A Bayesian Analysis of Prenatal Maternal Factors Predicting Nonadherence to Infant HIV Medication Regimen
Supplemental Digital Content
Prior Specification
A technique called “expert elicitation” was used to specify the prior information needed for the Bayesian analysis in this study. As suggested by the name, expert elicitation is the process of consulting authorities in the scientific discipline and summarizing their beliefs into prior probability distributions for the estimates of interest. Ultimately, for this analysis, prior information was needed for the vector of β parameters in the regression model (in a logistic regression, βi is the log odds ratio for Y = 1 comparing subpopulations defined by Xi = xi + 1 to those with Xi = xi). However, it is much more natural for researchers to think in terms of “success” probabilities than logistic regression coefficients. In this study, expert elicitation of prior information about the probabilities of infant nonadherence under certain combinations of predictor variables was likely to be more accurate than elicitation of prior information about log odds ratios. One solution is to use a “data augmentation” strategy to obtain priors, e.g. Bedrick, Christiensen, and Johnson, 1997 1. Utilizing this strategy, prior information about success probabilities can be mathematically transformed into prior information about βcoefficients.
For a model with p parameters, elicitation of this type of prior information begins by constructing a “dataset” of p hypothetical subjects/cases. These subjects aregiven values of the model covariates that are sufficiently meaningful and distinct in order to allow experts to estimate the probability of “success” (Y = 1) for each case. In this study, an optimal experimental design for the covariate matrixX was generated using the R package “AlgDesign”2and a subset of seven cases (one for the intercept plus six covariates) were selected from the twelve case solution (call this 7x7 matrix of hypothetical cases Xp). “Subjects” were selected based on difference from each other and ease of interpretation, i.e., the seven were chosen because they were judged to be the easiest to evaluate. Information about the cases was translated into a short questionnaire (see Figure 1 in the main paper for the “dataset” of hypothetical cases and an example item from the questionnaire), which was completed by six experts: two investigators from the University of Miami, two investigators from the Human Sciences Research Council (Pretoria, South Africa), and two South African staff “managers” overseeing implementation of the study in community health clinics. At the time of prior elicitation, no experts had seen the data utilized in the analysis, nor had the researcher performing the elicitation process. Experts were asked to provide an estimate of the probability that a woman with the specified set of characteristics would fail to provide at least one dose of HIV medication to her infant in seven days.
Hypothetical case ratings were averaged to obtain a point estimate for the prior probability of nonadherence for each case, and the minimum and maximum ratings (over all raters, changing 0% ratings to 1% and 100% to 99%) were used as lower and upper bounds for the probabilities. The probability of infant nonadherence for each hypothetical case, pi,, was assumed to follow a Beta distribution with parameters ai and bi.To find aiand bi,Beta distributions were numerically fit to these point estimates, minimums, and maximums using the BetaExpertfunction of the “Prevalence” R package3 with an “expert confidence” level of .95. Finally, these Beta distributions, each describing the probability of infant nonadherence under a certain combination of covariate values, were used to provide prior information for the vector of βcoefficients (i.e., likely values of the log odds ratios) by essentially solving the equation for β. Bedrick et al. 1 provide additional detail on the specification and estimation process.
Results of the elicitation process
Figure 1 shows density plots of prior and posterior probabilities of nonadherence for the seven hypothetical cases with lines corresponding to averaged point estimates from each set of experts. There was substantial heterogeneity in priors across prototypical cases. The experts tended to agree on cases one, two, four, and six, and the priors were relatively precise. Case three ended up with a fairly vague prior, and cases five and seven were rated with extreme positive and negative values, leading to bimodal priors. It was of interest to determine which group of experts was best at predicting posterior probability of nonadherence for the prototypical cases. As can be seen in the figure, there is not a clear “winner,” but the South Africainvestigators’ priors tended to be the closest to the posterior means.
References
1.Bedrick EJ, Christensen R, Johnson W. Bayesian Binomial Regression: Predicting Survival at a Trauma Center. The American Statistician. 1997;51(3):211-218.
2.AlgDesign: Algorithmic Experimental Design. R package version 1.1-7.3. [computer program]. 2014.
3.Prevalence: Tools for prevalence assessment studies. R package version 0.4.0. [computer program]. 2014.
Figure 1. Prior and posterior density plots of the probabilities of infant nonadherence for seven hypothetical subjects with lines corresponding to averaged point estimates from each pair of experts. Each prior density represents a combination of all experts’ opinions about the likelihood of infant nonadherence under a certain combination of predictor variables (see figure 1 in the main paper for specific combinations); individual lines are averages from a particular pair of experts. Each posterior density shows how the data update the probability of infant nonadherence under the same combination of predictors as the prior. Maximum density heights indicate the most likely probability of nonadherence, e.g., experts collectively predicted about a 15% probability of infant nonadherence for a woman with characteristics matching case one;after taking the data into account, the predicted probability lowered to about 10%.