ANOVA Randomized Block Design

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1 Biostatistics 301 ANOVA Randomized Block Design 1 ORIGIN 1 Data Structure: Let index i,j indicate the ith column (treatment class) and jth row (block). For each i,j combination, there are n replicates. Model: X i,j = + j + i + i j + i,j Restriction: i j ij 0 i,j is the error term specific to each object i,j PB Machines example: i j i ^ allows estimation of parameters j are a random sample ~ N(0, 2 ) ij are a random sample ~ N(0, 2 ), spherical j and ij are independent M READPRN ("c:/data/biostatistics/machines.txt" ) M Y M < measurement variable k 3 < number of treatments (machines) b 6 < number of blocks (workers) n 3 < number of replicates N 54 < total number of observations Assumptions: Number & Means: ANOVA Randomized Block Design ANOVA designs involving "randomized blocks with replicates", also termed ANOVA "repeated measures designs with replicates", are an extension of the approach in Biostatistics Worksheet 300 by adding replicate sets of mached observations. Whereas before, individuals (or objects) comprised "blocks" by themselves, now a block contains replicate sets that are considered more similar to each other (i.e., correlated) than to observations taken from other blocks. The objective of the ANOVA analysis is to control for these correlations when assessing treatment effect. This approach is an example of "mixed-model" (Type III) ANOVA. Shown here is the traditional approach to this problem using a standard two-way ANOVA table and modified Mean Squares F-ratios as specified by Zar's Table 12.3 p. 262 and Biostatistics Worksheet 310. More modern approaches to mixed model ANOVA, utilizing liklihood estimation and ANOVA full versus reduced model tests, have largely supplanted the methods presented here. Data is from JC Pinhiero & DM Bates 2004 Mixed-Effects Models in S and S-Plus p. 21. k treatments exactly matched within individuals (objects). Typically, the order in which specific treatments are presented to individuals is randomized with n replicates for each combination of treatment and block. < where: ANOVA Randomized Block Design Treatment Classes: Blocks #1 #2 #3 #k 1 n n n n 2 n n n n 3 n n n n b n n n n means: X1bar X2bar X3bar Xkbar is the grand mean of all objects. j is a random effect for each block j i is a constant effect for each treatment i. i j is the interaction between treatment and block

2 Biostatistics 301 ANOVA Randomized Block Design 2 GM 1 N j 1 k b mean A 1 A bar meana 2 Y mean A 3 GM A 1j M 3 j A 2j M 3 jkb A 3j M 3 j2k b A bar A 1 < grand mean - sample estimate of A < means for treatments A < treatments B B B B B B < blocks B bar mean B 1 mean B 2 mean B 3 mean B 4 mean B 5 mean B 6 B bar < means for blocks

3 Biostatistics 301 ANOVA Randomized Block Design < cells mean 11 mean 12 mean 13 mean 14 mean 15 mean 16 mean 21 mean 22 mean 23 mean 24 mean 25 mean 26 mean 31 mean 32 mean 33 mean 34 mean 35 mean < cell means Sums of Squares: SSA for Treatment: i 1 k SSA n b A bari GM SSA i SSB for factor B: j 1 b 2 2 SSB n k B barj GM SSB SS for Interaction : j 2 SS n A ji bari B barj GM SS i j SS Total: kk 1 N 2 SSTO Y GM SSTO kk SSE for Error: kk SSE SSTO SS SSA SSB SSE

4 Biostatistics 301 ANOVA Randomized Block Design 4 Randomized Block NOVA Table: Sum of Squares: Degrees of Freedom: Mean Squares: SSA SSA df A k 1 df A 2 MSA MSA df A SSB df B ( b 1) df B 5 SS df ( k 1) ( b 1) df 10 SSE df E k b ( n 1) df E 36 SSB MSB MSB df B SS MS MS df SSE MSE MSE df E SSTO df T n k b 1 df T 53 Omnibus F Test for Treatment Effect: Hypotheses: H 0 : i = 0 for all i H 1 : At least one i 0 Test Statistic: MSA F F MS Sampling Distribution of the test Statistic F: If Assumptions hold and H 0 is true then F ~F (dfa,df) Critical Value of the Test: 0.05 < All treatment class deviations from the grand mean are 0 < Two sided test < Probability of Type I error must be explicitly set C qf 1 df A df C Decision Rule: IF F > C, THEN REJECT H 0 OTHERWISE ACCEPT H 0 Probability Value: P A 1 pf F df A df P A Omnibus F Test for Block Effect: Hypotheses: H 0 : j = 0 for all j H 1 : At least one j 0 Test Statistic: < Ratio of "treatment" versus "interaction" Mean Squares < All treatment class deviations from the grand mean are 0 < Two sided test MSB F F MSE < Ratio of "block" versus "error" Mean Squares

5 Biostatistics 301 ANOVA Randomized Block Design 5 Sampling Distribution of the test Statistic F: If Assumptions hold and H 0 is true then F ~F (dfb,dfe) Critical Value of the Test: 0.05 < Probability of Type I error must be explicitly set C qf 1 df B df E C Decision Rule: IF F > C, THEN REJECT H 0 OTHERWISE ACCEPT H 0 Probability Value: P A 1 pf F df B df E P A 0 F Test for Treatment X Block Interaction: Hypotheses: H 0 : ij = 0 for all ij H 1 : At least one ij 0 Test Statistic: MS F MSE Sampling Distribution of the test Statistic F: If Assumptions hold and H 0 is true then F ~F (df,dfe) Critical Value of the Test: 0.05 C qf 1 df df E Decision Rule: F < Probability of Type I error must be explicitly set C IF F > C, THEN REJECT H 0 OTHERWISE ACCEPT H 0 < All treatment class deviations from the grand mean are 0 < Two sided test < Ratio of "interaction" versus "error" Mean Squares Probability Value: P A 1 pf F df df E P A 0

6 Biostatistics 301 ANOVA Randomized Block Design 6 Prototype in R: #ANOVA FOR RANDOMIZED BLOCKS WITH REPLICATES M=read.table("C:/DATA/Biostascs/machinesR.txt",header=TRUE) M aach(m) fmachine=factor(machine) fworker=factor(worker) LM=lm(score~fMachine*fWorker) anova(lm) alpha = 0.05 MSA=anova(LM)[1,3] MSB=anova(LM)[2,3] MS=anova(LM)[3,3] MSE=anova(LM)[4,3] dfa=anova(lm)[1,1] dfb=anova(lm)[2,1] df=anova(lm)[3,1] dfe=anova(lm)[4,1] > anova(lm) Analysis of Variance Table Response: score Df Sum Sq Mean Sq F value Pr(>F) fmachine < 2.2e-16 *** fworker < 2.2e-16 *** fmachine:fworker < 2.2e-16 *** Residuals Signif. codes: 0 *** ** 0.01 * #OMNIBUS F TEST FOR TREATMENT EFFECT: F=MSA/MS F C=qf(1 alpha,dfa,df) C P=1 pf(f,dfa,df) P #OMNIBUS F TEST FOR BLOCK EFFECT: F=MSB/MSE F C=qf(1 alpha,dfb,dfe) C P=1 pf(f,dfb,dfe) P #F TEST FOR TREATMENT BY BLOCK INTERACTION: F=MS/MSE F C=qf(1 alpha,df,dfe) C P=1 pf(f,df,dfe) P > F [1] > C [1] > P [1] > F [1] > C [1] > P [1] 0 > F [1] > C [1] > P [1] 0