Using fixed-effects model multivariate analysis of variance in marine biology and ecology

The robustness and power of four commonly used MANOVA statistics (the Pillai-Bartlett trace (V), Wilks' Lambda (W), Hotelling's trace (1), Roy's greatest root (R)) are reviewed and their behaviours demonstrated by Monte Carlo simulations using a one-way fixed effects design in which assumptions of the model are violated in a systematic way under different conditions of sample size (n), number of dependent variables (P), number of groups (k), and balance in the data. The behaviour of Box's M statistic, which tests for covariance heterogeneity, is also examined. The behaviours suggest several recommendations for multivariate design and for application of MANOVA in marine biology and ecology, viz. (1) Sample sizes should be equal. (2) p, and to a lesser extent k, should be kept to a minimum insofar as the hypothesis permits. (3) Box's M statistic is rejected as a test of homogeneity of covariance matrices. A suitable alternative is Hawkins' (1981) statistic that tests for heteroscedasticity and non-normality simultaneously. (4) To improve agreement with assumptions, and thus reliability of tests, reduction of p (e.g. by PCA or MDS methods) and/or transforming data to stabilise variances should be attempted. (5) The V statistic is recommended for general use but the others are more appropriate in particular circumstances. For Type I errors, the violation of the assumption of homoscedasticity is more serious than is nonnormality and the V statistic is clearly the most robust to variance heterogeneity in terms of controlling level. Kurtosis reduces the power of all statistics considerably. Loss of power is dramatic if assumptions of normality and homoscedasticity are violated simultaneously. (6) The preferred approach to multiple comparison procedures after MANOVA is to use Bonferroni-type methods in which the total number of comparisons is limited to the fewest possible. If all possible comparisons are required an alternative is to use the V statistic in the overall test and the R statistic in a follow-up simultaneous test procedure. We recommend following a significant MANOVA result with a canonical discriminant analysis. (7) Classical parametric MANOVA should not be used with data in which high levels of variance heterogeneity cannot be rectified or in which sample sizes are unequal and assumptions are not satisfied. We discuss briefly alternatives to parametric MANOVA.