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Lecture 11: One Way ANOVA Repeated Measures

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## Lecture 11: One Way ANOVA Repeated Measures

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**Lecture 11:One Way ANOVARepeated Measures**Laura McAvinue School of Psychology Trinity College Dublin**Analysis of Variance**One way ANOVA Factorial ANOVA More than One Independent Variable One Independent Variable Between subjects Repeated measures / Within subjects Two way Three way Four way Different participants Same participants**One Way Repeated Measures ANOVA**• A statistical technique for testing for differences between the means of several groups • Groups are related in some way • Usually same participants in each group • Similar to the paired samples T-Test • But no restriction on the number of groups**A few examples…**• What are the independent & dependent variables in each of the following studies? • Longitudinal study of annual growth of children over the first five years of life • The effects of three levels of methylphenidate on reaction time performance of the same participants • The measurement of depression before therapy, after therapy and at a nine month follow up • Same group of people is tested under different conditions or at different times**Repeated Measures ANOVA**Data points in each group are related Between Subjects ANOVA Data points in each group are unrelated**Population**draw one sample Did the manipulation alter the sample to such an extent that the sample represents another population during at least one condition? Manipulate the sample Cond1 Cond 2 Cond 3 Number of trials required to pass test µ1 µ2 µ3 measure effect of IV on a DV**Null Hypothesis Significance Testing**• Step 1: Halt – At least one condition mean is significantly different from the others • Step 2: Ho – All of the condition means are equal • Step 3: Collect your data • Step 4: Run the ANOVA • Step 5: Obtain the F statistic and associated p value • Step 6: Decide whether to reject or fail to reject Ho on the basis of the p value**Total Variance, Between Subjects ANOVA**Between Groups Within Groups Variance between the means Variance within each group Captures the effect of the manipulation / treatment Variance due to random error / chance The bigger the effect of your manipulation, the bigger the Between Groups Variance**F Ratio**Compares the variance due to the treatment / manipulation to the variance due to random error / chance MS between groups MS within groups Between Groups Variance Within Groups Variance Treatment Effect + Differences due to chance / error Differences due to chance / error If the treatment / manipulation has an effect, then… MS between groups > MS within groups F > 1**Total Variance, Between Subjects ANOVA**Between Groups Within Groups Variance due to random error / chance Variance due to individual differences between subjects Variance due to random error**Repeated Measures Design**• Powerful • Error Variance • Variance between participants • Unexplained Random Error • Because we test participants repeatedly with this design, we can quantify this variance between participants • We can remove this variance from the error term • Making the error term smaller, • Making the F ratio bigger, • Making it easier to obtain a statistically significant F**Four Kinds of Variance**• Total Variance • SStotal • ∑ (xij - Grand Mean )2 • Variance due to the manipulation • SStreatment • n∑ (Group meanj - Grand Mean )2**Four Kinds of Variance**• Variance due to individual differences between participants • SSparticipants • No. of conditions ∑ (Participant mean - Grand Mean )2 • Variance due to random error • SSrandom error = SStotal - SStreatment - SSparticipants MS Treatment MS Random error F**Condition / Treatment**SStotal : ∑ (xij - Grand Mean )2 ∑(2 – 18.67) 2 + (10 – 18.67)2 +…+ (34 – 18.67) 2**Condition / Treatment**SStreatment : n ∑ (Treatment Mean - Grand Mean )2 4 ∑(16 – 18.67) 2 + (19 – 18.67)2 + (21 – 18.67) 2**Condition / Treatment**SSparticipants : no. of conditions ∑ (Participant Mean - Grand Mean )2 3 ∑(4.33 – 18.67) 2 + (11.67 – 18.67)2 +…+ (31.67 – 18.67) 2**Condition / Treatment**SSrandom error : SStotal - SStreatment - SSparticipants**ANOVA table**N = Total no. of observations, n = no. of people in each condition, K = no. of conditions**Post hoc testing**• Significant F value • At least one condition mean is significantly different from the others • But which one? • Posthoc tests • Bonferroni • Tukey • Sidak**A few assumptions…**Data in each group should be… • Interval scale • Normally distributed • Histograms, box plots • Homogeneity of variance • Variance within each condition should be roughly equal**Extra Assumption**• Sphericity • Variance of the differences between conditions is the same • Variance t1-t2 Variance t1-t3 Variance t2-t3 • Correlation between pairs of groups is the same • corr t1-t2 corr t1-t3 corrt2-t3**Testing Sphericity**• Mauchly’s Test • If p > .05, assume equality of variances • If p < .05, then the data fails to meet the assumption of sphericity • Need to use one of the correction factors • E.g. Greenhouse-Geisser • A more conservative test, using different dfs**Advantages & Disadvantages of Repeated Measures Designs**• Major Advantage • Individual differences between participants can be removed from the analysis • More power • Disadvantage • Risk of carry-over effects from one condition to the next • Practice effects • Reduce by counterbalancing