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STAT22200 Chapter 13 Complete Block Designs STAT22200 Chapter 13 Complete Block Designs Yibi Huang 13.1-13.2 Randomized Complete Block Design (RCBD) 13.3 Latin Square Designs 13.3.1 Crossover Designs 13.3.4 Replicated Latin Square Designs 13.4 Graeco-Latin Squares Chapter 13 - 1 Block Designs I A block is a set of experimental units that are homogeneous in some sense. Hopefully, units in the same block will have similar responses (if applied with the same treatment.) I Block designs: randomize the units within each block to the treatments. Chapter 13 - 2 Randomized Complete Block Design (RCBD) We want to test g treatments. There are b blocks of units available, each block contains k = rg units. I Within each block, the k = rg units are randomized to the g treatments, r units each. I \Complete" means each of the g treatments appears the same number of times (r) in every block. I Mostly, block size k = # of treatments g, i.e., r = 1. I Matched-Pair design is a special case of RCBD in which the block size k = 2: Block 1 Block 2 ··· Block b Treatment 1 y11 y12 ··· y1b Treatment 2 y21 y22 ··· y2b . ··· . Treatment g yg1 yg2 ··· ygb Normally, data are shown arranged by block and treatment. Cannot tell from the data what was/was not randomized. Chapter 13 - 3 Advantage of Blocking I Blocking is the second basic principle of experimental design after randomization. \Block what you can, randomize everything else." I If units are highly variable, grouping them into more similar blocks can lead to a large increase in efficiency (more power to detect difference in treatment effects). I The choice of blocks is crucial Commonly used blocking I Block on batch (e.g., milk produced in a day) I Block spatially I Block on time I Block when you can identify a source of variation (e.g., age, gender, and history blocks) Caution: Blocking must be done at the time of randomization; you can't group units into blocks after the experiment has been run Chapter 13 - 4 Example 13.1 (Mealybugs on Cycads) I Treatment: water (control), fungal spores, and horticultural oil I 5 infested cycads, 3 branches are randomly chosen on each cycad, and 2 patches (3 cm 3 cm) are marked on each branch × I 3 branches on each cycad are randomly assigned to the 3 13.2treatments The Randomized Complete Block Design 317 I Response: difference of the # of mealybugs in the patches Tablebefore 13.1: andChanges 3 days in after mealybug treatments counts on are cycads applied after treatment. Treatments are water, Beauveria bassiana spores, and horticultural oil. Plant 1 2 3 4 5 Water -9 18 10 9 -6 -6 5 9 0 13 Spores -4 29 4 -2 11 7 10 -1 6 -1 Oil 4 29 14 14 7 11 36 16 18 15 Chapter 13 - 5 branches on each cycad are randomly assigned to the three treatments. After three days, the patches are counted again, and the response is the change in the number of mealybugs (before after). Data for this experiment are given in Table 13.1 (data from Scott Smith).− How can we decode the experimental design from the description just given? Follow the randomization! Looking at the randomization, we see that the treatments were applied to the branches (or pairs of patches). Thus the branches (or pairs) must be experimental units. Furthermore, the randomiza- tion was done so that each treatment was applied once on each cycad. There was no possibility of two branches from the same plant receiving the same treatment. This is a restriction on the randomization, with cycads acting as blocks. The patches are measurement units. When we analyze these data, we can take the average or sum of the two patcheson each branch as the response for the branch. To recap, there were g = 3 treatments applied to N = 15 units arranged in r = 5 blocks of size 3 according to an RCB design; there were two measurement units per experimental unit. Why did the experimenter block? Experience and intuition lead the ex- perimenter to believe that branches on the same cycad will tend to be more alike than branches on different cycads—genetically, environmentally, and perhaps in other ways. Thus blocking by plant may be advantageous. It is important to realize that tables like Table 13.1 hide the randomization that has occurred. The table makes it appear as though the first unit in every block received the water treatment, the second unit the spores, and so on. This is not true. The table ignores the randomization for the convenience of a readable display. The water treatment may have been applied to any of the three units in the block, chosen at random. You cannot determine the design used in an experiment just by looking at a table of results, you have to know the randomization. There may be many Example 13.1 (Mealybugs on Cycads) What are the experimental units, branches or patches? Why? > cycad = read.table("cycad.txt",h=T) > cycad trt plant change 1 1 1 -9 2 1 1 -6 3 2 1 -4 4 2 1 7 5 3 1 4 6 3 1 11 ... (omitted) ... 25 1 5 -6 26 1 5 13 27 2 5 11 28 2 5 -1 29 3 5 7 30 3 5 15 Chapter 13 - 6 Example 13.1 (Mealybugs on Cycads) As the patches are measurement units, we take the average of the two patches on each branch as the response > avechange = (cycad$change[2*(1:15)-1]+cycad$change[2*(1:15)])/2 > cycad = data.frame(trt = cycad$trt[2*(1:15)], plant = cycad$plant[2*(1:15)], avechange) > cycad trt plant avechange 1 1 1 -7.5 2 2 1 1.5 3 3 1 7.5 4 1 2 11.5 5 2 2 19.5 6 3 2 32.5 7 1 3 9.5 (... omitted ...) 14 2 5 5.0 15 3 5 11.0 Chapter 13 - 7 Example 13.1 (Mealybugs on Cycads) > cycad$trt = factor(cycad$trt,labels=c("Water","Spore", "Oil")) > interaction.plot(cycad$trt, cycad$plant, cycad$avechange, type="b", xlab="Treatments", ylab="Avg. changes in mealybug counts\n before and after treatment", legend=FALSE,ylim=c(-10,32)) > legend("top","Labels = Plant ID",bty="n",cex=.8) 2 Labels = Plant ID 30 20 2 34 2 5 10 3 1 54 5 134 0 before and after treatment before 1 −10 Avg. changes in mealybug counts changes in mealybug Avg. Water Spore Oil Treatments Chapter 13 - 8 Models for RCBDs yij = µ + αi + βj + "ij in which I yij = response of the unit receiving treatment i in block j I µ = the grand mean I αi = the treatment effects I βj = the block effects 2 I "ij : measurement errors, i.i.d. N(0; σ ) ∼ Question: In an RCBD model, which parameters are we more interested in? αi 's or βj 's? Just like the models for factorial design, the model above is over-parameterized. some constraints must be imposed. The commonly used constraints are X3 X5 αi = 0 and βi = 0: i=1 j=1 Chapter 13 - 9 Parameter Estimates for RCBD Models The model for a RCBD 2 yij = µ + αi + βj + "ij "ij 's are i.i.d. N(0; σ ): has the same format as the additive model for a balanced two-way factorial design. So the two models have the same least square estimate for parameters: µb = y •• αi = y µ = y y b i• − b i• − •• βbj = y µ = y y •j − b •j − •• for i = 1;:::; g and j = 1;:::; b: Questions: Why not including treatment-block interactions? Chapter 13 - 10 Fitted Values for RCBD The fitted values for RCBD is then ybij = µb + αbj + βbj = y + (y y ) + (y y ) •• i• − •• •j − •• = y + y y i• •j − •• = row mean + column mean grand mean − just like in an additive model for a balanced two-way design. Block 1 Block 2 Block b row mean ··· Treatment 1 y11 y12 y1b y ··· 1• Treatment 2 y21 y22 y2b y 2• . ··· . ··· Treatment g yg1 yg2 ygb y ··· g• column mean y y y grand mean y •1 •2 ··· •b •• Chapter 13 - 11 Sum of Squares and Degrees of Freedom The sum of squares and degrees of freedom for RCBD are just like those for the additive models: SST = SStrt + SSblock + SSE where Xg Xb 2 SST = (yij y ) i=1 j=1 − •• Xg Xb 2 Xg 2 SStrt = (y y ) = b (y y ) i=1 j=1 i• − •• i=1 i• − •• Xg Xb 2 Xb 2 SSblock = (y y ) = g (y y ) i=1 j=1 •j − •• j=1 •j − •• Xg Xb 2 SSE = (yij y y + y ) : i=1 j=1 − i• − •j •• Total Treatment Block Error dfT = bg 1 dftrt = g 1 dfblock = b 1 dfE = (g 1)(b 1) − − − − − Chapter 13 - 12 Expected Values for the Mean Squares Just like CRD, the mean squares for RCBD is the sum of squares divided by the corresponding d.f. SStrt SS SSE MS = ; MS = block ; MSE = : trt g 1 block b 1 (g 1)(b 1) − − − − Under the effects model for RCBD, 2 yij = µ + αi + βj + "ij "ij 's are i.i.d. N(0; σ ): one can show that g b X (MS ) = σ2 + α2 E trt g 1 i − i=1 b g X (MS ) = σ2 + β2 E block b 1 j − j=1 2 E(MSE) = σ Thus the MSE is again an unbiased estimator of σ2: Chapter 13 - 13 ANOVA F -Test for Treatment Effect To test whether there is an treatment effect H0 : α1 = α2 = = αg v.s.
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