Analysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionAnalysis – RegressionEffect SizeEffect SizeEffect SizeEffect SizeAnalysis – Regression• The ANOVA through regression approach is still the same, but expanded to include all IVs and the interaction• The number of orthogonal predictors needed for each main effect is simply the number of degrees of freedom for that effect• The interaction predictors are created by cross multiplying the predictors from the main effectsAnalysis – Regression• Example• You have 27 randomly selected subjects all of which suffer from depression.• You randomly assign them to receive either psychotherapy, Electroconvulsive therapy or drug therapy (IV A – therapy)• You further randomly assign them to receive therapy for 12 months, 24 months or 36 months (IV B – Length of Time)• At the end of there therapy you measure them on a 100 point “Life Functioning” scale where higher scores indicate better functioningAnalysis – Regression• Example• So, the design is a 3 (therapy, a1 = Psychotherapy, a2 = ECT, a3 = Drugs) * 3 (time, b1 = 12 months, b2 = 24 months, b3 = 36 months) factorial design• There are 3 subjects per each of the 9 cells, evenly distributing the 27 subjects• For regression, we will need:• 3 – 1 = 2 columns to code for therapy• 2 columns to code for time • and (3-1)(3-1) = 2*2 = 4 columns to code for the interaction• a total of 8 columns • Plus 8 more to calculate the sums of products (Y * X)• 16 columns in allAnalysis – RegressionA B A*B Sum of Products X1*X3X1*X4X2*X3X2*X4Y*X1 Y*X2 Y*X3 Y*X4 Y*X5 Y*X6 Y*X7 Y*X8 a b Y X1 X2 X3 X4X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 33 2 0 -1 1 -2 2 0 0 66 0 -33 33 -66 66 0 0 41 2 0 -1 1 -2 2 0 0 82 0 -41 41 -82 82 0 0 b1 33 2 0 -1 1 -2 2 0 0 66 0 -33 33 -66 66 0 0 63 2 0 0 -2 0 -4 0 0 126 0 0 -126 0 -252 0 0 85 2 0 0 -2 0 -4 0 0 170 0 0 -170 0 -340 0 0 b2 78 2 0 0 -2 0 -4 0 0 156 0 0 -156 0 -312 0 0 70 2 0 1 1 2 2 0 0 140 0 70 70 140 140 0 0 92 2 0 1 1 2 2 0 0 184 0 92 92 184 184 0 0 a1 b3 78 2 0 1 1 2 2 0 0 156 0 78 78 156 156 0 0 70 -1 1 -1 1 1 -1 -1 1 -70 70 -70 70 70 -70 -70 70 85 -1 1 -1 1 1 -1 -1 1 -85 85 -85 85 85 -85 -85 85 b1 78 -1 1 -1 1 1 -1 -1 1 -78 78 -78 78 78 -78 -78 78 70 -1 1 0 -2 0 2 0 -2 -70 70 0 -140 0 140 0 -140 78 -1 1 0 -2 0 2 0 -2 -78 78 0 -156 0 156 0 -156 b2 85 -1 1 0 -2 0 2 0 -2 -85 85 0 -170 0 170 0 -170 100 -1 1 1 1 -1 -1 1 1 -100 100 100 100 -100 -100 100 100 92 -1 1 1 1 -1 -1 1 1 -92 92 92 92 -92 -92 92 92 a2 b3 92 -1 1 1 1 -1 -1 1 1 -92 92 92 92 -92 -92 92 92 70 -1 -1 -1 1 1 -1 1 -1 -70 -70 -70 70 70 -70 70 -70 78 -1 -1 -1 1 1 -1 1 -1 -78 -78 -78 78 78 -78 78 -78 b1 92 -1 -1 -1 1 1 -1 1 -1 -92 -92 -92 92 92 -92 92 -92 100 -1 -1 0 -2 0 2 0 2 -100 -100 0 -200 0 200 0 200 100 -1 -1 0 -2 0 2 0 2 -100 -100 0 -200 0 200 0 200 b2 100 -1 -1 0 -2 0 2 0 2 -100 -100 0 -200 0 200 0 200 56 -1 -1 1 1 -1 -1 -1 -1 -56 -56 56 56 -56 -56 -56 -56 56 -1 -1 1 1 -1 -1 -1 -1 -56 -56 56 56 -56 -56 -56 -56 a3 b3 48 -1 -1 1 1 -1 -1 -1 -1 -48 -48 48 48 -48 -48 -48 -48 Sum 2023 0 0 0 0 0 0 0 0 -304 50 104 -254 295 -61 131 251 Sum of sq 161955 54 18 18 54 36 108 12 36 283710 121370 96468 358416 166389 635673 73161 265997 N 27Analysis – Regression• Formulas()()()( )[]22222()()()()(.)()iiiiiiiiiYSS Y YNXSS X XNYXSP YX YXNSP YXSS reg XSS X=−=−=−=∑∑∑∑∑∑∑Analysis – Regression• Formulas12345678() ()(.) (.)(.) (.)(.) (.) (.) (.)() ()Total YA reg X reg XB reg X reg XAB reg X reg X reg X reg Xresidual Total A B ABSS SSSS SS SSSS SS SSSS SS SS SS SSSS SS SS SS SS==+=+=+++=−−−Analysis – Regression• SST:• SSA:() ()4092529161955 161955 151575.15 10379.8527Total YSS SS== − = − =12(.) (.)22(304) (50) 92416 25001711.41 138.89 1850.354 18 54 18A reg X reg XSS SS SS=+=+= += + =Analysis – Regression• SSB:• SSAB:56 8(.) (.) (.7) (.)2222(295) ( 61) (131) (251) 87025 3721 17161 6300136 108 12 36 36 108 12 362417.36 34.45 1430.08 1750.03 5631.92A B reg X reg X reg X reg XSS SS SS SS SS=+++=−+++=+++=++ + =34(.) (.)22(104) ( 254) 10816 64516600.89 1194.74 1795.6318 54 18 54BregX regXSS SS SS=+=−+=+=+=Analysis – Regression• SSregression:• DF:• DFA=3 – 1 = 2• DFB=3 – 1 = 2• DFAB=(3 – 1)(3 – 1) = 2 * 2 = 4• DFS/AB= 9(3 – 1) = 27 – 9 = 18• DFtotal=27 – 1 = 26() ()10379.85 1850.3 1795.63 5631.92 1102residual Total A B ABSS SS SS SS SS= −−− =−− − =Analysis – Regression• Source Table• Fcrit(2,18) = 3.55; both main effects are significant• Fcrit(4,18) = 2.93; the interaction is significantSource SS df MS F Therapy 1850.30 2 925.148 15.111Time 1795.63 2 897.815 14.665Therapy * Time 5631.92 4 1407.981 22.998Error 1102.00 18 61.222 Total 10379.85 26Analysis – RegressionTIME36 months24 months12 monthsMean "Life Functioning" Score12010080604020THERAPYpsychotherapyECTDrug TherapyEffect Size• Eta Squared• Omega Squared22effecteffecttotalSSRSSη==/2/()effect effect S ABeffectTSABSS df MSSS MSω−=+LL)Effect Size• The regular effect size measure make perfect sense in one-way designs because the SStotal= SSeffect+ SSerror• However in a factorial design this is not true, so the effect size for each effect is often under estimated because the total SS is much higher than it would be in a one-way design• So, instead of using total we simply use the SSeffect+ SSerroras our denominatorEffect Size• Partial Eta Squared• Partial Omega Squared2/ effecteffecteffect S ABSSpartialSS SSη=+L/2//()/ ()effect effect S ABeffecteffect effect S AB S ABdf MS MS Npartialdf MS MS MSω−=−+LLL)Effect Size• Example22/2//2//1850.3.1810379.851850.3 .631850.3 1102()1850.3 2(61.22) 1727.86.16510379.85 61.22 10441.07()/ (AATAAASABAASABATSABAA SABAAA SASSSSSSpartialSS SSSS df MSSS M Sdf MS M S Npartialdf M S M Sηηωω== === =++−−====++−=−))[][]/2(925.15 61.22) / 27) 2(925.15 61.22) 61.2263.99.5163.99