Repeated-Measures ANOVA

Repeated-Measures ANOVAOld and NewOne-Way ANOVA looks at differences between different samples exposed to different manipulations (or different samples that may come from different groups) within a single factor.That is, 1 factor, k levels k separate groups are compared.NEW: Often, we can answer the same research questions looking at just 1 sample that is exposed to different manipulations within a factor.That is, 1 factor, k levels 1 sample is compared across k conditions.

Multiple measurements for each sample.That is, a single sample is measured on the same dependent variable once for each condition (level) within a factor.Investigate development over time (Quazi-Independent: time 1, time 2, time 3)Chart learning (manipulate different levels of practice)(Independent: 1 hour, 4 hours, 7 hours)Compare different priming effects in a LDT(Independent: Forward, backward, no-prime, non-word)Simply examine performance under different conditions with the same individuals.(Independent: suspect, equal, detective, audio format)

Extending t-testsT-testANOVA cousinComparing two independent samples?Independent-samples t-test!

Comparing the dependent measures of a single sample under two different conditions?Related- (or dependent- or paired-) sample t-test!

Comparing more than two independent samples within a single factor?One-way ANOVA!Comparing the dependent measures of a single sample under more than two different conditions?Repeated-Measures ANOVA!

R-M ANOVALike the Related-samples t, repeated measures ANOVA is more powerful because we eliminate individual differences from the equation.

In a One-way ANOVA, the F-ratio is calculated using the variance from three sources:F = Treatment(group) Effect + Individual differences + Experimenter error/Individual differences + Experimenter error.The denominator represents random error and we do not know how much was from ID and EE.This is error we expect from chance.

Why R-M ANOVA is COOLWith a Repeated-Measure ANOVA, we can measure and eliminate the variability due to individual differences!!!! So, the F ratio is conceptually calculated using the variance from two sources:F = Treatment(group) Effect + Experimenter error/ Experimenter error.The denominator represents truly random error that we cannot directly measurethe leftovers. This is error we expect just from chance.What does this mean for our F-value?

R-M vs. One-Way: PowerAll else being equal, Repeated-Measures ANOVAs will be more powerful because they have a smaller error term bigger Fs.Let me demonstrate, if you will.Assume treatment variance = 10, experimental-error variance = 1, and individual difference variance = 1000.In a One-Way ANOVA, F conceptually = (10 + 1 + 1000)/(1 + 1000) = 1.01In a Repeated-Measures ANOVA, F conceptually = (10 + 1)/(1) = 11

What is a R-M ANOVA doing?Again, SStotal = SSbetween + SswithinDifference: We break the SSwithin into two parts:SSwithin/error = SSsubjects/individual differences + SSerrorTo get SSerror we subtract SSsubjects from SSwithin/error.This time, we truly have SSerror , or random variability.Other than breaking the SSwithin into two components and subtracting out SSsubjects, repeated measures ANOVA is similar to One-Way ANOVA.

Lets learn through example.A Priming Experiment! 10 participants engage in an LDT, within which they are exposed to 4 different types of word pairs. RT to click Word or Non-Word recorded and averaged for each pair type.Forward-Prime pairs (e.g., baby-stork)Backward-Prime pairs (e.g., stork-baby)Unrelated Pairs (e.g., glass-apple)Non-Word Pairs (e.g., door-blug)HypothesesFor the overall RM ANOVA:Ho: f = b = u= nHa: At least one treatment mean is different from another.Specifically:Ho: f < bHo: f and b < u and nHo: u < n

The DataPartForwardBackwardUnrelatedNonwordSum Part.1.2.1.4.71.42.5.3.8.92.53.4.3.6.82.1F2= 1.364.4.2.8.92.3B2= .585.6.4.8.82.6U2= 3.826.3.3.5.81.9N2= 6.637.1.1.5.71.4P2= 3858.2.1.6.91.8Xt2= 12.39

9.3.2.4.71.610.4.2.6.92.13.42.268.119.7All = 19.7

Formula for SStotal Ians SSt =Remember, this is the sum of the squared deviations from the grand mean. So, SStotal = 12.39 (19.72/40) = 12.39 9.70225= 2.68775Importantly, SStotal = SSwithin/error + SSbetween

The DataPartForwardBackwardUnrelatedNonwordSum Part.1.2.1.4.71.42.5.3.8.92.53.4.3.6.82.1F2= 1.364.4.2.8.92.3B2= .585.6.4.8.82.6U2= 3.826.3.3.5.81.9N2= 6.637.1.1.5.71.4P2= 3858.2.1.6.91.8Xt2= 12.39

9.3.2.4.71.610.4.2.6.92.13.42.268.119.7All = 19.7

Within-Groups Sum of Squares: SSwithin/errorSSwithin/error = the sum of each SS with each group/condition.Measures variability within each condition, then adds them together.

So, SSwithin/error =(1.36 ([3.4]2/10)) + (.58 ([2.2]2/10)) + (3.82 ([6]2/10) + (6.63 ([8.1]2/10)= (1.36 1.156) + (.58 .484) + (3.82 3.6) + (6.63 6.561) = .204+ .096+ .22+ .069= .589

The DataPartForwardBackwardUnrelatedNonwordSum Part.1.2.1.4.71.42.5.3.8.92.53.4.3.6.82.1F2= 1.364.4.2.8.92.3B2= .585.6.4.8.82.6U2= 3.826.3.3.5.81.9N2= 6.637.1.1.5.71.4P2= 3858.2.1.6.91.8Xt2= 12.39

9.3.2.4.71.610.4.2.6.92.13.42.268.119.7All = 19.7

Breaking up SSwithin/errorWe must find SSSUBJECTS and subtract that from total within variance to get SSERRORSSSUBJECTS = K is generic for the number of conditions, as usual.SSSUBJECTS = (1.42/4 +2.52/4 +2.12/4 +2.32/4 +2.62/4 +1.92/4 +1.42/4 +1.82/4 +1.62/4 +2.12/4 +) 19.72/40 = .49+1.5625+1.1025+1.3225+1.69+.9025+.49+.81+.64+1.1025) -9.70225 = 10.1125-9.70225=.41025

Now for SSerror SSerror = SSwithin/error SSsubjectsSSerror = .589 - .41025 = .17875 or .179Weeeeeeeeee! We have pure randomness!The DataPartForwardBackwardUnrelatedNonwordSum Part.1.2.1.4.71.42.5.3.8.92.53.4.3.6.82.1F2= 1.364.4.2.8.92.3B2= .585.6.4.8.82.6U2= 3.826.3.3.5.81.9N2= 6.637.1.1.5.71.4P2= 3858.2.1.6.91.8Xt2= 12.39

9.3.2.4.71.610.4.2.6.92.13.42.268.119.7All = 19.7

SSbetween-group:The (same) Formula:

So = SSbetween = [((3.4)2/10) + ((2.2)2/10) + ((6)2/10) + ((8.1)2/10)] 19.72/40 = (1.156+ .484+ 3.6+ 6.561) 9.70225 = 11.801 9.70225 = 2.09875 or 2.099

Getting Variance from SSNeed? DEGREES OF FREEDOM!K = 4Ntotal = total number of scores = 40 (4x10)DfTOTAL = Ntotal 1 = 39DfBETWEEN/GROUP = k 1 = 3Dfwithin/error= N K = 40 4 = 36Dfsubjects = s-1 = 10-1 (where s is the # of subjects) = 9Dferror = (k-1)(s-1) = (3)(9) = 27OR Dfwithin/error - Dfsubjects = 36 9 = 27

Mean Squared (deviations from the mean)We want to find the average squared deviations from the mean for each type of variability.To get an average, you divide by n in some form (or k which is n of groups) and do a little correction with -1. That is, you use df.

MSbetween/group = = 2.099/3 = .7

MSwithin/error = = .179/27 = .007

How do we interpret these MSMS error is an estimate of population variance. Or, variability due to ___________?

F?F = = .7/.007 = 105.671

(looks like it should be 100, but there were rounding issues due to very small numbers.

OK, what is Fcrit? Do we reject the Null??

Pros and ConsAdvantagesEach participant serves as own control. Do not need as many participants as one-way. Why? More power, smaller error term. Great for investigating trends and changes.DisadvantagesPractice effects (learning)Carry-over effects (bias)Demand characteristics (more exposure, more time to think about meaning of the experiment).ControlCounterbalancingTime (greater spacingbut still have implicit memory).Cover Stories

SphericityLevels of our IV are not independent same participants are in each level (condition).Our conditions are dependent, or related.We want to make sure all conditions are equally related to one another, or equally dependent.We look at the variance of the differences between every pair of conditions, and assume these variances are the same.If these variances are equal, we have Sphericity

More SphericityTesting for SphericityMauchlys testSignificant, no sphericity, NS Sphericity!If no sphericity, we must engage in a correction of the F-ratio. Actually, we alter the degrees of freedom associated with the F-ratio.Four types of correction (see book)Estimate sphericity from 0 (no sphericity) to 1 (sphericity)Greenhouse-Geiser (1/k-1)Huynh-FeldtMANOVA (assumes measures are independent)non-parametric, rank-based Friedman test (one-factor only)Symmetry

Effect SizesThe issue is not entirely settled. Still some debate and uncertainty on how to best measure effect sizes given the different possible error terms.2 = See book for equation.

Specific testsCan use Tukey post-hoc for explorationCan use planned comparisons if you have a priori predictions.Sphericity not an issue

Contrast Formula

Same as one way, except error term is differentContrastsSome in SPSS:Difference: Each level of a factor is compared to the mean of the previous levelHelmert: Each level of a factor is compared to the mean of the next levelPolynomial: orthogonal polynomial contrastsSimple: Each level of a factor is compared to the last levelSpecific:GLM forward backward unrelate nonword/WSFACTOR = prime 4 special (1 1 1 1-1 -1 1 1-1 1 0 00 0 -1 1)2+ Within FactorsSet up.Have participants run on tread mill for 30min.Within-subject factors:Factor AMeasure Fatigue every 10min, 3 time points.Factor BDo this once after drinking water, and again (different day) after drinking new sports drink.3 (time) x 2 (drink) within-subject design.Much is the same, much differentWe have 2 factors (A and B) and an interaction between A and B.These are within-subjects factorsAll participants go through all the levels of each factor.Again, we will want to find SS for the factors and interaction, and eventually the respective MS as well.Again, this will be very similar to a one-way ANOVA.Like a 1-factor RM ANOVA, we will also compute SSsubject so we can find SSerror.What is very different?Again, we can parse up SS w/e into SSsubject and SSerror.NEW: We will do this for each F we calculate.For each F, we will calculate:SS Effect ; SS Subject (within that effect) ; and SS Error What IS SSError for each effect?(We will follow the logic of factorial ANOVAS)What are the SSErrors now?Lets start with main effects.Looking at Factor A, we haveVariability due to Factor A: SSFactor AVariability due to individual differences.How do we measure that?By looking at the variability due to a main effect of Participant (i.e., Subject): SSSubject (within Factor A)Variability just due to error.How do we calculate that!?!?!? Think about the idea that we actually have 2 factors here, Factor A and Subject.

The Error is in the INTERACTIONS with Subject.For FFactor A (Time)SSAS is the overall variability looking at Factor A and Subjects in Factor A (collapsing across Drink).To find SSerror for the FFactor A (Time) Calculate: SSAS; SSFactor A ; SSSubject (within Factor A) SSerror is: SSAS - (SSFactor A + SSSubject (within Factor A)) That is, SSerror for FFactor A (Time)is SS A*subj !!!!!!Which measures variability within factor A due to the different participants (i.e., error)

The Same for Factor B.For FFactor B (Drink)SSAS is the overall variability looking at Factor B and Subjects in Factor B (collapsing across Time).To find SSerror for the FFactor B (Drink) Calculate: SSBS; SSFactor B ; SSSubject (within Factor B) SSerror is: SSBS - (SSFactor B + SSSubject (within Factor B)) That is, SSerror for FFactor B (Drink)is SS B*subj !!!!!!SS B*subj: Which measures variability within factor B due to the different participants (i.e., error)

Similar for AxB InteractionFor FAxBSSBetween groups is the overall variability due to Factor A, Factor B, and Subjects.To find SSerror for the FAxB Calculate: SSBetween; SSFactor A ; SSFactor B ; SSSubjectSSerror is: SSBetween - (SSFactor A + SSFactor B + SSSubject) That is, SSerror for FAxB is SS A*B*subj !!!!!!SS A*B*subj: measures variability within the AxB interaction due to the different participants (i.e., error)

We are finely chopping SSW/ESS W/E

SSSub(A)SSSub(B)SSSub (AxB)Getting to FFactor A (Time)SSA = 91.2; dfA = (kA 1) = 3 1 = 2SSError(A*S) = 16.467; dfError(A*S) = (kA 1)(s 1) = (3-1)(10-1) = 18So, MSA = 91.2/ 2 = 45.6 So, MSError(A*S) = 16.467/ 18 = .915 FA = 45.6/.915 = 49.846

Snapshot of other FsFactor B (Drink)dfB = (kB 1) = 2 1 = 1dfError(B*S) = (kB 1)(s 1) = (2-1)(10-1) = 9AxB (Time x Drink)dfAxB = (kA 1)(kB 1) = 2 x 1= 2dfError(A*B*S) = (kA 1)(kB 1)(s 1) = 2 x 1 x 9 = 18Use SSs to calculate the respective MSeffect and MSerror for the other main effect and the Interaction F-values.