Tukey's Studentized Range (HSD) Test for logcountBonferroni (Dunn) t Tests for logcountScheffe's Test for logcountTukey's Studentized Range (HSD) Test for logcountBonferroni (Dunn) t Tests for logcountIES 612/STA 4-573/STA 4-576Spring 2005Week 6 – IES612-week06-lecture.docANOVA MODELS – models for comparing means of different treatments or populationsRecall your old friend the two-group pooled variance t-testH0: 1 = 2 [two populations do NOT differ in mean response]Ha: 1 ≠ 2 Assumptions/Data?Data from population 1: (y11, y12, . . . , y1n1)Data from population 2: (y21, y22, . . . , y2n2)Assume Yij ~ independent N(i, 2)i = 1,2j = 1, 2, …, niAnother way of writing this is Yij = i + ij with ij ~ independent N(, 2)In other words, the response of the “jth” observation in the “ith” population can be written in termsof the mean of the ith population + how this observation differs from the mean. Does this look familiar?Test Statistic?2122111nnsyytpobs where  22)1()1(2121 12212222112 nnyynnsnsnsinjiijpiThe pooled variance looks like something from regression. What?How about your new friend, regression?Can we test H0: 1 = 2 [two populations differ in mean response]? Vs. Ha: 1 ≠ 2 Assumptions/Data?Data from population 1: (y11, y12, . . . , y1n1)Data from population 2: (y21, y22, . . . , y2n2)Assume Yij ~ independent N(i, 2) i = 1,2, j = 1, 2, …, niAnother way of writing this is Yij = i + ij with ij ~ independent N(, 2)Let X = 1 (if group 2) and X=0 (if group 1) and Y = 0 + 1 X + ThenGroup 1: Y = 0 + Group 2: Y = 0 + 1 + Implying 1 = 0 and 2 = 0 + 1 so 1 = 2 - 1. Thus, H0: 1 = 2 AND H0: 1 = 0 test the same hypothesis.Example: Comparing Two-group T-test, Regression test and one-way ANOVA testoptions ls=80 formdlim=”-“ nocenter nodate;data meat; input condition $ logcount @@; ivac = (condition=”vacuum”); imix = (condition=”mixed”); datalines;vacuum 5.26 vacuum 5.44 vacuum 5.80 mixed 7.41 mixed 7.33 mixed 7.04;ods html;title “Log(bacteria count) for different packaging conditions”;proc boxplot;title2 “Boxplots of log(count)”; plot logcount*condition; run;proc ttest;title2 “T-test comparing mix to vacuum conditions”; class condition; var logcount; run;proc reg;title2 “Regression with indicator variable for mix condition”; model logcount = imix; run;proc glm;title2 “One-way anova model”; class condition; model logcount = condition;run;ods html close;Log(bacteria count) for different packaging conditionsT-test comparing mix to vacuum conditionsThe TTEST ProcedureStatisticsVariable condition N LowerCLMeanMean UpperCLMeanLowerCLStd DevStd Dev UpperCLStd DevStd Err Minimum Maximumlogcount mixed 3 6.7764 7.26 7.7436 0.1014 0.1947 1.2235 0.1124 7.04 7.41logcount vacuum 3 4.817 5.5 6.183 0.1432 0.275 1.728 0.1587 5.26 5.8logcount Diff (1-2) 1.22 1.76 2.3 0.1427 0.2382 0.6845 0.1945 T-TestsVariable Method Variances DF t Value Pr > |t|logcount Pooled Equal 4 9.05 0.0008logcount Satterthwaite Unequal 3.6 9.05 0.0013Equality of VariancesVariable Method Num DF Den DF F Value Pr > Flogcount Folded F 2 2 1.99 0.6678Log(bacteria count) for different packaging conditionsRegression with indicator variable for mix conditionThe REG ProcedureModel: MODEL1Dependent Variable: logcount Number of Observations Read 6Number of Observations Used 6Analysis of VarianceSource DF Sum ofSquaresMeanSquareF Value Pr > FModel 1 4.64640 4.64640 81.87 0.0008Error 4 0.22700 0.05675 Corrected Total 5 4.87340 Root MSE 0.23822 R-Square 0.9534Dependent Mean 6.38000 Adj R-Sq 0.9418Coeff Var 3.73390 Parameter EstimatesVariable DF ParameterEstimateStandardErrort Value Pr > |t|Intercept 1 5.50000 0.13754 39.99 <.0001imix 1 1.76000 0.19451 9.05 0.0008Log(bacteria count) for different packaging conditionsOne-way anova modelThe GLM ProcedureClass Level InformationClass Levels Valuescondition 2 mixed vacuumNumber of Observations Read 6Number of Observations Used 6Dependent Variable: logcount Source DF Sum of Squares Mean Square F Value Pr > FModel 1 4.64640000 4.64640000 81.87 0.0008Error 4 0.22700000 0.05675000 Corrected Total 5 4.87340000 R-Square Coeff Var Root MSE logcount Mean0.953421 3.733896 0.238223 6.380000Source DF Type I SS Mean Square F Value Pr > Fcondition 1 4.64640000 4.64640000 81.87 0.0008Source DF Type III SS Mean Square F Value Pr > Fcondition 1 4.64640000 4.64640000 81.87 0.0008Test statistic P-value CommentT-test tobs=9.05 0.0008 Test of “1 = 2“ – note unequal variance t-test has same value test statistic [b/c sample sizes are the same]; however, slight modification in degrees of freedomRegression tobs=9.05Fobs=81.870.0008 Test of “1=0” in model logcount =0 + 1 I[condition=”mix”] + One-way ANOVA Fobs=81.87 0.0008 Test of “1 = 2“A more general formulation …Numeric data – samples from “t’ populations obtained(y11, y12, . . . , y1n1) = {y1j} j=1, …, n1(y21, y22, . . . , y2n2) = {y2j} j=1, …, n2…(yt1, yt2, . . . , ytnt) = {ytj} j=1, …, ntAssume Yij ~ independent N(i, 2)ni = number of observations from the ith populationi = 1,2, …, t (populations or treatments)j = 1, 2, …, ni (observations)TtinjijinjijitTnyymeanOverallnyymeansampleiPopulationnnnnsobsrvationofNumberTotalii  1 1..121::...:Terminology* Designed experiments versus observational studies* Completely Randomized Designs (CRD)H0: 1 = 2= 3= … = tHa: i ≠ j [at least two population means differ]Assumptions/Data?Assume Yij ~ independent N(i, 2)i = 1,2, …, tj = 1, 2, …, niTest Statistic?22WBobsSSF  where the between(among) group variability is  1112..2tyyntSSBStiiiB and the within group variability is  tnnnyytnSSWSttinjiijTWi ...211 122Reject H0 if tntaobsTFF,1,AOV TableSource SS df MS FobsBetween SSB t-1 SSB/(t-1) MSB/MSWWithin SSW nT-t SSW/( nT-t)Totals TSS nT-1  tinjijiyyTSS1 12..Example Bacteria growth in meat under different packaging conditions (revisited)*--------------------------------------------------------------------;title “One-way ANOVA/ CRD example + contrasts + multiple