Randomized Blocks Designs Concepts of Experimental DesignReview: Completely Randomized Design• Homogeneous EU Concepts of Experimental DesignReview: Completely Randomized Design• Homogeneous EU• Completely Randomize TreatmentsBCCCCABAABABSetting for Randomized Blocks Designs Randomized block designs are used whenever the experimental units occur in groups. These groups of experimental units are called blocks. Units within the blocks are more nearly homogeneous than units in different blocks. Treatments are randomly assigned to units within the groups of units, with each treatment appearing the same number of times in each block, usually once. C B A B A C B A C A C B Concepts of Experimental Design Randomized Blocks DesignModel for Randomized Blocks Designs Measurements are made on each experimental unit. Denote yij = observation on EU in block j assigned to treatment i. The statistical model for the RBD consists of the equation together with a description of all terms in the equation. overall mean averaged over all treatments and all possible blocks effect of treatment i effect of block j random error associated with EU assigned to treatment i in block j Assumptions about model terms • The random errors eij are assumed independent and normally distributed with mean 0 and variance s2 • The block effects bij are normally distributed with mean 0 and variance sb2 yC2 yB2 yA2 yB1 y yA1 yC1 C yB3 yA3 yC3 yA4 yC4 yB4 ijijijybµαε=+++µ=iα=jb=ijε=Treatment Means in terms of Model Parameters The population mean for treatment i is The estimate of the population mean for treatment i is Benefits of Using Randomized Blocks Design The benefits of using a RCB design can be seen in terms of the model: Model Treatment mean Difference between treatment means The difference between treatment means does not contain block effects. In this sense, the RCB design controls variation due to differences between blocks. The benefit also is apparent in the variance of means and differences between means. Variance of a treatment mean: Variance of difference between treatment means: The variance of the difference between treatment means does not contain the block variance component. iiµµα=+.12(...)/iiiibyyyyb=+++ijijijybµαε=+++...iiiybµαε=+++....ABABAByyααεε−=−++22.()()/ibVybσσ=+2..()2/ABVyybσ−=Analysis of Variance for Randomized Blocks Design Sources of MS= Variation df SS SS/df F Blocks b-1 SS(B) MS(B) Treats t-1 SS(T) MS(T) MS(T)/MS(E) Error (b-1)(t-1) SS(E) MS(E) Total bt-1 SS(Tot) Computations of Sums of Squares in Randomized Blocks Design Motivation for the F-test The null hypothesis is H0: µ1 = µ2 = … = µt . It can be shown that: MS(T) estimates s2+ bSi(µi - µ)2 and MS(E) estimates s2 F=MS(T)/MS(E) tends to be large when H0 is false, around 1 when H0 is true. Therefore, large values of F infer that H0 is false. 222......11()()//bbjjjjSSBtyyytybt===−=−∑∑222......11()()//ttiiitSSTbyyybybt===−=−∑∑222....1111()()/btbtijijjijiSSTotyyyybt=====−=−∑∑∑∑22222........111111()()///()()()()btbtbtijijijjijijijiSSEyyyyyytybybtSSESSTotSSBSST=======−−−=−−+=−−∑∑∑∑∑∑Example: Analysis of Variance for Randomized Block Design An engineer conducted an experiment to compare three metals, nickel, iron and copper, as bonding agents for an alloy material. Components of the alloy were bonded using the metals as bonding agents, and the pressures required to break the bonds were measured. Seven ingots of the alloy material were available for the experiment, and the engineer recognized the possibility of variability between the ingots. Thus, the engineer constructed bonds of components from each ingot using each metal. Diagram of Bonding Agent ExperimentIngot 1 Ingot 2 Ingot 3 … Ingot 7…INCCNICNINICData for the breaking pressures: Bonding Agent (Metal) ______________________________ Ingot Nickel Iron Copper Mean Total __________________________________________ 1 67.0 71.9 72.2 70.4 211.1 2 67.5 68.8 66.4 67.6 202.7 3 76.0 82.6 74.5 77.7 233.1 4 72.7 78.1 67.3 72.7 218.1 5 73.1 74.2 73.2 73.5 220.5 6 65.8 70.8 68.7 68.4 205.3 7 75.6 84.9 69.0 76.5 229.5 Mean 71.1 75.9 70.2 72.6 Total 497.7 531.3 495.5 1524.5 This experiment is an example of a randomized blocks design. The “treatments” are the metals, and the “blocks” are the ingots of the alloy material. An experimental unit is a component from an ingot. The overall objective is to compare the treatment means, µN, µI, and µC. An analysis of variance provides statistical computations to perform a test of significance of differences between the means. The null hypothesis is H0: µN = µI = µC, and the alternative hypothesis is that at least two means differ.Following is an analysis of variance table for these data. Analysis of Variance Source of Variation DF SS MS F P ____________________________________________________________ Ingot 7-1=6 268.29 268.29/6=44.71 Metal 3-1=2 131.90 131.90/2=65.95 6.36 .0131 Error 6*2=12 124.46 124.46/12=10.37 Total 21-1=20 524.65 Conclusion: The value F=6.36 has p-value=0 .0131. This is highly significant evidence of differences between the treatment means. Sums of squares for the sources of variation are computed as follow: SS(Ingot) = 3(70.4-72.6)2 + 3(67.6-72.6)2 +...+ 3(76.5-72.6)2 = 211.12/3 + 202.72/3 +...+ 233.12/3 - 1524.52/21 = 268.29 SS(Metal) = 7(71.1-72.6)2 + 7(75.9-72.6)2 + 7(70.2-72.6)2 = 497.72/7 + 531.32/7 + 495.52/7 - 1524.52/21 = 131.90 SS(Total) = (67.0-72.6)2 + (71.9-72.6)2 +...+ (69.0-72.6)2 = 67.02 + 71.72 +...+ 69.02 - 1524.52/21 = 524.65 SS(Error) = SS(Total) - SS(Ingot) - SS(Metal) = 124.46Follow-up analyses. As a result of the ANOVA F-test, the engineer believes there are differences between the metal means, and next wants to investigate differences between individual means. This can be done with LSD multiple comparisons. LSD = t.025,12(2*MS(Error)/n)1/2 = 2.18(2*10.37/7)1/2 = 2.18*1.72 = 3.75 Nickel vs Iron: |71.1 - 75.9|
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