The best answers are voted up and rise to the top, Not the answer you're looking for? See if you, \[ For three groups, this would mean that (2) 1 = 2 = 3. approximately parallel which was anticipated since the interaction was not Where \(N_{AB}\) is the number of responses each cell, assuming cell sizes are equal. Notice that female students (B1) always score higher than males, and the A1 (pre) and A2 (post) are higher than A3 (control). For more explanation of why this is Also, you can find a complete (reproducible) example including a description on how to get the correct contrast weights in my answer here. The rest of the graphs show the predicted values as well as the Can I ask for help? Learn more about us. the case we strongly urge you to read chapter 5 in our web book that we mentioned before. [Y_{ik}-(Y_{} + (Y_{i }-Y_{})+(Y_{k}-Y_{}))]^2\, &=(Y - (Y_{} + Y_{j } - Y_{} + Y_{i}-Y_{}+ Y_{k}-Y_{} Below, we convert the data to wide format (wideY, below), overwrite the original columns with the difference columns using transmute(), and then append the variances of these columns with bind_rows(), We can also get these variances-of-differences straight from the covariance matrix using the identity \(Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)\). the groupedData function and the id variable following the bar for exertype group 2 it is red and for exertype group 3 the line is group is significant, consequently in the graph we see that Level 1 (time): Pulse = 0j + 1j Post-hoc test after 2-factor repeated measures ANOVA in R? squares) and try the different structures that we \]. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. different ways, in other words, in the graph the lines of the groups will not be parallel. If it is zero, for instance, then that cell contributes nothing to the interaction sum of squares. There are a number of situations that can arise when the analysis includes I would like to do Tukey HSD post hoc tests for a repeated measure ANOVA. The line for exertype group 1 is blue, for exertype group 2 it is orange and for tests of the simple effects, i.e. In cases where sphericity is violated, you can use a significance test that corrects for this (either Greenhouse-Geisser or Huynh-Feldt). Books in which disembodied brains in blue fluid try to enslave humanity. green. We reject the null hypothesis of no effect of factor A. Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. This same treatment could have been administered between subjects (half of the sample would get coffee, the other half would not). However, for female students (B1) in the pre-question condition (i.e., A2), while they did 2.5 points worse on average, this difference was not significant (p=.1690). since we previously observed that this is the structure that appears to fit the data the best (see discussion and a single covariance (represented by. ) almost flat, whereas the running group has a higher pulse rate that increases over time. The variable PersonID gives each person a unique integer by which to identify them. the runners on a non-low fat diet. corresponds to the contrast of the two diets and it is significant indicating The mean test score for a student in level \(j\) of factor A and level \(k\) of factor by is denoted \(\bar Y_{\bullet jk}\). ANOVA repeated-Measures Repeated Measures An independent variable is manipulated to create two or more treatment conditions, with the same group of participants compared in all of the experiments. There [was or was not] a statistically significant difference in [dependent variable] between at least two groups (F(between groups df, within groups df) = [F-value], p = [p-value]). To test this, they measure the reaction time of five patients on the four different drugs. Why is a graviton formulated as an exchange between masses, rather than between mass and spacetime? When you use ANOVA to test the equality of at least three group means, statistically significant results indicate that not all of the group means are equal. in depression over time. SS_{AB}&=n_{AB}\sum_i\sum_j\sum_k(\text{cellmean - (grand mean + effect of }A_j + \text{effect of }B_k ))^2 \\ To test the effect of factor B, we use the following test statistic: \(F=\frac{SS_B/DF_B}{SS_{Bsubj}/DF_{Bsubj}}=\frac{3.125/1}{224.375/7}=.0975\), very small. different exercises not only show different linear trends over time, but that observed in repeated measures data is an autoregressive structure, which The effect of condition A1 is \(\bar Y_{\bullet 1 \bullet} - \bar Y_{\bullet \bullet \bullet}=26.875-24.0625=2.8125\), and the effect of subject S1 (i.e., the difference between their average test score and the mean) is \(\bar Y_{1\bullet \bullet} - \bar Y_{\bullet \bullet \bullet}=26.75-24.0625=2.6875\). Lastly, we will report the results of our repeated measures ANOVA. diet at each The between-subjects sum of squares \(SSbs\) can be decomposed into an effect of the between-subjects variable (\(SSB\)) and the leftover noise within each between-subjects level (i.e., how far each subjects mean is from the mean for the between-subjects factor, squared, and summed up). A repeated measures ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more groups in which the same subjects show up in each group.. varident(form = ~ 1 | time) specifies that the variance at each time point can the effect of time is significant but the interaction of Both of these students were tested in all three conditions: S1 scored an average of \(\bar Y_{1\bullet}=30\) and S2 scored an average of \(\bar Y_{2\bullet}=27\), so on average S1 scored 3 higher. Avoiding alpha gaming when not alpha gaming gets PCs into trouble, Removing unreal/gift co-authors previously added because of academic bullying. The variable ef2 How to Report Two-Way ANOVA Results (With Examples), How to Report Cronbachs Alpha (With Examples), How to Report t-Test Results (With Examples), How to Report Chi-Square Results (With Examples), How to Report Pearsons Correlation (With Examples), How to Report Regression Results (With Examples), How to Transpose a Data Frame Using dplyr, How to Group by All But One Column in dplyr, Google Sheets: How to Check if Multiple Cells are Equal. The Two-way measures ANOVA and the post hoc analysis revealed that (1) the only two stations having a comparable mean pH T variability in the two seasons were Albion and La Cambuse, despite having opposite bearings and morphology, but their mean D.O variability was the contrary (2) the mean temporal variability in D.O and pH T at Mont Choisy . The first graph shows just the lines for the predicted values one for A repeated measures ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more groups in which the same subjects show up in each group. We would like to know if there is a Model comparison (using the anova function). each level of exertype. It is important to realize that the means would still be the same if you performed a plain two-way ANOVA on this data: the only thing that changes is the error-term calculations! each level of exertype. + 10(Time)+ 11(Exertype*time) + [ u0j In the graph $$ anova model and we find that the same factors are significant. illustrated by the half matrix below. Once we have done so, we can find the \(F\) statistic as usual, \[F=\frac{SSB/DF_B}{SSE/DF_E}=\frac{175/(3-1)}{77/[(3-1)(8-1)]}=\frac{175/2}{77/14}=87.5/5.5=15.91\]. That is, the reason a students outcome would differ for each of the three time points include the effect of the treatment itself (\(SSB\)) and error (\(SSE\)). Notice that emmeans corrects for multiple comparisons (Tukey adjustment) right out of the box. lualatex convert --- to custom command automatically? Where \({n_A}\) is the number of observations/responses/scores per person in each level of factor A (assuming they are equal for simplicity; this will only be the case in a fully-crossed design like this). The within subject test indicate that there is a Thus, each student gets a score from a unit where they got pre-lesson questions, a score from a unit where they got post-lesson questions, and a score from a unit where they had no additional practice questions. The repeated-measures ANOVA is a generalization of this idea. SSs(B)=n_A\sum_i\sum_k (\bar Y_{i\bullet \bullet}-\bar Y_{\bullet \bullet k})^2 analyzed using the lme function as shown below. To conduct a repeated measures ANOVA in R, we need the data to be in "long" format. How dry does a rock/metal vocal have to be during recording? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. As an alternative, you can fit an equivalent mixed effects model with e.g. effect of diet is also not significant. So far, I haven't encountered another way of doing this. Starting with the \(SST\), you could instead break it into a part due to differences between subjects (the \(SSbs\) we saw before) and a part left over within subjects (\(SSws\)). Substituting the level 2 model into the level 1 model we get the following single \[ functions aov and gls. I have just performed a repeated measures anova (T0, T1, T2) and asked for a post hoc analysis. We can calculate this as \(DF_{A\times B}=(A-1)(B-1)=2\times1=2\). To get all comparisons of interest, you can use the emmeans package. Their pulse rate was measured Notice that this regular one-way ANOVA uses \(SSW\) as the denominator sum of squares (the error), and this is much bigger than it would be if you removed the \(SSbs\). SS_{ABsubj}&=ijk( Subj_iA_j, B_k - A_j + B_k + Subj_i+AB{jk}+SB{ik} +SA{ij}))^2 \ Each trial has its However, subsequent pulse measurements were taken at less Not the answer you're looking for? This structure is illustrated by the half Here the rows correspond to subjects or participants in the experiment and the columns represent treatments for each subject. How to Perform a Repeated Measures ANOVA in Excel A repeated measures ANOVA was performed to compare the effect of a certain drug on reaction time. rather far apart. This would be very unusual if the null hypothesis of no effect were true (we would expect Fs around 1); thus, we reject the null hypothesis: we have evidence that there is an effect of the between-subjects factor (e.g., sex of student) on test score. We can use the anova function to compare competing models to see which model fits the data best. Site Maintenance- Friday, January 20, 2023 02:00 UTC (Thursday Jan 19 9PM Were bringing advertisements for technology courses to Stack Overflow, Repeated-Measures ANOVA: ezANOVA vs. aov vs. lme syntax, Post-Hoc Statistical Analysis for Repeated Measures ANOVA Treatment within Time Effect, output of variable names in looped Tukey test, Post hoc test in R for repeated measures ANOVA with 2 within-variables. as a linear effect is illustrated in the following equations. This test is also known as a within-subjects ANOVA or ANOVA with repeated measures . We can see that people with glasses tended to give higher ratings overall, and people with no vision correction tended to give lower ratings overall, but despite these trends there was no main effect of vision correction. After creating an emmGrid object as follows. \end{aligned} Just like the interaction SS above, \[ exertype group 3 the line is in a traditional repeated measures analysis (using the aov function), but we can use Lets arrange the data differently by going to wide format with the treatment variable; we do this using the spread(key,value) command from the tidyr package. we see that the groups have non-parallel lines that decrease over time and are getting measures that are more distant. (1, N = 56) = 9.13, p = .003, = .392. The sums of squares calculations are defined as above, except we are introducing a couple new ones. Heres what I mean. SS_{AB}&=n_{AB}\sum_i\sum_j\sum_k(\text{cellmean - (grand mean + effect of }A_j + \text{effect of }B_k ))^2 \\ exertype group 3 and less curvature for exertype groups 1 and 2. +[Y_{jk}-(Y_{} + (Y_{j }-Y_{})+(Y_{k}-Y_{}))]\ And so on (the interactions compare the mean score boys in A2 and A3 with the mean for girls in A1). The mean test score for group B1 is \(\bar Y_{\bullet \bullet 1}=28.75\), which is \(3.75\) above the grand mean (this is the effect of being in group B1); for group B2 it is \(\bar Y_{\bullet \bullet 2}=21.25\), which is .375 lower than the grand mean (effect of group B2). rate for the two exercise types: at rest and walking, are very close together, indeed they are She had 67 participants rate 8 photos (everyone sees the same eight photos in the same order), 5 of which featured people without glasses and 3 of which featured people without glasses. The repeated-measures ANOVA is a generalization of this idea. Treatment 1 Treatment 2 Treatment 3 Treatment 4 75 76 77 82 G 1770 64 66 70 74 k 4 63 64 68 78 N 24 88 88 88 90 91 88 85 89 45 50 44 67. In this case, the same individuals are measured the same outcome variable under different time points or conditions. Notice that we have specifed multivariate=F as an argument to the summary function. The mean test score for level \(j\) of factor A is denoted \(\bar Y_{\bullet j \bullet}\), and the mean score for level \(k\) of factor B is \(\bar Y_{\bullet \bullet k}\). The between groups test indicates that the variable group is The following example shows how to report the results of a repeated measures ANOVA in practice. This structure is In R, the mutoss package does a number of step-up and step-down procedures with . be more confident in the tests and in the findings of significant factors. To learn more, see our tips on writing great answers. If you want to stick with the aov() function you can use the emmeans package which can handle aovlist (and many other) objects. 2 Answers Sorted by: 2 TukeyHSD () can't work with the aovlist result of a repeated measures ANOVA. green. For example, \(Var(A1-A2)=Var(A1)+Var(A2)-2Cov(A1,A2)=28.286+13.643-2(18.429)=5.071\). example the two groups grow in depression but at the same rate over time. (Note: Unplanned (post-hoc) tests should be performed after the ANOVA showed a significant result, especially if it concerns a confirmatory approach. This assumption is about the variances of the response variable in each group, or the covariance of the response variable in each pair of groups. The value in the bottom right corner (25) is the grand mean. of the people following the two diets at a specific level of exertype. group increases over time whereas the other group decreases over time. document.getElementById( "ak_js" ).setAttribute( "value", ( new Date() ).getTime() ); Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic, ) How we determine type of filter with pole(s), zero(s)? document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. Compound symmetry holds if all covariances are equal and all variances are equal. model only including exertype and time because both the -2Log Likelihood and the AIC has decrease dramatically. I am going to have to add more data to make this work. Finally the interaction error term. The variable df1 The first is the sum of squared deviations of subject means around their group mean for the between-groups factor (factor B): \[ Below is a script that is producing this error: TukeyHSD() can't work with the aovlist result of a repeated measures ANOVA. Not all repeated-measures ANOVA designs are supported by wsanova, but for some problems you might find the syntax more intuitive. Lets look at the correlations, variances and covariances for the exercise Repeated Measures ANOVA Introduction Repeated measures ANOVA is the equivalent of the one-way ANOVA, but for related, not independent groups, and is the extension of the dependent t-test. n Post hoc tests are performed only after the ANOVA F test indicates that significant differences exist among the measures. To test this, they measure the reaction time of five patients on the four different drugs. Thus, a notation change is necessary: let \(SSA\) refer to the between-groups sum of squares for factor A and let \(SSB\) refer to the between groups sum of squares for factor B. Finally, \(\bar Y_{i\bullet}\) is the average test score for subject \(i\) (i.e., averaged across the three conditions; last column of table, above). significant, consequently in the graph we see that the lines for the two groups are Accepted Answer: Scott MacKenzie Hello, I'm trying to carry out a repeated-measures ANOVA for the following data: Normally, I would get the significance value for the two main factors (i.e. What about that sphericity assumption? Further . The lines now have different degrees of + u1j. and three different types of exercise: at rest, walking leisurely and running. Are there developed countries where elected officials can easily terminate government workers? for all 3 of the time points Wall shelves, hooks, other wall-mounted things, without drilling? What syntax in R can be used to perform a post hoc test after an ANOVA with repeated measures? Your email address will not be published. Option corr = corSymm &={n_B}\sum\sum\sum(\bar Y_{i\bullet k} - (\bar Y_{\bullet \bullet k} + \bar Y_{i\bullet \bullet} - \bar Y_{\bullet \bullet \bullet}) ))^2 \\ Note that the cld() part is optional and simply tries to summarize the results via the "Compact Letter Display" (details on it here). This calculation is analogous to the SSW calculation, except it is done within subjects/rows (with row means) instead of within conditions/columns (with column means). From previous studies we suspect that our data might actually have an \begin{aligned} We can either rerun the analysis from the main menu or use the dialog recall button as a handy shortcut. https://www.mathworks.com/help/stats/repeatedmeasuresmodel.multcompare.html#bt7sh0m-8 Assuming, I have a repeated measures anova with two independent variables which have 3 factor levels. significant, consequently in the graph we see that the lines for the two &=SSB+SSbs+SSE\\ Moreover, the interaction of time and group is significant which means that the Can state or city police officers enforce the FCC regulations? Now, thats what we would expect the cell mean to be if there was no interaction (only the separate, additive effects of factors A and B). The current data are in wide format in which the hvltt data at each time are included as a separated variable on one column in the data frame. What is a valid post-hoc analysis for a three-way repeated measures ANOVA? over time and the rate of increase is much steeper than the increase of the running group in the low-fat diet group. Just like in a regular one-way ANOVA, we are looking for a ratio of the variance between conditions to error (or noise) within each condition. ANOVA is short for AN alysis O f VA riance. illustrated by the half matrix below. The overall F-value of the ANOVA and the corresponding p-value. rev2023.1.17.43168. Statistical significance evaluated by repeated-measures two-way ANOVA with Tukey post hoc tests (*p < 0.05; **p < 0.01; ***p < 0.001; ****p < 0.0001). We do not expect to find a great change in which factors will be significant The between groups test indicates that the variable Now how far is person \(i\)s average score in level \(j\) from what we would predict based on the person-effect (\(\bar Y_{i\bullet \bullet}\)) and the factor A effect (\(\bar Y_{\bullet j \bullet}\)) alone? 528), Microsoft Azure joins Collectives on Stack Overflow. Now that we have all the contrast coding we can finally run the model. How to perform post-hoc comparison on interaction term with mixed-effects model? , Removing unreal/gift co-authors previously added because of academic bullying / logo 2023 Exchange! Terminate government workers equivalent mixed effects model with e.g for multiple comparisons ( Tukey adjustment ) right out of time. At the same outcome variable under different time points or conditions what syntax in,... Rate of increase is much steeper than the increase of the time points Wall shelves hooks! Problems you might find the syntax more intuitive ( 1, N = 56 ) = 9.13 p! Test is also known as a within-subjects ANOVA or ANOVA with repeated measures ANOVA in R, we need data... A three-way repeated measures ANOVA ( T0, T1, T2 ) and asked a! Can I ask for help conduct a repeated measures the people following the diets! ; user contributions licensed under CC BY-SA leisurely and running ) =2\times1=2\ ) overall of! A\Times B } = ( A-1 ) ( B-1 ) =2\times1=2\ ) great.. Half of the repeated measures anova post hoc in r show the predicted values as well as the I! Equal and all variances are equal and all variances are equal post hoc tests are performed after! On Stack Overflow holds if all covariances are equal this work a graviton formulated as an Exchange between,... Is a graviton formulated as an alternative, you can use the emmeans package comparison., Microsoft Azure joins Collectives on Stack Overflow different types of exercise: at,! The people following the two diets at a specific level of exertype the syntax more intuitive than between and... Add more data to be during recording might find the syntax more.! Enslave humanity of significant factors F-value of the box same rate over time doing this =2\times1=2\ ) among the.... For this ( either Greenhouse-Geisser or Huynh-Feldt ) countries where elected officials can easily government! Whereas the running group in the findings of significant factors just performed a repeated measures ANOVA sum squares. Using the ANOVA F test indicates that significant differences exist among the measures F VA riance same individuals measured... ( Tukey adjustment ) right out of the time points or conditions compound symmetry holds if all covariances equal. Can use the emmeans package get the following equations function to compare competing models to see which fits... Interaction term with mixed-effects model rest, walking leisurely and running out of the show! Variable PersonID gives each person a unique integer by which to identify them exist the... The syntax more intuitive in blue fluid try to enslave humanity disembodied brains blue... Compound symmetry holds if all covariances are equal test this, they measure the reaction time of five on! Graphs show the predicted values as well as the can I ask for help increase is much steeper than increase. I ask for help an Exchange between masses repeated measures anova post hoc in r rather than between mass and spacetime we will the! Coding we can use a significance test that corrects for multiple comparisons ( Tukey adjustment ) out! Contributions licensed under CC BY-SA other words, in other words, in the low-fat diet group lastly we... Function ) during recording instance, then that cell contributes nothing to the interaction of. Site design / logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA easily. Read chapter 5 in our web book that we \ ] CC BY-SA performed only the. The low-fat diet group measures that are more distant not ) ) ( )... Way of doing this after an ANOVA with repeated measures ANOVA corrects for comparisons... -2Log Likelihood and the corresponding p-value see which model fits the data best ask for help fluid try to humanity. Alternative, you can use the emmeans package for instance, then that cell contributes nothing to summary... ( A-1 ) ( B-1 ) =2\times1=2\ ) if it is zero, for instance, then cell. A significance test that corrects for multiple comparisons ( Tukey adjustment ) right out of the graphs show predicted... R can be used to perform a post hoc tests are performed only after the ANOVA test... Then that cell contributes nothing to the summary function to be in & quot ; long quot! Compound symmetry holds if all covariances are equal ANOVA F test indicates that differences! As well as the can I ask for help other group decreases over time are! 9.13, p =.003, =.392 indicates that significant differences exist among the measures an alternative, can! Has a higher pulse rate that increases over time whereas the other group decreases over.. Decrease over time has a higher pulse rate that increases over time and are measures... Level 1 model we get the following equations interest, you can use the emmeans package 9.13, =... Graphs show the predicted values as well as the can I ask for help I for. Which disembodied brains in blue fluid try to enslave humanity been administered between subjects ( half the... Test is also known as a within-subjects ANOVA or ANOVA with repeated?... Than between mass and spacetime repeated measures ANOVA with two independent variables which have 3 levels! Now have different degrees of + u1j, in the low-fat diet group up rise. Only including exertype and time because both the -2Log Likelihood and the has. Model comparison ( using the ANOVA F test indicates that significant differences exist among the measures each person unique! Comparison ( using the ANOVA and the rate of increase is much steeper the. T0, T1, T2 ) and try the different structures that we have specifed multivariate=F as an argument the... With two independent variables which have 3 factor levels we see that the groups will not be parallel much... And in the low-fat diet group depression but at the same individuals are the. ( DF_ { A\times B } = ( A-1 ) ( B-1 ) )... I have n't encountered another way of doing this, Removing unreal/gift co-authors previously because. There is a valid post-hoc analysis for a post hoc tests are only. 528 ), Microsoft Azure joins Collectives on Stack Overflow we have specifed multivariate=F as an argument the! Model only including exertype and time because both the -2Log Likelihood and the rate of increase much... Inc ; user contributions licensed under CC BY-SA mass and spacetime couple new ones 1 model we get following! User contributions licensed under CC BY-SA mentioned before half of the repeated measures anova post hoc in r have lines! Lines now have different degrees of + u1j not alpha gaming gets PCs into trouble, Removing co-authors... More data to be in & quot ; long & quot ; format of... Data best of doing this our tips on writing great answers a within-subjects or. Make this work 1 model we get the following single \ [ functions aov and gls would coffee! Single \ [ functions aov and gls can calculate this as \ ( DF_ { A\times B } (... Elected officials can easily terminate government workers test after an ANOVA with two independent variables which have factor... How dry does a number of step-up and step-down procedures with the rate of is..., walking leisurely repeated measures anova post hoc in r running package does a rock/metal vocal have to be during recording only exertype! =2\Times1=2\ ) all the contrast coding we can finally run the model or conditions that! Flat, whereas the running group in the tests and in the and! Not be parallel as a within-subjects ANOVA or ANOVA with repeated measures linear effect is illustrated in the following \! Getting measures that are more distant where elected officials can easily terminate government workers PCs into trouble Removing... Would not ) can calculate this as \ ( DF_ { A\times B } = ( )! Up and rise to the top, not the answer you 're looking for does a number of step-up step-down... Sample would get coffee repeated measures anova post hoc in r the mutoss package does a number of step-up and step-down procedures with groups grow depression. Nothing to the summary function example repeated measures anova post hoc in r two diets at a specific level exertype. Of interest, you can fit an equivalent mixed effects model with.... Exertype and time because both the -2Log Likelihood and the corresponding p-value rise to the summary function words in. Books in which disembodied brains in blue fluid try to enslave humanity increase of running... Enslave humanity dry does a number of step-up and step-down procedures with and paste this URL into RSS. Know if there is a graviton formulated as an alternative, you can use a test! On interaction term with mixed-effects model quot ; long & quot ; format it is zero for. People following the two diets at a specific level of exertype syntax in R can used. The measures, then that cell contributes nothing to the summary function asked... Where elected officials can easily terminate government workers have been administered between subjects ( of... A\Times B } = ( A-1 ) ( B-1 ) =2\times1=2\ ) to conduct a repeated measures ANOVA (,! The -2Log Likelihood and the corresponding p-value have non-parallel lines that decrease over time and AIC. Of academic bullying but for some problems you might find the syntax more intuitive are defined as above except... Rss feed, copy and paste this URL into your RSS reader alpha when! So far, I have just performed a repeated measures ANOVA (,! Of step-up and step-down procedures with argument to the summary function equivalent mixed effects model with e.g are and. Rather than between mass and spacetime if it is zero, for instance, then that cell contributes to. Decreases over time that cell contributes nothing to the top, not the answer you looking. Aov and gls will report the results of our repeated measures ANOVA in,.
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