgroups with similar baseline covariates through these coefficients. The model is specified as follows:

\begin{array}{rcl} \theta_g|X_g &=& \mu + \sum_{j=1}^P X'_{g,j} \gamma_j \\ \mu &\sim& N(MU,B) \\ \gamma_j &\sim& N(0, C) \qquad j=1,\ldots,P. \end{array}

Basic shrinkage model

This approach assumes all subgroups are exchangeable with regards to the treatment effect. The model is specified as follows:

\begin{array}{rcl} \theta_g &=& \mu + \phi_g \\ \mu &\sim& N(MU, B) \\ \phi_g &\sim& N(0, \omega^2) \\ \omega &\sim& {Half-}N(D). \end{array}

Simple regression and shrinkage model

This model combines basic regression with shrinkage, with a linear regression structure and a random effect term. Direct estimates are shrunken towards the regression surface. The model is specified as follows:

\begin{array}{rcl} \theta_g &=& \mu + \sum_{j=1}^P X'_{g,j} \gamma_j + \phi_g \\ \mu &\sim& N(MU,B) \\ \gamma_j &\sim& N(0, 1 C) \qquad j=1,\ldots,P\\ \phi_g &\sim& N(0, \omega^2) \\ \omega &\sim& {Half-}N(D). \end{array}

Dixon and Simon model

This model assumes that the elements in coefficient are exchangeable with each other, which allows information sharing among covariate effects. Similar to the simple regression model, only the first-order interactions are considered. The model is specified as follows:

\begin{array}{rcl} \theta_g &=& \mu + \sum_{j=1}^P X'_{g,j} \gamma_j \\ \mu &\sim& N(MU,B) \\ \gamma_j &\sim& N(0, \omega^2) \\ \omega &\sim& {Half-}N(D). \end{array}

Extended Dixon and Simon model

This approach extends the Dixon and Simon model by introducing the higher-order interactions, with the interaction effects exchangeable. The model is specified as follows:

\begin{array}{rcl} \theta_g &=& \mu + \sum_{k=1}^P \sum_{j \in \xi^{(k)}} X'_{\xi^{(k)},j} \gamma^{(k)}_{j} \\ \mu &\sim& N(MU, B) \\ \gamma^{(k)}_j &\sim& N(0, \omega_k^2) \qquad k=1,\ldots,P, \quad j\in \xi^{(k)} \\ \omega_k &\sim& {Half-}N(D), \end{array}

where \xi^{(k)} denotes the set of kth order interaction terms

Graphical user interface (GUI)

This package provides a web-based Shiny GUI. See bzShiny for details.

References

Jones HE, Ohlssen DI, Neuenschwander B, Racine A, Branson M (2011). Bayesian models for subgroup analysis in clinical trials. Clinical Trials, 8(2), 129-143.

Dixon DO, Simon R (1991). Bayesian subset analysis. Biometrics, 47(3), 871-881.

Wang C, Louis TA, Henderson NC, Weiss CO, Varadhan R (2018). beanz: An R Package for Bayesian Analysis of Heterogeneous Treatment Effects with a Graphical User Interface. Journal of Statistical Software, 85(7), 1-31.

Call STAN models

Description

Call STAN to draw posterior samples for Bayesian HTE models.

Usage

bzCallStan(
  mdls = c("nse", "fs", "sr", "bs", "srs", "ds", "eds"),
  dat.sub,
  var.estvar,
  var.cov,
  par.pri = c(B = 1000, C = 1000, D = 1, MU = 0),
  var.nom = NULL,
  delta = 0,
  prior.sig = 1,
  chains = 4,
  ...
)

Arguments

Value

A class beanz.stan list containing

mdl

name of the Bayesian HTE model

stan.rst

raw rstan sampling results

smps

matrix of the posterior samples

get.mus

method to return the posterior sample of the subgroup treatment effects

DIC

DIC value

looic

leave-one-out cross-validation information criterion

rhat

Gelman and Rubin potential scale reduction statistic

prior.sig

option for the informative prior on \sigma_g

delta

parameter for specifying the informative priors of \sigma_g

Examples

## Not run: 
var.cov    <- c("sodium", "lvef", "any.vasodilator.use");
var.resp   <- "y";
var.trt    <- "trt";
var.censor <- "censor";
resptype   <- "survival";
var.estvar <- c("Estimate", "Variance");

subgrp.effect <- bzGetSubgrpRaw(solvd.sub,
                                  var.resp   = var.resp,
                                  var.trt    = var.trt,
                                  var.cov    = var.cov,
                                  var.censor = var.censor,
                                  resptype   = resptype);

rst.nse    <- bzCallStan("nse", dat.sub=subgrp.effect,
                         var.estvar = var.estvar, var.cov = var.cov,
                         par.pri = c(B=1000, MU = 0),
                         chains=4, iter=600,
                         warmup=200, thin=2, seed=1000);

rst.sr     <- bzCallStan("sr", dat.sub=subgrp.effect,
                        var.estvar=var.estvar, var.cov = var.cov,
                        par.pri=c(B=1000, C=1000),
                        chains=4, iter=600,
                        warmup=200, thin=2, seed=1000);
## End(Not run)

Comparison of posterior treatment effects

Description

Present the difference in the posterior treatment effects between subgroups

Usage

bzSummaryComp(stan.rst, sel.grps = NULL, cut = 0, digits = 3, seed = NULL)

bzPlotComp(stan.rst, sel.grps = NULL, ..., seed = NULL)

bzForestComp(
  stan.rst,
  sel.grps = NULL,
  ...,
  quants = c(0.025, 0.975),
  seed = NULL
)

Arguments

Value

bzSummaryComp generates a data frame with summary statistics of the difference of treatment effects between the selected subgroups. bzPlotComp generates the density plot of the difference in the posterior treatment effects between subgroups. bzForestComp generates the forest plot of the difference in the posterior treatment effects between subgroups.

See Also

bzCallStan

Examples

## Not run: 
var.cov    <- c("sodium", "lvef", "any.vasodilator.use");
var.resp   <- "y";
var.trt    <- "trt";
var.censor <- "censor";
resptype   <- "survival";
var.estvar <- c("Estimate", "Variance");

subgrp.effect <- bzGetSubgrpRaw(solvd.sub,
                             var.resp   = var.resp,
                             var.trt    = var.trt,
                             var.cov    = var.cov,
                             var.censor = var.censor,
                             resptype   = resptype);

rst.sr     <- bzCallStan("sr", dat.sub=subgrp.effect,
                         var.estvar=var.estvar, var.cov = var.cov,
                         par.pri=c(B=1000, C=1000),
                         chains=4, iter=500,
                         warmup=100, thin=2, seed=1000);

sel.grps <- c(1,4,5);
tbl.sub <- bzSummaryComp(rst.sr, sel.grps=sel.grps);
bzPlot(rst.sr, sel.grps = sel.grps);
bzForest(rst.sr, sel.grps = sel.grps);
## End(Not run)

Gail-Simon Test

Description

Gail-Simon qualitative interaction test.

Usage

bzGailSimon(effects, sderr, d = 0)

Arguments

Examples

## Not run: 
var.cov    <- c("sodium", "lvef", "any.vasodilator.use");
var.resp   <- "y";
var.trt    <- "trt";
var.censor <- "censor";
resptype   <- "survival";
subgrp.effect <- bzGetSubgrp(solvd.sub,
                                  var.resp   = var.resp,
                                  var.trt    = var.trt,
                                  var.cov    = var.cov,
                                  var.censor = var.censor,
                                  resptype   = resptype);

gs.pval <- bzGailSimon(subgrp.effect$Estimate,
                       subgrp.effect$Variance); 
## End(Not run)

Get subgroup treatment effect estimation and variance

Description

Compute subgroup treatment effect estimation and variance for subgroup effect summary data. The estimation and variance are combined if there are multiple record of the same subgroup, defined by the covariates, in the data.

Usage

bzGetSubgrp(data.all, var.ey, var.variance, var.cov)

Arguments

Value

A dataframe with treatment effect estimation and variance for each subgroup

Get subgroup treatment effect estimation and variance

Description

Compute subgroup treatment effect estimation and variance from subject level data.

Usage

bzGetSubgrpRaw(
  data.all,
  var.resp,
  var.trt,
  var.cov,
  var.censor,
  resptype = c("continuous", "binary", "survival")
)

Arguments

Value

A dataframe with treatment effect estimation and variance for each subgroup

Examples

## Not run: 
var.cov    <- c("sodium", "lvef", "any.vasodilator.use");
var.resp   <- "y";
var.trt    <- "trt";
var.censor <- "censor";
resptype   <- "survival";
subgrp.effect <- bzGetSubgrpRaw(solvd.sub,
                                  var.resp   = var.resp,
                                  var.trt    = var.trt,
                                  var.cov    = var.cov,
                                  var.censor = var.censor,
                                  resptype   = resptype);
## End(Not run)

Predictive Distribution

Description

Get the predictive distribution of the subgroup treatment effects

Usage

bzPredSubgrp(stan.rst, dat.sub, var.estvar)

Arguments

Value

A dataframe of predicted subgroup treament effects. That is, the distribution of

\theta_g | \widehat{\theta}_1, \widehat{\sigma}^2_1, \ldots, \widehat{\theta}_G, \widehat{\sigma}^2_G.

Examples

## Not run: 
var.cov    <- c("sodium", "lvef", "any.vasodilator.use");
var.resp   <- "y";
var.trt    <- "trt";
var.censor <- "censor";
resptype   <- "survival";
var.estvar <- c("Estimate", "Variance");

subgrp.effect <- bzGetSubgrp(solvd.sub,
                                  var.resp   = var.resp,
                                  var.trt    = var.trt,
                                  var.cov    = var.cov,
                                  var.censor = var.censor,
                                  resptype   = resptype);

rst.nse    <- bzCallStan("nse", dat.sub=subgrp.effect,
                         var.estvar = var.estvar, var.cov = var.cov,
                         par.pri = c(B=1000),
                         chains=4, iter=4000,
                         warmup=2000, thin=2, seed=1000);

pred.effect <- bzPredSubgrp(rst.nes,
                            dat.sub = solvd.sub,
                            var.estvar = var.estvar);
## End(Not run)

Summary table of treatment effects

Description

Compare the DIC from different models and report the summary of treatment effects based on the model with the smallest DIC value

Usage

bzRptTbl(lst.stan.rst, dat.sub, var.cov, cut = 0, digits = 3)

Arguments

Value

A dataframe with summary statistics of the model selected by DIC

Run Web-Based BEANZ application

Description

Call Shiny to run beanz as a web-based application

Usage

bzShiny()

Posterior subgroup treatment effects

Description

Present the posterior subgroup treatment effects

Usage

bzSummary(
  stan.rst,
  sel.grps = NULL,
  ref.stan.rst = NULL,
  ref.sel.grps = 1,
  cut = 0,
  digits = 3
)

bzPlot(stan.rst, sel.grps = NULL, ref.stan.rst = NULL, ref.sel.grps = 1, ...)

bzForest(
  stan.rst,
  sel.grps = NULL,
  ref.stan.rst = NULL,
  ref.sel.grps = 1,
  ...,
  quants = c(0.025, 0.975)
)

Arguments

Value

bzSummary generates a dataframe with summary statistics of the posterior treatment effect for the selected subgroups. bzPlot generates the density plot of the posterior treatment effects for the selected subgroups. bzForest generates the forest plot of the posterior treatment effects.

See Also

bzCallStan

Examples

## Not run: 
sel.grps <- c(1,4,5);
tbl.sub <- bzSummary(rst.sr, ref.stan.rst=rst.nse, ref.sel.grps=1);
bzPlot(rst.sr, sel.grps = sel.grps, ref.stan.rst=rst.nse, ref.sel.grps=1);
bzForest(rst.sr, sel.grps = sel.grps, ref.stan.rst=rst.nse, ref.sel.grps=1);
## End(Not run)

Subject level data from SOLVD trial

Description

Dataset for use in beanz examples and vignettes.

Format

A dataframe with 6 variables:

trt

treatment assignment

y

time to death or first hospitalization

censor

censoring status

sodium

level of sodium

lvef

level of lvef

any.vasodilator.use

level of use of vasodilator

Details

Subject level data from SOLVD trial. SOLVD is a randomized controlled trial of the effect of an Angiotensin-converting-enzyme inhibitor (ACE inhibitor) called enalapril on survival in patients with reduced left ventricular ejection fraction and congestive heart failure (CHF).

References

Solvd Investigators and others, Effect of enalapril on survival in patients with reduced left ventricular ejection fraction and congestive heart failure. N Engl J Med. 1991, 325:293-302