© 2010 Pearson Prentice Hall. All rights reserved The Complete Randomized Block Design

13-2 In the completely randomized design, the researcher manipulates a single factor and fixes it at two or more levels and then randomly assigns experimental units to a treatment. This design is not always sufficient because the researcher may be aware of additional factors that cannot be fixed at a single level throughout the experiment.

13-3 The randomized block design is an experimental design that captures more information and therefore reduces experimental error.

13-4 “In Other Words” A block is a method for controlling experimental error. Blocks should form a homogenous group. For example, if age is thought to explain some of the variability in the response variable, we can remove age from the experimental error by forming blocks of experimental units with the same age.

13-5 CAUTION! When we block, we are not interested in determining whether the block is significant. We only want to remove experimental error to reduce the mean square error.

The response variable for each of the k populations is normally distributed. 2.The response variable for each of the k populations has the same variance; that is, each treatment group has population variance  2. Requirements for Analyzing Data from a Randomized Complete Block Design

13-7 A rice farmer is interested in the effect of four fertilizers on fruiting period. He randomly selects four rows from his field that have been planted with the same seed and divides each row into four segments. The fertilizers are then randomly assigned to the four segments. Assume that the environmental conditions are the same for each of the four rows. The data given in the next slide represent the fruiting period, in days, for each row/fertilizer combination. Is there sufficient evidence to conclude that the fruiting period for the four fertilizers differs at the  =0.05 level of significance? Parallel Example 2: Analyzing the Randomized Complete Block Design

13-8 Fertilizer 1Fertilizer 2Fertilizer 3Fertilizer 4 Row Row Row Row

13-9 Solution We wish to test H 0 :  1 =  2 =  3 =  4 versus H 1 : at least one of the means is different We first verify the requirements for the test: 1.Normal probability plots for the data from each of the four fertilizers indicates that the normality requirement is satisified.

13-10 Solution 2. s 1 = s 2 =1.775 s 3 =1.449 s 4 =1.509 Since the largest standard deviation is not more than twice the smallest standard deviation, the assumption of equal variances is not violated.

13-11 Solution ANOVA output from Minitab: Two-way ANOVA: Fruiting period versus Fertilizer, Row Source DF SS MS F P Fertilizer Row Error Total

13-12 Solution Since the P-value is < 0.001, we reject the null hypothesis and conclude that there is a difference in fruiting period for the four fertilizers after accounting for the row of planting. Note, we are not interested in testing whether the fruiting periods among the blocks or planting rows are equal.

13-13 Once the null hypothesis of equal population means is rejected, we can proceed to determine which means differ significantly using Tukey’s Intervals. The steps are identical to those for comparing means in the one-way ANOVA. The critical value is q ,,k using a familywise error rate of  with =(r-1)(c-1) = the error degrees of freedom (r is the number of blocks and c is the number of treatments) and k is the number of means being tested.

13-14 Parallel Example 3: Multiple Comparisons Using Tukey’s Intervals Use Tukey’s Intervals to determine which pairwise means differ for the data presented in Example 2 with a familywise error rate of  = 0.05, using MINITAB.

13-15 Tukey Simultaneous Tests Response Variable Fruiting period All Pairwise Comparisons among Levels of Fertilizer Fertilizer = 1 subtracted from: Fertilizer Lower Center Upper ( *------) ( *------) (------* ) Fertilizer = 2 subtracted from: Fertilizer Lower Center Upper (------* ) ( *------) Fertilizer = 3 subtracted from: Fertilizer Lower Center Upper (------* )

Results Fertilizers 1 and 2 are not significantly different. Fertilizers 2 and 3 are not significantly different. All other pairwise comparisons are significant. It seems that fertilizer 4 results in the greatest fruiting period of the rice.

The “Grand” Conclusion After accounting for the row the seeds were planted in, the farmer will maximize the fruiting period if the seeds are planted with fertilizer number 4.