Berkson's Minimum Chi-square Method
This method can be applied only when there are many observations of response variable having the same value of the vector of regressors (such situation may be referred to as "many observations per cell"). More specifically, the model can be formulated as follows.
Suppose among n observations there are only T distinct values of the regressors, which can be denoted as . Let be the number of observations with and the number of such observations with . We assume that there are indeed "many" observations per each "cell": for each .
Denote
Then Berkson's minimum chi-square estimator is a generalized least squares estimator in a regression of on with weights :
It can be shown that this estimator is consistent (as n→∞ and T fixed), asymptotically normal and efficient. Its advantage is the presence of a closed-form formula for the estimator. However, it is only meaningful to carry out this analysis when individual observations are not available, only their aggregated counts, and (for example in the analysis of voting behavior).
Read more about this topic: Probit Model
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