Stata Technical Bulletin
23
Saved results
mlogtest saves in r():
r(combine) contains results of tests to combine categories. Rows represent all contrasts among categories; columns indicate
the categories contrasted, the chi-squared value, the degrees of freedom, and the p-value.
r(iia) contains results of the Hausman test of IIA assumption. Each row is one test. Columns indicate the omitted category of
a given test, the chi-squared value, the degrees of freedom, and the p-value.
r(wald) contains results of the Wald test that all coefficients of an independent variable equal zero.
r(Irtest) contains results of the LR test that all coefficients associated with an independent variable equal zero.
Methods and formulas
This section provides brief descriptions of each of the tests. For further details, check the Stata manual for mlogit, test,
and hausman. Full details along with citations to original sources are found in Long (1997). To make our discussion of the tests
clear, we begin with a brief summary of the multinomial logit model (MNLM).
The multinomial logit model
For simplicity, we consider a model with three outcomes and three independent variables. The MNLM can be thought of as
simultaneously estimating binary logits among all pairs of the outcome categories. For example, with categories A, B, and C
and independent variables æɪ, a⅛ and a⅛, the MNLM is in effect simultaneously estimating three binary models:
In
'P(A∖x)'
ʃɔ(ɑlʃ).
In
'p(β∖xγ
,p⅛),
In
'P{A∖x)'
P(B∖x)
βθ,A∖C + βl,A∖Cxl + β2,A∖Cx2 + βs,A∖Cx3
βθ,B∖C + βl,B∖Cxl + β2,B∖cχ2 + β3,B∖cχ3
βθ,A∖B + βl,A∖Bxl + β2,A∖Bx2 + ∕¾,A∣Ba⅞
Note that three more equations could be listed, comparing C to A, C to B, and B to A. Given that the sum of the probabilities
for the outcomes must equal one, there is an implicit constraint on the three logits. Specifically,
In
'P{A∖x)
P(⅛
-In
~P(B∖x)~
,P⅛),
= In
'P{A∖x)
Р(В|ж)
in terms of the parameters
βk,A∖C - βk,B∖C — βk,A∖B
mlogit estimates and prints only the nonredundant coefficients, which are determined by the basecategory() option or,
by default, the category with the largest number of cases. The commands mcross (Rogers 1995) and Iistcoef (Long and
Freese 2000) list coefficients for all comparisons of outcome categories.
Testing the effect of an independent variable
With J dependent categories, there are J — 1 nonredundant coefficients associated with each independent variable x^. The
hypothesis that Xj. does not affect the dependent variable can be written as
Bals ∙ A,l∣Base — ’ ’ ’ — A,J∣Base — θ
where Base is the base category used in the comparison. Since ∕¾ιBase∣Base is necessarily zero, the hypothesis imposes constraints
on J — 1 parameters. This hypothesis can be tested with either a Wald or a LR test.
A LR test
First, estimate the full model Mp that contains all of the variables, with the resulting LR statistic LRF. Second, estimate
the restricted model Mp formed by excluding variable x>s, with the resulting LR statistic LRR. This model has J — 1 fewer
parameters. Finally, compute the difference LRrvsf = LRp — LRr which is distributed as chi-squared with J — 1 degrees of
freedom if the hypothesis that x⅛ does not affect the outcome is true. mlogtest, Ir computes this test for each of the K
independent variables by making repeated calls to Stata’s Irtest. Note that this requires estimating K additional multinomial
logit models.
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