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Omnibus Tests
Omnibus tests are statistical tests that are designed to detect any of a broad range of departures from a specific null hypothesis. For example, one might want to test that a random sample came from a population distributed as normal with unspecified mean and variance. A successful genuine omnibus test would lead one to reject this hypothesis if the data came from any other distribution. By contrast, a test of normality that was sensitive specifically to thick-tailed distributions such as Cauchy would not be considered an omnibus test. A genuine omnibus test is consistent for any departure from the null, rejecting a false null hypothesis with probability approaching unity as the sample size increases; however, some statisticians include in the omnibus category certain broad-spectrum tests that do not meet this stringent condition. This entry presents general types, general features, and examples of omnibus tests.
Categories
Here are some general types of omnibus tests: (a) tests for differences in means of groups of data distinguished by “treatments,” such as in one-way analysis of variance (ANOVA); (b) tests that combine the results of independent tests of various hypotheses to indicate whether one or more of them is false; (c) tests for time-series models, such as portmanteau tests for adequacy of autoregressive-integrated-moving-average (ARIMA) models and tests for structural breaks; (d) goodness-of-fit tests for hypotheses about marginal or joint distributions, such as tests for univariate or multivariate normality.
General Features
All of these procedures have the advantage of detecting whether all aspects of the data are consistent with the hypothesis in question, but this breadth of scope comes at a price: (a) Omnibus tests typically do not pinpoint the specific features of the data that are most at variance with the null; and (b) omnibus tests typically have less power against specific departures from the null hypothesis than do tests with narrower focus. Because of these considerations, conducting an omnibus test is most often just a first step in statistical analysis. It can help in deciding whether further study is appropriate; for example, if an ANOVA test rejects that all treatment means are equal, one might then seek to identify and explain specific sources of difference. An omnibus test can also help in selecting the best methods for further analysis; for example, if the data are consistent with the normality hypothesis, then one has available the usual minimum-variance estimators and uniformly most powerful tests for mean and variance. However, in applying a sequence of tests to given data, there is the danger of data snooping (performing repeated analytical tests on a data set in the hope of finding a significant test). Thus, when applied unconditionally, a valid .05-level test that two specific treatment means differ would reject a true “no-difference” null in 5% of repeated samples; yet the same test would reject more often if applied to the two of m > 2 treatments with the most divergent means in samples for which ANOVA rejects the full ra-variate null.
Examples
One-Way ANOVA
Given independent samples (vectors) Y1,…, Ym of sizes n1,…, nm, we test H0 that all m population means are equal, assuming that the populations are normal with the same variance. H0 is rejected at level α if Σjnj(Mj − MN)2/[(m − 1)S2p] exceeds the upper-a quantile of the F distribution with m − 1 and N = Σjnj degrees of freedom, where Mj is the mean of sample j, MN is the overall mean, and S2p is the weighted average of sample variances.
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