Chapter 8Comparing t > 2 Groups - Numeric ResponsesCompletely Randomized Design (CRD)1-Way ANOVA for Normal Data (CRD)Analysis of Variance - Sums of SquaresAnalysis of Variance Table and F-TestExpected Mean SquaresSlide 8Slide 9CRD with Non-Normal Data Kruskal-Wallis TestKruskal-Wallis TestPost-hoc Comparisons of TreatmentsFisher’s Least Significant Difference ProcedureTukey’s W ProcedureBonferroni’s Method (Most General)Slide 16Chapter 81-Way Analysis of Variance - Completely Randomized DesignComparing t > 2 Groups - Numeric Responses•Extension of Methods used to Compare 2 Groups•Independent Samples and Paired Data Designs•Normal and non-normal data distributions DataDesignNormal Non-normalIndependentSamples(CRD)F-Test1-WayANOVAKruskal-Wallis TestPaired Data(RBD)F-Test2-WayANOVAFriedman’sTestCompletely Randomized Design (CRD)•Controlled Experiments - Subjects assigned at random to one of the t treatments to be compared•Observational Studies - Subjects are sampled from t existing groups•Statistical model yij is measurement from the jth subject from group i:ijiijiijywhere is the overall mean, i is the effect of treatment i , ij is a random error, and i is the population mean for group i1-Way ANOVA for Normal Data (CRD)• For each group obtain the mean, standard deviation, and sample size:1)(2..ijiijiijijinyysnyy• Obtain the overall mean and sample sizeNyNynynynnNi jijttt ..11..1Analysis of Variance - Sums of Squares• Total Variation 1)(1 12.. NdfyyTSSTotalkinjiji• Between Group (Sample) Variation tinjtiTiiiitdfyynyySST1 1 12...2...1)()(• Within Group (Sample) VariationETTotalEtiiitinjiijdfdfdfSSESSTTSStNdfsnyySSEi 121 12.)1()(Analysis of Variance Table and F-TestSource ofVariation Sum of SquaresDegrres ofFreedom Mean Square FTreatments SST t-1 MST=SST/(t-1) F=MST/MSEError SSE N-t MSE=SSE/(N-t)Total TSS N-1• Assumption: All distributions normal with common variance•H0: No differences among Group Means (t =0)• HA: Group means are not all equal (Not all i are 0))(:)9(:..:..,1,obstNtobsobsFFPvalPTableFFRRMSEMSTFSTExpected Mean Squares•Model: yij = +i + ij with ij ~ N(0,2), i = 0:1)()( true),is ( otherwise1)()( true,is 0:When )1(11)()( 1)()(1021221221222MSEEMSTEHMSEEMSTEHtntnMSEEMSTEtnMSTEMSEEattiiitiiitiiiExpected Mean Squares•3 Factors effect magnitude of F-statistic (for fixed t)–True group effects (1,…,t)–Group sample sizes (n1,…,nt)–Within group variance (2)•Fobs = MST/MSE•When H0 is true (1=…=t=0), E(MST)/E(MSE)=1 •Marginal Effects of each factor (all other factors fixed)–As spread in (1,…,t) E(MST)/E(MSE) –As (n1,…,nt) E(MST)/E(MSE) (when H0 false)–As 2 E(MST)/E(MSE) (when H0 false)00.010.020.030.040.050.060.070.080.090 20 40 60 80 100 120 140 160 180 20000.010.020.030.040.050.060.070.080.090 20 40 60 80 100 120 140 160 180 20000.010.020.030.040.050.060.070.080.090 20 40 60 80 100 120 140 160 18000.010.020.030.040.050.060.070.080.090 20 40 60 80 100 120 140 160 180 200A)=100, 1=-20, 2=0, 3=20, = 20 B)=100, 1=-20, 2=0, 3=20, = 5C)=100, 1=-5, 2=0, 3=5, = 20 D)=100, 1=-5, 2=0, 3=5, = 5n A B C D4 9 129 1.5 98 17 257 2 1712 25 385 2.5 2520 41 641 3.5 41)()(MSEEMSTECRD with Non-Normal Data Kruskal-Wallis Test•Extension of Wilcoxon Rank-Sum Test to k > 2 Groups•Procedure:–Rank the observations across groups from smallest (1) to largest ( N = n1+...+nk ), adjusting for ties–Compute the rank sums for each group: T1,...,Tk . Note that T1+...+Tk = N(N+1)/2Kruskal-Wallis Test• H0: The k population distributions are identical (1=...=k)• HA: Not all k distributions are identical (Not all i are equal))(::..)1(3)1(12:..221,12HPvalPHRRNnTNNHSTkkiiiAn adjustment to H is suggested when there are many ties in the data. Formula is given on page 344 of O&L.Post-hoc Comparisons of Treatments•If differences in group means are determined from the F-test, researchers want to compare pairs of groups. Three popular methods include:–Fisher’s LSD - Upon rejecting the null hypothesis of no differences in group means, LSD method is equivalent to doing pairwise comparisons among all pairs of groups as in Chapter 6.–Tukey’s Method - Specifically compares all t(t-1)/2 pairs of groups. Utilizes a special table (Table 11, p. 701).–Bonferroni’s Method - Adjusts individual comparison error rates so that all conclusions will be correct at desired confidence/significance level. Any number of comparisons can be made. Very general approach can be applied to any inferential problemFisher’s Least Significant Difference Procedure•Protected Version is to only apply method after significant result in overall F-test•For each pair of groups, compute the least significant difference (LSD) that the sample means need to differ by to conclude the population means are not equal ijjiijjijijiijLSDyyLSDyytNnnMSEtLSD....2/ :Interval Confidence sFisher' if Conclude dfwith 11Tukey’s W Procedure•More conservative than Fisher’s LSD (minimum significant difference and confidence interval width are higher).•Derived so that the probability that at least one false difference is detected is (experimentwise error rate) tijjiijjijiijnntnWyyWyyN-tqnMSEtqW11 use unequal, are sizes sample When the :Interval Confidence sTukey' if Conclude with 701 p. 11, Tablein given ),(1....Bonferroni’s Method (Most General)• Wish to make C comparisons of pairs of groups with simultaneous confidence intervals or 2-sided tests •When all pair of treatments are to be compared, C = t(t-1)/2• Want the overall confidence level for all intervals to be “correct” to be 95% or the overall type I error rate for all tests to be 0.05• For confidence intervals, construct (1-(0.05/C))100% CIs for the difference in each pair of group means (wider than 95%
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