Multivariate analysis of variance (MANOVA) is an extremely powerful analytical tool to analyze
Multivariate data from multi-factorial
experiments where observations are partitioned into a priori groups, defined by multiple categorical independent variables. However, use of MANOVA requires the data to be continuous and normally distributed, the covariance matrices for all treatment groups to be homogeneous, the observations to be independent, and the number of variables not to exceed the number of observations. This paper is concerned with situations that do not meet these assumptions, specifically when the data are non-normal. Possible methods of handling such data are: (i) analysis of distance (AoD) or (ii)
permutational MANOVA. Steps in AoD include: (i) calculation of appropriate distances between observations, (ii) partition of the total sum of squared distances into appropriate components, (iii) permutation tests of hypotheses.
Here, we describe the AoD and the permutational MANOVA, and report the results of our investigation on the appropriateness of each technique for analyzing the significance of the interaction effects in multi-factorial experiments. We investigate the performance of each method and compare it with MANOVA whenever appropriate. We focus on testing interaction effects for various data types and the comparisons are conducted via Monte Carlo studies, using size and power of tests. To avoid complexity and extensive computer time, we focus on experiments having two cross-classified factors which can easily be extended to more complex designs. Here, we generate correlated response variables from
multivariate Gamma and multivariate Logit distributions to represent non-normal data. Analyses in AoD are based on both Euclidean and Mahalanobis distances. Our results indicate that, while Euclidean-based AoD tends to overestimate power, Mahalanobis-based AoD recorded better results. Interestingly, in situations with small samples from non-normal data, both permutational MANOVA and Mahalanobis-based AoD tend to perform slightly better than MANOVA. With no restriction on the number of variables or the nature of their individual distributions, both methods therefore provide good alternative to MANOVA.