Fixed effects models

Several uniform effects categories occur in Models 2 and 3. These categories are easily identified in the results by estimates and standard errors that are extremely

Table 3.4 Fixed effects estimates resulting from Model 2.

Standard

Wald

95%

Chi-

Parameter

DF

Estimate

Error

Confidence

Limits

Square

Pr>ChiSq

Intercept

1

-26

5793

2.1509

-30.7950

-22.3635

152.

70

0001

cf1

1

2.

6711

0.5548

1.5838

3.7584

23.

18

<.

0001

treat

A

1

1.

0470

0.6477

-0.2224

2.3164

2.

61

0.

1060

treat

B

1

2.

0302

0.6294

0.7967

3.2637

10.

41

0.

0013

treat

C

0

0.

0000

0.0000

0.0000

0.0000

centre

1

1

23.

4900

2.1070

19.3604

27.6197

124.

29

<.

0001

centre

2

1

23.

2142

2.2258

18.8516

27.5768

108.

77

<.

0001

centre

3

1

-0.

1467

110060.4

-215715

215714.3

0.

00

1.

0000

centre

4

1

23.

.2940

2.2582

18.8680

27.7201

106.

40

<.

0001

centre

5

1

24.

1606

2.1989

19.8508

28.4704

120.

72

<.

0001

centre

6

1

25.

3009

2.2938

20.8051

29.7966

121.

66

<.

0001

centre

7

1

21.

8392

2.3609

17.2120

26.4665

85.

57

<.

0001

centre

8

1

24.

7352

2.2914

20.2442

29.2262

116.

53

<.

0001

centre

9

1

1.

2139

322114.2

-631331

631333.5

0.

00

1.

0000

centre

11

1

22.

6362

2.3220

18.0852

27.1872

95.

04

<.

0001

centre

12

1

22.

1269

2.4024

17.4183

26.8355

84.

83

<.

0001

centre

13

1

24.

5423

2.4163

19.8066

29.2781

103.

17

<.

0001

centre

14

1

23.

1055

2.1751

18.8423

27.3687

112.

84

<.

0001

centre

15

1

22.

9148

2.3279

18.3521

27.4775

96.

89

<.

0001

centre

18

1

-0.

0486

126467.5

-247872

247871.8

0.

00

1.

0000

centre

23

1

24.

3186

2.7168

18.9938

29.6434

80.

12

<.

0001

centre

24

1

1.

2139

322114.2

-631331

631333.5

0.

00

1.

0000

centre

25

1

-0.

0211

117643.3

-230577

230576.5

0.

00

1.

0000

centre

26

1

22.

9963

2.3328

18.4242

27.5685

97.

18

<.

0001

centre

27

1

25.

5642

2.6333

20.4029

30.7254

94.

24

<.

0001

centre

29

1

24.

. 7202

2.9282

18.9811

30.4594

71.

27

<.

0001

centre

30

1

-0.

0857

137672.4

-269833

269832.9

0.

00

1.

0000

centre

31

1

-0.

0486

51630.15

-101193

101193.2

0.

00

1.

0000

centre

32

1

23.

3668

2.4567

18.5518

28.1817

90.

47

<.

0001

centre

35

1

-0.

2451

183098.8

-358867

358866.7

0.

00

1.

0000

centre

36

1

23.

2206

2.1447

19.0170

27.4241

117.

22

<.

0001

centre

37

1

23.

5944

2.3797

18.9302

28.2585

98.

30

<.

0001

centre

40

0

23.

7051

0.0000

23.7051

23.7051

centre

41

0

0.

0000

0.0000

0.0000

0.0000

large. The fixed effects estimates resulting from Model 2 are listed in Table 3.4. The uniform centre categories can be identified as centres 3, 9, 18, 24, 25, 30, 31 and 35. (Note that estimates for these centres become very large when the intercept term of -26.6 is added, while estimates for other centres are then more reasonable.) These are the centres where no patients had cold feet. Centre 40 has no DF. This has occurred because centre 41 (the usual reference category) is uniform and so centre 40 is used for reference. Standard errors for the other centre effects are based on comparisons with centre 40. However, their estimates in this output are still based on comparisons to centre 41! Thus, SAS does not produce useful centre estimates and standard errors when the reference category is uniform. Usually, centre estimates will not be of interest. However, if required they can be obtained by renumbering the centres so that the last one is nonuniform. Although we are clearly getting estimates and standard errors that are unstable, the likelihood still converges, since the uniform categories have little effect on it.

Although Models 2 and 3 are not recommended for estimating treatment effects, they can still be used to test the overall significance of the fixed centre and centre-treatment effects by using likelihood ratio tests. For example, to test centre effects in Model 2 we calculate twice the difference in the log likelihoods between Models 1 and 2, 178.17 - 147.78 = 30.39 - x228. This indicates that centre effects are non-significant. To test centre-treatment effects, twice the difference in the log likelihoods between Models 2 and 3 is taken, 147.78 - 100.05 = 47.73 — x427. This is also non-significant. However, it should be borne in mind that these tests have low power for detecting small centre or centre-treatment effects.

A preferable approach to fitting Model 2 may be to use an exact conditional logistic regression stratified by centre. This would avoid the loss of information on uniform centres. Or alternatively the smaller centres could be combined for the purposes of analysis as 'other centres'. However, the results from Models 2 and 3 indicate that centre effects are not important and therefore Model 1 is likely to be the most satisfactory fixed effects model.

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