beanz-package {beanz}

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.