Bayesian design for a small sample and how to include informed priors

Question

I am new to Bayesian approaches so thank you in advance for your patience.

I am looking to investigate the influence of a specific type of brain lesion on the lateralization of verbal memory in a data set of 38 individuals with epilepsy.

My outcome of interest is verbal memory lateralization which can be typical (left) or atypical(right or bilateral). I also have information about other covariates that may be related to the outcome, including participants' handedness and speech lateralization.

I am interested in using a Bayesian approach to analyze this data given the small sample. My questions are:

1. What would be the best Bayesian approach to answering my research question (is there substantial evidence that this lesion influences lateralization)?

2. What would be the best way to include informed priors? I may be able to find information about the covariates, e.g. incidence of handedness and/or speech lateralization in the epilepsy population, but there is not much research about the outcome (memory lateralization).

3. Can you recommend any programs in R that would be appropriate for the recommended approach?

Thank you so much for your responses!

Answer 1

The easiest approach is to let $\theta = P(\mathrm{Left\; Lat)},$ ignoring the distinction between Right and Bi-lateralization. Then find the number $x$ with Left lateralization in your sample of $n = 38.$

With an uninformative prior such as $\theta \sim \mathsf{Beta}(.5,.5),$ a Jeffreys prior distribution, and with data $x = 30,$ your posterior distribution on $\theta$ would be $\mathsf{Beta}(30.5, 8.5).$ Then a Bayesian posterior estimate of $\theta$ is $E(\theta) = \frac{30.5}{30.5+8.5} = 78.2.$ Also, a 95% Bayesian posterior interval estimate of $\theta$ ('credible interval') would be $(0.641,0.895),$ as computed in R below.

qbeta(c(.025,.975), 30.5, 8.5)
[1] 0.6416816 0.8951003

In that case, your prior distribution has very little effect on the result. Numerically, the Bayesian interval estimate is not substantially different from a frequentist 95% confidence interval based on $30$ binomial 'successes' in $n = 38$ trials.

An informative prior $\theta \sim \mathsf{Beta}(200, 300)$ would be consistent with belief that $\theta \approx 0.6$ and $P(0.36 \le \theta \le 0.44) \approx 0.95.$

qbeta(c(.025,.975), 200, 300)
[1] 0.3575005 0.4432573

With this prior and again data $x=30$ out of $n=38$ subjects, the posterior distribution is $\mathsf{Beta}(230, 308),$ the posterior mean is $44.2$ and a 95% Bayesian probability interval is $(0.386,0,470).$ In this case, the prior overwhelms the data, so that the result is mainly due to the prior.

qbeta(c(.025,.975), 230, 308)
[1] 0.3860167 0.4695124

How strong the prior distribution should be would depend upon how sure you are about $\theta$ after taking your past experience and your literature review into account. A productive approach is to seek an equitable balance between information from the prior and information from the data.

Bayes' Theorem. Both of the results above depend on using Bayes' Theorem to multiply the kernel of the beta prior (density function without norming constant) by the binomial likelihood to get eh kernel of the beta posterior distribution.