Presentation on theme: "Chapter 4 Randomized Blocks, Latin Squares, and Related Designs"— Presentation transcript:
1Chapter 4 Randomized Blocks, Latin Squares, and Related Designs
24.1 The Randomized Complete Block Design Nuisance factor: a design factor that probably has an effect on the response, but we are not interested in that factor.If the nuisance variable is known and controllable, we use blockingIf the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance (see Chapter 14) to remove the effect of the nuisance factor from the analysis
3If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experimentSometimes several sources of variability are combined in a block, so the block becomes an aggregate variable
4We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness testerAssignment of the tips to an experimental unit; that is, a test couponStructure of a completely randomized experimentThe test coupons are a source of nuisance variabilityAlternatively, the experimenter may want to test the tips across coupons of various hardness levelsThe need for blocking
5To conduct this experiment as a RCBD, assign all 4 tips to each coupon Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tipsVariability between blocks can be large, variability within a block should be relatively smallIn general, a block is a specific level of the nuisance factorA complete replicate of the basic experiment is conducted in each blockA block represents a restriction on randomizationAll runs within a block are randomized
6Suppose that we use b = 4 blocks: Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)
7Statistical Analysis of the RCBD Suppose that there are a treatments (factor levels) and b blocksA statistical model (effects model) for the RCBD is is an overall mean, i is the effect of the ith treatment, and j is the effect of the jth blockij ~ NID(0,2)
8Means model for the RCBD The relevant (fixed effects) hypotheses areAn equivalent way for the above hypothesisNotations:
10SST = SSTreatment + SSBlocks + SSE Total N = ab observations, SST has N – 1 degrees of freedom.a treatments and b blocks, SSTreatment and SSBlocks have a – 1 and b – 1 degrees of freedom.SSE has ab – 1 – (a – 1) – (b – 1) = (a – 1)(b – 1) degrees of freedom.From Theorem 3.1, SSTreatment /2, SSBlocks / 2 and SSE / 2 are independently chi-square distributions.
11The expected values of mean squares: For testing the equality of treatment means,
16Can also plot residuals versus the type of tip (residuals by factor) and versus the blocks. Also plot residuals v.s. the fitted values. Figure 4.5 and 4.6 in Page 137These plots provide more information about the constant variance assumption, possible outliers4.1.3 Some Other Aspects of the Randomized Complete Block DesignThe model for RCBD is complete additive.
17Interactions?For example:The treatments and blocks are random.Choice of sample size:Number of blocks , the number of replicates and the number of error degrees of freedom
18Estimating miss values: Approximate analysis: estimate the missing values and then do ANOVA.Assume the missing value is x. Minimize SSE to find xTable 4.8Exact analysis
194.1.4 Estimating Model Parameters and the General Regression Significance Test The linear statistical modelThe normal equations
20Under the constraints,the solution isand the fitted values,The sum of squares for fitting the full model:The error sum of squares
224.2 The Latin Square Design RCBD removes a known and controllable nuisance variable.Example: the effects of five different formulations of a rocket propellant used in aircrew escape systems on the observed burning rate.Remove two nuisance factors: batches of raw material and operatorsLatin square design: rows and columns are orthogonal to treatments.
23The Latin square design is used to eliminate two nuisance sources, and allows blocking in two directions (rows and columns)Usually Latin Square is a p p squares, and each cell contains one of the p letters that corresponds to the treatments, and each letter occurs once and only once in each row and column.See Page 145
24The statistical (effects) model is yijk is the observation in the ith row and kth column for the jth treatment, is the overall mean, i is the ith row effect, j is the jth treatment effect, k is the kth column effect and ijk is the random error.This model is completely additive.Only two of three subscripts are needed to denote a particular observation.
25Sum of squares:SST = SSRows + SSColumns + SSTreatments + SSEThe degrees of freedom:p2 – 1 = p – 1 + p – 1 + p – 1 + (p – 2)(p – 1)The appropriate statistic for testing for no differences in treatment means isANOVA table (Table 4-10) (Page 146)Example 4.3
26The residualsTable 4.13If one observation is missing,Replication of Latin Squares:Three different casesSee Table 4.14, 4.15 and 4.16Crossover design: Pages 150 and 151
274.3 The Graeco-Latin Square Design Two Latin SquaresOne is Greek letter and the other is Latin letter.Two Latin Squares are orthogonalTable 4.18Block in three directionsFour factors (row, column, Latin letter and Greek letter)Each factor has p levels. Total p2 runs
28The statistical model: yijkl is the observation in the ith row and lth column for Latin letter j, and Greek letter k is the overall mean, i is the ith row effect, j is the effect of Latin letter treatment j , k is the effect of Greek letter treatment k, l is the effect of column l.ANOVA table (Table 4.19)Under H0, the testing statistic is Fp-1,(p-3)(p-1) distribution.Example 4.4
294.4 Balance Incomplete Block Designs May not run all the treatment combinations in each block.Randomized incomplete block design (BIBD)Any two treatments appear together an equal number of times.There are a treatments and each block can hold exactly k (k < a) treatments.For example: A chemical process is a function of the type of catalyst employed. See Table 4.22
304.4.1 Statistical Analysis of the BIBD a treatments and b blocks. Each block contains k treatments, and each treatment occurs r times. There are N = ar = bk total observations. The number of times each pairs of treatments appears in the same block isThe statistical model for the BIBD is