24
Stata Technical Bulletin
STB-58
A Wald test
While the LR test is generally considered to be superior, if the model is complex or the sample is very large, the computational
costs of the LR test can be prohibitive. Alternatively, K Wald tests can be computed without estimating additional models. This
test is defined as follows. Let ∕⅜ be the J — 1 coefficients associated with a⅛. Let Var(∕¾.) be the estimated covariance matrix.
The Wald statistic for the hypothesis that all of the coefficients associated with x>s are simultaneously zero is computed as
Wfe = ⅞Var(⅛j⅛
If the null hypothesis is true, then Wj. is distributed as chi-squared with J-I degrees of freedom.
Testing multiple independent variables
This logic of the Wald or LR tests can be extended to simultaneously test that the effects of two or more independent
variables are zero. For example, the hypothesis to test that x⅛ and хц, have no effects is
-®0 ∙ Λ,l∣Base — ’ ’ ’ — A,J∣Base — ∕¾,l∣Base — ’ ’ ’ — ∕¾,J∣Base — θ
The set option in mlogtest specifies which variables are to be simultaneously tested. This is particularly useful when a series
of dummy variables are used to code a nominal or ordinal independent variable.
Testing that two outcomes can be combined
If none of the xβs significantly affect the odds of outcome m versus outcome n, we say that m and n are indistinguishable
with respect to the variables in the model. If βιtm∖n, ■ ■ ∙ , ∕‰,m∣n are the coefficients for xi through xκ from the logit of m
versus n, then the hypothesis that outcomes m and n are indistinguishable corresponds to
∙¾> ∙ βl,m∖n ’ βκ,m∖n θ
Note that if the base category used by Stata is not n, these coefficients are not directly available. However, this hypothesis can
be rewritten equivalently using the coefficients with respect to the base category
-®0 : (∕¾-,m∣Base β1 ,n∣Base) —
ase βκ,m ∣ Base) — θ
A Wald test for this hypothesis can be computed with Stata’s test command. mlogtest, combine executes and summarizes
the results of J × (J — 1) calls to test for all pairs of outcome categories.
An LR test of this hypothesis can be computed by first estimating the full model that contains all of the variables, with
the resulting LR statistic LRp. Then estimate a restricted model Mr in which category m is used as the base category and all
the coefficients (except the constant) in the equation for category n are constrained to 0, with the resulting chi-squared statistic
LRR. The test statistic is the difference LRRvsf = LRp — LRr, which is distributed as chi-squared with K degrees of freedom.
mlogtest, Ircomb summarizes the results of the J × (J — 1) LR tests for all pairs of outcome categories.
Independence of irrelevant alternatives
The MNLM assumes that the odds for any pair of outcomes are determined without reference to the other outcomes that might be
available. This is known as the independence of irrelevant alternatives property or simply IIA. Hausman and McFadden (1984)
proposed a Hausman-type test of this hypothesis. Basically, this involves the following steps.
1. Estimate the full model with all J outcomes included; these estimates are contained in ∕⅜.
2. Estimate a restricted model by eliminating one or more outcome categories; these estimates are contained in ∕‰.
3. Let ∕3F be a subset of βγ after eliminating coefficients not estimated in the restricted model. The Hausman test of IIA is
defined as
‰ = (⅛ - ⅛)' [var(⅛) - var(3p)] ^1 (⅛ - ⅛)
¾A is asymptotically distributed as chi square with degrees of freedom equal to the number of rows in ∕‰ if IIA is true.
Significant values of Hjja indicate that the IIA assumption has been violated.
Hausman and McFadden (1984, 1226) note that Hjja can be negative when Var(∕‰) — Var( JF ) is not positive semidefinite and
suggest that a negative Hjja is evidence that IIA holds.