Learn Posterior Estimation by Heart: An illustration in pharmacology
DA is a remedial tool for sparse data issues to provide unbiased estimation of parameters. To illustrate DA mechanism, we considered how to estimate posterior properties via inverse-variance weighting, and then show the influence of prior and likelihood components on estimating posterior distributions. Finally, we depict how to construct data from prior to observe its role as equivalent data augmented to actual data.
Result of a hypothetical data was reported Ln(RR)=Ln (6.3) =1.82, Variance(Ln(RR)) = 0.84 and 95% limits RR= ( 1.02 , 37.3) which was subjected to sparse data due to wide 95% limits . Suppose that, in a meta-analysis for side effect of a drug study we found prior information for RR with 95% limits between \(\frac{\mathbf{1}}{\mathbf{3}}\) and 3. Mean and variance of prior for Ln (RR) are estimated as Prior mean for ln (RR)=\(\frac{(\text{Ln}\left(\frac{1}{3}\right)+\text{Ln}(3))}{2}=\)0, and Prior Variance for ln (RR)=\(\ {(\frac{\left|\text{Ln}\left(\frac{1}{3}\right)-Ln\left(3\right)\right|}{2*1.96})}^{2}=0.10\). Inverse variances equaling \(\frac{1}{0.1}=10\) and\(\frac{1}{0.84}=1.2\) illustrating prior information dominated data information by nearly 8 times. Posterior mean and variance for Ln (RR) could be estimated as the following weighted averaging rule of thumb; Posterior mean for\(\ \ln\left(\text{RR}\right)=\frac{\frac{0}{0.10}+\frac{1.82}{0.84}}{\frac{1}{0.10}+\frac{1}{0.84}}=\ \ 0.19\)and Posterior variance for\(\ln{\left(\text{RR}\right)\approx\frac{1}{\frac{1}{0.10}+\frac{1}{0.84}}}=0.09\). Posterior RR and its 95%CI through DA provided unbiased estimation of RR with more reasonable values of RR and narrower 95%CI. In addition, the value of posterior mean which is closer to prior means showed the influence of the prior as well. Various prior ranges for Ln (RR), estimated posterior 95% CI, as well as the influence of prior and data, were illustrated in (Table 2). It was depicted that, for prior (\(\frac{\mathbf{1}}{\mathbf{6}}\), 6) data and prior had the same influence (equal weights) while for (\(\frac{\mathbf{1}}{\mathbf{10}}\), 10) it was data dominated.
Here, Compatibility of prior and data is a great issue as well. DA fails in case of incompatibility causes misleading results(13). For our example,\(\frac{(Ln\left(6.3\right)-0)}{{(0.84+0.10)}^{\frac{1}{2}}}=1.9P_{\text{value}}=0.057\)showed that compatibility hypothesis is not rejected.