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Lambda

Lambda, often called Wilks’ lambda, is a statistic used most often in multivariate analysis of variance, or MANOVA (e.g., discriminant analysis or canonical correlation). The comparison is to univariate analysis of variance, or ANOVA, where there is only one dependent variable. Essentially, the function of Wilks’ lambda is to serve as an omnibus test much in the same manner that the F test is used to examine the significance of any effect in univariate analysis. A second use involves the generation of a multidimensional vector or equation used to generate a means of predicting classification of individual elements. A final use of lambda involves the consideration of multiple dependent variables where correlation exists between the dependent variables and the desire is to take that into consideration. In communication research, lambda is important to the extent that it can be used to facilitate the identification of methods to identify and classify members of a particular group. This entry will examine how lambda can be adopted as an omnibus test to justify specific ANOVAs, to generate a multivariate vector to classify elements, and to consider relationships among dependent variables.

Using Lambda as an Omnibus Test Justifying Individual ANOVAs

Wilks’ lambda is usually used to respond to a concern that may exist when using multiple dependent variables that could be analyzed separately using the F test in an ANOVA. The use of multiple separate univariate F statistics could raise the level of Type I error (false positives) since each test has an independent probability of alpha (Type I error) of 5% (p < .05). The problem with using multiple tests, where each one separately tests a different dependent variable, is the potential for false positives or Type I errors. The more tests that are run, the greater the probability that one of the tests will randomly be significant. Considering the possibility of such errors, the term familywise error is used to apply to the use of multiple tests across a set of conditions. The goal of lambda was to create a means of handling multiple dependent variables and maintaining the same level of Type I error (usually set at 5%). Wilks’ lambda provides a distribution for the multivariate normal that creates an overall test with a total Type I error rate (often called familywise error) of 5% (p < .05).

Much of the suggested practice involves using the omnibus or overall test as Wilks’ lambda. If that test is significant, then the possibility for separate univariate ANOVAs becomes justified. In this respect, the test can function as an overall test to maintain Type I rates (it should be noted that such application will significantly increase the Type II error rate or false negatives). The investigator needs to consider whether maintaining Type I error rates is worth the cost in loss of statistical power for individual tests.

The focus of this use of lambda becomes not the results of the test but instead its use as an omnibus test simply provides the basis as a justification for the eventual use of univariate follow-up ANOVA tests that are employed. The only advantage of this use is maintaining the familywise Type I error rate because the actual statistic provides no other useful information.

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