one-way ANOVA for repeated measures

When using a repeated measures or matched design, the IV can have 3 or more levels .The same issues of experimental design as examined previously (i.e., for a repeated measures/matched IV with 2 levels) apply
1. Note that if you calculated correlation coefficients, the scores across the repeated experimental conditions would be correlated  This is because the same participants are in every condition – the scores are not independent!
  1. the f ratio
    1. MSbetween/ MSerror
    2. MS bettwen = SSbetween conditions/ DF between conditions
    3. MSerror = SSerror/ DFerror
      1. violations to normaility
        ANOVA is robust to minor to moderate violations . If non-normality found in a minority of conditions and not of high magnitude can continue to use the ANOVA . Note it as a limitation
        May modify data to improve normality E.g., transformations, Winsorisation . If outliers cause the non-normality, may exclude outliers . Use a non-parametric test (not discussed further in this course!)
        effect size for repeated measures- Partial eta squared
        S=.01 M= .09. L= .25
    4. SSbetwen subjects are partioned out
      1. Null hypothesis  The independent variable has no effect on the dependent variable  τj will not contribute anything towards the individual’s score  The condition means will be the same. mu1= mu2=m3..Any numerical differences between the means is due to sampling error
        Alternative hypothesis  The independent variable has an effect on the dependent variable  τj will influence the individual’s score  The conditions means are not equal  This statement is false: mu 1 = mu2 = mu 3 =. The observed difference between the means is not due to sampling error.
        multiple comparisons
        can do hand t test (need to look up bonferoni back of book)
        We’ll use pairwise Bonferroni comparisons via the “Compare main effects” option in SPSS  Output table gives the mean difference and significance associated with the difference for all possible pairwise comparisons
        to get T in pair wise = mean diffrences divided by std error ( DF n-1)
        INFERNECE BY EYE-Different rules will apply than for independent groups  For repeated measures, need the confidence interval (or standard error) of the mean differences between the conditions you want to compare  If the null hypothesis is true, the mean difference = 0  If the CI of the difference scores does not include zero, conclude a significant difference at that level of confidence (e.g., 95% interval gives p < .05)
when you swe 'tests of within subject effects' - tells us we have a least one repeared measures variable in the analysis
1. Using SPSS to obtain a repeated measures oneway ANOVA 1. Click on Analyze General Linear Model Repeated Measures 2. In the dialogue box, select the name of the independent variable in the Within-Subject Factor Name: box. Enter the Number of Levels: in the second box. Click on the [Add] button. Next, click on the [Define] button. 3. In the next dialogue box, move the levels of the repeated measures independent variable in the Within-Subjects Variables box. 4. Click on the [Options] button to obtain Estimates of effect size, Display Means, and Compare main effects. 5. Finish running the Univariate procedure. will need to speciy names ie conditions and levels
  1. structural model for repeated measures One factor
    1. Xij = μ + πj + τi + εij
    2. μ = grand mean πj = variability associated with the jth person (measuring how much they differ from the average person) τi = the effect of the condition (IV) εij = error variability
      1. The structural model isolates the variability associated with the individual participant (πj)  This can be removed from the error term  This will make the error term smaller  The resulting F ratio can be more sensitive to the effects of the IV
      2. Like the structural model, partitioning (dividing up) the total variance is an extension of the independent groups case
        partitioned out individual variablilty so it dosnt play a role
      3. Assumptions
        normality
        for each condition (cell)
        Homogeneity of variance
        Fmax = largest condition variance diveded by smallest condition variance. Get variance from descriptives and sqaure the SD
        Sphericity -mauchlys W
        The variability in the differences between any pair of conditions is the same as the variability in the differences between any other pair of conditions
        The assumption of sphericity  Only relevant when there are three or more levels of the independent variable  The variability in the differences between any pair of conditions is the same as the variability in the differences between any other pair of conditions OR, IN OTHER WORDS  There is equality in the correlations (or more strictly the covariances) between each pair of conditions  E.g., the correlation (covariance) between the No Sound and Repeated Tone conditions is the same as between the No Sound and Music conditions, and the Music and Backward Speech conditions, and the……
        Sphericity: There is equality in the correlations (or more strictly the covariances) between each pair of conditions
        If the sphericity assumption is violated  Recommended practice is to use an adjusted degrees of freedom to evaluate the F statistic  The adjustment will increase the p-value associated with the F statistic  The two main adjustments are:  Greenhouse-Geisser (more conservative)  Huynh-Feldt (less conservative)  Both adjustments are automatically provided when you run the repeated measures ANOVA in SPSS
    3. partioning variance
      1. total variance
        between subjects(individual variabilty
        within subjects variability
        between conditions (effect of experimental condition/IV)
        SStotal- deviations of each score from the grand mean (DFtotal =N-1) total variation in the scores
        SSbetween subjects- Deviation of each subject's (participantion) mean from the grand mean - DFbetween N-1 (variability due to individaul diffreence ( not relevent to F)
        SSbetween conditions -Deviation of each condition’s mean from the grand mean. DFbetween conditions k-1- Variability due to the effect of the experimental manipulation
        SSerror = SStotal + SSbetweensubjects + SSbetween conditions DF = (n-1)(k-1) Random experimental error
        error - effect of random experimental error

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one-way ANOVA for repeated measures

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