To the Editor:
Rothman’s sufficient-cause model is a useful construct for disease causation.1 It provides a synthesis of multiple interacting risk factors, jointly and collectively.1,2 It also helps to evaluate the impact of public-health interventions.1–5 Sufficient-cause modeling for unmatched case-control data2,5 and person-time data3,4 is simple and can be implemented through the generalized linear model (GENMOD) procedure in SAS (SAS Institute, Cary, NC); this approach has been applied successfully in cardiovascular2,5 and cancer epidemiology.3,4 For matched data, extra programming is needed. Here we present simple SAS codes (eAppendix, http://links.lww.com/EDE/A716) illustrated with two examples: a matched case-control study6 and a survival dataset7 requiring a time-matched risk-set analysis.
The unit of analysis is the matching set either confounder-matched (for matched case-control data) or time-matched (for survival data).5,8 For the former, the model is
is the disease odds for individuals at the
th matching set who have a risk-factor profile of
For the latter, the model is
is the disease rate of individuals at time
who have a risk-factor profile of
The “intercepts” of the models, the
, are treated as nuisance variables and will be eliminated in the model-fitting process (conditional likelihood for matched case-control data; partial likelihood for survival data).
Under the assumptions of no confounding, monotonicity, and independent competing causes, the β-coefficients of the models correspond directly to the completion-potential indices for the various classes of sufficient causes (one completion-potential index for one class of sufficient causes; the completion-potential index for the all-unknown class is 1.0 by definition).5 A small-scale simulation study (eAppendix, http://links.lww.com/EDE/A716) shows that the completion-potential estimates are approximately unbiased. With additional algebra, other sufficient-cause–related indices (such as the individual-based and the population-wide causal-pie weights) and the attributable-fraction indices (such as the population attributable fraction and the attributable fraction among the exposed) can all be calculated from these completion-potential indices (see eAppendix, http://links.lww.com/EDE/A716, for the definitions of these indices).5 Confidence intervals (CIs) for all estimates are based on the bootstrap method.2,5,8
The first example is Leisure World Study of Endometrial Cancer,6 a 1:4 matched case-control study with 63 matching sets. After model fitting, the main effect of estrogen use has a β-coefficient (which is also the completion-potential value for the class of sufficient causes containing estrogen use) of 7.0 (95% CI = 2.7–18). This implies that this particular class of sufficient causes is seven times as likely to cause the disease as the all-unknown class.
The second example is the Bone Marrow Transplant Patients Study,7 which followed 137 subjects for adverse outcomes after transplant surgery (leukemia relapse or death). The mean follow-up duration is 782 days with a total of 83 observed failures. The model shows a main effect of the French-American-British disease classification grade and an interactive effect of cytomegalovirus infection and methotrexate use. The β-coefficients are approximately the same for the completion-potential index for the French-American-British class (1.4 [95% CI = 0.6–3.0]) and for the interactive class between cytomegalovirus infection and methotrexate use (1.6 [0.6–3.8]). However, the population-wide causal-pie weights for these two are quite different (23% vs. 14%) (Figure). Other sufficient-cause–related indices and the attributable-fraction indices for these two examples are presented in eAppendix (http://links.lww.com/EDE/A716).
These easy-to-use SAS codes for sufficient-cause modeling with matched case-control and survival data should facilitate the use of sufficient-cause modeling.
Research Center for Genes, Environment and Human Health, Institute of Epidemiology and Preventive Medicine, College of Public Health, National Taiwan University, Taipei, Taiwan, firstname.lastname@example.org
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3. Liao SF, Lee WC, Chen HC, Chuang LC, Pan MH, Chen CJ. Baseline human papillomavirus infection, high vaginal parity, and their interaction on cervical cancer risks after a follow-up of more than 10 years. Cancer Causes Control. 2012;23:703–708
4. Liao SF, Yang HI, Lee MH, Chen CJ, Lee WC. Fifteen-year population attributable fractions and causal pies of risk factors for newly developed hepatocellular carcinomas in 11,801 men in Taiwan. PLoS One. 2012;7:e34–779
5. Lee WC. Completion potentials of sufficient component causes. Epidemiology. 2012;23:446–453
6. Mack TM, Pike MC, Henderson BE, et al. Estrogens and endometrial cancer in a retirement community. N Engl J Med. 1976;294:1262–1267
7. Klein JP, Moeschberger ML Survival Analysis: Techniques for Censored and Truncated Data. 1997 New York Springer-Verlag
8. Langholz B, Richardson DB. Fitting general relative risk models for survival time and matched case-control analysis. Am J Epidemiol. 2010;171:377–383