Recommendations for analysis of repeated‐measures designs: testing and correcting for sphericity and use of manova and mixed model analysis

Abstract

Purpose: A common experimental design in ophthalmic research is the repeatedmeasures design in which at least one variable is a within-subject factor. This design is vulnerable to lack of 'sphericity' which assumes that the variances of the differences among all possible pairs of within-subject means are equal. Traditionally, this design has been analysed using a repeated-measures analysis of variance (RM-ANOVA) but increasingly more complex methods such as multivariate ANOVA (MANOVA) and mixed model analysis (MMA) are being used. This article surveys current practice in the analysis of designs incorporating different factors in research articles published in three optometric journals, namely Ophthalmic and Physiological Optics (OPO), Optometry and Vision Science (OVS), and Clinical and Experimental Optometry (CXO), and provides advice to authors regarding the analysis of repeated-measures designs. Recent findings: Of the total sample of articles, 66% used a repeated-measures design. Of those articles using a repeated-measures design, 59% and 8% analysed the data using RM-ANOVA or MANOVA respectively and 33% used MMA. The use of MMA relative to RM-ANOVA has increased significantly since 2009/10. A further search using terms to select those papers testing and correcting for sphericity ('Mauchly's test', 'Greenhouse-Geisser', 'Huynh and Feld') identified 66 articles, 62% of which were published from 2012 to the present. Summary: If the design is balanced without missing data then MANOVA should be used rather than RM-ANOVA as it gives better protection against lack of sphericity. If the design is unbalanced or with missing data then MMA is the method of choice. However, MMA is a more complex analysis and can be difficult to set up and run, and care should be taken first, to define appropriate models to be tested and second, to ensure that sample sizes are adequate.

The interaction between homatropine and optical blur on choroidal thickness

“…Since all continuous variables were normally distributed (Kolmogorov–Smirnov test, all p > 0.05) a repeated‐measures analysis of variance ( anova ) with three within‐subject factors (defocus, drug and time) and one between‐subject factor (refractive error) was performed to detect any significant differences in each of the ocular parameters as a result of the interaction between the different blur conditions and pharmacological agents. The assumption of sphericity was tested with Mauchly's test of sphericity and if not met, then a Greenhouse‐Geisser correction to the degrees of freedom was applied . Post‐hoc analysis with a Bonferroni adjustment was used to investigate specific differences in variables with significant within–subject effects and interactions.…”

Do people prefer natural landscapes? An empirical study in Chile

“…Multivariate repeated measures variance analyses (MANOVA) were applied for testing landscape composition and scenic beauty (within-subject factors) effects on responses (individual ratings), as well as the influence of personal characteristics (inter-subject factors) on those ratings (interaction effects). Repeated measures MANO-VA was preferred over ANOVA, because the last rests on the assumption of sphericity and compound symmetry (Armstrong 2017). Inspection of residuals was carried out to check for normality and homoscedasticity and variables were log-transformed as necessary.…”

Assessment of Problem‐Solving Skills and Inhibitory Control in Mice Using Water Escape Detour Test

“…Data from repeated trials performed on the same subjects are usually analyzed with repeated-measures ANOVA, but this test should be applied only to datasets that meet the requirement of sphericity (Armstrong, 2017;O'Brien & Kaiser, 1985). If data meet the requirement of sphericity, go to step 4.…”

“…Step 3: Apply multivariate analysis of variance (MANOVA) that does not require sphericity of data (Armstrong, 2017;O'Brien & Kaiser, 1985). In the case of a significant effect of at least one factor or presence of significant interaction between factors, go to steps 5 and 6.…”