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5 EXAMPLE: BALANCED ONE-WAY TREATMENT STRUCTURE

5 EXAMPLE: BALANCED ONE-WAY TREATMENT STRUCTURE

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14



Analysis of Messy Data, Volume III: Analysis of Covariance



Table 12.3 contains the PROC MIXED code to fit the model in Equation 12.1

where the covariance between the slopes and intercepts is specified to be zero. Using

the statement

random variety n2*variety/solution;

specifies that there are variance components for the intercepts (variety) and for the

slopes (n2*variety), but does not specify a value for the covariance between the

slopes and intercepts, so the covariance sets to zero. The REML estimates of the

variance components are σˆ 2a = 5.818, σˆ 2b = 0.1785, and σˆ 2ε = 0.6944. The mixed

ˆ = 10.4821 and βˆ = 1.9379.

model estimates of the parameters of the model are α

Table 12.4 contains the PROC MIXED code to fit the model in Equation 12.1

where the random statement allows for the slopes and intercepts to be correlated.

The statement

random Int N2/subject=variety type=un solution;

specifies that the intercepts (Int) and slopes (N2) within each level of variety (subject =

variety) has an unstructured correlation structure (type = un). When the random

statement involves a “/ ”, the interpretation of the terms on the left side of the “/ ”

includes the term or terms on the right side of the “/ ”. For this statement, Int means

Variety and n2 means n2*variety. The REML estimates of the parameters are σˆ 2a =

5.8518 from UN(1,1), σˆ 2b = 0.1785 from UN(2,2), σˆ ab = 0.7829 from UN(2,1),

and σˆ 2ε = 0.6944. The mixed models estimates of the mean of the intercepts and

ˆ = 10.4821, βˆ = 1.9379, so the estimate of the population mean yield

slopes are α

at a given level of nitrogen is µˆ y|N2 = 10.4821 + 1.9379N2. Since the model is

balanced, all parameters in Table 12.4 have the same estimates in Table 12.3, except

σab = 0. The solutions and standard errors for the fixed effects are identical, but the

solutions and standard errors for the random effects are different. The information

criteria indicate that the model with the non-zero covariance fits the data better than

the one with the covariance set to zero (the smaller the information criteria the better

the covariance structure). Table 12.5 contains the solution for the individual intercepts and slopes for each of the varieties in the study based on the model with the

non-zero covariance between the slopes and intercepts. These two sets of solutions

satisfy the sum-to-zero restriction, i.e., the intercepts sum-to-zero and the slopes

sum-to-zero. These are the estimated best linear unbiased predictors of the intercepts

and slopes (Littel et al., 1996). The solution for the random effects based on the

model with a zero covariance between the slopes and intercepts are somewhat

different than the ones in Table 12.5 (results not shown here). The main difference

is the estimated standard errors for the independent slope and intercept model are

smaller than for the correlated slope and intercept model. The predicted values of

ˆ + âi and bˆ i(p) = βˆ

the slope and intercept for the ith variety are obtained by âi(p) = α

ˆ

ˆ

ˆ

+ bi, where α and β are the estimates of the population intercept and slope and âi

and bˆ i are the predicted values of the intercepts and slopes obtained from the solution

for the random effects. Estimate statements in Table 12.6 are used to obtain the

predicted slopes and intercepts for each of the varieties. The statements

estimate ‘slope 1’ N2 1 | N2 1/subject 1;

estimate ‘slope 2’ N2 1 | N2 1/subject 0 1;

© 2002 by CRC Press LLC



Random Effects Models with Covariates



15



TABLE 12.4

PROC MIXED Code to Fit the Random Coefficient Model with the

Correlated Slopes and Intercepts Covariance Structure and Results

Proc mixed data=ex_12_5 cl covtest ic;

class variety;

model yield=N2/solution ddfm=kr;

random Int N2/subject=variety type=un solution;

CovParm

UN(1,1)

UN(2,1)

UN(2,2)

Residual



Subject

Variety

Variety

Variety



Estimate

5.8518

0.7829

0.1785

0.6944



StdErr

3.4794

0.5624

0.1238

0.2624



ZValue

1.68

1.39

1.44

2.65



ProbZ

0.0463

0.1639

0.0747

0.0041



Alpha

0.05

0.05

0.05

0.05



Parameters

4



AIC

108.86



AICC

110.76



HQIC

106.19



BIC

108.64



CAIC

112.64



Effect

Intercept

N2



Estimate

10.4821

1.9379



StdErr

0.9278

0.1745



DF

6

6



tValue

11.30

11.10



Probt

0.0000

0.0000



Effect

N2



NumDF

1



DenDF

6



FValue

123.26



ProbF

0.0000



Neg2LogLike

100.86



Lower

2.3813

–0.3194

0.0650

0.3722



Upper

30.3058

1.8852

1.3886

1.7270



TABLE 12.5

Solution for the Random Effect Slopes and Intercepts

for Each Variety

Effect

Intercept

N2

Intercept

N2

Intercept

N2

Intercept

N2

Intercept

N2

Intercept

N2

Intercept

N2



Variety

1

1

2

2

3

3

4

4

5

5

6

6

7

7



Estimate

1.1497

0.3635

0.3980

–0.1311

2.4327

0.6020

–0.0397

–0.1125

–0.6342

–0.3895

1.5760

0.1471

–4.8826

–0.4796



StdErrPred

0.9935

0.2406

0.9935

0.2406

0.9935

0.2406

0.9935

0.2406

0.9935

0.2406

0.9935

0.2406

0.9935

0.2406



df

7.6

9.6

7.6

9.6

7.6

9.6

7.6

9.6

7.6

9.6

7.6

9.6

7.6

9.6



tValue

1.16

1.51

0.40

–0.54

2.45

2.50

–0.04

–0.47

–0.64

–1.62

1.59

0.61

–4.91

–1.99



Probt

0.2823

0.1632

0.6997

0.5983

0.0416

0.0323

0.9692

0.6507

0.5420

0.1379

0.1533

0.5552

0.0014

0.0755



provide predicted values for the slopes of varieties 1 and 2 (similar statements were

used for the other varieties). The fixed effects are on the left hand side of “ | ” and

the random effects are on the right hand side. On the left hand side using N2 1

© 2002 by CRC Press LLC



16



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 12.6

Estimate Statements Used to Obtain Predicted Values for Each of the

Varieties At N2 = –3 and N2 = 3 and the Predicted Slopes and Intercepts

estimate

estimate

estimate

estimate

estimate

estimate

estimate

estimate



‘Var 1 at N2=–3’ intercept 1 N2 –3|Int 1 N2 –3/subject

‘Var 2 at N2=–3’ intercept 1 N2 –3|Int 1 N2 –3/subject 0

‘Var 1 at N2= 3’ intercept 1 N2 3|Int 1 N2 3/subject

‘Var 2 at N2= 3’ intercept 1 N2 3|Int 1 N2 3/subject 0

‘slope 1’ N2 1 | N2 1/subject 1;

‘slope 2’ N2 1 | N2 1/subject 0 1;

‘Int 1’ Int 1 | Int 1/subject 1;

‘Int 2’ Int 1 | Int 1/subject 0 1;



Slopes

Variety

1

2

3

4

5

6

7



Value

2.3014

1.8067

2.5399

1.8254

1.5483

2.0850

1.4583



Stderr

0.1933

0.1933

0.1933

0.1933

0.1933

0.1933

0.1933



Intercepts

Value

11.6319

10.8802

12.9149

10.4425

9.8479

12.0582

5.5995



Stderr

0.4194

0.4194

0.4194

0.4194

0.4194

0.4194

0.4194



1;

1;

1;

1;



Predicted values

at N2 = –3



Predicted values

at N2 = –3



Value

4.7278

5.4600

5.2952

4.9662

5.2029

5.8032

1.2246



Value

18.5360

16.3004

20.5345

15.9187

14.4929

18.3131

9.9744



Stderr

0.7229

0.7229

0.7229

0.7229

0.7229

0.7229

0.7229



Stderr

0.7084

0.7084

0.7084

0.7084

0.7084

0.7084

0.7084



selects the value of βˆ and N2 1/subject 1 selects the value of bˆ 1. Since the random

statement includes “/subject = variety”, then the “/subject 1” on the estimate statement specifies to select the first level of variety. Using “/subject 0 0 0 1” would

select the fourth level of variety. The statements

estimate ‘Int 1’ Int 1 | Int 1/subject 1;

estimate ‘Int 2’ Int 1 | Int 1/subject 0 1;

provide predicted values for the intercepts of varieties 1 and 2 (similar statements

ˆ and Int 1/subject

were used for the other varieties). Using Int 1 selects the value of α

1 selects the value of â1.

The predicted values of the slopes and intercepts and the corresponding estimated

standard errors for each of the varieties are displayed in Table 12.6. The variance

of these predicted slopes and of these predicted intercepts are proportional to the

estimates of the variance components σa2 and σb2 . The estimate statements

estimate ‘Var 1 at N2=–3’ intercept 1 N2 –3|Int 1

N2 –3/subject 1;

estimate ‘Var 2 at N2=–3’ intercept 1 N2 –3|Int 1

N2 –3/subject 0 1;

estimate ‘Var 1 at N2= 3’ intercept 1 N2 3|Int 1

N2 3/subject 1;

estimate ‘Var 2 at N2= 3’ intercept 1 N2 3|Int 1

N2 3/subject 0 1;

© 2002 by CRC Press LLC



Random Effects Models with Covariates



17



Random Coefficient Models for Varieties



Yield Per Plot (lbs)



25.0000

20.0000

15.0000

10.0000

5.0000

0



-4



-3



-2



-1



0



1



2



3



4



Coded N2 Level

1

5



2

6



3

7



4

8



FIGURE 12.1 Graph of the predicted models for each variety where variety 8 corresponds

to the mean model.



provide predicted values for varieties 1 and 2 at N2 = –3 and N2 = 3. These are

predicted values from the regression equations using the intercepts and slopes for

each variety. The set of statements can be extended to include all of the varieties.

The predicted values at N2 = –3 and 3 for all of the varieties are included in

Table 12.6. Those predicted values were used to construct a graph of the set of

regression lines for these varieties which are displayed in Figure 12.1. The methods

of moments estimates of the variance components process described in Section 12.3

are summarized in Table 12.7. Since the model is balanced, the method of moments

estimates of the variance components are identical to the REML estimates displayed

in Table 12.4.



TABLE 12.7

Sums of Squares, the Cross Product and Their Expectations

for Computing the Methods of Moments Estimates of the

Variance and Covariance Components for Example 12.5

Sum of Squares

SS(INTERCEPTS) = 578.434286

SS(SLOPES) = 511.817143

SSCP = 375.788571

SS(ERROR) = 9.721000

σε2 = 0.6943571

σS2 = 0.1785393



© 2002 by CRC Press LLC



Expected Sum of Squares

24 σε2 + 96 σI2

120 σε2 + 2400 σS2

2

480 σIS

14 σε2

σI2 = 5.8517679

2 = 0.7828929

σIS



18



Analysis of Messy Data, Volume III: Analysis of Covariance



FIGURE 12.2 Fit model screen to fit the random coefficient model with independent slopes

and intercepts.



FIGURE 12.3 REML estimates of the variance components and tests for the fixed effects

and the random effects.



The JMP® software can fit the independent slope and intercept model to the data

set, but it cannot fit the model with non-zero covariance. Figure 12.2 contains the fit

model screen where yield is specified to be the response variable and the model

contains N2, Variety, and N2*Variety, where the last two terms are specified as being

random by using the attributes menu. Figure 12.3 contains the REML estimates of the

variance components as well as test for the fixed and random effects. The indication

is that the variance components are significantly different from zero, or they are

important components in describing the variability in the model. The custom test menu

can be used to provide predicted values for each variety’s slope, intercept, or model

evaluated at some value for N2. Those results were not included in this example. PROC

MIXED can perform similar tests for the independent slope and intercept model by

specifying method = type1, or type2, or type3 (an example is included in Section 12.7).



12.6 EXAMPLE: UNBALANCED ONE-WAY

TREATMENT STRUCTURE

The data in Table 12.8 are from randomly selected cities and then randomly selected

school districts with each city where the response variable is the amount of income

© 2002 by CRC Press LLC



Random Effects Models with Covariates



19



TABLE 12.8

Data for Example in Section 12.6 with One-Way Random Effects Treatment

Structure (Cities) Where the Response Variable is the Amount Spent on

Vocational Training and the Covariate is the Percent Unemployment

City 1

Spent

27.9

43.0

26.0

34.4

43.4

42.4

27.9

25.9

23.3

34.3



Unemp

4.9

9.4

4.2

6.6

9.5

8.8

4.6

4.6

4.2

6.8



City 6

Spent

28.0

18.2

29.2

25.0

22.6

34.9

39.3

37.3

30.5



Unemp

6.3

2.6

6.7

5.4

3.8

8.4

9.8

9.1

6.8



City 2

Spent

29.6

12.7

19.9

12.3

23.8

28.8

21.0

13.9

13.2



Unemp

8.1

0.9

4.6

1.9

5.9

8.0

4.4

2.1

1.3



City 7

Spent

37.6

34.7

38.6

33.8

10.1

29.8

15.1

35.0



Unemp

9.9

8.8

9.9

8.2

0.1

7.0

1.8

8.5



City 3

Spent

17.5

21.2

19.0

14.5

25.2

15.0

24.6



Unemp

3.1

6.4

4.0

2.0

8.4

2.4

8.7



City 8

Spent

10.6

21.4

32.1

28.3

38.7

19.8

40.0

36.3

29.0



Unemp

0.0

3.4

7.0

5.8

9.2

2.7

9.7

8.2

5.9



City 4

Spent

28.0

16.0

16.8

26.5

23.1

18.0

17.8

11.6

24.2



Unemp

9.6

3.3

4.2

8.6

6.2

3.1

3.9

0.5

7.4



City 9

Spent

10.9

12.0

14.1

15.4

11.2

17.9

26.2

14.2

21.4



Unemp

2.9

2.8

3.6

5.2

1.4

5.6

10.0

3.6

7.2



City 5

Spent

22.0

22.6

12.8

11.8

23.5

26.0

24.5

18.8

11.2



Unemp

6.7

6.9

2.2

1.4

8.4

9.3

8.0

5.7

1.8



City 10

Spent

8.6

9.1

13.6

13.7

13.4

15.3

23.1

22.2



Unemp

0.8

0.7

3.4

3.1

3.6

4.4

7.8

7.4



to be spent on vocational training (y) and the possible covariate is the level of

unemployment (x). The PROC MIXED code to fit Model 12.1 is in Table 12.9 where

ˆ a2 = 2.5001 from UN(1,1), σ

ˆ b2 = 0.4187 from

the estimates of the parameters are σ

ˆ ε2 = 0.6832. The mixed models estimates

UN(2,2), σˆ ab = 0.3368 from UN(2,1), and σ

ˆ = 9.6299, βˆ = 2.3956, so the estimate

of the mean of the intercepts and slopes are α

of the population mean spent income on vocational training at a given level of

ˆ y|EMP = 9.6299 + 2.3956 UNEMP. The solution for the random

unemployment is µ

effects that satisfy the sum-to-zero restriction within the intercepts and within the

slopes are in Table 12.10. The estimate statements in Table 12.11 provide predicted

values (estimated BLUPS) for each of the selected cities evaluated at 10% unemployment. Table 12.11 contains the predicted values for each of the cities evaluated

at 0, 8, and 10%. Tables 12.12, 12.13, and 12.14 contain the PROC MIXED code

and results for fitting Model 12.1 where the values of the covariate have been altered

by subtracting 2, then mean (5.4195402) and 8 from the percent unemployment.

The estimate of the variance component for the intercepts depends on the amount

subtracted from the covariate as described in Section 12.4. When X0 is subtracted

© 2002 by CRC Press LLC



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