Statistics for the Social Sciences Psychology 340 Fall 2013 Tuesday, October 22 Repeated Measures Analysis of Variance (ANOVA)

Last Time ANOVA in research articles ANOVA structural model Repeated Measures ANOVA Definitional formulas Computational formulas

This Time Brief review of Repeated Measures ANOVA & Computational Forumlas Assumptions in Repeated Measures ANOVA Effect sizes in Repeated Measures ANOVA Repeated Measures ANOVA in SPSS

Statistical analysis follows design More than 2 scores per subject One group The 1 factor within groups ANOVA: Repeated measures

Statistical analysis follows design The 1 factor within groups ANOVA: Repeated measures More than 2 groups One group Matched groups More than 2 scores per subject - OR - Matched samples

Example Suppose that you want to compare three brand name pain relievers. Give each person a drug, wait 15 minutes, then ask them to keep their hand in a bucket of cold water as long as they can. The next day, repeat (with a different drug) Dependent variable: time in ice water Independent variable: 4 levels, within groups Drug A Drug B Drug C Placebo

Within-subjects ANOVA MB MA MC MD Drug A Drug B Drug C Placebo 4 6 7 3 1 5 2 n = 5 participants Each participates in every condition (4 of these)

Partitioning the variance Drug A Drug B Drug C Placebo 4 6 7 3 1 5 2 What is ? The average score for each person Between subjects variance

Partitioning the variance Total variance Stage 1 Between cond. variance Within cond. variance Stage 2 Between subjects variance Error variance

Partitioning the variance Now we return to variance. But, we call it Means Square (MS) Mean Squares (Variance) Between conditions variance Error variance Recall:

Partitioning the variance Total variance Stage 1 Between cond. variance Within cond. variance Stage 2 Between subjects variance Error variance

Computational Formulas T = Group Total G = Grand Total P = Person Total n = number of participants k = number of conditions N = number of scores Drug A Drug B Drug C Placebo Person Totals 4 6 7 3 P1 = 20 P2 = 12 1 5 2 P3 = 12 P4 = 8 P5 = 4 TA = 10 TB = 20 TC = 25 TD = 5 n = 5 SSA = 8 SSB = 6 SSC = 10 SSD = 8 k = 4 N = 20 G = 60

Computational Formulas Drug A Drug B Drug C Placebo Person Totals 4 6 7 3 P1 = 20 P2 = 12 1 5 2 P3 = 12 P4 = 8 P5 = 4 TA = 10 TB = 20 TC = 25 TD = 5 n = 5 SSA = 8 SSB = 6 SSC = 10 SSD = 8 k = 4 N = 20 G = 60

From Before Drug A Drug B Drug C Placebo 4 6 7 3 1 5 2

Computational Formulas Drug A Drug B Drug C Placebo Person Totals 4 6 7 3 P1 = 20 P2 = 12 1 5 2 P3 = 12 P4 = 8 P5 = 4 TA = 10 TB = 20 TC = 25 TD = 5 n = 5 SSA = 8 SSB = 6 SSC = 10 SSD = 8 k = 4 N = 20 G = 60

From Before Drug A Drug B Drug C Placebo 4 6 7 3 1 5 2

Computational Formulas Drug A Drug B Drug C Placebo Person Totals 4 6 7 3 P1 = 20 P2 = 12 1 5 2 P3 = 12 P4 = 8 P5 = 4 TA = 10 TB = 20 TC = 25 TD = 5 n = 5 SSA = 8 SSB = 6 SSC = 10 SSD = 8 k = 4 N = 20 G = 60

Between subjects variance From Before Drug A Drug B Drug C Placebo 4 6 7 3 1 5 2 Between subjects variance

Computational Formulas Drug A Drug B Drug C Placebo Person Totals 4 6 7 3 P1 = 20 P2 = 12 1 5 2 P3 = 12 P4 = 8 P5 = 4 TA = 10 TB = 20 TC = 25 TD = 5 n = 5 SSA = 8 SSB = 6 SSC = 10 SSD = 8 k = 4 N = 20 G = 60

Between subjects variance From Before Total variance Stage 1 Between cond. variance Within cond. variance Stage 2 Between subjects variance Error variance

Assumptions in Repeated Measures ANOVA Observations within each treatment condition are independent (this is an assumption for 1-way ANOVA too) Populations within each treatment follow a normal curve Populations have equal variances 21

Effect sizes in Repeated Measures ANOVA The effect size for ANOVA is r2 Sometimes called η2 (“eta squared”) The percent of the variance in the dependent variable that is accounted for by the independent variable

Effect sizes in Repeated Measures ANOVA The effect size for ANOVA is r2 Sometimes called η2 (“eta squared”) The percent of the variance in the dependent variable that is accounted for by the independent variable

Repeated Measures ANOVA in SPSS Let’s see how to do a repeated measures 1-factor ANOVA in SPSS Analyze=>General Linear Model=>Repeated Measures In the “Repeated Measures Define Factors” box, give your factor a name (if desired), and enter the number of levels (number of different treatment conditions) and click on “Add” and Click “Define”. Choose the column labels (variable names) for the treatment conditions into the “within subjects variables” box (Highlight the column label on the left an click the arrow to move it into the box). Under “options,” you can request descriptive statistics (e.g., to see group means) Click OK To read the output, look at the “Test of within-subjects effects” table. Look at the top line (“Sphericity assumed”)

ANOVA in Research Articles F(3, 12) = 24.88, p < .01, η2 = 0.862. Means and standard deviations given in a table or in the text 25

Rest of Class Work on homework (see handout)