# Effect of modelling matched data via fixed effects

A simple approach to handling matched or clustered data would be fit a regression model that includes fixed effects corresponding to matched pair or cluster ID. For example, with 1:1 matching on $N$ total observations, the linear predictor would contain coefficients for $\frac{N}{2}$ matched pair ID indicators. The linear predictor would also contain $\widehat{\beta}_1$, which estimates the effect of interest (as well as the usual population intercept).

I have heard that if one does this with logistic regression, under 1:1 matching, $\widehat{\beta}_1 \to 2\beta_{1}$ as $n \to \infty$. That is, $\widehat{\beta}_1$ is biased upward by a stunning 100%. The explanation for this was that the number of parameters increases with the sample size, so the usual asymptotics of the MLE break down. This makes sense to me intuitively.

It is also my understanding that fitting such a model does NOT cause bias in $\widehat{\beta}_1$ for linear regression or Poisson regression. What is special about linear and Poisson links that makes them robust to this violation of the standard asymptotic assumptions? Can we say anything about how such a model would behave with other link functions?

• tag clustering changed to cluster-sample. Be careful assigning tags, please. – ttnphns Mar 29 '16 at 6:36