up:: [[Independent Measures ANOVA Tests]] Tags:: # Repeated Measures ANOVA Repeated measures ANOVA test are a type of [[Independent Measures ANOVA Tests]] test that are done on [[Types of Experimental Design#Repeated measures design|repeated measure design]]. They are different to independent measures ANOVA tests because all individual differences are taken away. Individual differences are therefore a part of the numerator and denominator in a independent measures ANOVA test but completely eliminated in the numerator of a RMANOVA test. To get them out of the denominator, however, we must remove them ourselves. By subtracting individual differences from both the numerator and the denominator we lesson influence of sampling error and therefore increase [[Statistical Power]]. ### What are the assumptions for a RMANOVA test? 1. Random sampling and independent samples from the populations. 2. Normalcy. Distributions of sample size from which samples are selected or the distributions of sample means of sample are normal because n is greater than 30 by [[Central Limit Theorem (CLT)]]. 3. Homogeneity of variance. Variance of distributions in populations are equal. Welch's corrections to df values can compensate if there are differences. 4. Homogeneity of co-variance. #### What is the f-ratio equation for the RMANOVA? The repeated measures ANOVA equation is found by taking the [[Independent Measures ANOVA Tests#Equation for F-ratio for Independent Measures ANOVA:]] and changing it slightly. ![[Pasted image 20221019154900.png]] It changes into this: ![[Pasted image 20221019154748.png|500]] ###### What does the denominator represent in this equation? It represents the residual variance or error variance and measures how much variance is expected if there are no individual differences and no systematic treatment effects contributing to the variability in scores. ### Main formulas for the repeated measures ANOVA tests: ![[Pasted image 20221021085314.png]] ### The two stages of the RMANOVA To do a RMANOVA test we use a RMANOVA summary table: ![[Pasted image 20221021151915.png]] ### Stage 1 Stage one is identical to the calculations we do in an independent ANOVA test. We find the SStotal and dftotal and split them into within treatment and between treatment components. (see: [[Independent Measures ANOVA Tests#What are the general 9 calculations needed for an independent measures ANOVA test?]]). ### Stage 2 The second stage involves measuring the individual differences and then removing them from the denominator of the f-ratio. To do this we calculate the SSerror which equals this: ![[Pasted image 20221021092034.png|500]] We find the SSbetween subjects using this equation: ![[Pasted image 20221019155450.png|500]] ###### What does P represent in repeated measures ANOVA tests? P represents the person totals. It's calculated by adding up all the individual scores for a person. ###### What does k represent in this calculation? The number of scores added up to get each persons total is denoted by k not to be mistaken for the added up number of treatment groups like in the independent measures ANOVA. Then we do the same thing for degrees of freedom by finding the df error. df between subjects equation: ![[Pasted image 20221021151457.png|500]] Df error equation: ![[Pasted image 20221021092437.png|500]] or ![[Pasted image 20221021092458.png|500]] Finally with SSerror and dferror we can calculate the MS error with: ![[Pasted image 20221021092655.png|500]] and then the f-ratio for the repeated measures RMANOVA test with: ![[Pasted image 20221019154748.png|500]] ###### How do you report a f test for the RMANOVA? You report the degrees of freedom by stating the between treatments degrees of freedom and then the df error. ### Post hoc testing with RMANOVA You can do post hoc testing with [[Post hoc pairwise comparisons#Scheffe's post hoc test]] and [[Post hoc pairwise comparisons#Tukey HSD Test]] in the exact same way as the independent measures ANOVA provided that you substitute MSerror in place of MSwithin treatments and us dferror in place of dfwithin treatments when locating the critical values on the [[F unit table]]. ![[Pasted image 20221021152934.png|500]] #### Effect Size for Repeated Measures ANOVA ![[Pasted image 20221021152649.png|500]] Whatever probability you get you can say that x percent of the variance between the samples is because of differences from treatment effects. Related: [[Hypothesis Testing]] ___ # Resources