A Comparison of the Polytomous Logistic Regression and Joint Cox Proportional Hazards Models for Evaluating Multiple Disease Subtypes in Prospective Cohort Studies

Xiaonan Xue; Mimi Y. Kim; Mia M. Gaudet; Yikyung Park; Moonseong Heo; Albert R. Hollenbeck; Howard D. Strickler; Marc J. Gunter

doi:10.1158/1055-9965.EPI-12-1050

DOI: 10.1158/1055-9965.EPI-12-1050 Published February 2013

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Figure 1.
Residual plot for ER⁻/PR⁻ breast cancer compared with ER⁺/PR⁺ cancer.
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Figure 2.
Residual plot for ER⁺/PR⁻ breast cancer compared with ER⁺/PR⁺ cancer.

Tables

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Table 1.

Parameters chosen for different scenarios considered in the simulation study

Scenario	Event 1				Event 2
			OR₁				OR₂
	X = 0	X = 1			X = 0	X = 1
(a)	0.030	0.075	2.62	2.56	0.030	0.030	1.00	1.00	2.56
(b)	0.010	0.020	2.02	2.01	0.015	0.030	2.03	2.01	1.00
(c)	0.150	0.225	1.65	1.57	0.150	0.150	1.00	1.00	1.57
(d)	0.150	0.195	1.37	1.33	0.202	0.260	1.39	1.33	1.00
(e)	0.020	0.060	3.13	3.06	0.030	0.030	1.00	1.00	3.06
(f)	0.030	0.060	2.06	2.03	0.040	0.080	2.08	2.03	1.00
(g)	0.100	0.200	2.25	2.12	0.300	0.300	1.00	1.00	2.12
(h)	0.200	0.280	1.56	1.47	0.300	0.408	1.61	1.47	1.00

NOTE: The OR for event k, that is, OR_k is defined as the ratio of odds of developing event k for the exposed (X = 1) and the odds of event k for the unexposed (X = 0), k = 1, 2; the HR for event k, that is, is defined as the HR for developing event k associated with exposure; scenarios (a)–(d) considered events 1 and 2 have proportional hazards functions and scenarios (e)–(h) considered events 1 and 2 do not have proportional hazards functions.

Table 2.

Performance of the polytomous model estimates [ and in Eq. (2)] of OR₂ and and

Scenario	Event 2					Comparison
		OR₂
	Bias^a	Coverage	Bias	Coverage	Power/type I error	Bias	Coverage	Power/type I error
(a)	1.5	95.6	1.5	95.6	4.4	2.4	94.8	78.8
(b)	0.6	95.8	1.3	95.6	59.6	2.0	96.2	3.8
(c)	4.6	94.2	4.6	94.2	5.8	−0.1	97.8	73.2
(d)	2.2	94.6	6.3	92.2	86.4	−0.0	97.0	3.0
(e)	0.1	96.6	0.1	96.6	3.4	5.3	95.4	87.8
(f)	−0.4	95.2	1.8	96.4	94.2	5.6	95.4	4.6
(g)	2.9	94.0	2.9	94.0	6.0	4.3	96.2	99.6
(h)	2.6	94.4	12.3	79.4	99.8	8.8	96.4	3.6

NOTE: Scenarios considered for event 1 are also considered for event 2, that is, both events include the scenarios of positive associations with the exposure and null associations with the exposure. For the same scenario, the results for event 2 are similar to those for event 1. Therefore, for simplicity, in Tables 2 and 3, we omit the result for event 1 and only report for results on parameters associated with event 2 and parameters associated with the contrast between events 1 and 2.

↵^aBias is defined as (estimated value-parameter of interest)/parameter of interest × 100; coverage is the percentage of times the CI included the parameter of interest; power is the percentage of times the CI did not include 1 when the alternative is true and type I error rate is the same percentage but when the null is true.

Table 3.

Performance of the joint Cox model estimate [Eq. (3)] of and


	Bias	Coverage	Power/type I error	Bias	Coverage	Power/type I error
(a)	0.9	95.2	4.8	2.4	94.2	80.4
(b)	0.0	95.8	59.2	1.9	96.0	4.0
(c)	−0.4	95.4	4.6	−0.1	95.4	77.6
(d)	−0.6	94.6	82.0	−0.0	94.4	5.6
(e)	0.9	96.6	3.4	3.8	95.5	87.8
(f)	−1.1	95.2	93.8	3.5	95.6	4.4
(g)	−0.1	94.8	5.2	0.7	94.4	99.8
(h)	−0.2	95.4	99.8	0.6	95.4	4.6

NOTE: Bias, coverage, and power/type I error are defined the same as in Table 2.

Table 4.

Polytomous model and the joint Cox model on the AARP data to evaluate current BMI with four subtypes of breast cancer defined by the tumor's ER and PR status

Polytomous model	BMI < 25.0	BMI 25.0–29.9	Relative OR (CI)	BMI ≥ 30.0	Relative OR (CI)	P_trend	P value difference in trend
ER⁺/PR⁺
Cases	286	345	1(ref)	366	1(ref)
OR (CI)	1 (ref)	1.589 (1.333–1.894)^a		2.100 (1.753–2.516)^a		<0.0001	Ref
ER⁻/PR⁻
Cases	84	91	0.901	71	0.577
OR (CI)	1 (ref)	1.424 (1.025–1.979)^a	(0.617–1.316)	1.192 (0.824–1.725)	(0.394–0.844)^a	0.283	0.012
ER⁻/PR⁺
Cases	9	4		5
OR (CI)	1(ref)
ER⁺/PR⁻
Cases	97	73	0.585	61	0.433	0.531	<0.0001
OR (CI)	1 (ref)	0.885 (0.635–1.233)	(0.385–0.890)	0.902 (0.626–1.299)	(0.286–0.654)^a
Joint Cox model			Relative HR (CI)		Relative HR (CI)		Relative HR and CI for trend
ER⁺/PR⁺	1 (ref)	1.551 (1.304–1.845)^a	1 (ref)	1.758 (1.471–2.101)	1 (ref)	<0.0001	Ref
ER⁻/PR⁻			0.925		0.593		0.783
	1 (ref)	1.434 (1.034–1.990)^a	(0.639–1.338)	1.043 (0.729–1.493)	(0.399–0.883)^a	0.693	(0.654–0.938)^a
ER⁻/PR⁺			0.342		0.428		0.631
	1 (ref)	0.531 (0.139–2.021)	(0.089–1.316)	0.756 (0.224–2.531)	(0.126–1.456)	0.599	(0.317–1.257)
ER⁺/PR⁻			0.555		0.430		0.659
	1 (ref)	0.861 (0.620–1.194)	(0.384–0.802)^a	0.756 (0.532–1.075)	(0.291–0.636)^a	0.113	(0.544–0.798)^a

NOTE: Model adjusted for age, age at menarche, age at first live birth, parity, smoking, educational level, race, family history of breast cancer, fat intake, alcohol consumption, oophorectomy, physical activity, and height (in polytomous model, all the adjusting variables have different coefficients across subtypes; in the joint Cox model, those adjusting variables with the similar coefficient across subtypes are set to have the same coefficient).