1ANCOVACombining Quantitative and Qualitative PredictorsANCOVA• In an ANCOVA we try to adjust for differences in the quantitative variable.• For example, suppose that we were to compare men’s average faculty income to women’s average faculty income here at OU faculty. – Looking for a difference involves an ANOVA– Explaining the difference (if one is found) involves an ANCOVAExplaining the Difference• In trying to explain the difference between men and women, we may want to control for certain variables:– Experience–Rank– Performance record–Etc.• That is, we would like to show that the difference is due to relevant performance criteria. If we can’t show that this is the case, then we have a serious discrimination problem. What if?• The ANCOVA is many ways is a what if analysis– what if men and women had the same amount of experience? Would we still see the difference in income?• Inherent in this analysis is the possibility that the what if question is relevant. For example, it would be silly to compare basketball teams adjusting for the heights of players. This would be a meaningless comparison. Difference Are Due to ExperienceIncomeExperiencewomen menThe regression lines for men and women are the same.Grand mean on experiencePredicted income for men and womenDifference Are Due to ExperienceIncomeExperiencewomen menThe regression lines for men and women are the same.Grand mean on experiencePredicted income for men and womenWomen’s observed meanMen’s observed meandifference2Difference Due to Something Besides ExperienceIncomeExperiencewomen menmenwomenGrand mean on experienceWomen’s predicted incomeMen’s predicted incomeDifference Due to Something Besides ExperienceIncomeExperiencewomen menmenwomenGrand mean on experienceWomen’s adjusted meanMen’s adjusted meanAncova compares the adjusted meansDifference Due to Something Besides ExperienceIncomeExperiencewomen menmenwomenGrand mean on experienceWomen’s adjusted meanMen’s adjusted meanAncova compares the adjusted meansAdjusted differencesUnadjusted differencesANOVA compares the unadjusted meansDifference Are Due to InteractionIncomeExperiencewomen menThe regression lines for men and women have different slopes.Grand mean on experiencePredicted income for men menwomenPredicted income for women Situational and Individual Differences• In the social sciences researcher use ANCOVA to adjust the results for individual differences.• Suppose that you are looking at ethical decisions under a variety of situations (personal gain, accountability, etc.)– You would also like to see if certain individual difference variables (introversion, conscientiousness, cognitive style, etc.) moderate the situational results, you can adjust for these individual difference variables using an ANCOVA design.SAS ANCOVA SetupAnnual income for three groupsproc glm; class race;model inc = educ race / solution;means race / tukey;lsmeans race / tdiff adj=tukey;/* Note that contrast and estimate statementsare based on the adjusted means */contrast 'black vs white' race 10-1;estimate 'black vs white' race 10-1; run;proc glm data=anc; class race;model inc= race;estimate 'black vs white' race 10-1;contrast 'black vs white' race 10-1;run;ANCOVAANOVA333.0019.008.007.006.643.3513.8712.251616IncomeeducationMaximumMinimumStd DevMeanNBlacksVariable29.0016.008.008.006.402.3015.5011.641414IncomeeducationMaximumMinimumStd DevMeanNHispanicsVariable60.0020.009.007.0011.432.8021.2413.125050IncomeeducationMaximumMinimumStd DevMeanNWhites VariableANOVA: Unadjusted MeansANOVA Results(Comparing the Unadjusted Means)8440.48779Corrected Total98.737602.37077Error0.01784.24419.05838.1172ModelPr > FF ValueMean SquareSum of SquaresDFSourceTukey on the Unadjusted Means7.065-10.315-1.6251 - 2***-0.544-14.186-7.3651 - 310.315-7.0651.6252 - 11.440-12.920-5.7402 - 3***14.1860.5447.3653 - 112.920-1.4405.7403 - 2Simultaneous 95% Confidence LimitsDifferenceBetweenMeansraceComparisonComparisons significant at the 0.05 level are indicated by ***.<.000151.233061.3073061.3071ed0.00167.01419.058838.1172racePr > FF ValueMean SquareType I SSDFSource<.000151.233061.3073061.3071ed0.05293.06182.572365.1452racePr > FF ValueMean SquareType III SSDFSourceANCOVA ResultsNotice that the Type I and Type III Sums of Squares are different320.28165173217.81475542114.84442751LSMEAN Numberinc LSMEANraceAdjusted Means for IncomeWe see that even after we adjust for education there is still a difference between the averages. 1.0358130.55672.431120.04533-1.035810.55671.0477060.54932-2.431120.0453-1.047710.54931321i/jLeast Squares Means for Effect racet for H0: LSMean(i)=LSMean(j) / Pr > |t|Dependent Variable: incAdjustment for Multiple Comparisons: Tukey-Kramer4<.00017.160.30952.215ed...B0.000race 3(Whites)0.3036-1.042.3816B-2.466race 2(Hispanics)0.0174-2.432.2365B-5.437race 1(Blacks)0.0665-1.864.2060B-7.831InterceptPr > |t|tValueStandard ErrorEstimateParameterxyxy22.227.1322.2)44.583.7(11+−=+−−=BlacksUsing the Solution option in SASRegression Equation for each Groupxyxyxy22.283.722.23.1022.227.13321+−=+−=+−=Adjusted Means)7.12(22.283.736.20)7.12(22.23.1089.17)7.12(22.227.1392.14+−=+−=+−=20.2817.8114.84To obtain the adjusted means we use the regression equation for each group and the overall x meanlsmeansThe Adjusted MeansWhiteHispanicBlackEducationGrand MeanIncomeLsmeansParameter EstimateStandard ErrortValue Pr > |t|black vs white-5.43722419 2.23651016 -2.43 0.0174Parameter EstimateStandard ErrortValue Pr > |t|black vs white-7.36500000 2.85401409 -2.58 0.0118ANOVAResultsANCOVAEstimate StatementsSAS Type Sum of Squaresfor unequal n’sSS(AB| µ ,A,B)SS(AB| µ ,A,B)SS(AB| µ ,A,B)A*BSS(B| µ ,A,AB)SS(B| µ ,A)SS(B| µ,A)BSS(A| µ, B,AB)SS(A| µ,B)SS(A| µ)ASS IIISS IISS ISource5Unequal n’s Designs and Ancova Models• Under the MCAR (Missing data complete at random) assumption:– SAS Type III Sum of Squares provides a test of the partial effects, all submodels are compared to the overall model,ijiojijijijexexy++=+++=βββτµSequential Sum of SquaresSAS Type I• SAS model statement: (testing the equality of slopes assumption in ancova)model y= trt cov trt*cov;SS(trt | µ)SS(cov | µ, trt)SS(trt*cov | µ, trt, cov)For Type I SS, the sum of all effects add up to the model SS:SS(trt)+SS(cov)+SS(trt*cov)+SS(error)=SS(total)SS’s are also independent SAS Type II SS• SAS model statement:model y= trt cov