...models the Chow test (Chow 1960). For testing the equality two models with nonlinear link functions, some asymptotic tests are discussed by Andrews and Fair (1988) and Liao (2002). This paper assumes dispersion homogeneity and fixed groups, and focuses on the flexibility of the Wald statistic for testing hypotheses comparing multiple groups.

Whereas all three are asymptotically equivalent, the Wald test differs from the Likelihood Ratio (LR) and the Lagrange Multiplier (LM) tests in at least two important aspects. First, Wald's test explicitly expresses the hypothesis to be tested as a functional restriction of the parameters of the model under examination, and this is sometimes a desirable property. For example, a parameter in one group may be expressed as three times as large as (or the logarithm, or any other function, of) the same parameter in another group. Second, the Wald test requires the unrestricted model represented by an alternative hypothesis to be estimated; in comparison the LM test requires the restricted model to be estimated, and the LR test requires both the restricted and the unrestricted models to be estimated, thus requiring two paths of computation. In the social sciences researchers often prefer to present results separately for the social groups being analyzed (the unrestricted model), thereby making the Wald test a natural choice.

Perhaps computational efficiency is no longer a gain with the presence of today's superfast personal computers. However, the Wald test proposed in the paper has yet another important advantage over the LR test. When functional restrictions are applied to multiple parameters in multiple groups jointly. the LR test as implemented in statistical software packages is no longer adequate. As shown later in the paper, the proposed Wald test can flexibly perform such tests.

The paper contains three substantive sections following the introduction. The first discusses the usual Wald test. The section also develops a comparison-invariant Wald test that is appropriate for testing functional restriction constraints across multiple groups jointly. The second and the third sections use an example to apply the Wald test for comparing multiple logit models. Although the examples are both logit models, the applicability of the test should naturally cover all models belonging to the family of generalized linear models (McCullagh and Nelder 1989; Nelder and Wedderburn 1972).

Comparing Coefficient Estimates in Multiple Groups

Consider the following logit models for G groups of observations (which are sorted into multiple sets, i = 1,..., [M.sub.1], i = [M.sub.1]+1, ... , [M.sub.2], ... , and [M.sub.G-1] +1, ... , N, respectively, to facilitate comparison). For testing equality among G (with g running from 1 to G) sets of coefficients, we have

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[l].sub.i] (*) is the contribution to the likelihood for the ith case. The first line in (1) gives the restriction that all [[beta].sub.g] are equal, and the second line relaxes the constraint (Liao 2002). That is, the model represented by the first line estimates only one vector of [beta] parameters for the entire sample, while the model represented by the second line estimates a vector of [beta] parameters for each group of observations in the sample. To test for parameter equality, an LR test statistic can be formed by taking the ratio of the second line to the first in (1). Pairwise comparison is simply a special case of (1) where two groups of observations are involved at a time.

Researchers often prefer to analyze and present separate results for the social groups under investigation. The Wald test can be more easily applied because it relies on the unrestricted model only. The goal here is to test a null hypothesis ([H.sub.0]) that compares two or more groups in one way or another. We may test the following types of hypotheses.

First, we may be interested in testing the equality of pairs of coefficients among G number of groups. This may be represented by,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where g[not equal to][g.sup.*]. The null hypothesis of Ia is the most widely tested because it assumes identity between groups of coefficients pairwise. A variation of this gives [[beta].sub.g] as a function of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Commonly researchers may want to see, for example, if the effects of certain variables for one group of observations are twice as large as those for another group...

NOTE: All illustrations and photos have been removed from this article.



Looking for additional articles?
Search our database of over 3 million articles.

Looking for more in-depth information on this industry?
Search our complete database of Industry & Market reports by text, subject, publication name or publication date.

About Goliath
Whether you're looking for sales prospects, competitive information, company analysis or best practices in managing your organization, Goliath can help you meet your business needs.

Our extensive business information databases empower business professionals with both the breadth and depth of credible, authoritative information they need to support their business goals. Whether it be strategic planning, sales prospecting, company research or defining management best practices - Goliath is your leading source for accurate information.