# Multinomial Logit - Assumptions

Assumptions

The multinomial logit model assumes that data are case specific; that is, each independent variable has a single value for each case. The multinomial logit 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, collinearity is assumed to be relatively low, as it becomes difficult to differentiate between the impact of several variables if they are highly correlated.

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.

If the multinomial logit is used to model choices, it may in some situations impose too much constraint on the relative preferences between the different alternatives. This point is especially important to take into account if the analysis aims to predict how choices would change if one alternative was to disappear (for instance if one political candidate withdraws from a three candidate race). Other models like the nested logit or the multinomial probit may be used in such cases as they need not violate the IIA.

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