We've completed a systematic review of % diagnosed among female sex workers (FSW) living with HIV across Sub-Saharan Africa (SSA). In our Objective 1, we model this proportion $p_s$ as: \begin{aligned} y_s &\sim \textrm{binom}(p_s,n_s) \\ \textrm{logit}(p_s) &= \epsilon_s + R_s (\alpha_r + \beta_r t) \end{aligned} where:
- $y_s$ is the observed count diagnosed among $n_s$ respondents in survey $s$
- $\epsilon_s$ is a random study effect;
- $R_s$ and $r$ denote 1 of 4 SSA region of study $s$
- $\alpha_r$ and $\beta_r$ are intercept and slope w.r.t. (centered) calendar year $t$
If relevant, we solve the model in a Bayesian framework with priors on $\epsilon_s$, $\alpha_r$, $\beta_r$.
In our Objective 2, we want to infer the relative proportions among FSW vs women overall (WO) within each country-year -- this could be as odds-ratio or risk ratio. We want to consider possible differences in the relative proportions w.r.t. year and region. So, we propose the following model:
\begin{aligned} y_s &\sim \textrm{binom}(p_s,n_s) \\ \textrm{logit}(p_s) &= \epsilon_s + R_s (\alpha_r + \beta_r t) + P_s R_s (\alpha_{pr} + \beta_{pr} t) \end{aligned}
Where $P_s$ and $p \in \{\textrm{FSW},\textrm{WO}\}$ denote the population of study $s$, and we now include studies of both FSW and WO. FYI, We are mostly interested in simply plotting the point (95% CI) of $p_r = p_s - \epsilon_s$ here.
Questions:
- Is our overall approach to Objective 2 reasonable?
- Could we interpret $\mathrm{exp}(\alpha_{pr} + \beta_{pr} t)$ as the time-varying odds ratio in % diagnosed among FSW vs women overall in region $r$? I think: yes
- Is it important to find women-overall studies with same years as FSW studies? I think: not important
- Is it important to find all available WO studies? I think: yes, but this means we need to do another systematic review...
- Should we add a country level error term too, beyond $\epsilon_s$? I think: not sure
