A Note on Sufficient Conditions for Valid Unmodified <i>t</i> Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data
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Résumé
Abstract Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. "Observation vectors" may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized. Keywords: Bivariate correlationCircularityCompound symmetryEffective sample sizeSphericityUnmodified vs. modified t tests of significanceVariance–covariance matrices and structuresMathematics Subject Classification: 15A0415A2462H2062F03 Acknowledgments The author gratefully acknowledges an Individual Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). He also thanks Professor Douglas P. Wiens (University of Alberta) and two anonymous reviewers for constructive comments on a former version of the manuscript. Notes †This structure corresponds to sphericity. ‡The intra-class correlation structure is also called "compound symmetry"; see text.
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