More sophisticated ANOVA applications Repeated measures and factorial PSY SP2003

Major Topics What are repeated-measures? An example Assumptions Advantages and disadvantages Review questions

Effects of Counseling For Post-Traumatic Stress Disorder Foa, et al. (1991) –Provided supportive counseling (and other therapies) to victims of rape –Do number of symptoms change with time? Point out lack of control group –Not a test of effectiveness of supportive counseling Foa actually had controls. Cont.

Effect of Counseling--cont. –9 subjects measured before therapy, after therapy, and 3 months later We are ignoring Foa’s other treatment conditions.

Therapy for PTSD Dependent variable = number of reported symptoms. Question--Do number of symptoms decrease over therapy and remain low? Data on next slide

The Data

Plot of the Data

Preliminary Observations Notice that subjects differ from each other. –Between-subjects variability Notice that means decrease over time –Faster at first, and then slower –Within-subjects variability

Partitioning Variability Total Variability Between-subj. variability Within-subj. variability Time Error This partitioning is reflected in the summary table.

Summary Table

Interpretation Note parallel with diagram Note subject differences not in error term Note MS error is denominator for F on Time Note SS time measures what we are interested in studying

Assumptions Correlations between trials are all equal –Actually more than necessary, but close –Matrix shown below Cont.

Assumptions--cont. Previous matrix might look like we violated assumptions –Only 9 subjects –Minor violations are not too serious. Greenhouse and Geisser (1959) correction –Adjusts degrees of freedom

Multiple Comparisons With few means: –t test with Bonferroni corrections –Limit to important comparisons With more means: –Require specialized techniques Trend analysis

Advantages of Repeated- Measures Designs Eliminate subject differences from error term –Greater power Fewer subjects needed Often only way to address the problem –This example illustrates that case.

Disadvantages Carry-over effects –Counter-balancing May tip off subjects

Major Points What is a factorial design? An example Main effects Interactions Simple effects Cont.

Major Points-cont. Unequal sample sizes Magnitude of effect Review questions

What is a Factorial At least two independent variables All combinations of each variable R X C factorial Cells

Video Violence Bushman study –Two independent variables Two kinds of videos Male and female subjects See following diagram

2 X 2 Factorial

Bushman’s Study-cont. Dependent variable = number of aggessive associates 50 observations in each cell We will work with means and st. dev., instead of raw data. –This illustrates important concepts.

The Data (cell means and standard deviations)

Plotting Results

Effects to be estimated Differences due to videos –Violent appear greater than nonviolent Differences due to gender –Males appear higher than females Interaction of video and gender –What is an interaction? –Does violence affect males and females equally? Cont.

Estimated Effects--cont. Error –average within-cell variance Sum of squares and mean squares –Extension of the same concepts in the one-way

Summary Table

Conclusions Main effects –Significant difference due to video More aggressive associates following violent video –Significant difference due to gender Males have more aggressive associates than females. Cont.

Conclusions--cont. Interaction –No interaction between video and gender Difference between violent and nonviolent video is the same for males (1.5) as it is for females (1.4) We could see this in the graph of the data.

Elaborate on Interactions Diagrammed on next slide as line graph Note parallelism of lines –Means video differences did not depend on gender A significant interaction would have nonparallel lines –Ordinal and disordinal interactions

Line Graph of Interaction

Simple Effects Effect of one independent variable at one level of the other. e.g. Difference between males and females for only violent video Difference between males and females for only nonviolent video

Unequal Sample Sizes A serious problem for hand calculations Can be computed easily using computer software Can make the interpretation difficult –Depends, in part, on why the data are missing.

Minitab Example Analysis of Variance for AGGASSOC Source DF SS MS F P GENDER VIDEO Interaction Error Total Cont.

Minitab--cont. Individual 95% CI GENDER Mean ( * ) ( * ) Individual 95% CI VIDEO Mean ( * ) ( * )