Detecting special Toeplitz error structure and testing for homogeneity of effects in one-way ANOVA models

Observations within the treatment groups/blocks in ANOVA models often exhibit correlations due to the presence of temporal or spatial factors. Standard ANOVA tests are often seriously affected by correlated error structure, leading to invalid conclusions. The problems of testing (a) whether the covariance matrix of a multivariate random variable has a tridiagonal Toeplitz structure and (b) the homogeneity of mean treatment effects in a heteroscedastic one-way ANOVA model with a tridiagonal Toeplitz error structure are studied under multinormality. The likelihood ratio tests for both the problems are developed. The critical points of these tests are computed using parametric bootstrap. The asymptotic accuracy of the bootstrap is established for both problems. Simulations show that these tests control the type-I error rates quite effectively regardless of choice of sample sizes and parameter values. Moreover, they have very good power performance under various parametric configurations under the alternatives. The asymptotic versions of these tests are seen to achieve the nominal size only for large samples. Simultaneous confidence intervals are obtained for pairwise differences of mean treatment effects. A user-friendly software package has been developed in ‘R’ for implementation of tests on real data sets. Robustness of the proposed tests is assessed under mild departures from normality and in scenarios where the assumption of a tridiagonal Toeplitz error structure is violated for some groups. Finally, the practical usefulness of our proposed tests is illustrated using blood pressure and clinical trial data sets.