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I came across the scale parameter used in the logit and probit models. Does any one know what that is and what it is used for? What would go wrong if I did not use it?

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The basic form of the model is location-only component. It predicts the level of response. If some predictors are categorical, variability of response (more precisely, in the underlying continuous variable which is linked by a link function with the observed response) in different groups might be different (nonhomogeneity of variance, heteroscedasticity), but location-only model ignores this circumstance. In such situation adding scale component to the model - the component that accounts for the above nonhomogeneity and also estimates it - can improve the location prediction. So, the combined location+scale model is more general and in a sense is preferable.

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