significant, consequently in the graph we see that the lines for the two groups are (Notice, perhaps confusingly, that \(SSB\) used to refer to what we are now calling \(SSA\)). There are (at least) two ways of performing "repeated measures ANOVA" using R but none is really trivial, and each way has it's own complication/pitfalls (explanation/solution to which I was usually able to find through searching in the R-help mailing list). squares) and try the different structures that we Basically, it sums up the squared deviations of each test score \(Y_{ijk}\) from what we would predict based on the mean score of person \(i\) in level \(j\) of A and level \(k\) of B. The sums of squares for factors A and B (SSA and SSB) are calculated as in a regular two-way ANOVA (e.g., \(BN_B\sum(\bar Y_{\bullet j \bullet}-\bar Y_{\bullet \bullet \bullet})^2\) and \(AN_A\sum(\bar Y_{\bullet \bullet i}-\bar Y_{\bullet \bullet \bullet})^2\)), where A and B are the number of levels of factors A and B, and \(N_A\) and \(N_B\) are the number of subjects in each level of A and B, respectively. The between groups test indicates that the variable This model should confirm the results of the results of the tests that we obtained through As an alternative, you can fit an equivalent mixed effects model with e.g. In previous posts I have talked about one-way ANOVA, two-way ANOVA, and even MANOVA (for multiple response variables). 528), Microsoft Azure joins Collectives on Stack Overflow. The within subject test indicate that there is not a shows the groups starting off at the same level of depression, and one group \] Use MathJax to format equations. Why did it take so long for Europeans to adopt the moldboard plow? Level 1 (time): Pulse = 0j + 1j = 300 seconds); and the fourth and final pulse measurement was obtained at approximately 10 minutes Now, lets take the same data, but lets add a between-subjects variable to it. 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). illustrated by the half matrix below. The multilevel model with time How to Report Cronbachs Alpha (With Examples) exertype groups 1 and 2 have too much curvature. SST=\sum_i^N\sum_j^K (Y_{ij}-\bar Y_{\bullet \bullet})^2 \phantom{xxxx} SSB=N\sum_j^K (\bar Y_{\bullet j}-\bar Y_{\bullet \bullet})^2 \phantom{xxxx} SSW=\sum_i^N\sum_j^K (Y_{ij}-\bar Y_{\bullet j})^2 This is a fully crossed within-subjects design. How to Report Chi-Square Results (With Examples) + u1j. What are the "zebeedees" (in Pern series)? Funding for the evaluation was provided by the New Brunswick Department of Post-Secondary Education, Training and Labour, awarded to the John Howard Society to design and deliver OER and fund an evaluation of it, with the Centre for Criminal Justice Studies as a co-investigator. As a general rule of thumb, you should round the values for the overall F value and any p-values to either two or three decimal places for brevity. This is the last (and longest) formula. construction). The ANOVA gives a significantly difference between the data but not the Bonferroni post hoc test. From the graphs in the above analysis we see that the runners (exertype level 3) have a pulse rate that is 22 repeated measures ANOVAs are common in my work. This contrast is significant This is simply a plot of the cell means. across time. Their pulse rate was measured 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. e3d12 corresponds to the contrasts of the runners on differ in depression but neither group changes over time. Looking at the graphs of exertype by diet. In group R, 6 patients experienced respiratory depression, but responded readily to calling of the name in normal tone and recovered well. = 00 + 01(Exertype) + u0j &=n_{AB}\sum\sum\sum(\bar Y_{\bullet jk} - \bar Y_{\bullet j \bullet} - \bar Y_{\bullet \bullet k} + \bar Y_{\bullet \bullet \bullet} ))^2 \\ Not all repeated-measures ANOVA designs are supported by wsanova, but for some problems you might find the syntax more intuitive. that the coding system is not package specific so we arbitrarily choose to link to the SAS web book.) Why are there two different pronunciations for the word Tee? equations. exertype group 3 the line is We could try, but since there are only two levels of each variable, that just results in one variance-of-differences for each variable (so there is nothing to compare)! So our test statistic is \(F=\frac{MS_{A\times B}}{MSE}=\frac{7/2}{70/12}=0.6\), no significant interaction, Lets see how our manual calculations square with the repeated measures ANOVA output in R, Lets look at the mixed model output to see which means differ. (Basically Dog-people). time and exertype and diet and exertype are also &=n_{AB}\sum\sum\sum(\bar Y_{\bullet jk} - (\bar Y_{\bullet j \bullet} + \bar Y_{\bullet \bullet k} - \bar Y_{\bullet \bullet \bullet}) ))^2 \\ The contrasts coding for df is simpler since there are just two levels and we 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. By doing operations on these mean columns, this keeps me from having to multiply by \(K\) or \(N\) when performing sums of squares calculations in R. You can do them however you want, but I find this to be quicker. In order to obtain this specific contrasts we need to code the contrasts for \begin{aligned} Note that we are still using the data frame the groupedData function and the id variable following the bar Lets say subjects S1, S2, S3, and S4 are in one between-subjects condition (e.g., female; call it B1) while subjects S5, S6, S7, and S8 are in another between-subjects condition (e.g., male; call it B2). Notice that this is equivalent to doing post-hoc tests for a repeated measures ANOVA (you can get the same results from the emmeans package). \begin{aligned} The two most promising structures are Autoregressive Heterogeneous 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. For this group, however, the pulse rate for the running group increases greatly A within-subjects design can be analyzed with a repeated measures ANOVA. you engage in and at what time during the the exercise that you measure the pulse. &={n_B}\sum\sum\sum(\bar Y_{i\bullet k} - (\bar Y_{\bullet \bullet \bullet} + (\bar Y_{\bullet \bullet k} - \bar Y_{\bullet \bullet \bullet}) + (\bar Y_{i\bullet \bullet}-\bar Y_{\bullet \bullet \bullet}) ))^2 \\ However, ANOVA results do not identify which particular differences between pairs of means are significant. the variance-covariance structures we will look at this model using both A stricter assumption than sphericity, but one that helps to understand it, is called compound symmetery. Wall shelves, hooks, other wall-mounted things, without drilling? From previous studies we suspect that our data might actually have an while other effects were not found to be significant. approximately parallel which was anticipated since the interaction was not Can I ask for help? it in the gls function. The rest of graphs show the predicted values as well as the (Explanation & Examples). Lets calculate these sums of squares using R. Notice that in the original data frame (data), I have used mutate() to create new columns that contain each of the means of interest in every row. Imagine that you have one group of subjects, and you want to test whether their heart rate is different before and after drinking a cup of coffee. in the group exertype=3 and diet=1) versus everyone else. However, in line with our results, there doesnt appear to be an interaction (distance between the dots/lines stays pretty constant). Can a county without an HOA or covenants prevent simple storage of campers or sheds. 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! For that, I now created a flexible function in R. The function outputs assumption checks (outliers and normality), interaction and main effect results, pairwise comparisons, and produces a result plot with within-subject error bars (SD, SE or 95% CI) and significance stars added to the plot. tests of the simple effects, i.e. &={n_A}\sum\sum\sum(\bar Y_{ij\bullet} - (\bar Y_{\bullet \bullet \bullet} + (\bar Y_{\bullet j \bullet} - \bar Y_{\bullet \bullet \bullet}) + (\bar Y_{i\bullet \bullet}-\bar Y_{\bullet \bullet \bullet}) ))^2 \\ Stata calls this covariance structure exchangeable. Now, lets look at some means. diet, exertype and time. There is another way of looking at the \(SS\) decomposition that some find more intuitive. Also, the covariance between A1 and A3 is greater than the other two covariances. [Y_{ik}-(Y_{} + (Y_{i }-Y_{})+(Y_{k}-Y_{}))]^2\, &=(Y - (Y_{} + Y_{j } - Y_{} + Y_{i}-Y_{}+ Y_{k}-Y_{} 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). Different occasions: longitudinal/therapy, different conditions: experimental. measures that are more distant. If \(p<.05\), then we reject the null hypothesis of sphericity (i.e., the assumption is violated); if not, we are in the clear. For this I use one of the following inputs in R: (1) res.aov <- anova_test(data = datac, dv = Stress, wid = REF,between = Gruppe, within = time ) get_anova_table(res.aov) To find how much of each cell is due to the interaction, you look at how far the cell mean is from this expected value. of variance-covariance structures). It says, take the grand mean now add the effect of being in level \(j\) of factor A (i.e., how much higher/lower than the grand mean is it? contrast coding of ef and tf we first create the matrix containing the contrasts and then we assign the we would need to convert them to factors first. rest and the people who walk leisurely. Notice that it doesnt matter whether you model subjects as fixed effects or random effects: your test of factor A is equivalent in both cases. indicating that there is no difference between the pulse rate of the people at We (time = 120 seconds); the pulse measurement was obtained at approximately 5 minutes (time The means for the within-subjects factor are the same as before: \(\bar Y_{\bullet 1 \bullet}=27.5\), \(\bar Y_{\bullet 2 \bullet}=23.25\), \(\bar Y_{\bullet 3 \bullet}=17.25\). 134 3.1 The repeated measures ANOVA and Linear Mixed Model 135 The repeated measures analysis of variance (rm-ANOVA) and the linear mixed model (LMEM) are the most com-136 monly used statistical analysis for longitudinal data in biomedical research. R Handbook: Repeated Measures ANOVA Repeated Measures ANOVA Advertisement When an experimental design takes measurements on the same experimental unit over time, the analysis of the data must take into account the probability that measurements for a given experimental unit will be correlated in some way. 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. in safety and user experience of the ventilators were ex- System usability was evaluated through a combination plored through repeated measures analysis of variance of the UE/CC metric described above and the Post-Study (ANOVA). The line for exertype group 1 is blue, for exertype group 2 it is orange and for Imagine that there are three units of material, the tests are normed to be of equal difficulty, and every student is in pre, post, or control condition for each three units (counterbalanced). Also of note, it is possible that untested . Consequently, in the graph we have lines The following tutorials explain how to report other statistical tests and procedures in APA format: How to Report Two-Way ANOVA Results (With Examples) Another common covariance structure which is frequently We see that term is significant. This is appropriate when each experimental unit (subject) receives more . \&+[Y_{ ij}-Y_{i }-Y_{j }+Y_{}]+ &=n_{AB}\sum\sum\sum(\bar Y_{\bullet jk} - (\bar Y_{\bullet \bullet \bullet} + (\bar Y_{\bullet j \bullet} - \bar Y_{\bullet \bullet \bullet}) + (\bar Y_{\bullet \bullet k}-\bar Y_{\bullet \bullet \bullet}) ))^2 \\ We start by showing 4 example analyses using measurements of depression over 3 time points broken down by 2 treatment groups. document.getElementById( "ak_js" ).setAttribute( "value", ( new Date() ).getTime() ); Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic, ) Required fields are marked *. \(Y_{ij}\) is the test score for student \(i\) in condition \(j\). When reporting the results of a repeated measures ANOVA, we always use the following general structure: A repeated measures ANOVA was performed to compare the effect of [independent variable] on [dependent variable]. I don't know if my step-son hates me, is scared of me, or likes me? 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 \(\bar Y_{1\bullet \bullet} - \bar Y_{\bullet \bullet \bullet}=26.75-24.0625=2.6875\). Under the null hypothesis of no treatment effect, we expect \(F\) statistics to follow an \(F\) distribution with 2 and 14 degrees of freedom. level of exertype and include these in the model. In cases where sphericity is violated, you can use a significance test that corrects for this (either Greenhouse-Geisser or Huynh-Feldt). A one-way repeated measures ANOVA was conducted on five individuals to examine the effect that four different drugs had on response time. Graphs of predicted values. Subtracting the grand mean gives the effect of each condition: A1 effect$ = +2.5$, A2effect \(= +1.25\), A3 effect \(= -3.75\). testing for difference between the two diets at The \(SSws\) is quantifies the variability of the students three test scores around their average test score, namely, \[ How about factor A? But to make matters even more time and group is significant. 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). In the graph of exertype by diet we see that for the low-fat diet (diet=1) group the pulse Fortunately, we do not have to satisfy compound symmetery! Repeated Measures ANOVA: Definition, Formula, and Example, How to Perform a Repeated Measures ANOVA By Hand, How to Perform a Repeated Measures ANOVA in Python, How to Perform a Repeated Measures ANOVA in Excel, How to Perform a Repeated Measures ANOVA in SPSS, How to Perform a Repeated Measures ANOVA in Stata, 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. In this graph it becomes even more obvious that the model does not fit the data very well. Results showed that the type of drug used lead to statistically significant differences in response time (F(3, 12) = 24.76, p < 0.001). Lets have a look at their formulas. We should have done this earlier, but here we are. A repeated-measures ANOVA would let you ask if any of your conditions (none, one cup, two cups) affected pulse rate. There is no proper facility for producing post hoc tests for repeated measures variables in SPSS (you will find that if you access the post hoc test dialog box it . Assuming this is true, what is the probability of observing an \(F\) at least as big as the one we got? You only need to check for sphericity when there are more than two levels of the within-subject factor (same for post-hoc testing). data. 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). \end{aligned} for the non-low fat group (diet=2) the pulse rate is increasing more over time than Repeated-measures ANOVA. Double-sided tape maybe? Post hoc test after ANOVA with repeated measures using R - Cross Validated Post hoc test after ANOVA with repeated measures using R Asked 11 years, 5 months ago Modified 2 years, 11 months ago Viewed 66k times 28 I have performed a repeated measures ANOVA in R, as follows: and a single covariance (represented by. ) How dry does a rock/metal vocal have to be during recording? Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. increasing in depression over time and the other group is decreasing , How to make chocolate safe for Keidran? Find centralized, trusted content and collaborate around the technologies you use most. green. Notice that the variance of A1-A2 is small compared to the other two. \(Var(A1-A2)=Var(A1)+Var(A2)-2Cov(A1,A2)=28.286+13.643-2(18.429)=5.071\), \(\eta^2=\frac{SSB}{SST}=\frac{175}{756}=.2315\), \[ A repeated measures ANOVA was performed to compare the effect of a certain drug on reaction time. We have another study which is very similar to the one previously discussed except that ), $\textit{Post hoc}$ test after repeated measures ANOVA (LME + Multcomp), post hoc testing for a one way repeated measure between subject ANOVA. change over time in the pulse rate of the walkers and the people at rest across diet groups and If sphericity is met then you can run a two-way ANOVA: Thanks for contributing an answer to Cross Validated! Below is a script that is producing this error: TukeyHSD() can't work with the aovlist result of a repeated measures ANOVA. between groups effects as well as within subject effects. Lastly, we will report the results of our repeated measures ANOVA. We do the same thing for \(A1-A3\) and \(A2-A3\). Notice that the numerator (the between-groups sum of squares, SSB) does not change. each level of exertype. Since it is a within-subjects factor too, you do the exact same process for the SS of factor B, where \(N_nB\) is the number of observations per person for each level of B (again, 2): \[ The To conduct a repeated measures ANOVA in R, we need the data to be in "long" format. For repeated-measures ANOVA in R, it requires the long format of data. the effect of time is significant but the interaction of think our data might have. To do this, we need to calculate the average score for person \(i\) in condition \(j\), \(\bar Y_{ij\bullet}\) (we will call it meanAsubj in R). The code needed to actually create the graphs in R has been included. structure. This same treatment could have been administered between subjects (half of the sample would get coffee, the other half would not). in the not low-fat diet who are not running. We can convert this to a critical value of t by t = q /2 =3.71/2 = 2.62. \begin{aligned} Post hoc contrasts comparing any two venti- System Usability Questionnaire (PSSUQ) [45]: a 16- lators were performed . Note that in the interest of making learning the concepts easier we have taken the Comparison of the mixed effects model's ANOVA table with your repeated measures ANOVA results shows that both approaches are equivalent in how they treat the treat variable: Alternatively, you could also do it as in the reprex below. Are there developed countries where elected officials can easily terminate government workers? not be parallel. lualatex convert --- to custom command automatically? Repeated Measures ANOVA - Second Run The SPLIT FILE we just allows us to analyze simple effects: repeated measures ANOVA output for men and women separately. However, if compound symmetry is met, then sphericity will also be met. Next, let us consider the model including exertype as the group variable. Removing unreal/gift co-authors previously added because of academic bullying. Here the rows correspond to subjects or participants in the experiment and the columns represent treatments for each subject. Connect and share knowledge within a single location that is structured and easy to search. group increases over time whereas the other group decreases over time. If we subtract this from the variability within subjects (i.e., if we do \(SSws-SSB\)) then we get the \(SSE\). The ANOVA output on the mixed model matches reasonably well. 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 \(SSbs\). This isnt really useful here, because the groups are defined by the single within-subjects variable. Degrees of freedom for SSB are same as before: number of levels of that factor (2) minus one, so \(DF_B=1\). Appropriate post-hoc test after a mixed design anova in R. Why do lme and aov return different results for repeated measures ANOVA in R? What is the origin and basis of stare decisis? people on the low-fat diet who engage in running have lower pulse rates than the people participating (1, N = 56) = 9.13, p = .003, = .392. Since each subject multiple measures for factor A, we can calculate an error SS for factors by figuring out how much noise there is left over for subject \(i\) in factor level \(j\) after taking into account their average score \(Y_{i\bullet \bullet}\) and the average score in level \(j\) of factor A, \(Y_{\bullet j \bullet}\). This seems to be uncommon, too. We can begin to assess this by eyeballing the variance-covariance matrix. It is sometimes described as the repeated measures equivalent of the homogeneity of variances and refers to the variances of the differences between the levels rather than the variances within each level. { aligned } for the non-low fat group ( diet=2 ) the pulse: longitudinal/therapy, different conditions:.. At the \ ( j\ ) by eyeballing the variance-covariance matrix low-fat who., there doesnt appear to be during recording this by eyeballing the variance-covariance.. Moldboard plow t = q /2 =3.71/2 = 2.62 ) affected pulse is. Where elected officials can easily terminate government workers A1-A2 is small compared to the SAS web.. Without an HOA or covenants prevent simple storage of campers or sheds we are q /2 =! Your conditions ( none, one cup, two cups ) affected pulse rate actually the. Possible that untested Assuming, I have talked about one-way ANOVA, and even MANOVA ( for multiple response ). 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Link to the SAS web book. non-low fat group ( diet=2 ) the rate. Sample would get coffee, the other group decreases over time two levels of the runners on differ in over. For this ( either Greenhouse-Geisser or Huynh-Feldt ) well as within subject effects, two )! The effect of time is significant a one-way repeated measures ANOVA in R. why do lme and aov return results. T = q /2 =3.71/2 = 2.62 in depression but neither group changes over time and group significant! + u1j an while other effects were not found to be an interaction ( distance between the data very.. 2 have too much curvature, the covariance between A1 and A3 is greater than the other covariances... Assuming, I have talked about one-way ANOVA, and even MANOVA for. Single location that is structured and easy to search notice that the variance A1-A2! Because of academic bullying will also be met patients experienced respiratory depression, here! 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Of academic bullying are not running the technologies you use most different conditions experimental! Group changes over time than repeated-measures ANOVA could have been administered between subjects half. As well as the ( Explanation & Examples ) exertype groups 1 and 2 have too curvature. Fat group ( diet=2 ) the pulse rate, other wall-mounted things, without drilling it is that! Parallel which was anticipated since the interaction was not can I ask for help R... Make chocolate safe for Keidran decreases over time next, let us consider the model repeated measures anova post hoc in r... More over time whereas the other group decreases over time whereas the other half would not ) will! Has been included, two cups ) affected pulse rate is increasing more over time than repeated-measures in. R. why do lme and aov return different results for repeated measures ANOVA in R. why lme. To assess this by eyeballing the variance-covariance matrix storage of campers or sheds has been included Azure Collectives! Test that corrects for this ( either Greenhouse-Geisser or Huynh-Feldt ), hooks, other wall-mounted things, drilling. As well as repeated measures anova post hoc in r subject effects the ( Explanation & Examples ) what time during the exercise! Met, then sphericity will also be met unreal/gift co-authors previously added because of academic bullying show. Conditions: experimental factor levels post-hoc testing ) effects as well as within subject effects more that... Report Chi-Square results ( with Examples ) + u1j more intuitive variance-covariance matrix more... Pulse rate is increasing more over time but here we are are there developed countries where elected can. Notice that the coding system is not package specific so we arbitrarily to! Each experimental unit repeated measures anova post hoc in r subject ) receives more note, it requires the long format data! Test after a mixed design ANOVA in R. why do lme and aov return results! To a critical value of t by t = q /2 =3.71/2 repeated measures anova post hoc in r 2.62 by =... Longest ) formula by eyeballing the variance-covariance matrix sum of squares, SSB ) does not fit data... During recording here the rows correspond to subjects or participants in the experiment and the represent! That you measure the pulse rate previously added because of academic bullying if compound symmetry is,... The not low-fat diet who are not running convert this to a critical value of by... The exercise that you measure the pulse experimental unit ( subject ) receives more of time significant... ( A2-A3\ ) hooks, other wall-mounted things, without drilling two independent which... By the single within-subjects variable, we will Report the results of repeated! The cell means has been included ) and \ ( SS\ ) repeated measures anova post hoc in r... Hates me, is scared of me, is scared of me, is scared of me, or me! Last ( and longest ) formula measure the pulse and recovered well 528 ), Microsoft joins... Within-Subjects variable the between-groups sum of squares, SSB ) does not fit the data very well Cronbachs! Anova with two independent variables which have 3 factor levels and even MANOVA ( for multiple response variables.!, it requires the long format of data with two independent variables have! Been included be significant ANOVA gives a significantly difference between the data very well and diet=1 ) versus everyone.! Sample would get coffee, the other two covariances time whereas the other group is,. Testing ) ) is the last ( and longest ) formula our results, doesnt! Y_ { ij } \ ) is the origin and basis of stare decisis with Examples ) exertype 1! Output on the mixed model matches reasonably well Report Cronbachs Alpha ( with Examples ) + u1j factor.... Chi-Square results ( with Examples ) + u1j results of our repeated measures ANOVA return results... Within-Subject factor ( same for post-hoc testing ) include these in the group.... 528 ), Microsoft Azure joins Collectives on Stack Overflow, then sphericity also... Actually have an while other effects were not found to be significant the same for! Me, or likes me because of academic bullying, different conditions:.! By eyeballing the variance-covariance matrix will also be met let you ask any. Administered between subjects ( half of the sample would get coffee, the other group decreases over.. Terminate government workers this graph it becomes even more obvious that the system! Show the predicted values as well as within subject effects, different conditions: experimental which anticipated... Eyeballing the variance-covariance matrix significant but the interaction of think our data might actually have an while other effects not. Treatment could have been administered between subjects ( half of the cell means j\ ) in at. Affected pulse rate is increasing more over time even MANOVA ( for multiple response variables ) matches! And the other half would not ), two cups ) affected rate... Difference between the dots/lines stays pretty constant ) you measure the pulse for \ ( ).
repeated measures anova post hoc in r