Randomized Complete Block Designs and Latin Squares Randomized

Randomized Complete Block Designs and Latin Squares

Randomized Complete Block Designs Design and Statistical Analyses The randomized complete block design is an extension of the paired t-test to situations where the factor of interest has more than two levels. Figure 1 A randomized complete block design.

Randomized Complete Block Designs Design and Statistical Analyses For example, consider the situation where three different methods were used to predict the shear strength of steel plate girders. Say we use four girders as the experimental units.

General procedure for a randomized complete block design: 2

We may describe the observations in Table 2 by the linear statistical model: = the j th observation taken under treatment i. = the overall mean = the i th treatment effect = the j th block effect = A random error component The appropriate linear statistical model:

We assume • treatments and blocks are initially fixed effects • blocks do not interact • We are interested in testing: or for at least one pair We may also be interested in comparing block means by testing: for at least one pair

The mean squares are:

The expected values of these mean squares are: If there are no differences in treatment means ( then MSTreatments will estimates. If there are no differences in block means ( MSBlocks will estimates also. =0), = 0 ) , then

Definition The appropriate test statistic is for Treatments for Blocks

3 For treatments, H 0 will be rejected if F 0 > F , a-1, (a-1)(b-1) For blocks, H 0 will be rejected if F 0 > F , b-1, (a-1)(b-1)

Example 1 4 4

Randomized Complete Block Designs Example 1 5

Minitab Output for Example 1 6 1

Multiple Comparisons If ANOVA indicates a significant difference in treatment means, then the multiple comparisons may be applied to discover which treatment means differ. The Least Significant Difference (LSD) Method Fisher’s Least Significant Difference for this analysis is It is concluded that the population means if > LSD and differ

Duncan’s Multiple Range Test The same procedure of the Duncan’s multiple range test used in the single factor ANOVA can be applied. To find a set of a-1 least significant ranges , the standard error of a treatment mean is for = 2, 3, …, a. can be found from the Duncan’s table.

Residual Analysis and Model Checking The residuals for the randomized complete block design are the difference between the observed and estimated values. where Chemical Type 1 1 2 2 Fabric Sample 2 3 4 5 -0. 180. 02 0. 18 - 0. 44 0. 10 0. 00 0. 28 - 0. 08 0. 10 0. 02 - 0. 12 - 0. 30 0. 24 - 0. 08

Residual Analysis and Model Checking Figure 2 Normal probability plot of residuals from the randomized complete block design.

Randomized Complete Block Designs Figure 3 Residuals by treatment.

Randomized Complete Block Designs Figure 4 Residuals by block.

Randomized Complete Block Designs Figure 5 Residuals versus ŷij.

Choice of sample size Operating Characteristic Curves (OC Curves) can be used to guide the experimenter in selecting the sample size or the number of blocks. Method 1 The parameter can be computed from We need to choose the actual values of treatment means that leads to reject the null hypothesis with high probability 1 - and also estimate.

Method 2 This approach is to select a sample size or the number of blocks such that if the difference between any two treatment means exceeds a specific value, the hypothesis should be rejected. If the difference between any two treatment means is as large as , then the minimum can be found from

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 operators • Latin square design: rows and columns are orthogonal to treatments.

• The 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 Latin letters that corresponds to the treatments, and each letter occurs once and only once in each. row and column. 7

Examples of Latin Squares

• The statistical (effects) model is yijk is the observation in the ith row and kth column for the jth treatment. m is the overall mean. i is the ith row effect. j is the jth treatment effect. k is the kth column effect ijk is the random error. Only two of three subscripts are needed to denote a particular observation.

• Sum of squares: SST = SSRows + SSColumns + SSTreatments + SSE • The degrees of freedom: (p 2 – 1) = (p – 1) + (p – 2)(p – 1) • The appropriate statistic for testing for no differences in treatment means is For treatments, H 0 will be rejected if F 0 > The residuals can be determined from

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