Background
Methods
IRT parameterization of the linear predictor
The probability ratio: structure of the models
Models | |||
---|---|---|---|
Characteristics | Adjacent | Cumulative | Sequential |
Ordinal scale |
yes
|
yes
|
no
|
Reversibility |
yes
|
yes
|
no
|
Interpretation using the latent variable |
no
|
yes
|
yes
|
Always defined |
yes
|
yes(no
a) |
yes
|
The cumulative distribution function
IRT models |
\(\eta _{im}^{(j)}\)
|
(r,F, Z
q
,U
a
)
|
---|---|---|
Rating scale model |
\(\theta _{i}-\left (\delta _{m}+\tau _{j}\right)\)
|
(adjacent,logistic,
Z
2,U
1
)
|
Partial credit model |
θ
i
−δ
jm
|
(adjacent,logistic,
Z
1,U
1
)
|
Sequential Rasch model |
θ
i
−δ
jm
|
(sequential,logistic,
Z
1,U
1
)
|
Graded response model |
\(\alpha _{j}\left (\theta _{i}-\delta _{jm}\right)\)
|
(cumulative,logistic,nl)
|
Generalized partial credit model |
\(\alpha _{j}\left (\theta _{i}-\delta _{jm}\right)\)
|
(adjacent,logistic,nl)
|
Results
Simulation study
Design
Difficulty parameters | ||
---|---|---|
Models |
\(\delta _{1}^{ne}=(-1.6,1,1.45)\)
|
\(\delta _{1}^{fa}=(-2.1,1,2.75)\)
|
(r,F,Z
1,U
a
)
a=1,2
|
\(\delta _{2}^{ne}=(-0.8,1.15,1.9)\)
|
\(\delta _{2}^{fa}=(-1.25,1.4,3.3)\)
|
(adjacent,logistic,
Z
1,U
a
)
a=1,2
| Balanced responses | Focus on center categories (1 and 2) |
(cumulative,logistic,
Z
1,U
a
)
a=1,2
| Focus on extreme categories (0,1 and 3) | Balanced responses |
Simulation results
Parameter | Scenarios | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Values | AM using δ
ne
| CM using δ
fa
| CM using δ
ne
| AM using δ
fa
| |||||||||
\(\sigma _{1}^{2}\)
|
β
1
| LMM | AM | CM | LMM | AM | CM | LMM | AM | CM | LMM | AM | CM |
0.2 | −0.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.2 | 0.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | −0.5 | 97.7 | 99.3 | 56.49 | 100 | 94.6 | 93.0 | 100 | 61.3 | 95.7 | 100 | 99.7 | 89.5 |
0 | −0.3 | 99.0 | 100 | 33.0 | 100 | 88.6 | 93.3 | 100 | 36.3 | 94.9 | 100 | 100 | 83.3 |
0 | −0.2 | 100 | 99.6 | 49.3 | 100 | 94.6 | 93.8 | 100 | 71.7 | 95.8 | 100 | 99.6 | 79.0 |
0 | −0.1 | 98.7 | 95.7 | 94.8 | 100 | 98.7 | 89.6 | 100 | 99.0 | 90.4 | 100 | 100 | 88.1 |
0 | 0.0 | 95.6 | 100 | 94.6 | 99.0 | 99.7 | 91.8 | 99.0 | 99.7 | 89.7 | 97.0 | 99.7 | 94.4 |
0 | 0.1 | 83.0 | 100 | 94.8 | 93.3 | 100 | 92.6 | 97.0 | 100 | 90.9 | 87.3 | 100 | 94.7 |
0 | 0.3 | 98.3 | 99.6 | 90.6 | 100 | 99.6 | 89.1 | 100 | 100 | 93.7 | 100 | 99.6 | 93.8 |
0 | 0.5 | 100 | 100 | 94.3 | 100 | 99.3 | 94.7 | 100 | 100 | 97.6 | 100 | 100 | 97.2 |
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 |
Application to a real dataset
HRQoL | ||||
---|---|---|---|---|
Dimensions | Coefficient | Standard error |
P-value
| |
Global Health Status | ||||
β
2
| 0.098 | 0.070 | 0.166 | |
β
3
| 0.130 | 0.085 | 0.128 | |
Physical functioning | ||||
β
2
| -0.150 | 0.077 | 0.051 | |
β
3
| 0.122 | 0.098 | 0.212 | |
Role functioning | ||||
β
2
| -0.011 | 0.081 | 0.892 | |
β
3
| 0.157 | 0.103 | 0.131 | |
Emotional functioning | ||||
β
2
| 0.335 | 0.070 | <.001∗∗∗
| |
β
3
| 0.001 | 0.086 | 0.992 | |
Cognitive functioning | ||||
β
2
| -0.002 | 0.054 | 0.972 | |
β
3
| 0.088 | 0.067 | 0.189 | |
Social functioning | ||||
β
2
| 0.010 | 0.073 | 0.888 | |
β
3
| 0.116 | 0.093 | 0.211 | |
Fatigue | ||||
β
2
| -0.087 | 0.085 | 0.308 | |
β
3
| -0.033 | 0.107 | 0.761 | |
Nausea and vomiting | ||||
β
2
| -0.052 | 0.060 | 0.393 | |
β
3
| -0.069 | 0.072 | 0.336 | |
Pain | ||||
β
2
| -0.330 | 0.076 | <.001∗∗∗
| |
β
3
| -0.188 | 0.092 | 0.040 ∗
| |
Dyspnea | ||||
β
2
| -0.060 | 0.075 | 0.420 | |
β
3
| -0.093 | 0.088 | 0.295 | |
Insomnia | ||||
β
2
| -0.359 | 0.080 | <.001∗∗∗
| |
β
3
| 0.046 | 0.083 | 0.627 | |
Appetite loss | ||||
β
2
| -0.354 | 0.072 | <.001∗∗∗
| |
β
3
| -0.026 | 0.080 | 0.747 | |
Constipation | ||||
β
2
| -0.325 | 0.077 | <.001∗∗∗
| |
β
3
| 0.003 | 0.083 | 0.974 | |
Diarrhea | ||||
β
1
| 0.739 | 0.272 | 0.007 ∗∗
| |
β
2
| 0.018 | 0.067 | 0.792 | |
β
3
| -0.026 | 0.076 | 0.786 | |
Financial difficulties | ||||
β
2
| -0.522 | 0.282 | 0.066 | |
β
3
| 0.302 | 0.208 | 0.146 |
Discussion
Conclusions
Acknowledgements
Funding
www.iresp.net
) under the 2012 call for projects as part of the 2009-2013 Cancer Plan.