Randomized Complete Block Designs

13.8 Randomized Complete Block Designs

A typical layout for the randomized complete block design using 3 measurements in 4 blocks is as follows:

Block 4 t 2 t 1 t 3 t 2

The t’s denote the assignment to blocks of each of the 3 treatments. Of course, the true allocation of treatments to units within blocks is done at random. Once the experiment has been completed, the data can be recorded in the following 3 × 4 array:

534 Chapter 13 One-Factor Experiments: General

Treatment Block:

1 y 11 y 12 y 13 y 14

2 y 21 y 22 y 23 y 24

3 y 31 y 32 y 33 y 34 where y 11 represents the response obtained by using treatment 1 in block l, y 12

represents the response obtained by using treatment 1 in block 2, . . . , and y 34 represents the response obtained by using treatment 3 in block 4.

Let us now generalize and consider the case of k treatments assigned to b blocks. The data may be summarized as shown in the k × b rectangular array of Table

13.7. It will be assumed that the y ij , i = 1, 2, . . . , k and j = 1, 2, . . . , b, are values of independent random variables having normal distributions with mean μ ij and

common variance σ 2 . Table 13.7: k × b Array for the RCB Design

b Total Mean

Let μ i. represent the average (rather than the total) of the b population means for the ith treatment. That is,

Similarly, the average of the population means for the jth block, μ .j , is defined by

and the average of the bk population means, μ, is defined by

To determine if part of the variation in our observations is due to differences among the treatments, we consider the following test:

13.8 Randomized Complete Block Designs 535

Hypothesis of Equal Treatment

H 0 :μ 1. =μ 2. =···μ k. = μ,

Means

H 1 : The μ i. are not all equal.

Model for the RCB Design

Each observation may be written in the form

y ij =μ ij +ǫ ij ,

where ǫ ij measures the deviation of the observed value y ij from the population mean μ ij . The preferred form of this equation is obtained by substituting

μ ij =μ+α i +β j ,

where α i is, as before, the effect of the ith treatment and β j is the effect of the jth block. It is assumed that the treatment and block effects are additive. Hence, we may write

y ij =μ+α i +β j +ǫ ij .

Notice that the model resembles that of the one-way classification, the essential difference being the introduction of the block effect β j . The basic concept is much like that of the one-way classification except that we must account in the analysis for the additional effect due to blocks, since we are now systematically controlling variation in two directions. If we now impose the restrictions that

(μ + α i +β j )=μ+α i , for i = 1, . . . , k,

(μ + α i +β j )=μ+β j , for j = 1, . . . , b.

k i=1

The null hypothesis that the k treatment means μ i· are equal, and therefore equal to μ, is now equivalent to testing the hypothesis

H 0 :α 1 =α 2 =···=α k = 0,

H 1 : At least one of the α i is not equal to zero. Each of the tests on treatments will be based on a comparison of independent

estimates of the common population variance σ 2 . These estimates will be obtained

536 Chapter 13 One-Factor Experiments: General by splitting the total sum of squares of our data into three components by means

of the following identity.

Theorem 13.3: Sum-of-Squares Identity

The proof is left to the reader. The sum-of-squares identity may be presented symbolically by the equation

SST = SSA + SSB + SSE,

where

SST =

(y ij − ¯y .. ) 2 = total sum of squares,

= treatment sum of squares,

= block sum of squares,

j=1 k

SSE =

(y ij − ¯y i. − ¯y .j +¯ y .. ) 2 = error sum of squares.

i=1 j=1

Following the procedure outlined in Theorem 13.2, where we interpreted the sums of squares as functions of the independent random variables Y 11 ,Y 12 ,...,Y kb , we can show that the expected values of the treatment, block, and error sums of squares are given by

E(SSA) = (k − 1)σ 2 +b

α 2 2 i 2 , E(SSB) = (b − 1)σ +k β j ,

i=1

j=1

E(SSE) = (b − 1)(k − 1)σ 2 . As in the case of the one-factor problem, we have the treatment mean square

SSA

k−1

If the treatment effects α 1 =α 2 =···=α k = 0, s 2 1 is an unbiased estimate of σ 2 . However, if the treatment effects are not all zero, we have the following:

13.8 Randomized Complete Block Designs 537

Expected

Treatment Mean

In this case, s 2 1 overestimates σ 2 . A second estimate of σ 2 , based on b − 1 degrees of freedom, is

SSB

b−1

The estimate s 2 2 is an unbiased estimate of σ 2 if the block effects β 1 =β 2 =···= β b = 0. If the block effects are not all zero, then

and s 2 2 will overestimate σ 2 . A third estimate of σ 2 , based on (k − 1)(b − 1) degrees

of freedom and independent of s 2 1 and s 2 2 , is SSE

(k − 1)(b − 1)

which is unbiased regardless of the truth or falsity of either null hypothesis. To test the null hypothesis that the treatment effects are all equal to zero, we compute the ratio f 1 =s 2 1 /s 2 , which is a value of the random variable F 1 having an F-distribution with k − 1 and (k − 1)(b − 1) degrees of freedom when the null hypothesis is true. The null hypothesis is rejected at the α-level of significance when

f 1 >f α [k − 1, (k − 1)(b − 1)].

In practice, we first compute SST , SSA, and SSB and then, using the sum- of-squares identity, obtain SSE by subtraction. The degrees of freedom associated with SSE are also usually obtained by subtraction; that is,

(k − 1)(b − 1) = kb − 1 − (k − 1) − (b − 1). The computations in an analysis-of-variance problem for a randomized complete

block design may be summarized as shown in Table 13.8. Example 13.6: Four different machines, M 1 ,M 2 ,M 3 , and M 4 , are being considered for the assem-

bling of a particular product. It was decided that six different operators would be used in a randomized block experiment to compare the machines. The machines were assigned in a random order to each operator. The operation of the machines requires physical dexterity, and it was anticipated that there would be a difference among the operators in the speed with which they operated the machines. The amounts of time (in seconds) required to assemble the product are shown in Table

13.9. Test the hypothesis H 0 , at the 0.05 level of significance, that the machines perform at the same mean rate of speed.

538 Chapter 13 One-Factor Experiments: General

Table 13.8: Analysis of Variance for the Randomized Complete Block Design

Source of

Sum of

Degrees of

Squares Freedom

Square

SSA

Treatments SSA

(k − 1)(b − 1) s 2 =

(k − 1)(b − 1)

Table 13.9: Time, in Seconds, to Assemble Product

Solution : The hypotheses are

H 0 :α 1 =α 2 =α 3 =α 4 =0

(machine effects are zero),

H 1 : At least one of the α i is not equal to zero. The sum-of-squares formulas shown on page 536 and the degrees of freedom

are used to produce the analysis of variance in Table 13.10. The value f = 3.34 is significant at P = 0.048. If we use α = 0.05 as at least an approximate yardstick, we conclude that the machines do not perform at the same mean rate of speed.

Table 13.10: Analysis of Variance for the Data of Table 13.9 Source of

Sum of

Degrees of

Further Comments Concerning Blocking

In Chapter 10, we presented a procedure for comparing means when the observa- tions were paired. The procedure involved “subtracting out” the effect due to the

13.8 Randomized Complete Block Designs 539 homogeneous pair and thus working with differences. This is a special case of a

randomized complete block design with k = 2 treatments. The n homogeneous units to which the treatments were assigned take on the role of blocks.

If there is heterogeneity in the experimental units, the experimenter should not

be misled into believing that it is always advantageous to reduce the experimental error through the use of small homogeneous blocks. Indeed, there may be instances where it would not be desirable to block. The purpose in reducing the error variance is to increase the sensitivity of the test for detecting differences in the treatment means. This is reflected in the power of the test procedure. (The power of the analysis-of-variance test procedure is discussed more extensively in Section 13.11.) The power to detect certain differences among the treatment means increases with

a decrease in the error variance. However, the power is also affected by the degrees of freedom with which this variance is estimated, and blocking reduces the degrees of freedom that are available from k(b − 1) for the one-way classification to (k − 1)(b − 1). So one could lose power by blocking if there is not a significant reduction in the error variance.

Interaction between Blocks and Treatments

Another important assumption that is implicit in writing the model for a random- ized complete block design is that the treatment and block effects are additive. This is equivalent to stating that

μ ij −μ i ′ j =μ ij ′ −μ i ′ j ′ , for every value of i, i ′ , j, and j ′ . That is, the difference between the population

μ ij −μ ij ′ =μ i ′ j −μ i ′ j ′

or

means for blocks j and j ′ is the same for every treatment and the difference between the population means for treatments i and i ′ is the same for every block. The parallel lines of Figure 13.6(a) illustrate a set of mean responses for which the treatment and block effects are additive, whereas the intersecting lines of Figure 13.6(b) show a situation in which treatment and block effects are said to interact. Referring to Example 13.6, if operator 3 is 0.5 second faster on the average than operator 2 when machine 1 is used, then operator 3 will still be 0.5 second faster on the average than operator 2 when machine 2, 3, or 4 is used. In many experiments, the assumption of additivity does not hold and the analysis described in this section leads to erroneous conclusions. Suppose, for instance, that operator 3 is 0.5 second faster on the average than operator 2 when machine 1 is used but is 0.2 second slower on the average than operator 2 when machine 2 is used. The operators and machines are now interacting.

An inspection of Table 13.9 suggests the possible presence of interaction. This apparent interaction may be real or it may be due to experimental error. The analysis of Example 13.6 was based on the assumption that the apparent interaction was due entirely to experimental error. If the total variability of our data was in part due to an interaction effect, this source of variation remained a part of the

error sum of squares, causing the mean square error to overestimate σ 2 and thereby increasing the probability of committing a type II error. We have, in fact, assumed an incorrect model. If we let (αβ) ij denote the interaction effect of the ith treatment and the jth block, we can write a more appropriate model in the

540 Chapter 13 One-Factor Experiments: General

Block 1

Block 1

Block 2

Block 2 Population Mean

Population Mean

t 1 t 2 t 3 t 1 t 2 t 3 Treatments

Treatments

(a)

(b)

Figure 13.6: Population means for (a) additive results and (b) interacting effects.

form

y ij =μ+α i +β j + (αβ) ij +ǫ ij ,

on which we impose the additional restrictions

(αβ) ij =

(αβ) ij = 0, for i = 1, . . . , k and j = 1, . . . , b.

i=1

j=1

We can now readily verify that

(b − 1)(k − 1)

(b − 1)(k − 1) i=1 j=1

Thus, the mean square error is seen to be a biased estimate of σ 2 when existing interaction has been ignored. It would seem necessary at this point to arrive at

a procedure for the detection of interaction for cases where there is suspicion that it exists. Such a procedure requires the availability of an unbiased and independent estimate of σ 2 . Unfortunately, the randomized block design does not lend itself to such a test unless the experimental setup is altered. This subject is discussed extensively in Chapter 14.