Maximum likelihood vs. minimum chi-square.

In minimum chi-square logit or probit analysis of quantal bioassay data, a requirement for proper asymptotic behavior of the estimates made is that test-group sizes get indefinitely large. Inconsistent estimates result if group sizes are small, however numerous the groups. Maximum likelihood estimates do not show this inconsistent behavior, even if all the many group sizes are only unity. The inconsistent behavior for minimum chi-square results from a bias toward 0.5 for response probabilities. At 0.5 the binomial variance is at a maximum of 0.25, so tending to minimize the calculated value of chi square. The principle of minimum chi-square should not be confused with the principle of least squares.