Chapter 9Randomized Block Design (RBD)Randomized Complete Block DesignsRBD - ANOVA F-Test (Normal Data)Slide 5Pairwise Comparison of Treatment MeansExpected Mean Squares / Relative EfficiencyExample - Caffeine and EnduranceSlide 9Slide 10Slide 11Slide 12Slide 13RBD -- Non-Normal Data Friedman’s TestSlide 15Latin Square DesignLatin Square Design - ModelLatin Square Design - ANOVA & F-TestSlide 19Slide 202-Way ANOVA2-Way ANOVA - ModelFixed Effects ModelExample - Thalidomide for AIDSANOVA ApproachAnalysis of VarianceSlide 27Example - Thalidomide for AIDSSlide 29Comparing Main Effects (No Interaction)Miscellaneous TopicsChapter 9More Complicated Experimental DesignsRandomized Block Design (RBD)•t > 2 Treatments (groups) to be compared•b Blocks of homogeneous units are sampled. Blocks can be individual subjects. Blocks are made up of t subunits•Subunits within a block receive one treatment. When subjects are blocks, receive treatments in random order.•Outcome when Treatment i is assigned to Block j is labeled Yij•Effect of Trt i is labeled i •Effect of Block j is labeled j•Random error term is labeled ij•Efficiency gain from removing block-to-block variability from experimental errorRandomized Complete Block Designs• Model:21)(0)(0ijijtiiijjiijjiijVEY• Test for differences among treatment effects:• H0: 1t 0 (1t )• HA: Not all i = 0 (Not all i are equal)Typically not interested in measuring block effects (although sometimes wish to estimate their variance in the population of blocks). Using Block designs increases efficiency in making inferences on treatment effectsRBD - ANOVA F-Test (Normal Data)• Data Structure: (t Treatments, b Subjects)• Mean for Treatment i:• Mean for Subject (Block) j:• Overall Mean:• Overall sample size: N = bt • ANOVA:Treatment, Block, and Error Sums of Squares .iyjy...y )1)(1(1112....12...12...1 12.. tbdfSSBSSTTSSyyyySSEbdfyytSSBtdfyybSSTbtdfyyTSSEjiijBbjjTtiitibjTotalijRBD - ANOVA F-Test (Normal Data)• ANOVA Table:Source SS df MS FTreatments SST t-1 MST = SST/(t-1) F = MST/MSEBlocks SSB b-1 MSB = SSB/(b-1)Error SSE (b-1)(t-1) MSE = SSE/[(b-1)(t-1)]Total TSS bt-1•H0: 1t 0 (1t )• HA: Not all i = 0 (Not all i are equal))(::..:..)1)(1(,1,obstbtobsobsFFPvalPFFRRMSEMSTFSTPairwise Comparison of Treatment Means•Tukey’s Method- q in Table 11, p. 701 with = (b-1)(t-1) ijjiijjijiijWyyWyybMSEvtqW.... :Interval Confidence sTukey' if Conclude),(• Bonferroni’s Method - t-values from table on class website with = (b-1)(t-1) and C=t(t-1)/2 ijjiijjijivCijByyByybMSEtB....,,2/ :Interval Confidence s'Bonferroni if Conclude2Expected Mean Squares / Relative Efficiency •Expected Mean Squares: As with CRD, the Expected Mean Squares for Treatment and Error are functions of the sample sizes (b, the number of blocks), the true treatment effects (1,…,t) and the variance of the random error terms (2)•By assigning all treatments to units within blocks, error variance is (much) smaller for RBD than CRD (which combines block variation&random error into error term)•Relative Efficiency of RBD to CRD (how many times as many replicates would be needed for CRD to have as precise of estimates of treatment means as RBD does):MSEbtMSEtbMSBbMSEMSECRRCBRERCBCR)1()1()1(),(Example - Caffeine and Endurance•Treatments: t=4 Doses of Caffeine: 0, 5, 9, 13 mg•Blocks: b=9 Well-conditioned cyclists•Response: yij=Minutes to exhaustion for cyclist j @ dose i•Data:Dose \ Subject 1 2 3 4 5 6 7 8 90 36.05 52.47 56.55 45.20 35.25 66.38 40.57 57.15 28.345 42.47 85.15 63.20 52.10 66.20 73.25 44.50 57.17 35.059 51.50 65.00 73.10 64.40 57.45 76.49 40.55 66.47 33.1713 37.55 59.30 79.12 58.33 70.54 69.47 46.48 66.35 36.20Plot of Y versus Subject by Dose0.0010.0020.0030.0040.0050.0060.0070.0080.0090.000 1 2 3 4 5 6 7 8 9 10CyclistTime to Exhaustion0 mg5 mg9mg13 mgExample - Caffeine and EnduranceSubject\Dose 0mg 5mg 9mg 13mgSubj MeanSubj Dev Sqr Dev1 36.05 42.47 51.50 37.55 41.89 -13.34 178.072 52.47 85.15 65.00 59.30 65.48 10.24 104.933 56.55 63.20 73.10 79.12 67.99 12.76 162.714 45.20 52.10 64.40 58.33 55.01 -0.23 0.055 35.25 66.20 57.45 70.54 57.36 2.12 4.516 66.38 73.25 76.49 69.47 71.40 16.16 261.177 40.57 44.50 40.55 46.48 43.03 -12.21 149.128 57.15 57.17 66.47 66.35 61.79 6.55 42.889 28.34 35.05 33.17 36.20 33.19 -22.05 486.06Dose Mean 46.44 57.68 58.68 58.15 55.24 1389.50Dose Dev -8.80 2.44 3.44 2.91Squared Dev 77.38 5.95 11.86 8.48 103.68TSS 7752.773 24)19)(14(653.1261555812.933773.7752)24.5515.5819.3320.36()24.5544.4689.4105.36(81900.5558)50.1389(4)24.5519.33()24.5589.41(431412.933)68.103(9)24.5515.58()24.5544.46(9351)9(4773.7752)24.5520.36()24.5505.36(22222222EBTTotaldfSSBSSTTSSSSEdfSSBdfSSTdfTSSExample - Caffeine and EnduranceSource df SS MS FDose 3 933.12 311.04 5.92Cyclist 8 5558.00 694.75Error 24 1261.65 52.57Total 35 7752.77equal allnot are means that trueConcludeEXCEL) (From 0036.)92.5( :value01.3:0.05).(.92.557.5204.311:..Doses AmongExist sDifference :)0(Effect Dose Caffeine No :24,3,05.410FPPFFRRMSEMSTFSTHHobsobsAExample - Caffeine and Endurance83.99257.52875.2875.2 :B s'Bonferroni43.99157.5290.390.3: sTukey'24,6,2/05.24,4,05.BtWqWDoses High Mean Low Mean Difference Conclude5mg vs 0mg 57.6767 46.4400 11.2367 9mg vs 0mg 58.6811 46.4400 12.2411 13mg vs 0mg 58.1489 46.4400 11.7089 9mg vs 5mg 58.6811 57.6767 1.0044 NSD13mg vs 5mg 58.1489 57.6767 0.4722 NSD13mg vs 9mg 58.1489 58.6811 -0.5322 NSDExample - Caffeine and Endurance79.395.183939.6977)57.52)(1)4(9()57.52)(3(9)75.694(8)1()1()1(),(57.5275.69494:Design Randomized Completely Block to Randomized of Efficiency
View Full Document
Unlocking...