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Table 5 Percentages of selecting \(\mathcal {M}_{2}\) according to the BIC on N=500 datasets, given t v =(0,1,2,4,6,8,10,12) and \(\sigma _{0}^{2}=1.5\)

From: Item response models for the longitudinal analysis of health-related quality of life in cancer clinical trials

Parameter

Scenarios

 

Values

AM using δ ne

CM using δ fa

CM using δ ne

AM using δ fa

β 1

\(\sigma _{1}^{2}\)

LMM

AM

CM

LMM

AM

CM

LMM

AM

CM

LMM

AM

CM

1

0.01

0

2.3

24.9

0

5.0

6.9

0

2.7

3.7

0.3

6.4

24.8

 

0.02

0

21.4

54.7

0

37.6

44.1

0

17.7

18.1

0

50.0

77.0

 

0.03

0

61.0

91.0

0

75.7

80.0

0

41.3

45.6

0

86.3

98.3

 

0.05

0

97.7

99.7

0

100

100

0.3

89.0

90.0

0

99.3

100

 

0.2

39.3

100

100

40.7

100

100

10.7

100

100

57.7

100

100

 

0.5

100

100

100

100

100

100

100

100

100

100

100

100

0.5

0.005

0.2

25.5

56.4

0

41.2

25.7

0

14.9

11.0

0

41.2

53.6

 

0.008

0.8

73.8

89.4

0

85.8

73.4

0

42.9

38.8

0.2

91.6

93.6

 

0.01

2.0

91.2

97.0

0

97.0

91.6

0

66.6

63.4

0.6

99.2

99.2

 

0.02

26.4

100

100

4.8

100

100

0

100

100

51.8

100

100

 

0.03

77.0

100

100

64.8

100

100

0.8

100

100

96.6

100

100

 

0.05

99.8

100

100

100

100

100

62.3

100

100

100

100

100

0.3

0.002

16.7

6.3

21.4

0

2.1

4.0

0

3.1

3.9

11.0

11.0

15.3

 

0.005

72.3

86.3

92.7

30.7

55.3

59.0

0

32.3

46.0

85.7

87.3

91.7

 

0.008

97.7

100

100

86.0

97.3

98.0

4.0

76.3

88.3

99.3

99.7

100

 

0.01

100

100

100

96.3

99.7

99.3

17.3

94.0

97.0

100

100

100

 

0.02

100

100

100

100

100

100

96.7

100

100

100

100

100

0

0.001

24.8

2.8

5.7

6.8

0.4

1.6

4.8

0.6

1.9

15.2

1.4

3.7

 

0.002

70.2

32.0

37.3

26.4

6.6

8.2

20.6

2.4

5.1

47.6

15.2

21.4

 

0.005

99.8

99.4

99.6

92.2

70.4

77.2

88.2

61.8

72.4

99.6

97.8

97.8

 

0.008

100

100

100

99.8

98.0

98.8

99.8

98.4

99.2

100

100

100

 

0.01

100

100

100

100

100

100

100

100

100

100

100

100

 

0.02

100

100

100

100

100

100

100

100

100

100

100

100

−0.3

0.002

0.7

4.4

61.4

0

54.0

5.1

0

93.3

1.8

0

2.1

18.6

 

0.005

5.7

62.3

79.0

0

95.7

40.4

0

99.7

33.2

0

56.0

48.3

 

0.008

23.7

96.3

97.3

0

100

86.7

0

100

82.7

1.7

96.3

86.3

 

0.01

45.2

100

99.6

0

100

98.2

0

100

92.3

6.8

99.8

98.4

 

0.02

98.8

100

100

61.4

100

100

26.6

100

100

96.0

100

100

−0.5

0.005

2.6

12.1

48.6

0

57.6

13.2

0

84.8

9.8

0

41.2

53.6

 

0.008

3.8

43.5

70.7

0

85.8

41.6

0

96.1

33.5

0.2

91.6

93.6

 

0.01

5.6

70.0

84.8

0

95.6

61.5

0

98.4

49.6

0.6

99.2

99.2

 

0.02

12.8

100

100

0

100

100

0

100

97.8

51.8

100

100

 

0.03

36.8

100

100

0

100

100

0

100

100

96.6

100

100

 

0.05

93.0

100

100

43.4

100

100

17.6

100

100

100

100

100

−1

0.01

0

0.6

34.5

0

5.8

5.2

0

8.4

6.3

0

1.6

15.1

 

0.02

0

5.8

46.6

0

21.2

18.4

0

20.8

11.0

0

8.8

34.3

 

0.03

0

30.4

73.2

0

50.4

44.8

0

50.0

37.4

0

36.4

58.0

 

0.05

0

83.6

95.2

0

92.6

91.4

0

85.6

80.1

0

90.0

96.0

 

0.2

46.4

100

100

21.4

100

100

12.0

100

100

41.8

100

100

 

0.5

100

100

100

100

100

100

100

100

100

100

100

100

  1. For the (adjacent,logistic, Z 1,U a ) a=1,2 models and the (cumulative,logistic, Z 1,U a ) a=1,2 models are denoted by AM and CM, respectively. For the random component, U 1 if \(\sigma _{1}^{2}=0\) and U 2 if \(\sigma _{1}^{2}>0\)