Statistical Analysis of the RCBD

Cochran’s Theorem

4.1 The Randomized Complete Block Design

4.1.1 Statistical Analysis of the RCBD

Suppose we have, in general,atreatments that are to be compared and bblocks. The random- ized complete block design is shown in Figure 4.1. There is one observation per treatment in each block, and the order in which the treatments are run within each block is determined ran- domly. Because the only randomization of treatments is within the blocks, we often say that the blocks represent a restriction on randomization.

The statistical modelfor the RCBD can be written in several ways. The traditional model is an effects model:

(4.1) whereis an overall mean,iis the effect of the ith treatment,jis the effect of the jth block, andijis the usual NID (0,²) random error term. We will initially consider treatments and blocks to be fixed factors. The case of random blocks, which is very important, is considerd in Section 4.1.3. Just as in the single-factor experimental design model in Chapter 3, the effects model for the RCBD is an overspecified model. Consequently, we usually think of the treat- ment and block effects as deviations from the overall mean so that

It is also possible to use a means modelfor the RCBD, say

whereijij. However, we will use the effects model in Equation 4.1 throughout this chapter.

In an experiment involving the RCBD, we are interested in testing the equality of the treatment means. Thus, the hypotheses of interest are

Because the ith treatment mean i(1/b) (ij)i, an equivalent way to write the above hypotheses is in terms of the treatment effects, say

The analysis of variance can be easily extended to the RCBD. Let y_i.be the total of all observations taken under treatment i,y_.jbe the total of all observations in block j,y_..be the

H₁⬊i Z 0 at least one i H0⬊12Áa0

兺^bj1

H1⬊at least one i Z j

H₀⬊12Áa

y_ijijij

ⁱ1, 2, . . . , a j1, 2, . . . , b

^a

i1i0 and

_j1^{b j0}

y_ijijij

ⁱ1, 2, . . . ,a j1, 2, . . . ,b

4.1 The Randomized Complete Block Design

141

■ F I G U R E 4 . 1 The randomized complete block design

grand total of all observations, and Nabbe the total number of observations. Expressed mathematically,

(4.2)

(4.3) and

(4.4) Similarly, is the average of the observations taken under treatment i, is the average of the observations in block j, and is the grand average of all observations. That is,

(4.5) We may express the total corrected sum of squares as

(4.6) By expanding the right-hand side of Equation 4.6, we obtain

Simple but tedious algebra proves that the three cross products are zero. Therefore,

(4.7) represents a partition of the total sum of squares. This is the fundamental ANOVA equation for the RCBD. Expressing the sums of squares in Equation 4.7 symbolically, we have

(4.8) Because there are Nobservations,SS_ThasN1 degrees of freedom. There are atreat- ments and bblocks, so SS_TreatmentsandSS_Blockshave a1 and b1 degrees of freedom, respec- tively. The error sum of squares is just a sum of squares between cells minus the sum of squares for treatments and blocks. There are abcells with ab1 degrees of freedom between them, soSS_Ehasab1(a1)(b1)(a1)(b1) degrees of freedom. Furthermore, the degrees of freedom on the right-hand side of Equation 4.8 add to the total on the left; there- fore, making the usual normality assumptions on the errors, one may use Theorem 3-1 to show

SS_TSS_TreatmentsSS_BlocksSS_{E i1}

^a _j1

^{b(yijy.jyi.y..)2}

^a

i1_j1

^b(yijy..)2b_i1

^a(yi.y..)2a

_j1^b(y.jy..)2

2

_i1^a

_j1^{b (yi.y..)(yijyi.y.jy..)}

2_i1

^a _j1

^{b (y.jy..)(yijyi.y.jy..)}

_i1

^a _j1

^{b(yijyi.y.jy..)22}

_i1^a

_j1^{b(yi.y..)(y.jy..)}

^a

i1 _j1

^{b (yijy..)2b}_i1

^a(yi.y..)2a

_j1^{b (y.jy..)2}

(y_.jy_..)(y_ijy_i.y_.jy_..]²

^a

i1_j1

^{b (yijy..)2}_i1

^a _j1

^{b [(yi.y..)}

y_i.y_i./b y_.jy_.j/a y_..y_../N y_..

y_.j y_i.

y_..

_i1^a

_j1^{b yij}_i1

^{a yi.}_j1

^{b y.j}

y_.ji1

^{a yij j}1, 2, . . . ,b y_i.j1

^{b yij i}1, 2, . . . ,a

thatSS_Treatments/²,SS_Blocks/², and SS_E/²are independently distributed chi-square random vari- ables. Each sum of squares divided by its degrees of freedom is a mean square. The expected value of the mean squares, if treatments and blocks are fixed, can be shown to be

Therefore, to test the equality of treatment means, we would use the test statistic

which is distributed as Fa1,(a1)(b1)if the null hypothesis is true. The critical region is the upper tail of the Fdistribution, and we would reject H₀if F₀ F ,a1,(a1)(b1). A P-value approach can also be used.

We may also be interested in comparing block means because, if these means do not differ greatly, blocking may not be necessary in future experiments. From the expected mean squares, it seems that the hypothesis H₀:j 0 may be tested by comparing the statistic F₀MS_Blocks/MS_E toF,b1,(a1)(b1). However, recall that randomization has been applied only to treatments withinblocks; that is, the blocks represent a restriction on randomiza- tion. What effect does this have on the statistic F₀MS_Blocks/MS_E? Some differences in treat- ment of this question exist. For example, Box, Hunter, and Hunter (2005) point out that the usual analysis of variance Ftest can be justified on the basis of randomization only,²without direct use of the normality assumption. They further observe that the test to compare block means cannot appeal to such a justification because of the randomization restriction; but if the errors are NID(0,²), the statistic F₀MS_Blocks/MS_Ecan be used to compare block means.

On the other hand, Anderson and McLean (1974) argue that the randomization restriction pre- vents this statistic from being a meaningful test for comparing block means and that this Fratio really is a test for the equality of the block means plus the randomization restriction [which they call a restriction error; see Anderson and McLean (1974) for further details].

In practice, then, what do we do? Because the normality assumption is often question- able, to view F₀MS_Blocks/MS_Eas an exact Ftest on the equality of block means is not a good general practice. For that reason, we exclude this Ftest from the analysis of variance table.

However, as an approximate procedure to investigate the effect of the blocking variable, examining the ratio of MS_BlockstoMS_Eis certainly reasonable. If this ratio is large, it implies that the blocking factor has a large effect and that the noise reduction obtained by blocking was probably helpful in improving the precision of the comparison of treatment means.

The procedure is usually summarized in an ANOVA table, such as the one shown in Table 4.2. The computing would usually be done with a statistical software package.

However, computing formulas for the sums of squares may be obtained for the elements in Equation 4.7 by working directly with the identity

y_ijy_..(y_i.y_..)(y_.jy_..)(y_ijy_i.y_.jy_..) F₀ MS_Treatments

MS_E E(MS_E)² E(MS_Blocks)²

a_j1

^{b 2j}

b1 E(MS_Treatments)²

b_i1

^{a 2i}

a1

4.1 The Randomized Complete Block Design

143

2Actually, the normal-theory Fdistribution is an approximation to the randomization distribution generated by calculating F0from every possible assignment of the responses to the treatments.

These quantities can be computed in the columns of a spreadsheet (Excel). Then each column can be squared and summed to produce the sum of squares. Alternatively, computing formu- las can be expressed in terms of treatment and block totals. These formulas are

(4.9)

(4.10)

(4.11) and the error sum of squares is obtained by subtraction as

(4.12) SS_ESS_TSS_TreatmentsSS_Blocks

SS_Blocks 1

a_j1

^{b y2.jy}N^2..

SS_Treatments 1

b_i1

^{a y2i.y}N^2..

SS_Ti1

^a _j1

^{b y2ijy}N^2..

■ T A B L E 4 . 2

Analysis of Variance for a Randomized Complete Block Design

Source Degrees

of Variation Sum of Squares of Freedom Mean Square F₀

Treatments SS_Treatments a1

Blocks SS_Blocks b1

Error SS_E (a1)(b1)

Total SS_T N1

SS_E (a1)(b1)

SS_Blocks b1

MS_Treatments MS_E SS_Treatments

a1

E X A M P L E 4 . 1

A medical device manufacturer produces vascular grafts (artificial veins). These grafts are produced by extruding billets of polytetrafluoroethylene (PTFE) resin combined with a lubricant into tubes. Frequently, some of the tubes in a production run contain small, hard protrusions on the external surface. These defects are known as “flicks.” The defect is cause for rejection of the unit.

The product developer responsible for the vascular grafts suspects that the extrusion pressure affects the occur- rence of flicks and therefore intends to conduct an experi- ment to investigate this hypothesis. However, the resin is manufactured by an external supplier and is delivered to the medical device manufacturer in batches. The engineer also suspects that there may be significant batch-to-batch varia-

tion, because while the material should be consistent with respect to parameters such as molecular weight, mean par- ticle size, retention, and peak height ratio, it probably isn’t due to manufacturing variation at the resin supplier and nat- ural variation in the material. Therefore, the product devel- oper decides to investigate the effect of four different levels of extrusion pressure on flicks using a randomized com- plete block design considering batches of resin as blocks.

The RCBD is shown in Table 4.3. Note that there are four levels of extrusion pressure (treatments) and six batches of resin (blocks). Remember that the order in which the extru- sion pressures are tested within each block is random. The response variable is yield, or the percentage of tubes in the production run that did not contain any flicks.

4.1 The Randomized Complete Block Design

145

To perform the analysis of variance, we need the follow- ing sums of squares:

(514.6)²](2155.1)²

24 178.17

1

6 [(556.9)²(550.1)²(533.5)² SS_Treatments1

b

_i1^{4 y2i.y}N^2..

193,999.31(2155.1)²

24 480.31

SS_T

_i1⁴ _j1

^{6 y2ijy}N^2..

The ANOVA is shown in Table 4.4. Using 0.05, the critical value of FisF_{0.05, 3,15}3.29. Because 8.11 3.29, we conclude that extrusion pressure affects the mean yield.

TheP-value for the test is also quite small. Also, the resin batches (blocks) seem to differ significantly, because the mean square for blocks is large relative to error.

480.31178.17192.25109.89 SS_ESS_TSS_TreatmentsSS_Blocks

(2155.1)²

24 192.25

1

4 [(350.8)²(359.0)²Á(377.8)²] SS_Blocks1

a_j1

^{6 y2.jy}N^2..

■ T A B L E 4 . 3

Randomized Complete Block Design for the Vascular Graft Experiment Batch of Resin (Block)

Extrusion Treatment

Pressure (PSI) 1 2 3 4 5 6 Total

8500 90.3 89.2 98.2 93.9 87.4 97.9 556.9

8700 92.5 89.5 90.6 94.7 87.0 95.8 550.1

8900 85.5 90.8 89.6 86.2 88.0 93.4 533.5

9100 82.5 89.5 85.6 87.4 78.9 90.7 514.6

Block Totals 350.8 359.0 364.0 362.2 341.3 377.8 y_..2155.1

It is interesting to observe the results we would have obtained from this experiment had we not been aware of randomized block designs. Suppose that this experiment had been run as a completely randomized design, and (by chance) the same design resulted as in Table 4.3.

The incorrect analysis of these data as a completely randomized single-factor design is shown in Table 4.5.

Because the P-value is less than 0.05, we would still reject the null hypothesis and con- clude that extrusion pressure significantly affects the mean yield. However, note that the mean

■ T A B L E 4 . 4

Analysis of Variance for the Vascular Graft Experiment

Source of Sum of Degrees of Mean

Variation Squares Freedom Square F₀ P-Value

Treatments (extrusion pressure) 178.17 3 59.39 8.11 0.0019

Blocks (batches) 192.25 5 38.45

Error 109.89 15 7.33

Total 480.31 23

square for error has more than doubled, increasing from 7.33 in the RCBD to 15.11. All of the variability due to blocks is now in the error term. This makes it easy to see why we some- times call the RCBD a noise-reducing design technique; it effectively increases the signal-to- noise ratio in the data, or it improves the precision with which treatment means are compared.

This example also illustrates an important point. If an experimenter fails to block when he or she should have, the effect may be to inflate the experimental error, and it would be possible to inflate the error so much that important differences among the treatment means could not be identified.

Sample Computer Output. Condensed computer output for the vascular graft exper- iment in Example 4.1, obtained from Design-Expert and JMP is shown in Figure 4.2. The Design-Expert output is in Figure 4.2aand the JMP output is in Figure 4.2b. Both outputs are very similar, and match the manual computation given earlier. Note that JMP computes an F-statistic for blocks (the batches). The sample means for each treatment are shown in the out- put. At 8500 psi, the mean yield is , at 8700 psi the mean yield is , at 8900 psi the mean yield is , and at 9100 psi the mean yield is . Remember that these sample mean yields estimate the treatment means 1,2,3, and 4. The model residuals are shown at the bottom of the Design-Expert output. The residuals are calculated from

and, as we will later show, the fitted values are , so

(4.13) In the next section, we will show how the residuals are used in model adequacy checking.

Multiple Comparisons. If the treatments in an RCBD are fixed, and the analysis indicates a significant difference in treatment means, the experimenter is usually interested in multiple comparisons to discover whichtreatment means differ. Any of the multiple compar- ison procedures discussed in Section 3.5 may be used for this purpose. In the formulas of Section 3.5, simply replace the number of replicates in the single-factor completely random- ized design (n) by the number of blocks (b). Also, remember to use the number of error degrees of freedom for the randomized block [(a1)(b1)] instead of those for the com- pletely randomized design [a(n1)].

The Design-Expert output in Figure 4.2 illustrates the Fisher LSD procedure. Notice that we would conclude that 12, because the P-value is very large. Furthermore, 1differs from all other means. Now the P-value for H0:23is 0.097, so there is some evidence to conclude that 2 3, and 2 4because the P-value is 0.0018. Overall, we would conclude that lower extrusion pressures (8500 psi and 8700 psi) lead to fewer defects.

Z Z

eijyijyi.y.jy..

yˆ_ijyi.y.jy..

eijyijyˆ_ij

y4.85.77 y3.88.92

y2.91.68 y1.92.82

■ T A B L E 4 . 5

Incorrect Analysis of the Vascular Graft Experiment as a Completely Randomized Design

Source of Sum of Degrees of Mean

Variation Squares Freedom Square F₀ P-Value

Extrusion pressure 178.17 3 59.39 3.95 0.0235

Error 302.14 20 15.11

Total 480.31 23

4.1 The Randomized Complete Block Design

147

.

(a)

■ F I G U R E 4 . 2 Computer output for Example 4.1. (a) Design-Expert; (b) JMP

We can also use the graphical procedure of Section 3.5.1 to compare mean yield at the four extrusion pressures. Figure 4.3 plots the four means from Example 4.1 relative to a scaledtdistribution with a scale factor 1.10. This plot indicates that the two lowest pressures result in the same mean yield, but that the mean yields for 8700 psi and

7.33/6 MS_E/b

■ F I G U R E 4 . 2 (Continued)

80 85 90

2 3

4 1

95 Yield

■ F I G U R E 4 . 3 Mean yields for the four extrusion pressures relative to a scaledtdistribution with a scale factor

MS_E/b7.33/61.10 Oneway Analysis of Yield By Pressure

Block Batch

Oneway Anova Summary of Fit

0.771218 Rsquare

0.649201 Adj Rsquare

2.706612 Root Mean Square Error

89.79583 Mean of Response

24 Observations (or Sum Wgts)

Analysis of Variance

Mean Square Sum of Squares

DF

Source FRatio Prob > F

0.0019 8.1071

59.3904 178.17125

3 Pressure

0.0055 5.2487

38.4504 192.25208

5 Batch

7.3257 109.88625

15 Error

480.30958 23

C.Total

Means for Oneway Anova

Upper 95%

Lower 95%

Error Std.

Mean Number

Level

95.172 90.461

1.1050 92.8167

6 8500

94.039 89.328

1.1050 91.6833

6 8700

91.272 86.561

1.1050 88.9167

6 8900

88.122 83.411

1.1050 85.7667

6 9100

Error uses a pooled estimate of error variance Std.

Block Means

Number Mean

Batch

4 87.7000

1

4 89.7500

2

4 91.0000

3

4 90.5500

4

4 85.3250

5

4 94.4500

6

(b)

8900 psi (2and 3) are also similar. The highest pressure (9100 psi) results in a mean yield that is much lower than all other means. This figure is a useful aid in interpreting the results of the experiment and the Fisher LSD calculations in the Design-Expert output in Figure 4.2.