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7 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN STRUCTURE WITH FOUR TREATMENTS USING JMP®

20

Analysis of Messy Data, Volume III: Analysis of Covariance

TABLE 10.21

Estimate Statements Used to Provide Predicted Values for Treatment 1

for Each of the 16 Blocks

estimate ‘11’ intercept 1 trt 1 0 0 0 x*trt 100|block 1;

estimate ‘12’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 1;

estimate ‘13’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 1;

estimate ‘14’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 1;

estimate ‘15’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 1;

estimate ‘16’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 1;

estimate ‘17’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 1;

estimate ‘18’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 1;

estimate ‘19’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 0 1;

estimate ‘110’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 0 0 1;

estimate ‘111’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 0 0 0 1;

estimate ‘112’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 1;

estimate ‘113’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 1;

estimate ‘114’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 0 1;

estimate ‘115’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1;

estimate ‘116’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1;

Treatment 1

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Estimate

108.1873

106.6044

111.8443

111.3143

109.8052

112.3372

108.0669

108.1085

107.9095

103.9985

113.2101

109.8421

114.0777

113.4230

107.5628

112.3774

Means

109.9168

Treatment 2

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Estimate

109.5241

107.9412

113.1811

112.6511

111.1420

113.6740

109.4037

109.4453

109.2463

105.3353

114.5469

111.1789

115.4145

114.7598

108.8996

113.7142

111.2536

Treatment 3

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Estimate

108.5245

106.9416

112.1815

111.6515

110.1424

112.6743

108.4041

108.4456

108.2466

104.3357

113.5472

110.1793

114.4148

113.7602

107.9000

112.7146

110.2540

Treatment 4

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Estimate

112.8872

111.3043

116.5442

116.0142

114.5051

117.0370

112.7668

112.8083

112.6093

108.6984

117.9099

114.5420

118.7775

118.1229

112.2627

117.0773

114.6167

the model specification menu to enable the fitted model to match the models fit by

PROC MIXED. Click on the Run Model button to carry out the analysis. The

parameter estimates are in Figure 10.3 where the ones of interest correspond to

Intercept, TRT[1], TRT[2], TRT[3], X, TRT[1]*X, TRT[2]*X, and TRT[3]*X. The

© 2002 by CRC Press LLC

More Than Two Treatments in a Blocked Design Structure

21

FIGURE 10.1 JMP® data screen for the incomplete block example.

FIGURE 10.2 Fit model table where block is declared as a random effect and no center is

selected from model specification.

estimates of the slopes and intercepts can be constructed as described in Chapter 3.

The estimates of the variance components and tests for the fixed effects are in

Figure 10.4. The estimates of the variance components are the same as obtained by

PROC MIXED, displayed in Table 10.19, as is the test for equal intercepts, TRT.

Since the model included X and TRT*X, a test of the slopes equal to zero hypothesis

is not obtained, but a test of the equal slopes hypothesis is provided through the

TRT*X term. The significance level is 0.0897. The custom test screen needs to be

used to provide the test of the slopes equal to zero hypothesis. Such a screen is in

Figure 10.5 where there are four columns used to construct the estimates of the

slopes. The value line provides the estimates of the slopes, which are identical to

© 2002 by CRC Press LLC

22

Analysis of Messy Data, Volume III: Analysis of Covariance

FIGURE 10.3 Parameter estimates from JMP®.

FIGURE 10.4 Estimates of the variance components and tests for the effects.

those from PROC MIXED; thus JMP® is providing the combined estimates of the

model parameters (Table 10.19). The F Ratio provides the test of the slopes equal

to zero hypothesis and is the same as obtained by PROC MIXED in Table 10.19.

Finally, the custom test screen is used to provide estimates of the models for each

of the treatments evaluated at X = 100, as displayed in Figure 10.6. The results of

the custom test to provide the least squares means are in Figure 10.7. These are the

same least squares means as obtained by PROC MIXED and have the same interpretation (Table 10.20). The least squares means are the mean of the predicted values

at X = 100 from all of the blocks, whether or not a treatment occurred in all blocks.

10.8 SUMMARY

When the design structure involves blocking, information about slopes and intercepts

can be extracted from within block comparisons and between block comparisons.

The type of information available depends on the type of blocking. If the designs

are connected, the same within block information is available from all designs, but

© 2002 by CRC Press LLC

More Than Two Treatments in a Blocked Design Structure

23

FIGURE 10.5 Custom tests screen used to provide estimates of the slopes and a test of the

slopes equal to zero hypothesis.

FIGURE 10.6 Coefficients used to provide least squares means evaluated at X = 100.

the between block information depends on the type of blocking used. RCB designs

have different information than incomplete block designs and these differences are

evident when the within block and between block estimates are combined. The Type I

SS BLOCKS from the within block analysis contains variation due to covariates;

thus biased estimates of variance components occur from the within block comparisons. Most of the difficulties occurring because of blocking are avoided when a

mixed model approach is used to provide the analysis. The details of the mixed

model approach are discussed in Chapter 13.

The final topic discussed was that of using an incomplete block design structure.

The example in Section 10.6 indicates the interpretation of the least squares means

© 2002 by CRC Press LLC

24

Analysis of Messy Data, Volume III: Analysis of Covariance

FIGURE 10.7 Least squares means evaluated at X = 100 for each of the treatments.

is the means of the predicted values of the treatments from all blocks, whether the

treatments were observed in all blocks or not. This is another reason that the choice

of the blocking factor is very important. The choice of the blocking factor must not

be able to interact with the treatment or else the least squares means are likely not

meaningful.

REFERENCES

Kenward, M. G. and Roger, J. H. (1997). Small sample inference for fixed effects from

restricted maximum likelihood, Biometrics 54:983.

Milliken, G. A. and Johnson, D. E. (1992). Analysis of Messy Data, Volume I: Design

Experiments, London, Chapman & Hall.

EXERCISES

EXERCISE 10.1: For the data set in Section 10.6, obtain the within block and the

between block estimates of the model’s parameters. The process is to show that there

is between block information about the intercepts when an incomplete block design

structure is used while it does not exist when a complete block design structure is

used for the data set in Section 10.5.

EXERCISE 10.2: For the data in Section 10.6, use a common slope model and

obtain the within block and between block estimates of the parameters.

EXERCISE 10.3: The data in the following table consist of eight incomplete blocks

where Treatments 1 and 2 occur in blocks 1 through 4 and Treatments 3 and 4 occur

in blocks 5 through 8. Use a common slopes model to describe the data for each

treatment. Obtain the within block estimates and the between block estimates of the

parameters. Since the design is not connected, care must be taken in constructing

the parameters for each of the models. Use a mixed models code such as PROC

MIXED to obtain predicted values for each of the treatments at each of the blocks

evaluated at X = 40. Show that the means of these predicted values are the same as

the least squares means evaluated at X = 40.

© 2002 by CRC Press LLC

More Than Two Treatments in a Blocked Design Structure

25

Data for Exercise 10.3

block

1

2

3

4

5

6

7

8

© 2002 by CRC Press LLC

treat

1

1

1

1

3

3

3

3

x

40.7

41.9

41.2

38.3

40.7

39.3

39.5

39.0

y

62.1

61.1

62.6

57.7

69.3

70.2

66.0

64.0

treat

2

2

2

2

4

4

4

4

x

37.5

40.2

39.0

40.0

41.0

39.0

39.1

40.8

y

63.1

62.9

60.6

63.3

70.6

71.7

63.5

68.5

11

Covariate Measured

on the Block in RCB

and Incomplete Block

Design Structures

11.1 INTRODUCTION

Measuring the value of the covariate on the block, i.e., setting up blocks where all

experimental units within a block have the same value of the covariate, is often used

as a method of constructing blocks of experimental units. Animals are grouped by

age, weight, or stage of life. Students are grouped by class, age, or by IQ. The

grouping of the experimental units by the value of some covariate forms more

homogeneous groups on which to compare the treatments than by not blocking.

However, one must be much more concerned that there is no interaction between

the levels of the treatments and the levels of the factor used to construct the blocks.

An assumption of the RCB (randomized complete block), or as a matter of fact any

blocked design structure, is that there is no interaction between the factors in the

treatment structure and the factors in the design structure (Milliken and Johnson,

1992). It is not unusual to see researchers construct blocks by using a factor such

as initial age or weight or current thickness. However, some thought should be taken

into account about the possibility of interaction with the levels of the treatments.

The usual approach to the analysis of such data sets is to remove the block to block

variation by the analysis of variance and not consider doing an analysis of covariance,

i.e., ignore the fact that a covariate was measured. That strategy is appropriate if the

slopes of the treatments’ regression lines are equal, but if the slopes of the lines are

unequal, a model with a covariate is required in order to extract the necessary

information from the data. A block total or mean model must be used to make

decisions about the slopes of the model before the analysis can continue.

There are some changes in the form of the analysis when the covariate is

measured on the block rather than being measured on the experimental units within

each block. The data in Table 11.1 are from a blocked experiment with two treatments

where the covariate is measured on the block, i.e., the covariate has the same value

for each of the two experimental units within the block. The within block analysis

does not provide information about the magnitudes of the slopes. The sum of squares

due to the slopes being zero after the block effects have been removed does not test

the hypothesis that the slopes are zero, in fact the sum of squares is equal to zero

providing no test. If the slopes are assumed to be equal, then the block effects

© 2002 by CRC Press LLC

2

Analysis of Messy Data, Volume III: Analysis of Covariance

TABLE 11.1

Data for Example in Section 11.7

BLK

1

2

3

4

5

6

7

8

X

23.2

26.9

29.4

22.7

30.6

36.9

17.6

28.5

y1

60.4

59.9

64.4

63.5

80.6

75.9

53.7

66.3

y2

76.0

76.3

77.8

75.6

94.6

96.1

62.3

81.6

y_sum

136.4

136.2

142.2

139.1

175.2

172.0

116.0

147.9

y_dif

–15.6

–16.4

–13.4

–12.1

–14.0

–20.2

–8.6

–15.3

Note: Involves two treatments in a one-way treatment

structure in a RCB design structure where y1 is the

response for Treatment 1 and y2 is the response for Treatment 2; x is the covariate; BLK is the block; y_sum and

y_dif are the sum and difference, respectively, of the

observations within each block.

removes all information about the slopes. However, the information about the slopes

can be extracted by constructing and analyzing the proper models. The proper models

are the within block model and the between block model. The above described

example with two treatments is analyzed in detail in Sections 11.2 to 11.7. Two

additional examples involving more than two treatments, one in a RCB design

structure and the other in a balanced incomplete block design structure, are included

to demonstrate additional complications in the analyses.

11.2 THE WITHIN BLOCK MODEL

The basic model to describe the data in Table 11.1 is

y ij = α i + βi x j + b j + ε ij , i = 1, 2, j = 1, 2, …, 6

(11.1)

where bj ~ iid N(0, σ 2b), εij ~ iid N(0, σ 2ε ) and the value of the covariate is the same

for both experimental units in each block denoted by xj (only one subscript on X).

The within block model is constructed by subtracting the values of the two

observations within each block of Model 11.1, say the observation for Treatment 1

minus the observation for Treatment 2, yielding the model:

y1 j − y 2 j = α1 − α 2 + (β1 − β2 ) x j + ε1 j − ε 2 j

(

)

= α d + βd x j + ε dj , where ε dj ~ N 0, 2σ ε2 .

(11.2)

This is a simple linear regression model where y1j – y2j is the dependent variable

and xj is the independent variable with intercept α1 – α2 and slope β1 – β2. Fitting

© 2002 by CRC Press LLC

Covariate Measured on the Block in RCB

3

the model to the data provides estimates of α1 – α2, β1 – β2, and σ 2ε. If β1 = β2, the

appropriate analysis for comparing the treatments is a paired t-test or a RCB analysis

of variance with two treatments. The within block analysis provides statistics to test

H0: α1 – α2 and H0: β1 – β2, but it does not enable the individual parameters (α1,

α2, β1, and β2) to be estimated.

11.3 THE BETWEEN BLOCK MODEL

The between block or block sum or block total model is:

y1 j + y 2 j = α1 + α 2 + (β1 + β2 ) x j + 2 b j + ε1 j + ε 2 j

( (

))

= α s + βs x j + esj* , where esj* ~ N 0, 2 σ ε2 + 2σ b2 .

(11.3)

Model 11.3 is a simple linear regression model where where y1j + y2j is the dependent

variable and xj is the independent variable with intercept α1 + α2 and slope β1 + β2.

Fitting this model to the data provides estimates of α1 + α2, β1 + β2 and σ 2ε + 2σ 2b = σ2e*.

11.4 COMBINING WITHIN BLOCK AND BETWEEN

BLOCK INFORMATION

By combining the estimates from the within block model and the between block

model, estimates of all of the parameters can be obtained. The estimates are

αˆ 1 = (α s + α d ) 2 , α 2 = (αˆ s − αˆ d ) 2 ,

(

)

(

)

βˆ 1 = βˆ s + βˆ d 2 , βˆ 2 = βˆ s − βˆ d 2

(

(11.4)

)

σˆ b2 = σˆ e2* − σˆ ε2 2 .

The estimates of α1, α2, β1, and β2 involve both between block information and

within block information. Assuming the data are normally distributed, the estimators

from within blocks are independently distributed of the estimators from between

blocks.

As in Chapters 9 and 10, the beta-hat model can be used to combine the within

block and the between block information to obtain the estimators of the model’s

parameters and the corresponding covariance matrix. Let 1 = {(α1 – α2), (β1 – β2)] ′,

2 = (α1 + α2, β1 + β2) ′, Var(ˆ 1) = Σw , Var(ˆ 2) = ⌺b and = (α1, α2, β1, β2)′. The

beta-hat models relating ˆ 1 and ˆ 2 to are

ˆ 1 = H1 + e1, e1 ~ N(0, Σ w ) and

ˆ 2 = H2 + e2 , e2 ~ N(0, Σ b )

© 2002 by CRC Press LLC

(11.5)

4

Analysis of Messy Data, Volume III: Analysis of Covariance

where

1

H1 =

0

−1

0

0

1

0

1

and H2 =

−1

0

1

0

0

1

0

.

1

The combined estimator of θ is

[

θˆ c = H1′ Σˆ w−1 H1 + H′2 Σˆ −b1 H2

] [H′ Σˆ

−1

1

−1

w

θˆ 1 + H′2 Σˆ −b1 θˆ 2

]

with estimated approximate covariance matrix

[

Σˆ θ = H1′ Σˆ w−1 H1 + H′2 Σˆ −b1 H2

]

−1

.

(11.6)

To demonstrate the above described process, the data in Table 11.1 are used

where the within block analysis is in Table 11.5 and the between block analysis is

in Table 11.6. From Table 11.5, the within block information is

−1.585320 ˆ

2

ˆ 1 =

⌺2 = (2.05389)

−

0

476912

.

3.153408

−0.112267

−0.112267

0.00416

and from Table 11.6 the between block information is

66.442387 ˆ

2

θˆ b =

⌺b = (10.09140)

.

2

935407

3.153408

−0.112267

−0.112267

.

0.00416

The combined estimator of θ and the approximate covariance matrix are

83.608516

32.4285

34.0138

ˆ = 76.957248

⌺

θˆ c =

θ

−2.976619

1.2292

−2.739822

1.7062

76.957248

83.608516

−2.739822

2.976619

−2.976619

−2.739822

0.110347

0.101569

−2.739822

−2.976619

0.101569

0.113047

A matrix manipulation software such as PROC IML described in Chapter 9 can be

used to carry out the above computations. PROC MIXED was used to provide the

computations of the combined estimator, which are displayed in the fourth part of

Table 11.7. To test equality of slopes, test H0: β1 – β2 = 0 vs. Ha: (not Ha). Let

a = [0 0 1 –1] ′, then

βˆ 1 − βˆ 2 = a ′ˆ c = −0.4769

© 2002 by CRC Press LLC

Covariate Measured on the Block in RCB

5

with variance Var( βˆ 1 – βˆ 2 ) = a′ Σˆ θ a = (.132502) 2 . The test statistic is t c =

–0.4769/.132502 = –3.5993, and when compared to a t-distribution with 6 d.f., the

significance level is 0.0114. The contrast statement was used in Table 11.7 to provide

the test for equal slopes, labeled as b1 = b2.

11.5 COMMON SLOPE MODEL

When the slopes are equal, Model 11.1 becomes

y1 j = α i + βX j + b j + ε ij .

(11.7)

y1 j − y 2 j = α1 − α 2 + ε1 j − ε 2 j ,

(11.8)

The within block model is

a model with intercept α1 – α2 and does not contain any information about the

covariate. The between block model is

y1 j + y 2 j = (α1 + α 2 ) + 2βX j + 2 b j + ε ij ,

(11.9)

a simple linear regression model with intercept α1 + α2 and slope 2β. All information

about the covariate is contained in the between block model.

In order to compute adjusted means or LSMEANS, the estimate of = [α1, α2, β] ′

needs to be computed. Let 1 = (α1 – α2), 2 = (α1 + α2, β), Var (ˆ 1) = Σw Var (ˆ 2) =

Σb, then the beta-hat models relating 1 and 2 to θ are

1 = H1 + e1

e1 ~ N(0, Σ 2 )

2 = H2 + e2 e2 ~ N(0, Σ b )

[ ].

1 1 0

0 0 1

where H1 = [1 –1 0] and H2 =

The weight least squares estimate of is

[

ˆ c = H1′ Σˆ 2 H1 + H′2 Σˆ b H2

] [H′ Σˆ

−1

1

w

ˆ 1 + H′2 Σˆ b ˆ 2

with estimated approximate covariance matrix

[

Σˆ θ = H1′ Σˆ w H1 + H′2 Σˆ b H2

© 2002 by CRC Press LLC

]

−1

.

]

## Analysis of messy data volume III analysis of covariance

## 4 EFFECT OF DIET ON CHOLESTEROL LEVEL: AN EXCEPTION TO THE BASIC ANALYSIS OF COVARIANCE STRATEGY

## 4 EXAMPLE: DRIVING A GOLF BALL WITH DIFFERENT SHAFTS

## 5 EXAMPLE: EFFECT OF HERBICIDES ON THE YIELD OF SOYBEANS — THREE COVARIATES

## 6 EXAMPLE: MODELS THAT ARE QUADRATIC FUNCTIONS OF THE COVARIATE

## 7 EXAMPLE: COMPARING RESPONSE SURFACE MODELS

## 4 EXAMPLE: AVERAGE DAILY GAINS AND BIRTH WEIGHT Û COMMON SLOPE

## 5 EXAMPLE: ENERGY FROM WOOD OF DIFFERENT TYPES OF TREES - SOME UNEQUAL SLOPES

## 7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH MISSING CELLS

## 3 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH EQUAL SLOPES MODEL

## 6 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH ONE COVARIATE

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7 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN STRUCTURE WITH FOUR TREATMENTS USING JMP®