I am a PhD student using logistic regression to investigate mental health epidemiology. Since participants in my cohort study have a diagnosis or not (coded 1 or 0) - I'm using logistic regression to estimate the assocation of mental health disorders with my exposures (continuous in nature). I'm also using hierarchical model due to the multilevel sampling structure of my data. I've already decided to use a Bayesian framework, and this has worked beautifully for my analysis of continuous outcomes. However, I've found that setting informative and weakly informative priors very challenging for hierarchical logistic regression.
Reading Gelman et al. (2008), I understand one approach to Bayesian logistic regression (not hierarchical) is to standardize the input variables. They say, scale variables before setting priors by doing the following:
- "Binary inputs shifted to have a mean of 0 and to differ by 1 in their lower and upper conditions. (For example, if a population is 10% African-American and 90% other, we would define the centered “African-American” variable to take on the values 0.9 and −0.1.)"
- "Other inputs are shifted to have a mean of 0 and scaled to have a standard deviation of 0.5. This scaling puts continuous variables on the same scale as symmetric binary inputs (which, taking on the values ±0.5, have standard deviation 0.5)."
Once data is scaled in this way, Gelman et al. (2008) and Gelman again in this webpage go on to explain that one can use a Cauchy distribution or more recently a Student's t distribution with $3<\nu<7$ for as a weakly informative coefficients in the predictor.
Now my question - will I illustrate with by expanding on Gelman's example:
Say, I am investigating the association between being African-American in the United States, $x_1$, and the prevalence $p$ of depression. Socio-economic income is confounding variable, therefore I condition on household income $x_2$. This data has come from 50 different sources, each representing data from a US state.
Without taking into account the clustered nature of this data, it is relatively simple to follow Gelman et al.'s (2008) advice and just scale the data based on the nationwide means of the variables ($x_1$ and $x_2$). Assuming $x_1$ and $x_2$ are scaled, and $i = 1, ..., N$ represents each subject, the model would be:
$$log(\frac{p_i}{1-p_i}) = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} $$
and I could set weakly informative priors:
$$ \boldsymbol{\beta} \sim student(\nu=5) $$
But since I have data in a multilevel, for multiple reasons I wish to look at the association of depression with being African-American using a random-intercept model for each state.
This model, for the $i^{th}$ subject in the $j^{th}$ sttate could now be written as:
$$log(\frac{p_{ij}}{1-p_{ij}}) = \beta_0 + u_j + \beta_1x_{1i} + \beta_2x_{2i} $$
where $u_j$ represents the random-intercept term for the level 2 unit, in this case the US state, and is normally distributed: $u_j \sim N(0, \sigma^2)$
But now I am stuck! My questions:
Do I still scale predictor variables based on nationwide means of the variables ($x_1$ and $x_2$)? Or do I instead use the within-state means and standard deviations? I wish to go with the former, but I worry that this will unduly affect my output estimates.
Using either method from (1), would I still use the same weakly informative prior for $\beta$ coefficients in the same way?
Finally, what are your approaches for weakly informative or informative priors for hierarchical logistic regression, if different from the paper I've referenced?
I'm sorry for the long intro! Would really love to hear your thoughts on any of the above three questions!
References
Gelman, A., Jakulin, A., Pittau, M. G., & Su, Y. S. (2008). A weakly informative default prior distribution for logistic and other regression models. Annals of applied Statistics, 2(4), 1360-1383. http://www.stat.columbia.edu/~gelman/research/published/priors11.pdf
