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Factorial Design

Factorial Design. One Between-Subject Variable. One Within-Subject Variable. SS Total. SS between subjects. SS within subjects. Treatments by Groups. Groups. Subjects within groups. Treatments. Treatments by Subjects within groups. Differences Between Subjects.

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Factorial Design

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  1. Factorial Design One Between-Subject Variable One Within-Subject Variable SSTotal SSbetween subjects SSwithin subjects Treatments by Groups Groups Subjects within groups Treatments Treatments by Subjects within groups Differences Between Subjects Differences Within Subjects Groups – differences between groups of subjects SS w/in Groups – differences between subjects w/in a group Treatment – differences between subject’s scores across treatments Treat x Groups – interaction between Treatments and Groups Treats x Ss w/in Groups – interaction between Subjects and Treatments hold Groups factor constant

  2. Example Speed (Repeated Measure) Group 1 Group 2 =GT =GM

  3. Calculate MS Divide SS by appropriate df SSbs by #Ss - 1 SSgrp by #Grps - 1 SSss w/in grps by (#Singrp-1) x (# of grps) SSws by #Ss (# Treatments – 1) SStreat by # Treatments - 1 SSTxG by (#grp – 1) (#Treats -1) SSTxS w/in grpsby (#Treats -1) x (n-1) x (# of grps)

  4. Prepare Summary Table 7 What are the appropriate error terms? (the denominators for the Fratios) Interpolation? 8

  5. Repeated Measures Assumptions 1) normality 2) homogeneity of variance 3) compound symmetry - constant variances on diagonal - constant covariances off diagonal A variance / covariance matrix for each group and overall 4) T X Ss interactions are constant across groups - test with Fmax

  6. Example No STRAT Var/Covar Matrix Speed The assumption of compound symmetry is usually replaced by the assumption of sphericity = a constant across all pairs of conditions = .574 = .853

  7. Simple Effects Factorial Design One B-S variable One W-S variable The W-S variable - Separate One-Way ANOVAs (repeated measures) ∙ Error terms pooled = MS T X Ss w/in groups ∙ Or, use the MST X Ss for each separate analysis No STRAT STRAT SSTotal = 67.44 SSTotal = 67.44 SSTotal = 168.04 SSbs = 7.14 SSbs = 5.29 SSTreat = 54.69 SSTreat = 159.19 SSerror = 3.56 SSerror = 5.56

  8. No STRAT STRAT + = 67.44 168.04 235.48 SSTotal SSTotal (overall) + = 12.45 7.19 5.29 SSbs SSbs (overall) + 159.19 = 213.88 SSTreat 54.69 SSTreat + SST X G Why? (overall) = SSerror 5.56 + 3.56 9.12 SST X S w/in group (overall)

  9. Between-Subjects Simple Effects We could do a separate analysis of each level - unnecessary loss of df SSgrp at 5 = = 8.0 SSgrp at 15 = = 4.5 SSgrp at 25 = = 2.0 = SSgrp at 35 = 8.0 MS all 1 df SSerror term = SSw/cells = SSSs w/in grp + SS T X Ss w/in grps Why? MSerror =

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