1-Way Analysis of VarianceSlide 2Within and Between Group VariationExample: Policy/Participation in European ParliamentSlide 5F-Test for Equality of MeansSlide 7Analysis of Variance TableEstimating/Comparing MeansMultiple Comparisons of GroupsBonferroni Multiple ComparisonsInterpretationsSlide 13Slide 14Regression Approach To ANOVATest Comparisons2-Way ANOVAExample - Thalidomide for AIDSANOVA ApproachSlide 20Example - Thalidomide for AIDSSlide 22Regression ApproachSlide 24Slide 25Regression with InteractionSlide 27Slide 281- Way ANOVA with Dependent Samples (Repeated Measures)Slide 30ANOVA & F-TestPost hoc Comparisons (Bonferroni)Repeated Measures ANOVASlide 34Slide 351-Way Analysis of Variance•Setting: –Comparing g > 2 groups–Numeric (quantitative) response–Independent samples•Notation (computed for each group):–Sample sizes: n1,...,ng (N=n1+...+ng)–Sample means:–Sample standard deviations: s1,...,sg NYnYnYYYggg111,...,1-Way Analysis of Variance•Assumptions for Significance tests:–The g distributions for the response variable are normal–The population standard deviations are equal for the g groups ()–Independent random samples selected from the g populationsWithin and Between Group Variation•Within Group Variation: Variability among individuals within the same group. (WSS)•Between Group Variation: Variability among group means, weighted by sample size. (BSS) 1)1()1(22112211gdfYYnYYnBSSgNdfsnsnWSSBggWgg• If the population means are all equal, E(WSS/dfW ) = E(BSS/dfB) = 2Example: Policy/Participation in European Parliament•Group Classifications: Legislative Procedures (g=4): (Consultation, Cooperation, Assent, Co-Decision)•Units: Votes in European Parliament•Response: Number of Votes CastLegislative Procedure (i) # of Cases (ni)Mean iYStd. Dev (si)Consultation 205 296.5 124.7Cooperation 88 357.3 93.0Assent 8 449.6 171.8Codecision 133 368.6 61.175.3334345.144845434)6.368(133)6.449(8)3.357(88)5.296(205434133888205 YNSource: R.M. Scully (1997). “Policy Influence and Participation in the European Parliament”, Legislative Studies Quarterly, pp.233-252.Example: Policy/Participation in European Parliamenti n_i Ybar_i s_i YBar_i-Ybar BSS WSS1 205 296.5 124.7 -37.25 284450.313 31722182 88 357.3 93.0 23.55 48805.02 7524633 8 449.6 171.8 115.85 107369.78 206606.74 133 368.6 61.1 34.85 161531.493 492783.7602156.605 462407243044344624072)1.61)(1133()7.124)(1205(3146.602156)75.3336.368(133)75.3335.296(2052222WBdfWSSdfBSSF-Test for Equality of Means•H0: g•HA: The means are not all equal)(:..)/()1/(..,1,obsgNgobsobsFFPPFFRRWMSBMSgNWSSgBSSFST• BMS and WMS are the Between and Within Mean SquaresExample: Policy/Participation in European Parliament•H0: 4•HA: The means are not all equal001.)42.5()67.18(60.2:..67.18430/46240723/6.602156)/()1/(..430,3,05.,1,FPFFPPFFFRRgNWSSgBSSFSTobsgNgobsobsAnalysis of Variance Table•Partitions the total variation into Between and Within Treatments (Groups)•Consists of Columns representing: Source, Sum of Squares, Degrees of Freedom, Mean Square, F-statistic, P-value (computed by statistical software packages)Source ofVariation Sum of SquaresDegrres ofFreedom Mean Square FBetween BSS g-1 BMS=BSS/(g-1) F=BMS/WMSWithin WSS N-g WMS=WSS/(N-g)Total TSS N-1Estimating/Comparing Means•Estimate of the (common) standard deviation:gNdfWMSgNWSS^• Confidence Interval for i: igNintY^,2/• Confidence Interval for ij jigNjinntYY11^,2/Multiple Comparisons of Groups•Goal: Obtain confidence intervals for all pairs of group mean differences. •With g groups, there are g(g-1)/2 pairs of groups.•Problem: If we construct several (or more) 95% confidence intervals, the probability that they all contain the parameters (i-j) being estimated will be less than 95%•Solution: Construct each individual confidence interval with a higher confidence coefficient, so that they will all be correct with 95% confidenceBonferroni Multiple Comparisons•Step 1: Select an experimentwise error rate (E), which is 1 minus the overall confidence level. For 95% confidence for all intervals, E=0.05.•Step 2: Determine the number of intervals to be constructed: g(g-1)/2•Step 3: Obtain the comparisonwise error rate: C= E/[g(g-1)/2]•Step 4: Construct (1- C)100% CI’s for i-j: jigNjinntYYC11^,2/Interpretations•After constructing all g(g-1)/2 confidence intervals, make the following conclusions:–Conclude i > j if CI is strictly positive–Conclude i < j if CI is strictly negative–Do not conclude i j if CI contains 0•Common graphical description.–Order the group labels from lowest mean to highest–Draw sequence of lines below labels, such that means that are not significantly different are “connected” by linesExample: Policy/Participation in European Parliament•Estimate of the common standard deviation:7.1034304624072^gNWSS• Number of pairs of procedures: 4(4-1)/2=6• Comparisonwise error rate: C=.05/6=.0083• t.0083/2,430 z.0042 2.64Example: Policy/Participation in European ParliamentComparisonjiYYjinnt11^Confidence IntervalConsult vs Cooperate 296.5-357.3 = -60.8 2.64(103.7)(0.13)=35.6 (-96.4 , -25.2)*Consult vs Assent 296.5-449.6 = -153.1 2.64(103.7)(0.36)=98.7 (-251.8 , -54.4)*Consult vs Codecision 296.5-368.6 = -72.1 2.64(103.7)(0.11)=30.5 (-102.6 , -41.6)*Cooperate vs Assent 357.3-449.6 = -92.3 2.64(103.7)(0.37)=101.1 (-193.4 , 8.8)Cooperate vs Codecision 357.3-368.6 = -11.3 2.64(103.7)(0.14)=37.6 (-48.9 , 26.3)Assent vs Codecision 449.6-368.6 = 81.0 2.64(103.7)(0.36)=99.7 (-18.7 , 180.7)Consultation Cooperation Codecision AssentPopulation mean is lower for consultation than all other procedures, no other procedures are significantly different.Regression Approach To ANOVA•Dummy (Indicator) Variables: Variables that take on the value 1 if observation comes from a particular group, 0 if not. •If there are g groups, we create g-1 dummy variables.•Individuals in the
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