Psych 610 Handout #29, p. 1Prof Colleen MooreFollow-up tests for Mixed designs (some Grouping factors, and some within-subjectfactors)An example with contrived data fictionalized from a paper by Surber & Gzesh,1984,Journal of Experimental Child Psychology. The task is a two arm balance scale with aconstant weight on one arm. On each trial a weight is placed at some distance on oneside. The participant says where to put the constant weight on the other side in order tomake the two arms balance. The task can be done without feedback by fixing the arms.Inhelder and Piaget used this device in their studies of the development of logicalreasoning, and so did Robert Siegler. The others used a choice task, whereas Surber &Gzesh had a continuous measure. The expectation is that there will be developmentaldifferences in use of the Weight and Distance cues to make the scale balance. Thesedevelopmental differences should show up as significant Grade x Wt or Grade x Distinteractions. Siegler claimed that children start by attending first to the #weights, thenlater learn to attend to distance, and finally combine them in the proper way. Siegler’schoice task was insensitive to the difference between adding weight and distance versusmultiplying. By using the continuous response measure, Surber & Gzesh had a better testof the multiplying model for subjective combination of weight and distance.The grouping variables are Grade and Gender, and the trial factors are Weight(2)and Distance (3). Weight 1 Weight 2D1 D2 D3 D1 D2 D3P grade gend _________________________________ 1 1 1 1 1 2 2 3 52 1 1 1 2 2 3 4 63 1 1 3 4 4 2 5 54 1 2 2 3 2 4 6 75 1 2 1 3 3 4 5 46 1 2 2 2 3 5 6 77 2 1 3 3 5 4 7 88 2 1 1 1 3 3 8 79 2 1 2 2 4 3 5 710 2 2 3 4 5 3 5 711 2 2 1 2 4 2 5 612 2 2 1 3 5 2 6 81. The overall anova (from my ‘legacy’ DOS software, BMDP): Because Grade andGender both begin with G, I named Grade “radeg” (moved the first letter to end).We have sig main effects of Weight and Distance, and a Weight x Dist interaction. Thenthere are significant interactions of Grade x Gender x Weight, Distance x Grade.Handout #29, Followup tests in Mixed Designs Psychology 610Prof Moore2SOURCE SUM OF D.F. MEAN F TAIL SQUARES SQUARE PROB. MEAN 1027.55556 1 1027.55556 435.20 0.0000 radeg (grade) 8.00000 1 8.00000 3.39 0.1029 gend 1.38889 1 1.38889 0.59 0.4651 rg 4.50000 1 4.50000 1.91 0.2048 1 ERROR 18.88889 8 2.36111 w 102.72222 1 102.72222 71.12 0.0000 wr 0.05556 1 0.05556 0.04 0.8494 wg 0.00000 1 0.00000 0.00 1.0000 wrg 8.00000 1 8.00000 5.54 0.0464 2 ERROR 11.55556 8 1.44444 dist 78.69444 2 39.34722 92.89 0.0000 dr 9.25000 2 4.62500 10.92 0.0010 dg 0.19444 2 0.09722 0.23 0.7975 drg 1.75000 2 0.87500 2.07 0.1592 3 ERROR 6.77778 16 0.42361 wd 10.02778 2 5.01389 13.13 0.0004 wdr 2.69444 2 1.34722 3.53 0.0538 wdg 1.08333 2 0.54167 1.42 0.2710 wdrg 0.75000 2 0.37500 0.98 0.3961 4 ERROR 6.11111 16 0.38194SOURCE GREENHOUSE HUYNH GEISSER FELDT PROB. PROB. dist 0.0000 0.0000 dr 0.0016 0.0010 dg 0.7767 0.7975 drg 0.1649 0.1592 wd 0.0005 0.0004 wdr 0.0559 0.0538 wdg 0.2714 0.2710 wdrg 0.3942 0.39612A. Interaction contrast on the within factors. If the task is done correctly, then the Wtx Dist interaction should be Linear x Linear.Step 1: Generate contrast coeff’sStep 2: Apply contrast coeff’s to indiv data, find psi-hats for individuals.Step 3: Analyze the psi-hats in a Grade x Gender between-groups anova. The W-linear x D-linear is the test of the grand mean.Step 1: generate contrast coefficients by multiplying them together:D1 D2 D3-1 0 1W1 1 -1 0 1W2 -1 1 0 -1Handout #29, Followup tests in Mixed Designs Psychology 610Prof Moore3Step 2: Multiply contrast coefficients x the individual data to make individual psi-hats.grade gender psi-hat1 1 -21 1 -21 1 -21 2 -31 2 21 2 -12 1 -22 1 -22 1 -22 2 -22 2 -12 2 -2Step 3: Analyze the psi-hats in a between group 2 way between-participants anova. Thisgives the error term we need. It is a partitioned error. Notice that we have 8 df for errorfor this test, whereas in the original anova there were 16 df for error.The test of the grand mean is significant – this says the W-linear x D-linearinteraction is significant.We need to normalize the SS to make sure it is part of the SS WxD in the originalanova. ∑cj-squared = 4. So proper SS is 30.0833/4 = 7.521. This is a pretty goodproportion of the SS W x D from the original anova: 7.521/10.028 = .75. We can alsonormalize the SS error = 13.3333 / 4 = 3.3333.DEPENDENT VARIABLE - LxL SOURCE SUM OF D.F. MEAN F TAIL SQUARES SQUARE PROB. MEAN 30.08333 1 30.08333 18.05 0.0028 radeg (grade) 0.75000 1 0.75000 0.45 0.5212 gend 2.08333 1 2.08333 1.25 0.2960 rg 0.75000 1 0.75000 0.45 0.5212 1 ERROR 13.33333 8 1.66667Other things to note:-- The Grade effect in this anova is the Grade x Wt-linear x Dist-linear interactioncontrast (yes, a 3-way interaction contrast).-- The Gender effect is the Gender x Wt-linear x Dist-linear