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A note on bias due to fitting prospective multivariate generalized linear models to categorical outcomes ignoring retrospective sampling schemes


  • Mukherjee, Bhramar
  • Liu, Ivy


Outcome-dependent sampling designs are commonly used in economics, market research and epidemiological studies. Case-control sampling design is a classic example of outcome-dependent sampling, where exposure information is collected on subjects conditional on their disease status. In many situations, the outcome under consideration may have multiple categories instead of a simple dichotomization. For example, in a case-control study, there may be disease sub-classification among the "cases" based on progression of the disease, or in terms of other histological and morphological characteristics of the disease. In this note, we investigate the issue of fitting prospective multivariate generalized linear models to such multiple-category outcome data, ignoring the retrospective nature of the sampling design. We first provide a set of necessary and sufficient conditions for the link functions that will allow for equivalence of prospective and retrospective inference for the parameters of interest. We show that for categorical outcomes, prospective-retrospective equivalence does not hold beyond the generalized multinomial logit link. We then derive an approximate expression for the bias incurred when link functions outside this class are used. Most popular models for ordinal response fall outside the multiplicative intercept class and one should be cautious while performing a naive prospective analysis of such data as the bias could be substantial. We illustrate the extent of bias through a real data example, based on the ongoing Prostate, Lung, Colorectal and Ovarian (PLCO) cancer screening trial by the National Cancer Institute. The simulations based on the real study illustrate that the bias approximations work well in practice.

Suggested Citation

  • Mukherjee, Bhramar & Liu, Ivy, 2009. "A note on bias due to fitting prospective multivariate generalized linear models to categorical outcomes ignoring retrospective sampling schemes," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 459-472, March.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:3:p:459-472

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    References listed on IDEAS

    1. Chatterjee, Nilanjan, 2004. "A Two-Stage Regression Model for Epidemiological Studies With Multivariate Disease Classification Data," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 127-138, January.
    2. N. E. Breslow & N. Chatterjee, 1999. "Design and analysis of two-phase studies with binary outcome applied to Wilms tumour prognosis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 48(4), pages 457-468.
    3. Shaun R. Seaman, 2004. "Equivalence of prospective and retrospective models in the Bayesian analysis of case-control studies," Biometrika, Biometrika Trust, vol. 91(1), pages 15-25, March.
    4. Zhang, Biao, 2006. "Prospective and retrospective analyses under logistic regression models," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 211-230, January.
    5. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    6. Wang, C. Y. & Wang, Suojin & Carroll, R. J., 1997. "Estimation in choice-based sampling with measurement error and bootstrap analysis," Journal of Econometrics, Elsevier, vol. 77(1), pages 65-86, March.
    7. John M. Neuhaus, 2002. "Bias due to Ignoring the Sample Design in Case-Control Studies," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 44(3), pages 285-293, September.
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