One-Way BG ANOVAAndrew AinsworthPsy 420Topics• Analysis with more than 2 levels• Deviation, Computation, Regression, Unequal Samples• Specific Comparisons• Trend Analysis, Planned comparisons, Post-Hoc Adjustments• Effect Size Measures• Eta Squared, Omega Squared, Cohen’s d• Power and Sample Size EstimatesDeviation Approach• When the n’s are not equal222/T ijAjS A ij jSS Y GMSS n Y GMSS Y Y2A j jSS n Y GMAnalysis - Traditional• The traditional analysis is the same2222YTYTSS SS Y YN an22jAaTSSn an22/jSAaSS YnAnalysis - Traditional• Traditional Analysis – Unequal Samples22222222/jikAi j kjikSAi j kAAATSSn n n NAAASS Yn n nUnequal N and DFs1 2 3/ 1 2 31 ( ) 11( 1) ( 1) ( 1) ( 1)total kAS A kdf N n n n ndf adf n n n nAnalysis - Regression• In order to perform a complete analysis of variance through regression you need to cover all of the between groups variance• To do this you need to:• Create k – 1 dichotomous predictors (Xs)• Make sure the predictors don’t overlapAnalysis – RegressionAnalysis – Regression• One of the easiest ways to ensure that the comps do not overlap is to make sure they are orthogonal• Orthogonal (independence) • The sum of each comparison equals zero• The sum of each cross-product of predictors equals zeroAnalysis – RegressionLevel of A Case Y X1 X2 Y2 X12 X22 YX1 YX2 s1 8 s2 7 s3 9 s4 9 Control s5 7 s6 8 s7 8 s8 6 s9 7 Ativan s10 7 s11 4 s12 5 s13 4 s14 7 Scruital s15 4 100 15 Sum N Mean 6.67Analysis - RegressionLevel of A Case Y X1 X2 Y2 X12 X22 YX1 YX2 s1 8 2 s2 7 2 s3 9 2 s4 9 2 Control s5 7 2 s6 8 -1 s7 8 -1 s8 6 -1 s9 7 -1 Ativan s10 7 -1 s11 4 -1 s12 5 -1 s13 4 -1 s14 7 -1 Scruital s15 4 -1 100 15 Sum N Mean 6.67Analysis - RegressionLevel of A Case Y X1 X2 Y2 X12 X22 YX1 YX2 X1X2 s1 8 2 0 0 s2 7 2 0 0 s3 9 2 0 0 s4 9 2 0 0 Control s5 7 2 0 0 s6 8 -1 1 -1 s7 8 -1 1 -1 s8 6 -1 1 -1 s9 7 -1 1 -1 Ativan s10 7 -1 1 -1 s11 4 -1 -1 1 s12 5 -1 -1 1 s13 4 -1 -1 1 s14 7 -1 -1 1 Scruital s15 4 -1 -1 1 100 0 0 0 15 Sum N Mean 6.67 0 0Analysis – RegressionLevel of A Case Y X1 X2 Y2 X12 X22 YX1 YX2 X1X2 s1 8 2 0 64 4 0 16 0 0 s2 7 2 0 49 4 0 14 0 0 s3 9 2 0 81 4 0 18 0 0 s4 9 2 0 81 4 0 18 0 0 Control s5 7 2 0 49 4 0 14 0 0 s6 8 -1 1 64 1 1 -8 8 -1 s7 8 -1 1 64 1 1 -8 8 -1 s8 6 -1 1 36 1 1 -6 6 -1 s9 7 -1 1 49 1 1 -7 7 -1 Ativan s10 7 -1 1 49 1 1 -7 7 -1 s11 4 -1 -1 16 1 1 -4 -4 1 s12 5 -1 -1 25 1 1 -5 -5 1 s13 4 -1 -1 16 1 1 -4 -4 1 s14 7 -1 -1 49 1 1 -7 -7 1 Scruital s15 4 -1 -1 16 1 1 -4 -4 1 100 0 0 708 30 10 20 12 0 15 Sum N Mean 6.67 0 0Analysis – Regression• Formulas2222()()()()iiiiiiiji j i jYSS Y YNXSS X XNYXSP YX YXNXXSP X X X XNAnalysis – Regression• Formulas( ) ( )22( ) ( . ) ( . )( ) ( ) ( )()()( ) ( )[ ( )][ ( )] [ ( )] [ ( )] [ ( )][ ( )][ ( )] [ ( )] [ (ijTotal Yjiregression reg X reg Xijresidual Total regressioni j k j kii j kSS SSSP YXSP YXSS SS SSSS X SS XSS SS SSSP YX SS X SS X SP YX SP YXbSS X SS X SS X SP X2 2 2)] [ ( )] [ ( )]i j i k j kX SP X X SP X XAnalysis – Regression• Example22122(100) 10,000( ) 708 708 708 666.67 41.3315 15(0)( ) 30 3015(0)( ) 10 1015SS YSS XSS XAnalysis - Regression• Example1212(100)(0)( ) 20 2015(100)(0)( ) 12 1215(0)(0)( ) 0 015SP YXSP YXSP X XAnalysis - Regression• Example()22()()41.3320 12 400 14413.33 14.4 27.7330 10 30 1041.33 27.73 13.6TotalregresSSSSSSAnalysis - Regression• Example()()()1 15 1 14#215 3 12Totalregresdf Ndf predictorsdf N aAnalysis - Regression• Example• Fcrit(2,12) = 3.88, since 12.253 is greater than 3.88 you reject the null hypothesis.• There is evidence that drug type can predict level of anxiety Source SS df MS F Reg 27.73 2 13.867 12.235 Res 13.60 12 1.133 Total 41.33 14Analysis - Regression• Example12221 1 2 21220(10) 12(0) 200 0.6730(10) (0) 300 012(30) 20(0) 360 01.230(10) (0) 300 0( ) ( ) 6.67 .67(0) 1.2(0) 6.67' 6.67 .67( ) 1.2( )bba Y b X b XY X XAnalysis - Regression• SPSSModel Summary.819a.671.6161.06458Model1RR SquareAdjusted R SquareStd. Error ofthe EstimatePredictors: (Constant), X2, X1a.Analysis - Regression• SPSSANOVAb27.733213.86712.235.001a13.600121.13341.33314RegressionResidualTotalModel1Sum of SquaresdfMean SquareFSig.Predictors: (Constant), X2, X1a. Dependent Variable: Yb.Analysis - Regression• SPSSCoefficientsa6.667.27524.254.000.667.194.5683.430.0051.200.337.5903.565.004(Constant)X1X2Model1BStd. ErrorUnstandardized CoefficientsBetaStandardizedCoefficientstSig.Dependent Variable: Ya.Specific Comparisons• F-test for Comparisons• n = number of subjects in each group• = squared sum of the weighted means• = sum of the squared coefficients• MSS/A= mean square error from overall ANOVA22( . )/ ( )( ) /jreg Xj j jS A residSSn w Y wFMS MS2jjwYjwSpecific Comparisons• If each group has a different sample size…22/( ) / ( / )j j j jSAw Y w nFMSSpecific Comparisons• Example122222()222 2 2()25[(2)(8)+(-1)(7.2)+(-1)(4.8)][2 ( 1) ( 1) ]1.135[16 7.2 4.8] /6 13.3311.81.13 1.135[(0)(8)+(1)(7.2)+(-1)(4.8)][0 (1) ( 1) ]1.135[0 7.2 4.8] / 2 14.412.741.13 1.13XXFFSpecific Comparisons• Trend Analysis• If you have ordered groups (e.g. they differ in amount of Milligrams given; 5, 10, 15, 20)• You often will want to know whether there is a consistent trend across the ordered groups (e.g. linear trend)• Trend analysis comes in handy too because there are orthogonal weights already worked out depending on the number of groups (pg. 703)Specific Comparisons• Different types of trend and coefficients for 4 groupsSpecific Comparisons• Mixtures of Linear and Quadratic TrendSpecific Comparisons• Planned comparisons - if the comparisons are planned than you test them without any correction• Each F-test for the comparison is treated like any other F-test• You look up an F-critical value in a table with dfcompand dferror.Specific Comparisons• Example – if the comparisons are planned than you test them without any correction…• Fx1, since 11.8 is larger than 4.75 there is evidence that the subjects in the control group had higher anxiety than the treatment groups• Fx2, since 12.75 is larger than 4.75 there is evidence that subjects in the