Table 2 Mathematical description of statistical models used for studying model variation
| Model | Model description | Description |
|---|---|---|
| Ignoring temporal and geographic variation | ||
| Model 1 | logit(p ij ) = α 0 + βX ij where p ij denotes the probability of the outcome for the ith patient at the ith hospital. From this model, we extracted the linear predictor (LP ij ) | Fixed effects model, ignoring temporal and geographic heterogeneity |
| Models accounting for geographic heterogeneity | ||
| Model 2 | logit(p ij ) = α 0j + βX ij where the hospital-specific random effects α 0j ~ N(α 0, σ 2) | Random intercept model, allowing for variation in baseline risk, but assuming common prognostic effects |
| Model 3 | logit(p ij ) = α 0j + α 1LP ij where the hospital-specific random effects α 0j ~ N(α 0, σ 2) | Rank 1 model, allowing for common effect of the linear predictor |
| Model 4 |
logit(p
ij
) = α
0j
+ α
1j
LP
ij
where \( \left(\begin{array}{l}{\alpha}_{0 j}\\ {}{\alpha}_{1 j}\end{array}\right)\sim \mathrm{M}\mathrm{V}\mathrm{N}\left(\left(\begin{array}{l}{\alpha}_0\\ {}{\alpha}_1\end{array}\right),\left(\begin{array}{l}{\sigma}_1^2\kern0.5em {\sigma}_{12}\\ {}{\sigma}_{12}\kern0.5em {\sigma}_2^2\end{array}\right)\right) \)
The distribution of the random effects was estimated to be \( \left(\begin{array}{l}{\alpha}_{0 j}\\ {}{\alpha}_{1 j}\end{array}\right)\sim \mathrm{M}\mathrm{V}\mathrm{N}\left(\left(\begin{array}{l}0.005\\ {}1.008\end{array}\right),\left(\begin{array}{l}0.0444\kern1em 0.0139\\ {}0.0139\kern1em 0.0162\end{array}\right)\right) \) | Rank 1 model, allowing for heterogeneity in the effect of the linear predictor |
| Model 5 | logit(p ij ) = α 0j + α 1j LP ij + α 2j X 1ij where \( \left(\begin{array}{l}{\alpha}_{0 j}\\ {}{\alpha}_{1 j}\\ {}{\alpha}_{2 j}\end{array}\right)\sim \mathrm{M}\mathrm{V}\mathrm{N}\left(\left(\begin{array}{l}{\alpha}_0\\ {}{\alpha}_1\\ {}{\alpha}_2\end{array}\right),\left(\begin{array}{l}{\sigma}_1^2\kern0.5em {\sigma}_{12}\kern0.5em {\sigma}_{13}\\ {}{\sigma}_{12}\kern0.5em {\sigma}_2^2\kern0.5em {\sigma}_{23}\\ {}{\sigma}_{13}\kern0.5em {\sigma}_{23}\kern0.5em {\sigma}_3^2\end{array}\right)\right) \) and X 1ij denote an individual predictor (e.g., age) | Fully stratified model, allowing for differential prognostic effects (one model per covariate) |
| Models accounting for temporal heterogeneity | ||
| Model 6 | logit(p ij ) = α 0j + α 1 T ij + α 2LP ij where the hospital-specific random effects α 0j ~ N(α 0, σ 2) and the fixed effect for LP ij are defined as in Model 3, and T ij denotes the temporal period (T = 0 for phase 1 vs T = 1 for phase 2) | Random intercept model with a fixed main effect for phase 2 vs phase 1 |
| Model 7 | logit(p i ) = α 0j + α 1 T ij + α 2LP ij + α 3 T ij × LP ij | Random intercept model with a fixed interaction effect for phase 2 vs phase 1. The prognostic effect differs between time periods |
| Model 8 | logit(p i ) = α 0j + α 1 X ij + α 2 T ij + α 3 T ij × X ij | Random intercept model that allowed effect of each predictor to vary between time periods |
| Simultaneous exploration of geographic and temporal heterogeneity of predictor effects | ||
| Model 9 | logit(p ij ) = α 0j + α 1j LP ij + α 2j T ij + α 3j T ij × LP ij where \( \left(\begin{array}{l}{\alpha}_{0 j}\\ {}{\alpha}_{1 j}\\ {}{\alpha}_{2 j}\\ {}{\alpha}_{3 j}\end{array}\right)\sim \mathrm{M}\mathrm{V}\mathrm{N}\left(\left(\begin{array}{l}{\alpha}_0\\ {}{\alpha}_1\\ {}{\alpha}_2\\ {}{\alpha}_3\end{array}\right),\left(\begin{array}{l}{\sigma}_1^2\kern0.5em {\sigma}_{12}\kern0.5em {\sigma}_{13}\kern0.5em {\sigma}_{14}\\ {}{\sigma}_{12}\kern0.5em {\sigma}_2^2\kern0.5em {\sigma}_{23}\kern0.5em {\sigma}_{24}\\ {}{\sigma}_{13}\kern0.5em {\sigma}_{23}\kern0.5em {\sigma}_3^2\kern0.5em {\sigma}_{34}\\ {}{\sigma}_{14}\kern0.5em {\sigma}_{24}\kern0.5em {\sigma}_{34}\kern0.5em {\sigma}_4^2\end{array}\right)\right) \) | The effect of the linear predictor varies between hospitals; the effect of temporal period varies across hospitals; and the effect of temporal period on the predictor effects varies across hospitals |