Abstract
Logistic regression is usually used to model probabilities of categorical responses as functions of covariates. However, the link connecting the probabilities to the covariates is non-linear. We show in this paper that when the cross-classification of all the covariates and the dependent variable have no empty cells, then the probabilities of responses can be expressed as linear functions of the covariates. We demonstrate this for both the dichotmous and polytomous dependent variables.
Highlights
The probability of a dichotomous response is usually modelled as functions of covariates using the following: Pr Y 1 X1 x1, X p xpA feature of the above formulation is that the quantity on the right-hand side of the above equation is a fraction, and so the rule that probabilities have to lie in the interval [0, 1] is not violated assuming the estimates of, 1, p exist
We show in the remaining paper that the above, linear formulation will yield estimates of probabilities lying in [0, 1] if the cross-classification of all the covariates and the dependent variable has no empty cells
We demonstrated that probability estimates lying in the interval [0, 1] can be obtained if the probabilities themselves are modelled as linear functions of covariates, provided that the cross-classification of the covariates and the response has no empty cells
Summary
Let Y be a categorical variable with possible values 0, , q. The covariates, X1, , X p , may be categorical or continuous. Let y j ; x j1, , x jp , 1 j n , denote a data set with n outcomes of Y and of each of the p covariates. Theorem 1: Suppose that the cross-classification of the data y j ; x j1, , x jp , 1 j n , has no empty cells. If the mle’s obtained by specifying the likelihood using (1.1) and (1.2) exist, the estimates of probabilities of the response given the covariates are constrained to lie in the interval (0, 1). All the estimates of the probabilities in (1.1) and (1.2) are constrained to lie in the interval (0, 1)
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