Title : Parametric and non-parametric one-way multivariate analysis of variance with ordinal data: Power and type I error rates
Abstract:
Using R software, a simulation study was developed to evaluate and compare the performance of four parametric (Pillai’s trace, Wilks’ lambda, Hotelling’s trace, and Roy’s largest root) and two non-parametric tests (Pillai’s Trace and Wilks’ Lambda applied to the rank transformed data). In the data generation process, scales with three, four, five, and seven points were considered and different marginal distributions, correlation structures, numbers of groups, and sample sizes were accounted for. A Monte Carlo resampling simulation (1000 replications) was carried out to estimate the power and type I error rate from the proportion of rejected null hypotheses at a 5% significance level of parametric and non-parametric one-way MANOVA with ordinal data.
The analysis of the simulation results allows us to conclude that Roy’s largest root was the statistic that presented the highest values of the probability of type I error, and was also one of the statistics with the biggest statistical power rate. The power presents different behaviours depending on the distribution, the magnitude of correlation between items, the sample dimension, and the points of the scale.
Based on the frequency distribution, the data analysis of the power results allows us to identify three distinct situations. In the first situation, for different scenarios, the power is of low magnitude because MANOVA does not detect differences between groups due to their similarity. In the second situation, the magnitude of the power rate is similar in parametric and non-parametric tests, and it depends on the sample size and the number of scale points, and in different scenarios, the power has a higher magnitude if the sample size is larger and the points of scale are lower. In the third situation, the magnitude of the power rate of MANOVA depends on a combination of the correlation between dependent variables, the sample size, and the number of scale points. As the correlations and number of scale points increased and the sample size decreased, the power rate of MANOVA decreased, and Wilks’ lambda applied to the rank transformed data had a higher power rate than the other statistics.
In order to carry out the analysis of these scales, three practical applications that fitted the study simulation scenarios are presented. Weight and level of spinal cord injuries were the independent variables chosen for the selection of groups, whereas new-borns were grouped by “weight classes” and children and young people with spina bifida were grouped by “level of spinal cord injuries” and age group. There was a good framework of the results of practical applications. The results with real data (practical applications) are similar to the results that emerged from the simulation study, so researchers and health professionals who work with scales should take these results into account.
