chapter12 - 1-Way Analysis of VarianceSetting: Comparing...

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Unformatted text preview: 1-Way Analysis of VarianceSetting: Comparing g> 2 groupsNumeric (quantitative) responseIndependent samplesNotation (computed for each group): Sample sizes: n1,...,ng(N=n1+...+ng)Sample means: Sample standard deviations: s1,...,sg++=NYnYnYYYggg111,...,1-Way Analysis of VarianceAssumptions for Significance tests:The gdistributions for the response variable are normalThe population standard deviations are equal for the ggroups ()Independent random samples selected from the gpopulationsWithin and Between Group VariationWithin Group Variation: Variability among individuals within the same group. (WSS)Between Group Variation: Variability among group means, weighted by sample size. (BSS)( 29( 291)1()1(22112211-=-++-=-=-++-=gdfYYnYYnB SSgNdfsnsnW SSBggWgg If the population means are all equal, E(WSS/dfW) = E(BSS/dfB) = 2Example: Policy/Participation in European ParliamentGroup Classifications: Legislative Procedures (g=4):(Consultation, Cooperation, Assent, Co-Decision)Units: Votes in European ParliamentResponse: Number of Votes CastLegislative Procedure (i)# of Cases (ni)Mean ( 29iYStd. Dev (si)Consultation205296.5124.7Cooperation88357.393.0Assent8449.6171.8Codecision133368.661.175.3334345.144845434)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 Parliamentin_iYbar_is_iYBar_i-YbarBSSWSS1205296.5124.7-37.25284450.3133172218288357.393.023.5548805.0275246338449.6171.8115.85107369.78206606.74133368.661.134.85161531.493492783.7602156.605462407243044344624072)1.61)(1133()7.124)(1205(3146.602156)75.3336.368(133)75.3335.296(2052222=-==-++-==-==-++-=WBdfWSSdfBSSF-Test for Equality of MeansH: 1= 2= = gHA: The means are not all equal)(:..)/()1/(..,1,obsgNgobsobsFFPPFFRRWMSBMSgNWSSgBSSFST==--=--BMS and WMS are the Between and Within Mean SquaresExample: Policy/Participation in European ParliamentH: 1= 2= 3= 4HA: 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 TablePartitions 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 ofVariationSum of SquaresDegrres ofFreedomMean SquareFBetweenBSSg-1BMS=BSS/(g-1)F=BMS/WMSWithinWSSN-gWMS=WSS/(N-g)TotalTSSN-1Estimating/Comparing MeansEstimate of the (common) standard deviation:gNdfWMSgNWSS-==-=^ Confidence Interval for i: igNintY^,2/- Confidence Interval for i-j( 29jigNjinntYY11^,2/+--Multiple Comparisons of GroupsGoal: Obtain confidence intervals for all pairs of...
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chapter12 - 1-Way Analysis of VarianceSetting: Comparing...

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