Stats Lunch: Day 8 Repeated-Measures ANOVA and Analyzing Trends (It’s Hot)

Repeated Measures/Within Subjects Designs  Measure the same variable (e.g., test scores) in the same subjects more than once… Benefits: 1)Sensitive: Unsystematic variance is reduced  Increased Effect Sizes, Increased Power 2)Economical: Fewer subjects needed Problems: 1) Experimental Issues: Carry-over effects, fatigue, drop-out, etc. 2)Some statistical issues:  Same subjects across all levels of IV, so there are correlations between levels (violates ANOVA’s assumption of independence)  Which means we need to be concerned with sphericity

Sphericity Problem: ANOVA assumes homogeneity of variance, but is robust against violations of this assumption UNLESS the different levels are correlated… We know that this is an issue in rANOVA, so there is an additional assumption… Assumption of Sphericity: that correlations between treatment levels are the same, or more specifically:  Assumes that the variances of the DIFFERENCES of pairs of treatment means are equal.  Violations of Sphericity affect the quality of the inferences we can draw from our stats…makes a Type I error more likely… Sphericity is tested with Mauchly’s Test of Sphericity:  SPSS spits it out by default when you use the repeated measures module…  If the test is significant (p <.05) it’s bad: can’t assume sphericity Can’t get enough of sphericity? Then go to for a nice discussion …

rANOVA in SPSS 1)Click on “Analyze” 2)Choose GLM and then “Repeated Measures” 3)You’ll need to define (create) your repeated measures variable. a.First, give it a name b.Then enter how many levels it has. c.Click on “Add”

rANOVA in SPSS 4) You should now see your variable (number of levels) in this window. Click on “Define” 5) Select the options (effect sizes, descriptives, etc.), plots, and contrasts you’d want, and then Click on “Ok”

rANOVA in SPSS So, we’ve violated the assumption of sphericity If epsilon is <.7, it’s a large violation…

Correcting for Sphericity More Bad News: We damn near always violate the assumption of sphericity. Also, Mauchly’s test can be problematic (e.g., when we have very small or large N’s). So, we need to correct for the errors we might make… Three Corrections in SPSS 1)Huynh-Feldt (Least Conservative) 2)Greenhouse-Geisser (Conservative) 3)Lower Bound (Very Conservative)  All work my adjusting the DF by multiplying by an estimate of sphericity.  Makes it harder to reject null, decreases chance of Type I error  Greenhouse-Geisser most common (Safe bet: always use it)  In your paper, you usually report original DF, but adjusted p values.

rANOVA in SPSS Again, we’ve violated the assumption of Sphericity… So, we’d use one of the corrections (not the differences in the DF)

rANOVA in SPSS Another way to deal with violations of sphericity is to use a hypothesis test that doesn’t assume it… Multivariate Analysis of Variance (MANOVA)  Which SPSS also helpfully spits out in the repeated measures module.  rMANOVA tends to be less powerful than rANOVA  But can be a useful option when samples sizes are LARGE (n > k+10) and epsilons are small (<.7)  Or when G-G and H-F do not agree.

Trend Analyses A set of planned or post-hoc contrasts that test the overall pattern (distribution) of the data from the different levels of your IV, rather than testing differences between means:  Does a linear (straight) trend significantly account for variance?  Does a quadratic/cubic/quartic (and whatever comes after quartic) significantly account for variance? Linear TrendQuadratic Trend Cubic Trend  Can make a –1 contrasts, from least to most complicated: So if your IV has 3 levels, you can make (3-1) two contrasts (Linear and quadratic).  Called “Orthogonal Polynomial Contrasts”  Could be used predictively or descriptively.

Things to Consider  Levels of your IV should be evenly spaced: e.g., you measure symptoms ratings on Day 5, Day 10, and Day 15, not Day 5, Day 7, and Day 22.  The levels of your IV should span a large enough range (high to low) of the variable of interest, otherwise you could miss things  Another way to put this would be to say that you have measured enough intervals (levels of your IV) What would your conclusions be if you only recorded data from the first four time points?

A Brief Look Under the Hood  Remember, the way contrasts are performed in ANOVA is by multiplying the data from different levels of your IV by contrast coefficients, getting a SS and seeing if it’s significant…  So if you your IV (Time) has four levels and you wanted to compare Time 1 vs Time 3, your contrasts would look like: Time1Time2Time3Time4 010  Polynomial Contrasts (trend analyses) work the same way: By using contrasts that represent idealized versions of the trend (linear, quadratic, etc.) you’re looking for. Note that this set of comparisons is ORTHOGONAL

A Brief Look Under the Hood 1st2nd3rd4th

A Brief Look Under the Hood

rANOVA in SPSS 1)Set up everything like you did before, only this time click on “contrasts” and select “polynomial”, in the Change Contrast GUI. 2) Polynomial is the default option, but if you’ve previously run other types of contrasts, you’ll need to hit “Change” before you continue.

Output Have we violated Sphericity? Were the univariate effects significant? Do we need to correct? If so, what should we correct with?

Contrast Output  So, both the linear and quadratic trends are significant?  Which should we report:  Depends on the data, and who you ask:  Report Both  Contrast with higher effect size  The highest order contrast that is significant  What makes the most sense with your design and hypotheses.