We have a study to predict the disease event outcome: We used the mixed effect logistic regression.
Outcome: binary disease yes(1) or no(0)| Fixed effect: complication1(binary, yes or no), size of hospital (small/medium/big, ref=small), gender (ref=male)| Random effect: hospital (different hospitals)|
Results were:
Random intercept: hospital, covariance estimate=0.03, SE=0.02, p=0.06
Fixed intercept, estimate=-1.32; Fixed effect:
| effect | Estimate (SE) |
|---|---|
| Complication1 | 0.20(0.11) |
| hospital size (M) | -0.07(0.06) |
| hospital size (L) | 1.33(1.10) |
| gender female | 0.14(0.07) |
Reviewer asked us to write the formula/equation based on this results above.
Anyone can let me know how to write it based on this outcome which included a random intercept?
This should probably go on Cross Validated, where it would be much easier to write an answer using TeX.
A logistic regression models the probability of disease, π, in the log odds scale:
The equation for your model then becomes:
The dummy indicators and matching parameters can be allocated as you like (as long as they match), for example x1 = Complication1 (so B1 = 0.20), x2 = hospital M (B2 = -0.07), x3 = hospital L, x4 = female. Gamma_i is your per-hospital random effect, which is distributed as follows:
The important parameter is sigma_h or the across-hospital standard deviation or variance, which from your output I'm guessing is 0.03 ('hospital covariance estimate').