Correction of Selection Bias in Survey Data: Is the Statistical Cure Worse Than the Bias?

摘要

In previous articles in the American Journal of Epidemiology ( Am J Epidemiol . 2013;177(5):431–442) and American Journal of Public Health ( Am J Public Health . 2013;103(10):1895–1901), Masters et al. reported age-specific hazard ratios for the contrasts in mortality rates between obesity categories. They corrected the observed hazard ratios for selection bias caused by what they postulated was the nonrepresentativeness of the participants in the National Health Interview Study that increased with age, obesity, and ill health. However, it is possible that their regression approach to remove the alleged bias has not produced, and in general cannot produce, sensible hazard ratio estimates. First, we must consider how many nonparticipants there might have been in each category of obesity and of age at entry and how much higher the mortality rates would have to be in nonparticipants than in participants in these same categories. What plausible set of numerical values would convert the (“biased”) decreasing-with-age hazard ratios seen in the data into the (“unbiased”) increasing-with-age ratios that they computed? Can these values be encapsulated in (and can sensible values be recovered from) one additional internal variable in a regression model? Second, one must examine the age pattern of the hazard ratios that have been adjusted for selection. Without the correction, the hazard ratios are attenuated with increasing age. With it, the hazard ratios at older ages are considerably higher, but those at younger ages are well below one. Third, one must test whether the regression approach suggested by Masters et al. would correct the nonrepresentativeness that increased with age and ill health that I introduced into real and hypothetical data sets. I found that the approach did not recover the hazard ratio patterns present in the unselected data sets: the corrections overshot the target at older ages and undershot it at lower ages. This article was jointly published in the American Journal of Epidemiology ( Am J Epidemiol . 2017;185(6):409–411) and the American Journal of Public Health ( Am J Pub Health . 2017;107(4):503–505). The Editors of the American Journal of Public Health ( AJPH ) and the American Journal of Epidemiology ( AJE ) asked me to comment on hazard ratio calculations in two articles by Masters et al. 1,2 At issue are age-specific hazard ratios for the differences in mortality rates between obesity categories. The hazard ratio estimates in the AJE article 2 were derived using Cox regression models that allowed for different hazard ratios for different age intervals; those in the subsequent AJPH article 1 were obtained using smooth-in-age hazard ratio models. These estimates deserve scrutiny because they were derived from a statistical approach that the authors claimed corrected the observed hazard ratios for selection bias, 3 that is, a distortion caused by what they postulated was the nonrepresentativeness of the participants in the National Health Interview Study (NHIS) that increased with age, obesity, and ill health. In the AJPH article, the authors used these corrected hazard ratios to arrive at population attributable fractions (a secondary topic that is addressed in Appendices 1 and 2, available as supplements to the online version of this article at http://www.ajph.org ). The way Masters et al. arrived at the age-specific hazard ratios was addressed in their response to a letter to the editor in the AJE 4 and was also addressed in an unanswered letter to the editor in the AJPH . 5 Appendix 3 and Figure A (available as supplements to the online version of this article at http://www.ajph.org ) show my simple reanalyses of some of the NHIS data and the age-specific hazard ratios (which were very similar to those of Masters et al.) that I obtained using their correction for selection bias. However, it is possible that the way they removed the alleged bias has not produced, and in general cannot produce, sensible hazard ratio estimates. Masters et al. postulated that the nonrepresentativeness of the NHIS participants, which increased with age and ill health, distorted the age-specific mortality gradients across the body mass index (BMI) categories: Participants who were older at entry and in the higher-risk BMI categories would have been healthier than their same-age counterparts who did not participate. Ideally, to directly and accurately adjust the observed age-specific hazard ratios for this discrepancy, one would (1) add appropriate (specific to age at entry) numbers of nonparticipants to each of the BMI categories, (2) assign them higher mortality rates than those seen in persons in the same category who did participate, and (3) calculate a new (higher) rate for each category. It is difficult to see how estimates of these additional (age-at-entry–specific sets of) quantities implicated in the selection bias can be extracted from the NHIS data simply by using a regression model. Masters et a