Example: piglet diet experiment with three litters SST is the “sum of squares treatments”SSB is the “sum of squares blocks”SSE is the “sum of squares error”.F*=MST/MSETopic 23 – ANOVA (III) 23-1 Topic 23 - Randomized Complete Block Designs (RCBD) Defn: A Randomized Complete Block Design is a variant of the completely randomized design (CRD) that we recently learned. In this design, blocks of experimental units are chosen where the units within are block are more similar to each other (homogeneous) than to units in other blocks. In a complete block design, there are at least t experimental units in each block where t is the number of treatments in the factor(s) of interest. Examples of blocks: 1) a litter of animals could be considered a block since they all have similar genetic structure, similar prenatal/parental care, etc. 2) a field or pasture that can be divided into quadrants since soil properties, environmental conditions, etc are similar within a field 3) a greenhouse with multiple benches since environmental conditions are usually more similar within a greenhouse than between greenhouses 4) a year in which the experiment is performed since environmental conditions are similar within a year Example of a RCBD: A nutritionist is interested in comparing the effect of three diets on weight gain in piglets. In order to perform the experiment, the researcher chooses 10 litters, each with at least three healthy and similarly sized piglets that have just been weaned. In each litter, three piglets are selected and one treatment is randomly assigned to each piglet. Diets are labeled A, B or C.Topic 23 – ANOVA (III) 23-2 Litter Piglet 1 2 3 1 A C B 2 B C A … 10 C B A In a design without blocking, the researcher would pick 30 piglets from different litters and randomly assign treatments to them. This is known as unrestricted randomization. Blocking designs have restricted randomization since the treatments are randomly assigned WITHIN each block. Another Example of a RCBD: An animal behaviorist is interested in habitat use by gopher tortoises. There are eleven conservation management areas within the distribution of the species that have the four habitats of interest for comparative study. The response variable is the density of active burrows in each habitat. In this example, the blocks are the conservation management areas and each has all four levels of the treatment (habitat). Here we do not assign treatments to individual experimental units. The units are the habitat areas within each block. An RCBD has two factors: the factor of interest that includes the treatments to be studied and the “Blocking Factor” that identifies the blocks used in the experiment. There are several forms of Blocking Designs: 1) the RCBD that we will studyTopic 23 – ANOVA (III) 23-3 2) incomplete block designs in which not every block has t experimental units 3) block designs in which the blocks have more than t experimental units that are used in the experiment 4) Latin square designs which have very specific forms of randomization of treatments within blocks (example is usually relates to time ordering of treatments) Assumptions of the RCBD: 1) Sampling: a. The blocks are independently chosen b. The treatments are randomly assigned to the experimental units within a block. 2) Homogeneous Variance: the treatments all have the same variability, i.e. they all have the same variance 3) Approximate Normality: each population is normally distributed Hypotheses As we will see, the blocking factor is included in the study only as a way of explaining some of the variation in responses (Y) of the experimental units. As such, we are not interested in testing hypotheses about the blocking factor. Instead, just like in a one-way ANOVA, we restrict our attention to the other factor (“research” factor). So, hypothesis testing proceeds similar to the techniques we learned for the one-way ANOVA. The big differences are 1) we won’t test for blocking effect and 2) the variability assigned to the error term isTopic 23 – ANOVA (III) 23-4 broken down into 2 parts, variability among blocks and the left-over unexplained variability (still associated with the error term). Hence the MSE from a 1-way ANOVA can be decomposed into two parts: calculation of a new error variance (MSE) and a calculation of the effect of the blocking factor (MSB). Notation t the number of treatments of interest in the “research” factor b the number of blocks containing exactly t experimental units N = t × b, the total sample size yij observed value for the experimental unit in the jth block assigned to the ith treatment, j = 1,2,…,b and i = 1,2,…,t •iy bybjij∑==1, the sample mean of the ith treatment jy• tytiij∑==1, the sample mean of the jth block ••y tbytibjij∑∑===11, the overall sample mean of the combined treatmentsTopic 23 – ANOVA (III) 23-5 Example: piglet diet experiment with three litters Diet Block Litter A B C Mean 1 yA1 = 54.3 yB1 = 53.1 yC1 = 59.7 7.551=•y2 yA2 = 53.6 yB2 = 52.4 yC2 = 59.7 2.552=•y3 yA3 = 55.2 yB3 = 57.1 yC3 = 67.2 2.623=•y Treatment Mean 4.54=•Ay 2.55=•By 8.59=•Cy Grand Mean 9.56=••y Model: ijjiijYεβαμ+++= where • μ is the overall (grand) mean, • αi is the effect due to the ith treatment, • βj is the effect due to the jth block, and, • εij is the error term where the error terms, are independent observations from an approximately Normal distribution with mean = 0 and constant variance 2εσ Total variability of all of the Yij, is TSS ∑∑••−=ijijyy2)( which can be broken up into three parts: TSS = SST + SSB + SSETopic 23 – ANOVA (III) 23-6 SST∑∑=−=•••iiiibyyb22ˆ)(α is the “sum of squares treatments” SSB∑∑=−=•••jjjjtyyt22ˆ)(β is the “sum of squares blocks” SSE∑∑∑∑=+−−=••••ijijijjiijyyyy22ˆ)(ε is the “sum of squares error”. Like before, we are interested in the Mean Squares: MST 1−=tSST, the Mean Square Treatments MSB 1−=bSSB, the Mean Square Blocks MSE )1)(1( −−=btSSE, the Mean Square Error Here ∑−+=iitbMSTE)1()(22ασε and 2)(εσ=MSEE. ANOVA Table for a Randomized Complete Block Design Source Sum of Squares Degrees of Freedom Mean Square F-stat Treatment SST t – 1 MST F*=MST/MSEBlock SSB b – 1 MSB
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