**by Karin Dahmström**

**Research Report
1999:2**

Department of Statistics, Stockholm University, S-106 91 Stockholm, Sweden

**Abstract**

The parameters aand
b
in
the logistic model are estimated by maximum likelihood when the dependent
variable Y are observed for three fixed values of the predictor X. The
total sample is varied mainly between 30 and 90 and the properties of the
estimators in this small-sample situtation are studied. By means of complete
enumeration of all possible combinations of values Y=1 at different X-levels,
the exact distribution of the estimators are determined. Furthermore, the
exact variance is compared with two appoximations based on the asymptotic
variance including terms up to order *n ^{-}*

The exact distribution of the estimators shows a doubtful approximation to the normal distribution even for n=60 in some "nice" cases. The exact confidence level of an interval is also computed and compared with the interval based on the standard normal approximation. The different designs of the X-values are characterized by the first three central moments of the variable, and it is found that mainly the variance and the skewness of the chosen levels will influence the difference between the exact and asymptotic variance. Furthermore, a concentration of the levels to the tails of the true model will increase the uncertainty of the small-sample inference.

**Key words:** Logistic model,
small-sample inference, ML-estimation, complete enumeration, exact distribution,
relative bias, asymptotic variance, variance of order *n ^{-}*

*Last update: 1999-09-16/CE*