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Cancer Epidemiology, Biomarkers & Prevention

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Research Articles

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 and Marc J. Gunter
Xiaonan Xue
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Mimi Y. Kim
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Mia M. Gaudet
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Yikyung Park
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Moonseong Heo
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Albert R. Hollenbeck
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Howard D. Strickler
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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Marc J. Gunter
Authors' Affiliations: Department of Epidemiology & Population Health, Albert Einstein College of Medicine, Bronx, New York; Epidemiology Research Program, American Cancer Society, Atlanta, Georgia; Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, Maryland; AARP, Washington, District of Columbia; and Department of Epidemiology and Biostatistics, School of Public Health, Imperial College, London, United Kingdom
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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

    ScenarioEvent 1Event 2Embedded Image
    Embedded ImageOR1Embedded ImageEmbedded ImageOR2Embedded Image
    X = 0X = 1X = 0X = 1
    (a)0.0300.0752.622.560.0300.0301.001.002.56
    (b)0.0100.0202.022.010.0150.0302.032.011.00
    (c)0.1500.2251.651.570.1500.1501.001.001.57
    (d)0.1500.1951.371.330.2020.2601.391.331.00
    (e)0.0200.0603.133.060.0300.0301.001.003.06
    (f)0.0300.0602.062.030.0400.0802.082.031.00
    (g)0.1000.2002.252.120.3000.3001.001.002.12
    (h)0.2000.2801.561.470.3000.4081.611.471.00

    NOTE: The OR for event k, that is, ORk 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, Embedded Image 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 [Embedded Image and Embedded Image in Eq. (2)] of OR2 and Embedded Image and Embedded Image

      ScenarioEvent 2Comparison
      OR2Embedded ImageEmbedded Image
      BiasaCoverageBiasCoveragePower/type I errorBiasCoveragePower/type I error
      (a)1.595.61.595.64.42.494.878.8
      (b)0.695.81.395.659.62.096.23.8
      (c)4.694.24.694.25.8−0.197.873.2
      (d)2.294.66.392.286.4−0.097.03.0
      (e)0.196.60.196.63.45.395.487.8
      (f)−0.495.21.896.494.25.695.44.6
      (g)2.994.02.994.06.04.396.299.6
      (h)2.694.412.379.499.88.896.43.6

      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 Embedded Image and Embedded Image

      Embedded ImageEmbedded Image
      BiasCoveragePower/type I errorBiasCoveragePower/type I error
      (a)0.995.24.82.494.280.4
      (b)0.095.859.21.996.04.0
      (c)−0.495.44.6−0.195.477.6
      (d)−0.694.682.0−0.094.45.6
      (e)0.996.63.43.895.587.8
      (f)−1.195.293.83.595.64.4
      (g)−0.194.85.20.794.499.8
      (h)−0.295.499.80.695.44.6

      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 modelBMI < 25.0BMI 25.0–29.9Relative OR (CI)BMI ≥ 30.0Relative OR (CI)PtrendP value difference in trend
        ER+/PR+
         Cases2863451(ref)3661(ref)
         OR (CI)1 (ref)1.589 (1.333–1.894)a2.100 (1.753–2.516)a<0.0001Ref
        ER−/PR−
         Cases84910.901710.577
         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)a0.2830.012
        ER−/PR+
         Cases945
         OR (CI)1(ref)
        ER+/PR−
         Cases97730.585610.4330.531<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 modelRelative HR (CI)Relative HR (CI)Relative HR and CI for trend
        ER+/PR+1 (ref)1.551 (1.304–1.845)a1 (ref)1.758 (1.471–2.101)1 (ref)<0.0001Ref
        ER−/PR−0.9250.5930.783
        1 (ref)1.434 (1.034–1.990)a(0.639–1.338)1.043 (0.729–1.493)(0.399–0.883)a0.693(0.654–0.938)a
        ER−/PR+0.3420.4280.631
        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−0.5550.4300.659
        1 (ref)0.861 (0.620–1.194)(0.384–0.802)a0.756 (0.532–1.075)(0.291–0.636)a0.113(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).

        • ↵aP < 0.05.

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      Cancer Epidemiology Biomarkers & Prevention: 22 (2)
      February 2013
      Volume 22, Issue 2
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      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 and Marc J. Gunter
      Cancer Epidemiol Biomarkers Prev February 1 2013 (22) (2) 275-285; DOI: 10.1158/1055-9965.EPI-12-1050

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      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 and Marc J. Gunter
      Cancer Epidemiol Biomarkers Prev February 1 2013 (22) (2) 275-285; DOI: 10.1158/1055-9965.EPI-12-1050
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        • Abstract
        • Introduction
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        • Appendix I: Proof of the Validity of a Polytomous Model to Compare Exposure–Disease Associations
        • Appendix II: A Method to Simulate an Independent Exponential Censoring Time
        • Appendix III: Data Structure and Program Code
        • Disclosure of Potential Conflicts of Interest
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