chapter12 - 1-Way Analysis of Variance•Setting...

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Unformatted text preview: 1-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 gdistributions for the response variable are normal–The population standard deviations are equal for the ggroups (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)( 29( 291)1()1(22112211-=-++-=-=-++-=gdfYYnYYnB SSgNdfsnsnW SSBggWgg• 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 ( 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 Means•H: 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•H: 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 ofVariationSum of SquaresDegrres ofFreedomMean SquareFBetweenBSSg-1BMS=BSS/(g-1)F=BMS/WMSWithinWSSN-gWMS=WSS/(N-g)TotalTSSN-1Estimating/Comparing Means•Estimate of the (common) standard deviation:gNdfWMSgNWSS-==-=^σ• Confidence Interval for μi: igNintY^,2/σα-±• Confidence Interval for μi-μj( 29jigNjinntYY11^,2/+±--σαMultiple Comparisons of Groups•Goal: Obtain confidence intervals for all pairs of...
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This note was uploaded on 07/08/2011 for the course STA 6127 taught by Professor Mukherjee during the Fall '08 term at University of Florida.

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chapter12 - 1-Way Analysis of Variance•Setting...

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