Multinomial logistic regression: Difference between revisions

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==Assumptions==

The multinomial logistic model assumes that data are case -specific; that is, each independent variable has a single value for each case. The multinomial logistic model also assumes that the dependent variable cannot be perfectly predicted from the independent variables for any case. As with other types of regression, there is no need for the independent variables to be [[statistically independent]] from each other (unlike, for example, in a [[naive Bayes classifier]]); however, [[multicollinearity|collinearity]] is assumed to be relatively low, as it becomes difficult to differentiate between the impact of several variables if this is not the case.<ref>{{cite book | last = Belsley | first = David | title = Conditioning diagnostics : collinearity and weak data in regression | publisher = Wiley | ___location = New York | year = 1991 | isbn = 9780471528890 }}</ref>

If the multinomial logit is used to model choices, it relies on the assumption of [[independence of irrelevant alternatives]] (IIA), which is not always desirable. This assumption states that the odds of preferring one class over another do not depend on the presence or absence of other "irrelevant" alternatives. For example, the relative probabilities of taking a car or bus to work do not change if a bicycle is added as an additional possibility. This allows the choice of ''K'' alternatives to be modeled as a set of ''K''-1 independent binary choices, in which one alternative is chosen as a "pivot" and the other ''K''-1 compared against it, one at a time. The IIA hypothesis is a core hypothesis in rational choice theory; however numerous studies in psychology show that individuals often violate this assumption when making choices. An example of a problem case arises if choices include a car and a blue bus. Suppose the odds ratio between the two is 1 : 1. Now if the option of a red bus is introduced, a person may be indifferent between a red and a blue bus, and hence may exhibit a car : blue bus : red bus odds ratio of 1 : 0.5 : 0.5, thus maintaining a 1 : 1 ratio of car : any bus while adopting a changed car : blue bus ratio of 1 : 0.5. Here the red bus option was not in fact irrelevant, because a red bus was a [[perfect substitute]] for a blue bus.