1-Way Analysis of Variance1-Way Analysis of VarianceWithin and Between Group VariationExample: Policy/Participation in European ParliamentExample: Policy/Participation in European ParliamentF-Test for Equality of MeansExample: Policy/Participation in European ParliamentAnalysis of Variance TableEstimating/Comparing MeansMultiple Comparisons of GroupsBonferroni Multiple ComparisonsInterpretationsExample: Policy/Participation in European ParliamentExample: Policy/Participation in European ParliamentRegression Approach To ANOVATest Comparisons2-Way ANOVAExample - Thalidomide for AIDSANOVA ApproachANOVA ApproachExample - Thalidomide for AIDSExample - Thalidomide for AIDSRegression ApproachExample - Thalidomide for AIDSExample - Thalidomide for AIDSRegression with InteractionExample - Thalidomide for AIDSExample - Thalidomide for AIDS1- Way ANOVA with Dependent Samples (Repeated Measures)1- Way ANOVA with Dependent Samples (Repeated Measures)ANOVA & F-TestPost hoc Comparisons (Bonferroni)Repeated Measures ANOVARepeated Measures ANOVARepeated Measures ANOVA1-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⎟⎟⎠⎞⎜⎜⎝⎛++=NYnYnYYYgggL111,...,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 (σ)– Independent random samples selected from the gpopulationsWithin 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(22112211−=−++−=−=−++−=gdfYYnYYnBSSgNdfsnsnWSSBggWggLL• If the population means are all equal, E(WSS/dfW) = E(BSS/dfB) = σ2Example: 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(2052222=−==−++−==−==−++−=WBdfWSSdfBSSLLF-Test for Equality of Means• H0: µ1= µ2= ⋅⋅⋅ = µg• HA: The means are not all equal)(:..)/()1/(..,1,obsgNgobsobsFFPPFFRRWMSBMSgNWSSgBSSFST≥=≥=−−=−−α• BMS and WMS are the Between and Within Mean SquaresExample: Policy/Participation in European Parliament• H0: µ1= µ2= µ3= µ4• HA: The means are not all equal001.)42.5()67.18(60.2:..67.18430/46240723/6.602156)/()1/(..430,3,05.,1,=≥<=≥=≈=≥==−−=−−FPFFPPFFFRRgNWSSgBSSFSTobsgNgobsobsαAnalysis 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 SquareFBetweenBSS g-1BMS=BSS/(g-1)F=BMS/WMSWithinWSS N-g WMS=WSS/(N-g)TotalTSS N-1Estimating/Comparing Means• Estimate of the (common) standard deviation:gNdfWMSgNWSS−==−=^σigNintY^,2/σα−±• Confidence Interval for µi: • Confidence Interval for µi−µj:()jigNjinntYY11^,2/+±−−σαMultiple 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 (µi-µj) 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 (αE), which is 1 minus the overall confidence level. For 95% confidence for all intervals, αE=0.05.• Step 2: Determine the number of intervals to be constructed: g(g-1)/2• Step 3: Obtain the comparisonwise error rate: αC= αE/[g(g-1)/2]• Step 4: Construct (1-αC)100% CI’s for µi-µj:()jigNjinntYYC11^,2/+±−−σαInterpretations• After constructing all g(g-1)/2 confidence intervals, make the following conclusions:– Conclude µi> µjif CI is strictly positive– Conclude µi< µjif CI is strictly negative– Do not conclude µi≠µjif 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^==−=gNWSSσ• Number of pairs of procedures: 4(4-1)/2=6• Comparisonwise error rate: αC=.05/6=.0083•t.0083/2,430 ≈z.0042≈ 2.64Example: Policy/Participation in European ParliamentComparisonjiYY −jinnt11^+σConfidence 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: