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help for ^iia^ (Jeroen Weesie/ICS Aug 11, 1997)
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Hausman's test for independence of irrelevant alternatives
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^iia^ depvar varlist [^if^ exp] [^in^ exp] [weight]^, g^roup^(^varname^)^
^c^ode^(^varname^)^ [ clogit-options ]
Description
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Estimates McFadden's discrete choice model (with ^clogit^) and subsequently
performs Hausman tests for the assumption of "independence of irrelevant
alternatives" (IIA) for each of the alternatives.
^iia^ can also be used in more general contexts where the conditional
maximum-likelihood estimator for the logistic regression model (e.g., the ^rasch^
model of item-response theory).
Options
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^group(^varname^)^ is not optional and should be non-missing; it specifies the
identifier variable (numeric or string) for the matched groups.
^code(^varname^)^ is not optional and should be non-missing; specifies the
identifier variable (numeric or string) for the alternatives.
Example
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We have data on choices of means of transportation by a number of subjects. For
each subject and for each of his feasible alternatives (e.g., car, public
transportation, bike/walk) our data comprises measures of the monetary costs
(money), the time costs (time), and the "comfort costs" (comfort). ^clogit^ and
hence ^iia^ requires a data organization in which each (subject,alternative)
combination is described in a seperate record. In your data you have to have
variables that identify subjects (say respnr) and alternative (say means). Then
. ^iia choice money time comfort, group(respnr) code(means)^
Details
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IIA is the notorious assumption in individual decision theories and in social
choice theory that the choice (preference) between a collection of alternatives
is not affected if non-chosen alternatives are made unavailable. As a nice
example in which IIA is violated: in a restaurant Goucho Marx is offered the
choice between a Steak, Chicken, and Salmon. He chooses Steak. Then the
waitress tells him that the Chief has informed her that chicken is not
available afterall. Then Goucho Marx decides to take the Salmon rather than
the steak.
Hausman's test compares the maximum-likelihood estimator of beta based on all
data with mle's of beta that are based on data in which one alternative B is
dropped, while cases in which B was actually selected are fully dropped. Under
IIA, b(restricted) and b(overall) should be approximately equal, while IIA is
violated if the two estimates of b are different.
Formally, Hausmann has shown that the test statistic
H = (b_r - b_f)' inv(V_r - V_f) (b_r - b_f)
is approximately chi-square distributed under Ho: IIA, where b and V denote
the estimate and approximate variance matrix based on the full (f) and
restricted (r) data.
References
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Hausman, J.A. and D. McFadden (1984)
Specification Tests for the Multinomial Logit Model.
Econometrica 52: 1377-1398.
Also See
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Manual: [R] clogit
On-line: help for @clogit@; @hausman@; @rasch@.