B AYESIAN “ FEEDBACK ”

Bayesian feedback occurs in Bayesian joint models when estimation of the parameters of the outcome model affects estimation of the parameters in subsidiary models, such as the measurement error and exposure models. This concept was explored by Lunn et al. in the context of pharmacokinetic/pharmacodynamic models [135]; more recently, Plummer [129] illustrated

178 the problem of feedback specifically for the Bayesian measurement error example, using the conditional independence structure of Richardson and Gilks [35].

Bayesian “feedback” in the measurement error correction methods presented The challenge of this thesis was finding methods that would perform well with a replicate study, high measurement error, or a binary outcome. These attributes may lead to a likelihood that is empirically unidentifiable (or a flat likelihood). It was felt that the use of prior information in Bayesian analysis would enable the model to overcome these shortcomings to arrive at informative posterior estimates. However, without strong priors for parameters in the measurement error and exposure models or a strong likelihood, feedback resulted in uninformative posteriors (or a flat posterior).

Feedback has been observed in a number of the methods implemented in this thesis. In Chapter 3, the use of the SMC-FCS implementation of MI ensured that imputations were drawn from the full joint posterior distribution. In doing so, when there was insufficient support from the likelihood, the feedback resulted in erratic parameter estimates. If MI had been performed without inclusion of the outcome or consideration of the form of the substantive model, more precise estimates might have been obtained but it is hard to say exactly what posterior distribution they would represent and whether they would be unbiased.

In Chapter 5, DIC was not useful as a means of discerning the best model. This was because by specifying either the quadratic or linear models, the parameters of the substantive model played a role in determining the parameter estimates in the measurement error and exposure models. As a result, the plausibility of each form of the substantive model was inflated.

The most severe effects of feedback were observed in Chapter 6 in the context of trying to achieve the flexibility of the fractional polynomial method while relying on a replicate study for information about the measurement error. The use of a joint model (BTS with measurement error correction) in this setting resulted in an uninformative posterior (i.e. erratic curve estimates). However, even when a validation study was present, the use of the joint model of this type resulted in higher empirical type I error as a result of feedback.

It is important to note that Bayesian feedback is not necessarily a drawback. In Chapter 4, when the substantive model was specified to be a quadratic model, the feedback of this information to estimating the parameter estimates of the subsidiary models (measurement error and exposure) is an element of Bayesian analysis which improves the estimates of the association resulting in unbiased and more efficient estimates [149].

179 Future directions: Potential methods for addressing Bayesian “feedback”

The Bayesian RC methods presented in this thesis separate the estimation of the measurement error and exposure model parameters from the estimation of the substantive model parameters. The substantive model includes only a one-dimensional summary measure of the measurement error and exposure models. This approach is not unlike propensity score analysis wherein a logistic regression model is used to estimate the binary probability of being in the treated or untreated group in an observational study; a one-dimensional summary measure of that probability can be adjusted for in the exposure-outcome model [136]. This may be referred to broadly as “sequential analysis”. In 2009, McCandless introduced a Bayesian propensity score analysis approach which used the joint likelihood of the propensity score model and the outcome model for estimates of causal inference [151]. The feedback of information was noted in that publication as a feature resulting in more precise estimate of causal effect. However, Zigler et al demonstrated via simulation study that the Bayesian joint estimation of propensity scores and causal effects can result in biased estimates of the causal effects [136].

Stronger priors for the parameters of the subsidiary models and weaker priors for the parameters of the substantive model may improve estimation when the substantive model is unknown by effectively weighting the subsidiary models over the substantive model. In the implementations of BTS in Chapters 5 and 6, I have done the opposite by introducing stronger priors in the substantive model (in order to improve convergence for those methods). To my knowledge, there is no direct manner of introducing a prior that weights the faith in each model.

The most common solution to Bayesian “feedback” when the effects are undesirable is to “cut” the flow of information between the subsidiary models and the outcome model [135,137]. Posterior estimates that result from a cut model no longer represent draws from the full joint posterior; it is undefined what density they do represent, and therefore, these estimates are only approximately Bayesian.

The option to create a cut in the Bayesian model is available in OpenBUGS [93] and WinBUGS [94] but not in the JAGS software [62]. Plummer has demonstrated that the use of cuts as they are implemented in OpenBUGS results in different posterior distributions depending on the sampler used [129]. As an alternative, Plummer proposes an improved cut algorithm (“tempered cut algorithm”).

It has been suggested that MI would also achieve a separation of the joint model in a fashion similar to cut models were the outcome to be excluded from the imputation model [129]. For missing data, it has been shown that exclusion of the outcome from the imputation model results in greater bias than MI with the outcome included [103]. However, in that study no difference in the standard errors or the coverage was observed and both methods resulted in much less bias

180 than complete case analysis. For correction of additive measurement error where a validation study is available, it is possible that by using 𝑊 as a (highly correlated) auxiliary variable for 𝑋 the bias from ignoring the outcome would be minimal. Additionally, if the measurement error model were correctly specified, MI with a replicate study available which ignores the outcome would be similarly feasible. Further work should be done to compare 1) ignoring the outcome entirely in the imputation model to 2) incorporating the outcome into the imputation model in a way that incorrectly represents the relationship between exposure and outcome, i.e. assumes a linear relationship when, in fact, they are non-linearly related.