Bayesian evaluation of behavior change interventions: a brief introduction and a practical example

Abstract

Introduction: Evaluating effects of behavior change interventions is a central interest in health psychology and behavioral medicine. Researchers in these fields routinely use frequentist statistical methods to evaluate the extent to which these interventions impact behavior and the hypothesized mediating processes in the population. However, calls to move beyond the exclusive use of frequentist reasoning are now widespread in psychology and allied fields. We suggest adding Bayesian statistical methods to the researcher's toolbox of statistical methods.

Objectives: We first present the basic principles of the Bayesian approach to statistics and why they are useful for researchers in health psychology. We then provide a practical example on how to evaluate intervention effects using Bayesian methods, with a focus on Bayesian hierarchical modeling. We provide the necessary materials for introductory-level readers to follow the tutorial.

Conclusion: Bayesian analytical methods are now available to researchers through easy-to-use software packages, and we recommend using them to evaluate the effectiveness of interventions for their conceptual and practical benefits.

Keywords: Bayes; Bayesian estimation; health behavior change; intervention evaluation; tutorial.

Figure 1.

Three alternative priors, with varying informativeness. Dotted line depicts N(0, 1), solid N(0, 2.5), and dashed a uniform distribution.

Figure 2.

Prior (dotted) and likelihood (dashed) distributions.

Figure 3.

Prior (dotted), likelihood (dashed) and posterior (solid).

Figure 4.

The R console. This figure shows how to assign (R uses the left arrow, <-, for assignment) all whole numbers from 0 to 100 to a variable called numbers. Computer code can often be read from right to left, so the first line here could be read as ‘integers 0 through 100, assign to numbers’. We then calculated the mean of those numbers by using R’s built in function, mean().

Figure 5.

RStudio with its text editor and R console (upper and lower left panels, respectively). The three lines of code saved into the R script ‘t-test-kids-grownups. R’ shows how to save numbers into variables, and then conduct a t-test between the variables.

Figure 6.

Prior probability distributions for Model 1 in the tutorial. The left panel shows the prior distribution which is assigned to all regression coefficients $\beta$. Middle panel shows the prior distribution of the standard deviation parameter of the person-specific intercepts. Right panel shows the prior distribution for the residual standard deviation.

Figure 7.

Left panel: Density curves of the posterior distributions of the four population-level regression parameters. The shaded area indicates the 95% Credible Interval, and the vertical line indicates the posterior mean. The density curves are estimated from MCMC samples, and slightly smoothed for the figure. Right panel: Trajectories of change across time for the two intervention groups (blue: control group, red: intervention group). Each line denotes the posterior mean regression line for that group, and the surrounding shades are the 95% Credible Intervals for the regression lines. The code for creating these two figures can be found in the complete code listing for this tutorial.

Figure 8.

Posterior distributions of the three main population-level regression coefficients, and the transformed parameter $\delta$ (delta), which denotes the effect of time in the intervention group only.

Figure 9.

Traceplot of the model’s intercept. Each of the four chains is plotted in a different hue. The posterior samples (x-axis) are connected with a line; y-axis are the samples’ values. The four chains’ traces look highly similar, suggesting to us that the MCMC approximation has worked well. If the chains looked very dissimilar, we would be prompted to further investigate the model’s performance.

Figure 10.

Graphical comparison of the actual data set to replicated data sets should reveal a very similar shape of the densities, if the model fits the data well. Here, we do not see serious problems with how the model seems to replicate the data (but note that we have not taken into account the natural 1–5 limits of the response scale, or that the raw responses are ordered categories).

Figure A1.

Posterior predictive check for the ordinal logistic model.