One-Way Analysis of Variance

Repeated Measures ANOVAStarting with One-Way RMMore fascinating than a bowl of porridge1.

KNR 445FACTORIALANOVA IISlide 2One-Way Repeated Measures ANOVAData considerationsOne interval/ratio dependent variableOne categorical independent variable of > 2 levelsAnalogous to dependent t-test, but for more than 2 levels of the independent variableSo here people are measured more than two timesAll participants participate in all levels of the IVE.g. midterm, final, quiz score for this class in SP 20091.

KNR 445FACTORIALANOVA IISlide 3One-Way Repeated Measures ANOVAAdvantages of repeated measuresAgain, as per paired t-test...SensitivityReduction in error variance (subjects serve as own controls)So, more sensitive to experimental effectsEconomyNeed less participants With many levels, this might be even more important for ANOVA than t-test(need to be careful of fatigue effects, though)1.

KNR 445FACTORIALANOVA IISlide 4One-Way Repeated Measures ANOVAPossible serious disadvantage of RMOrder effects and treatment carry-over effects (goes for paired t-test too)E.g. three chocolatesones bad, the others goodShould counterbalance (randomly assign to treatment order)E.g. (for 2 levels of RM: A & B)Order of administration12% of subjects50AB50BA1. 2. 4. 3.

KNR 445FACTORIALANOVA IISlide 5One-Way Repeated Measures ANOVAPossible serious disadvantage of RMOrder effects (goes for paired t-test too)E.g. (for 3 levels of RM: A, B & C)Order of administration123% of subjects33ABC33BCA33CAB1. This type of control for order effects is known as a Latin Square design

KNR 445FACTORIALANOVA IISlide 6One-Way Repeated Measures ANOVAPossible serious disadvantage of RMTreatment carry-over effects (goes for paired t-test too)Even if order effects are controlled for, there must be sufficient time between treatments so that you can be sure that the score on each level of the RM is due to only one treatment (not a combination of two or more)Note order & treatment carry-over effects are design rather than statistical issues, but very important nevertheless1.

KNR 445FACTORIALANOVA IISlide 7One-Way Repeated Measures ANOVAExample, with chat about variance partitioning and assumptions...Remember the one-way between subjects ANOVA? Data looked like this in SPSSAnd the trick was to make variance due to treatments bigger than variance due to everything else (& everything else included variance due to individual differences)Well, what if you could take out variance due to individual differences?

1. 2. 4. 3.

KNR 445FACTORIALANOVA IISlide 8One-Way Repeated Measures ANOVAThats what the one-way RM ANOVA doesData now looks like this, as each person is measured on all levels of the IVVariation due to individual differences can then be separated from variation due to chance, as the same people are present within each condition.

1. 2.

KNR 445FACTORIALANOVA IISlide 9One-Way Repeated Measures ANOVANow a pause before we consider variance partitioning in RM ANOVA, as we see how to conduct the test.Heres the first step

1. Choose this...

KNR 445FACTORIALANOVA IISlide 10One-Way Repeated Measures ANOVA1. Type in the variable name (drug) where it says factor1, and the number of levels it has (4)

2. Then click add and proceed by clicking define

KNR 445FACTORIALANOVA IISlide 11One-Way Repeated Measures ANOVA1. ...next you choose all the levels of the repeated measures factor (i.v.)...

2. And slide them over to the within subjects variables box just another name for repeated measures variable...or factor

3.

KNR 445FACTORIALANOVA IISlide 12One-Way Repeated Measures ANOVAOutput

2. This first bit is from the multivariate (more than one dependent variable 4 here) approach to repeated measures. It has some potential advantages (essentially that one does not have to meet the sphericity assumption...see next slide)1.

KNR 445FACTORIALANOVA IISlide 13One-Way Repeated Measures ANOVAAnd...more output...

This bit is important. Its a test of one of the more important assumptions of RM ANOVA sphericity. Its kind of like the homogeneity of variance test, but its the variance of the difference scores between the levels of the independent variable that are being testedyou really have to adjust for it, & we see how on the next slide (if this is NOT significant, its good)Another important bitthe Huynh-Feldt Epsilon...see next slide1. 2. What to do 3. sphericity

KNR 445FACTORIALANOVA IISlide 14One-Way Repeated Measures ANOVAAnd...still more output...

Finally, the bit that counts. Note there are FOUR (count em) separate versions of each effect. Heres the rule (Schutz & Gessaroli, 1993): If Huynh-Feldt Epsilon (see previous slide) is > .7, use Huynh-Feldt adjusted F (third line). If it is less than .7, use G-G (second line)1. 2.

KNR 445FACTORIALANOVA IISlide 15One-Way Repeated Measures ANOVASame bit once again...

1. Here, you can see that, as the epsilon is 1, there is no correction, and the F statistic stays the same throughout.

2.

KNR 445FACTORIALANOVA IISlide 16One-Way Repeated Measures ANOVAOne last bit (that you can ignore)...

Lets just look at this first. In the top box, you can see a bunch of stuff like linear, quadratic, & cubic thats to do with the shape of the difference that the change in scores might take as they progress from drug 1 to drug 4, and only really makes sense in trend analysis, which is again beyond our scope.Finally, down here you see between subjects effects. There are none here (just one I.V., and its RM). The error variance here is essentially a measure of individual differences, as well see in a minute...1. 2.

KNR 445FACTORIALANOVA IISlide 17One-Way Repeated Measures ANOVASo, how does the variance thing work?Lets compare the two methods (between subjects and repeated measures) directly, bearing in mind where the variances in the output tables come fromIn this way, my goal is simply to indicate the benefit of taking out variation due to individual differencesWell start with the between subjects method...(see next slide)1. 2. Simply put3.

KNR 445FACTORIALANOVA IISlide 18One-Way Repeated Measures ANOVA

Here the between groups variance is 698.2 this is just variation of mean scores on the different treatments about the overall mean...so this is the bit that is essentially the treatment effectAnd here is the within subjects variation...it is calculated from the sum of the variation within each of the treatments about each of the treatment means26.425.615.6321. 2. 4. 3.

5.

KNR 445FACTORIALANOVA IISlide 19One-Way Repeated Measures ANOVANow for the repeated measures version:

Note that the average score for each subject across the four treatments is different. This is due to individual differences...and is the between subjects error variance2716233424.5The thing that makes repeated measures powerful is that this variation is taken out of the within subjects error term...see next slide1. 2. 4. 3. Sum of squares = 680.8(= sum of squared deviations from the mean of these 5 scores, which is 24.9, multiplied by the # levels of the I. V.))

One-Way Repeated Measures ANOVAKNR 445FACTORIALANOVA IISlide 20

Error SS in non-RM ANOVA = 793.6Error SS in RM ANOVA = 793.6 680.8 = 112.8 1. 2. 4. 3.

KNR 445FACTORIALANOVA IISlide 21

One-Way Repeated Measures ANOVA

individual differencesError variation in between subjects ANOVANow what you have to see is that the SS for the denominator in the F test in RM is now 112.8, which is derived from 793.6 680.8 = 112.81. 2. 4. 3.

KNR 445FACTORIALANOVA IISlide 22

One-Way Repeated Measures ANOVA

And finally, as a direct consequence of all this, the numerator in the F-test is unchanged (698.2), but the denominator has been reduced from 49.6 to 9.4, resulting in an increase in F from 4.69 to 24.76!which of course means...more significance, more power2. 1.

KNR 445FACTORIALANOVA IISlide 23One-Way Repeated Measures ANOVASo, to summarizeBecause of the way RM ANOVA partitions variance for the RM factors, we have a far more powerful test for the RM factorsBut you have to make sure you control for spurious effects by controlling for order effects and carryover effectsAlso crucial that you adjust for violations of sphericity1.

KNR 445FACTORIALANOVA IISlide 24Example of interpretation of resultsInterpretation:A one-way repeated measures ANOVA was conducted to students confidence in statistics prior to the class, immediately following the class, and three months after the class. Due to a mild violation of the sphericity assumption ( = .82), the Huynh-Feldt adjusted F was used. There was a significant difference in confidence levels across time, F (1.421, 41.205) = 33.186, p < .001, partial 2 = .86. Dependent t-tests were used as post-hoc tests for significant differences with Bonferroni-adjusted = .017. Confidence levels after three months (M = 25.03, SD = 5.20) were significantly higher than immediately following the class (M = 21.87, SD = 5.59), which in turn were significantly higher than pre-test levels (M = 19.00, SD = 5.37).Note partial 2 is reported too 1.

2.

KNR 445FACTORIALANOVA IISlide 25ANOVA/Inferential Statistics Wrap-upInferential tests to compare differences in groups:Independent t-testsDependent t-testsOne-way ANOVAFactorial ANOVAOne-way repeated measures ANOVAFactorial repeated measures ANOVAMixed between-within groups ANOVA (split-plot)Analysis of covariance (ANCOVA)Multivariate analysis of variance (MANOVA)Nonparametric tests (next)1. 2.

KNR 445FACTORIALANOVA IISlide 26Factorial RM ANOVASame notions as for factorial ANOVA main effects, interactions and so onData set up a bit tricky KNR 445FACTORIALANOVA IISlide 27Two-way ANOVA with repeated measures on one factorSometimes referred to as a split plot or Lindquist type 1 or (most commonly in my experience) a Two-way ANOVA with repeated measures on one factor.Research question: Which diet (traditional, low carb, exercise only) is more effective in weight loss across three time periods (before diet, three months later, six months later)? Is there a weight loss across time?KNR 445FACTORIALANOVA IISlide 28Two-way ANOVA with repeated measures on one factorDiet RQ (continued)Looks like a 3x3 two-factor ANOVA, except that one of the factors is a repeated measure (one group of subjects tested three times)As such, a two-factor between-groups ANOVA is not appropriate; rather, we have one factor that is between diet types and another that is within a single group of subjectsKNR 445FACTORIALANOVA IISlide 29Two-way ANOVA with repeated measures on one factorUse a mixed design ANOVA when:A nominal between-subjects IV with 2+ levelsA nominal within-subjects IV with 2+ levelsA continuous interval/ratio DVNote: you can add additional IVs to this test, but just as with Factorial ANOVA, when you get 3+ IVs, interpreting findings gets really nasty due to all of the interactionsKNR 445FACTORIALANOVA IISlide 30Two-way ANOVA with repeated measures on one factorInteractions: like a two-factor between-subjects ANOVA, there may be both main effects for each of the two IVs plus an interaction between the two IVsKNR 445FACTORIALANOVA IISlide 31Analysis of CovarianceAn extension of ANOVA that allows you to explore differences between groups while statistically controlling for an additional continuous variableCan be used with a nonequivalent groups pre-test/post-test design to control for differences in pre-test scores with pre-existing groupsYou could use a mixed design ANOVA here, but with small sample size, ANCOVA may be a better alternative due to increased statistical powerBe careful of regression towards the mean as a cause of post-test differences (after using the covariate to adjust pre-test scores)KNR 445FACTORIALANOVA IISlide 32Multivariate Analysis of VarianceAn extension of ANOVA for use with multiple dependent variablesWith multiple DVs, you could simply use multiple ANOVAs (one per DV), but risk inflated Type 1 errorSame reason we didnt conduct multiple t-tests instead of an ANOVAEx. Do differences exist in GRE scores and grad school GPA based on race?There are such things as Factorial MANOVAs, RM MANOVAs, and even MANCOVAsFinito!KNR 445FACTORIALANOVA IISlide 33