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In normal multivariate regression you can leave out covariates (assuming they aren't correlated with any covariates you leave in) and the variance of covariates left out gets subsumed into the error term.

In survival analysis poisson models are used. In these models the variance is a function of the mean. What then happens to the variance of covariates left out of the model since there is no free parameter for estimating the error?

My, maybe naive, thought is that the variance estimate for poisson models is only correct if the model is fully specified. But that would mean that the variance for any model leaving out covariates is underestimated. Which, in turn, would mean that the p-values for any such would be very optimistic. But such models are very common in epidemiology and I can't believe they would all be this flawed. What am I missing?

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When you drop a covariate, the means and variances for each remaining covariate pattern are re-estimated. E.g., suppose there is a 2*2 factorial design with 3 observations in each of 4 cells. First, consider a full model with interaction, which will fit 4 cell means with variances set to be equal to those fitted means.

Next, you decide to drop one factor and it becomes a two-sample test with 6 observations per sample. The cell means are recomputed, and so are the variances. Apparently, the two-sample model will have more bias because now it has to use one cell mean to fit twice as many observations. That's about the same as saying that the variance of response explained by the dropped covariate was added to the "error term".

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