How do you perform post-hoc tests and pairwise comparisons after Wilks' lambda?

When conducting a statistical analysis like those listed in this article, certain assumptions in the data must be met. Additionally, we have to ensure that we pick the correct analysis to run. If we fail to adhere to those assumptions or we use the wrong analysis, we run the risk of providing inaccurate/incomplete insights to our clients which has downstream effects that are almost never good. This is also true when using Post-hoc testing to identify group differences. As such I always recommend doing your research so that you can utilize the right Post-hoc test for your data.

Suppose you conducted a MANOVA assessing the impact of teaching methods on student performance across multiple subjects. After obtaining a significant result, you decide to use Tukey's HSD for post-hoc tests. Pairwise comparisons reveal that students taught with Method A significantly outperform those taught with Method B in Mathematics, providing valuable insights for educators. Remember to tailor the choice of post-hoc test and interpretation to your specific research goals and context.

Here is a 448 character contribution on post-hoc tests for MANOVA: After obtaining an overall significant MANOVA result, conduct follow-up analysis to pinpoint the source of group differences. Assess each DV independently via ANCOVA to control covariates while testing adjusted means - applying Bonferroni corrections for multiple tests. Or use discriminant analysis to uncover the dimensions maximally separating groups, gauging the percentage of variance explained. Cross-validate the discriminant functions on out-of-sample data to evaluate generalizability. Plot factor score means visually to observe group distinctions. Alternatively, model the DVs jointly with canonical correlation analysis to reveal multifaceted relationships.

While omnibus tests determine overall group differences, pairwise comparisons deliver granular insights into contrasts between specific conditions. Simple t-tests identify divergence but inflate familywise Type I error rates. Apply protections like Holm or False Discovery Rate control adjusting for multiplicity. Visualize confidence intervals to gauge precision and standard errors - wide bounds indicate unreliable separation. Plot effect sizes comparing true score means rather than just testing null hypotheses. Contrast analysis facilitates flexible querying of custom linear combinations tailored to research questions. Evaluate patterns in the full cross-tabulation of pairwise outputs to qualify higher order relationships.

The emmeans package streamlines post-hoc analysis after MANOVA. The pairs() function tests all pairwise comparisons between groups with automatic multiplicity correction. Apply further protections like Holm or FDR to tighten Type I error control. Use contrast() to probe custom linear combinations of means. The testInteractions() function parses higher order interaction effects. To decompose multivariate tests, utilize the regrid() function to conduct univariate post-hocs per response variable. Visualize marginal means with CIs through plot() arrangements. For discriminate analysis, use the candisc package. Automate reporting by templating R Markdown documents capturing code, output, and interpretations.

After performing a test like Wilks' lambda in a multivariate analysis of variance (MANOVA) to assess overall group differences, post-hoc tests and pairwise comparisons are essential for pinpointing where specific differences lie. Start by using tests such as Bonferroni or Tukey’s HSD to adjust for multiple comparisons and control the family-wise error rate. Conduct pairwise comparisons between groups on each dependent variable individually to identify which pairs of groups differ significantly. Ensure that the assumptions of normality and homogeneity of variance are met for each comparison.