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In: Statistics and Probability

What is logit model? How it can be responses for multicategory responses?

What is logit model? How it can be responses for multicategory responses?

Solutions

Expert Solution

The logit model uses something called the cumulative distribution function of the logistic distribution.the logistic model (or logit model) is used to model the probability of a certain class or event existing such as pass/fail, win/lose, alive/dead or healthy/sick. This can be extended to model several classes of events such as determining whether an image contains a cat, dog, lion, etc... Each object being detected in the image would be assigned a probability between 0 and 1 and the sum adding to one.

Logit models for multinomial responses

When the dependent variable is not binary but has multiple outcome cate-

gories we use the multinomial instead of binomial for modeling.

Two types of multicategory outcome can be considered: nominal and ordinal.

Nominal response variable

In logistic regression the log odds of ’success’ is predicted.

When there are J outcome categories, the multinomial logit model predicts

simultaneously all pairs of log odds. Not all are necessary: a good choice of

J − 1 provides all information.

Ordinal response variable

Cumulative logit models define cumulative probabilities

P(Y ≤ j|x) = π1(x) + π2(x) + . . . + πj(x), j = 1, . . . , J − 1.

and from these the cumulative logits

logit[P(Y ≤ j|x)] = log (P(Y ≤ j|x) /1 − P(Y ≤ j|x))

A model for a single logit[P(Y ≤ j|x)] alone is a standard logit model. It is

better to simultaneously model all cumulative logits (i.e. for different j.)


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