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3 INCOMPLETE BLOCK DESIGN STRUCTURE — WITHIN AND BETWEEN BLOCK INFORMATION

3 INCOMPLETE BLOCK DESIGN STRUCTURE — WITHIN AND BETWEEN BLOCK INFORMATION

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More Than Two Treatments in a Blocked Design Structure



3



block design provides the same within block information as the RCB, i.e., estimates







of α1 – α., α2 – α., …, αt – α., β1, β2, …, βt.

The between block information from the incomplete block design is different

than that of the RCB. The reason is that not all treatments occur in each block; thus

the block total models are different. Let Rs = {indices of treatments in block s}.

Then the block total model for block s is



∑ y = ∑α + ∑β X

is



i



iεRs



iεRs



i



is



+ rs bs +



iεRs



∑ε



is



(10.3)



iεRs



where rs is number of treatments in block s. The variance of a block total is





Var 

y is  = rs σ ε2 + rsσ b2 .





 iεRs 







(



)



For simplicity, assume the rs are all equal to r (equal block sizes); then





Var 

y is  = r σ ε2 + rσ b2 .





 iεRs 







(



)



(10.4)



If there is a sufficient number of blocks (more than 2t), then between block

information provides estimates of α1, α2, …, αt , β1, β2, …, βt . Table 10.1 contains

an incomplete block design with three treatments in eight blocks. Let Tj denote the

total for block j; then the between block or block total model is

 T1  1

 T  0

 2 

 T3  1

  

T4  = 0

 T5  1

  

T6  1

 T  0

 7 

 T8  0



1

1

0

1

1

1

1

1



0

1

1

1

0

0

1

1



X11

0

X13

0

X15

X16

0

0



X 21

X 22

0

X 24

X 25

X 26

X 27

X 28



0 

X 32 



X 33 



X 34 

0 



0 

X 37 



X 38 



 α1 

α 

 2

α 3 

 +e

 β1 

 β2 

 

 β3 



The model is full rank, thus all parameters are estimable and therefore the between

block model provides estimates of α1, α2, α3, β1, β2, and β3. The between block

information needs to be combined with the within block information to obtain better

estimators. There are some incomplete block designs that do not allow all intercepts

to be estimated from the between block model. Those types of designs are not

considered here.



© 2002 by CRC Press LLC



4



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 10.1

An Incomplete Block Arrangement with

Three Treatments in Eight Blocks

Block

Treatments



1

1

2



2

2

3



3

3

1



4

3

2



5

2

1



6

1

2



7

2

3



8

3

2



10.4 COMBINING BETWEEN BLOCK AND WITHIN

BLOCK INFORMATION

When combining between and within block information, the functions of the parameters need to be consistent for both models. For example, the RCB provides within





block estimates of αi – α. and βi and between block estimates of α. and βi. The goal







is to obtain a combined estimate of the vector of parameters θ = (α., α1 – α ., α2 – α .,



…, αt – α., β1, β2, …, βt)′.

An additional complication occurs when the solution to the normal equations

does not yield estimates of the desired functions of the parameters. That is the case

for PROC GLM of the SAS® system where the within block information provides

estimates of

α1 − α t , α 2 − α t , …, α t − α t = 0, β1, β2 , …, β t .

Let α*i = (αi – α t ), then

α i* − α.* = α i − α. .

Thus, before continuing with the combining process, the estimates of estimable



functions of the αi’s need to be transformed into estimates of αi – α..

For three treatments in a RCB design structure, the within block estimates are



(



)





θˆ 1 = α1 − α., α 2 − α., α 3 − α., βˆ 1, βˆ 2 , βˆ 3



(10.5)



and the between block estimates are



(



)





θˆ 2 = αˆ ., βˆ 1, βˆ 2 , βˆ 3 .

The within block estimates can be expressed as the beta-hat model



© 2002 by CRC Press LLC



(10.6)



More Than Two Treatments in a Blocked Design Structure



0



0



0

␪ˆ 1 = H1 ␪ + ε1 where H1 = 

0



0



0





5



1



0



0



0



0



0



1



0



0



0



0



0



1



0



0



0



0



0



1



0



0



0



0



0



1



0



0



0



0



0



0



0



0

,

0



0



1 



(10.7)



ε1 ~ N(0, Σ1 ), Σ1 = σ ε2 G1,

G1 is a matrix of constants from the respective partition of the inverse of X′ X, and

X is the design matrix including, treatment effects, block effects, and covariates.

The between block information from the RCB can be expressed as the beta-hat

model

␪ˆ 2 = H 2␪ + ε 2

1



0

where H = 

0



0



0



0



0



0



0



0



0



0



1



0



0



0



0



0



1



0



0



0



0



0



0



0

,

0



1 



(10.8)



ε2 ~ N(0, Σ2), Σ2 = σ B2 G2, σ B2 = r (σ ε2 + rσ b2 ), and G2 is the inverse of WW′ where

W is the between block design matrix.

The between block estimates and the within block estimates are distributed

independently (under normality and the independence of the b’s and the ε’s); thus

the joint within/between beta-hat model is

␪ˆ 1  H1 

 ε1 

ˆ  =  ␪ +  

␪2  H 2 

ε 2 

where

0 



⌺2  



 0  ⌺1

 ε1 

ε  ~ N  0 ,  0





 2



The best linear unbiased estimator (assuming σ ε2 and σ b2 are known) of θ is



(



␪ˆ = H1′ ⌺1− H1 + H′2 ⌺2− H 2



© 2002 by CRC Press LLC



) (H′ ⌺ ␪ˆ

−1



1





1 1



)



+ H′2 ⌺2−␪ˆ 2 .



6



Analysis of Messy Data, Volume III: Analysis of Covariance



The weighted least squares estimate of θ or combined between/within block estimate

of θ (assuming σ ε2 and σ b2 are unknown) is



(



ˆ − H + H′ ⌺

ˆ−

␪ˆ c = H1′ ⌺

2 2 H2

1

1



) (H′ ⌺ˆ ␪ˆ + H′ ⌺ˆ ␪ˆ )

−1



1





1



2



1





2



2



(10.9)



ˆ ε2 G1 and Σˆ 2 = r(σ

ˆ ε2 + rσ

ˆ b2 )G2, σ

ˆ ε2 is the within block residual mean

where Σˆ 1 = σ

2

2

square and r(σˆ ε + r σˆ b ) is the between block residual mean square. (If the block sizes

are not equal, then the function of σε2 and σb2 cannot be factored out of Σ2 as well as

the block totals will have unequal variances. (See Chapter 13 when you have unequal

block sizes.) The estimated approximate variance of θˆ c is



( ) (



ˆ − H + H′ ⌺

ˆ−

Var ␪ˆ c = H1′ ⌺

2 2 H2

1

1



)







(10.10)



and the variance of an estimable linear combination of θ, say a′ θˆ c, is Var = (a′ θˆ c ) =

a′Var (θˆ c )a. The number of degrees of freedom associated with this estimated variance needs to be approximated. The Satterthwaite or Kenward and Roger approximation or a weighted average of t-values can be used. An approximate (1 – α)100%

LSD value can be computed (using a weighted t-value similar to Chapter 24 of

Milliken and Johnson, 1992) by replacing Σˆ 1 by (tα/2,df1)Σˆ 1 and Σˆ 2 by (tα/2,df2 )Σˆ 2 in

the Var (θˆ c) where df1 is the degrees of freedom of the residual mean square from

the within block analysis and df2 is the degrees of freedom of the residual mean

square from the within block analysis. The resulting value of Var (θˆ c) is the approximate LSD value. The approximate t-value used in this LSD computation is



t ␣* 2 =



( (



)



)



( )

ˆ

a ′ Var(␪ ) a







ˆ − H + H′ t

ˆ

a ′ H1′ t ␣ 2,df1 ⌺

2 ␣ 2,df2 ⌺ 2 H 2 a

1

1



.



(10.11)



c



* to the t

Approximate degrees of freedom can be computed by matching tα/2

α/2,df values

in the t-table. A Satterthwaite approximation and a Kenward-Roger (Kenward and

Roger, 1997) approximation to the degrees of freedom are available as options in

PROC MIXED. The statistic to test the parallelism hypothesis can be computed by

constructing a beta-hat model for the slopes. Let ␤ˆ c = W ␪ˆ c where W = [0, 0, I]; then

the asymptotic sampling distribution of ␤ˆ c is N(␤, ⌺␤ˆ ) where ⌺␤ˆ = W Var(␪ˆc )W′′.

The beta-hat model under the equal slope hypothesis is



␤ˆ c = j ␤ + ε*

where j is a t × 1 vector of ones. The residual mean square from the beta-hat model

is the test statistic, i.e.,

ˆ −1 − ⌺

ˆ −1 j j′ ⌺

ˆ −1

uc = ␤ˆ ′c  ⌺

␤ˆ

␤ˆ

 ␤ˆ



( j′⌺ˆ j) ␤ˆ

−1

␤ˆ



c



(t − 1)



which has an approximate small sample size distribution of F(t – 1, df1 ).

© 2002 by CRC Press LLC



(10.12)



More Than Two Treatments in a Blocked Design Structure



7



Another small sample approximation can be obtained by recomputing uc

ˆ and ⌺

ˆ are replaced with (F

ˆ

ˆ

where ⌺

1

2

α,(t – 1),df1)⌺ 1 and (Fα,(t – 1),df2)⌺ 2, respectively.

Denote this value of uc by u*c . An approximate F value is

F *α,t −1,df = u c u c* .

(



c



(10.13)



)



*

The approximate degrees of freedom dfc are determined by matching F(α,t

– 1,dfc) to an

F-table with t – 1 degrees of freedom in the numerator and significance level α. The

approximate small sampling distribution for uc is F(t – 1,dfc) .

For the three-treatment incomplete block design structure in Table 10.1, the

within block estimates and corresponding beta-hat model are the same as above for

the RCB (Equations 10.5 through 10.7). The between block estimates are



(



)





␪ˆ 2 = αˆ 1, αˆ 2 , αˆ 3 , βˆ 1, βˆ 2 , βˆ 3 .

Next transform the intercepts to ␪ˆ 3 where

␪ˆ 3 = T ␪ˆ 2 where



(



)





␪ˆ 3 = αˆ ., α1 − α., α 2 − α., α 3 − α., βˆ 1, βˆ 2 , βˆ 3

and

 13

 23



− 1 3



T = − 1 3

 0



 0

 0





13

−1 3

23

−1 3

0

0

0



13

−1 3

−1 3

23

0

0

0



0

0

0

0

1

0

0



0

0

0

0

0

1

0



0

0



0



0

0



0

1



The beta-hat model for θˆ 3 is

␪ˆ 3 = H 3␪ + ε 3



(10.14)



where H3 = I7 and Var(ε 3 ) = Σ3 = T Σ2T ′.

The combined estimate of ␪ can be computed from Equation 10.9 where ␪ˆ 2, H2

and Σˆ 2 are replaced by ␪ˆ 3, H3, and Σˆ 3, respectively. Two examples are used to

demonstrate the above ideas.

© 2002 by CRC Press LLC



8



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 10.2

Data for the Example in Section 10.5 Involving Five Treatments

in a RCB Design Structure Where Y is the Response Variable

and X is the Covariate

BLOCK

1

2

3

4

5

6

7

8

9

10

11

12



Treatment 1



Treatment 2



Treatment 3



Treatment 4



Treatment 5



Y1

55.1

50.7

58.4

60.5

60.6

60.4

62.1

59.5

68.1

66.7

68.7

64.9



Y2

59.6

59.1

65.5

57.9

65.6

64.8

64.7

64.0

64.9

75.9

68.5

63.6



Y3

63.5

65.7

66.1

72.7

70.6

62.4

66.0

76.1

80.5

82.8

75.3

75.5



Y4

66.1

59.9

77.8

73.5

91.1

69.7

76.2

78.3

79.8

84.0

87.8

80.2



Y5

80.5

78.6

76.4

78.7

82.2

67.0

86.8

81.2

96.4

79.0

76.1

88.5



X1

14.0

15.4

13.5

19.0

12.0

24.1

28.9

18.7

25.6

22.7

25.3

22.6



X2

11.4

20.6

21.2

9.0

16.9

21.3

23.6

16.0

13.2

28.5

14.8

11.1



X3

14.4

20.3

15.6

20.8

16.5

14.2

17.8

24.8

27.2

28.5

19.2

18.4



X4

12.2

11.9

21.6

17.0

31.4

16.2

22.9

19.1

19.4

22.4

26.1

19.8



X5

21.1

24.3

16.1

17.5

19.0

12.5

26.9

19.8

28.6

16.7

14.9

22.4



10.5 EXAMPLE: FIVE TREATMENTS IN RCB

DESIGN STRUCTURE

The data in Table 10.2 are yields (Y) (kilograms per hectare) of winter wheat where

the treatments are herbicides applied during the spring and the covariate is the depth

of adequate moisture (centimeters) measured on each plot. The design structure is

a RCB.

Table 10.3 contains the SAS® system code used to extract the within block

information from the data set. The estimate of σε2 is 0.9965 and the parameter

estimates provide estimates of the slopes (denoted by x*trt 1, …, x*trt 5) and

estimates of αi – α5 i = 1, 2, …, 5, (denoted by trt 1, … , trt 5). Estimate statements



have been included in Table 10.4 to provide estimates of αi – α

., quantities that are

needed in the combined estimator process. Table 10.5 contains the PROC GLM code

to provide the statistic to test the equal slopes hypothesis using the within block

information. The value of the F statistic is 32.23 with a significance level less than

0.0001, indicating there is sufficient within block information to conclude the slopes

are not all equal. Using the expected mean square for BLOCKS and ERROR, the

ˆ b2 = (59.8718 – 0.99650/4.5455 =

estimate of the block variance component is σ

12.9524. The block totals for Y are listed in Table 10.6 and are denoted by SY, where

SX1, …, SX5 are the sums of the covariates within each block. The PROC GLM

code to fit the between block model is in Table 10.7. Also included are estimate



statements to provide estimates of α and the five slopes. The contrast statement

provides a between block test of the equal slopes hypothesis. The resulting F statistic

is 5.50 with significance level 0.0330; again there is sufficient between block information to conclude the slopes are not all equal. The between block analysis of

© 2002 by CRC Press LLC



More Than Two Treatments in a Blocked Design Structure



9



TABLE 10.3

PROC GLM Code to Fit the Within Block Model with

Parameter Estimates

PROC GLM DATA=LONG10; CLASSES TRT BLOCK;

MODEL Y=BLOCK TRT X*TRT/SOLUTION INVERSE;

ESTIMATE “A1-Abar” TRT .8 –.2 –.2 –.2 –.2;

ESTIMATE “A2-Abar” TRT –.2 .8 –.2 –.2 –.2;

ESTIMATE “A3-Abar” TRT –.2 –.2 .8 –.2 –.2;

ESTIMATE “A4-Abar” TRT –.2 –.2 –.2 .8 –.2;

ESTIMATE “A5-Abar” TRT –.2 –.2 –.2 –.2 .8;

Source

Model

Error

Corr Total



df

20

39

59



SS

5724.9054

38.8640

5763.7693



MS

286.2453

0.9965



FValue

287.25



ProbF

0.0000



Source

BLOCK

trt

x*trt



df

11

4

5



SS(III)

658.5902

6.5423

1040.1444



MS

59.8718

1.6356

208.0289



FValue

60.08

1.64

208.76



ProbF

0.0000

0.1834

0.0000



Estimate

–1.3249

0.5019

0.9823

3.0808

0.0000

0.5100

0.6734

0.9135

1.0798

1.4309



Biased

1

1

1

1

1

0

0

0

0

0



StdErr

1.8508

1.7831

1.9378

1.9275



tValue

–0.72

0.28

0.51

1.60



Probt

0.4783

0.7798

0.6151

0.1180



0.0613

0.0579

0.0708

0.0611

0.0693



8.32

11.63

12.90

17.67

20.66



0.0000

0.0000

0.0000

0.0000

0.0000



Parameter

trt 1

trt 2

trt 3

trt 4

trt 5

x*trt 1

x*trt 2

x*trt 3

x*trt 4

x*trt 5



TABLE 10.4

PROC GLM Code for the Within Block Model

Parameter Estimates from Estimate Statements

ESTIMATE

ESTIMATE

ESTIMATE

ESTIMATE

ESTIMATE

Parameter

A1-Abar

A2-Abar

A3-Abar

A4-Abar

A5-Abar



© 2002 by CRC Press LLC



“A1-Abar”

“A2-Abar”

“A3-Abar”

“A4-Abar”

“A5-Abar”

Estimate

–1.9730

–0.1461

0.3343

2.4328

–0.6480



TRT

TRT

TRT

TRT

TRT



.8 –.2 –.2 –.2 –.2;

–.2 .8 –.2 –.2 –.2;

–.2 –.2 .8 –.2 –.2;

–.2 –.2 –.2 .8 –.2;

–.2 –.2 –.2 –.2 .8;

StdErr

1.0980

0.9878

1.2054

1.1397

1.2343



tValue

–1.80

–0.15

0.28

2.13

0.53



Probt

0.0801

0.8832

0.7830

0.0391

–0.6026



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