Intro to Statistics for the Behavioral Sciences PSYC 1900 Lecture 17: Repeated-Measures ANOVA

Repeated-Measures ANOVA Use is similar to one-way ANOVA Use is similar to one-way ANOVA Examine influence of an independent variable with 3 or more levels on a dependent variable Examine influence of an independent variable with 3 or more levels on a dependent variable If only 2 levels, use paired/related sample t-test If only 2 levels, use paired/related sample t-test Primary difference is that all subjects are in all conditions Primary difference is that all subjects are in all conditions Common example is pre-post design Common example is pre-post design

Benefits The perfect control group! The perfect control group! Each subject is his/her own control Each subject is his/her own control Groups are balanced for everything except effects of manipulations Groups are balanced for everything except effects of manipulations As subjects are measured more than once, we can find subject differences (main effect for subjects) As subjects are measured more than once, we can find subject differences (main effect for subjects) Reduced error variance = more power Reduced error variance = more power Smaller number of subjects needed for “same” sample sizes as between groups designs. Smaller number of subjects needed for “same” sample sizes as between groups designs.

Example Effect of relaxation training on migraines. Effect of relaxation training on migraines. A baseline measure is taken, followed by measures every week for 4 weeks during training. A baseline measure is taken, followed by measures every week for 4 weeks during training. Null is that relaxation training did not reduce migraines. Null is that relaxation training did not reduce migraines.

Logic of Analysis To answer this question, we treat the one-way ANOVA like a factorial, in that we calculate a main effect for subjects. To answer this question, we treat the one-way ANOVA like a factorial, in that we calculate a main effect for subjects. The SS for subjects removes variability associated with individuals The SS for subjects removes variability associated with individuals Some people have more migraines than others irrespective of the manipulation. Some people have more migraines than others irrespective of the manipulation. The MS error term is reduced by removing these individual differences The MS error term is reduced by removing these individual differences In between-groups ANOVA’s, we rely on randomization to balance such differences, but the variability is still present In between-groups ANOVA’s, we rely on randomization to balance such differences, but the variability is still present

Partitioning Variability Total Variability Between-subj. variability Within-subj. variability Time Error

Calculation Formulae for ANOVA SS total is the sum of squared deviations of all observations from the grand mean. SS total is the sum of squared deviations of all observations from the grand mean. SS subjects is the sum of squared deviations of the subjects’ means from the grand mean. m is the number of times a subj is measured. SS subjects is the sum of squared deviations of the subjects’ means from the grand mean. m is the number of times a subj is measured. SS time is the sum of squared deviations within each time period. SS time is the sum of squared deviations within each time period.

Calculation Formulae for ANOVA Now we convert SS into Mean Squares (MS). Now we convert SS into Mean Squares (MS). We divide SS by associated df’s. We divide SS by associated df’s. df total =N-1 df total =N-1 df weeks = #weeks-1 df weeks = #weeks-1 df error =(#weeks-1)X(n-1) df error =(#weeks-1)X(n-1) F is the ratio of these two estimates of population variance. F is the ratio of these two estimates of population variance. Critical values of F are found using F(df group, df error ), Critical values of F are found using F(df group, df error ), Note that subjects SS is only used to calculate subj. variance and remove it from error term. Note that subjects SS is only used to calculate subj. variance and remove it from error term.

Fisher’s Least Significant Difference (LSD) Test We use the LSD to test individual means again. We use the LSD to test individual means again. MS error here is adjusted for correlation between scores. MS error here is adjusted for correlation between scores. Test using t distribution with df error degrees of freedom. Test using t distribution with df error degrees of freedom.

Assumptions Usual assumptions of ANOVA hold, plus one additional. Usual assumptions of ANOVA hold, plus one additional. Correlations among pairs of scores are constant. Correlations among pairs of scores are constant. If violated, use Greenhouse and Geisser correction. If violated, use Greenhouse and Geisser correction. More stringent F-value through adjusting df’s for effect to be 1 and n-1 for effect and error respectively. More stringent F-value through adjusting df’s for effect to be 1 and n-1 for effect and error respectively.

Effect Size We can again calculate Cohen’s d. We can again calculate Cohen’s d. Standard deviation (s) is calculated as the square root of MS error. This reduces error term for individual differences. Standard deviation (s) is calculated as the square root of MS error. This reduces error term for individual differences.

Disadvantages of Repeated Measures Designs Prone to biases stemming from practice or fatigues effects Prone to biases stemming from practice or fatigues effects Deal with through counterbalancing. Deal with through counterbalancing. Subjects receive conditions in random order (if variable is manipulated). Subjects receive conditions in random order (if variable is manipulated). What do you do with time? What do you do with time? Control group. Control group.