We investigated bias produced by incorrectly identifying the group with zero exposure prevalence. The observed difference in bias sensitivity between the two AF estimators was small and may not be of practical importance. Nevertheless, the bias in FIGURE T (Eq 14) can be viewed as the upper limit of the (negative) bias in FIGURE L in the absence of confounding and effect modification. This is true unless errors in the estimated exposure prevalences reverse the direction of the estimated effect. Even small proportions of exposed subjects within the group regarded as truly unexposed produced severe bias of both AF estimates and decreased coverage of corresponding CIs. As highlighted by the empirical example, it is crucial when establishing exposure databases that the exposure prevalence is assessed also for groups in which the exposure is expected to be rare, and not regard these simply as zero.
When estimating the AF based on the exposure proportion of the cases, lacking representativeness of the case series may produce attenuations as well as exaggerations of the AF estimate even if the OR estimate is unbiased. 27 Confounding across groups can be adjusted for in partially ecologic settings if individual data on confounders are available. It should be noted, however, that the additive-relative OR model (Eq 13) provides a valid confounder adjustment only if each exposure probability x is constant across various levels of the confounders, ie, when there is no confounding within groups. Any residual confounding across or within groups may hide, alter, or even spuriously create a nonzero AF. However, the partially ecologic case-control design facilitates a more detailed grouping of the population, which may reduce such ecologic bias 43 and makes the design far more attractive than the traditional pure ecologic design.
Maria Albin and Timo Kauppinen gave access to the empirical dataset presented in the text.
Appendix A: Derivation of vâr(AF L)
Let S ca and S co denote the sum of the observed exposure probabilities among the cases and controls, respectively, and let MATH
The mean exposure probability among the controls is MATHwhere n is the number of controls. The logit transformation of AF L (Eq 6) is MATH
Thus, by assuming that the control selection is such that is a valid estimate of the exposure prevalence JOURNAL/epide/04.02/00001648-200207000-00015/ENTITY_OV0335/v/2017-07-26T080015Z/r/image-png in the population, it follows that MATH
If lnˆ;β is an unbiased estimator of the true value lnβ and U( lnβ) is the efficient score evaluated at lnβ, then it can be shown that asymptotically, subject to certain regularity conditions, 31,44 MATH
For binary regression models, 45 such as the linear OR model MATH
Thus, MATHand hence MATH
Using the delta method, 32 it follows that MATHand, furthermore, MATHand MATH
Accordingly, var(FIGURE L) can be estimated as MATH
Appendix B: The Distribution of the Exposure Probabilities Estimated from the Case Series
The exposure probability for the cases in group j, x ′j, satisfies 27 MATHand MATHsuch that MATH
Thus, x ′j >x j if β > 0 and 0 <x j < 1, ie, when there is an harmful effect of exposure, cases are more likely to have been exposed than controls within the same group. As a result, the overall exposure prevalence among the cases is MATHwhere η′j is the proportion of cases that belongs to group j.
Under the linear OR model (Eq 4), and as a reasonable approximation under the additive-relative OR model (Eq 13), the group-specific proportions satisfy 12 MATHwhich is equivalent with MATH
Given that ∑j = 0 J − 1 ηj = 1, some algebraic manipulations yield MATH
Thus, ηj (j = 0, 1, ..., J − 1) and hence var(x) can be estimated on the basis of the observed distribution of the cases in the various groups.
1. Björk J, Strömberg U. Effects of systematic exposure assessment errors in partially ecologic case-control studies. Int J Epidemiol 2002; 31: 154–160.
2. Künzli N, Tager IB. The semi-individual study in air pollution epidemiology: a valid design as compared to ecologic studies. Environ Health Perspect 1997; 105: 1078–1083.
3. Webster T. Can semi-individual studies have ecologic bias? Epidemiology 2000; 11: S95.
4. Cocco P, Dosemeci M, Heineman EF. Occupational risk factors for cancer of the central nervous system: a case-control study on death certificates from 24 U.S. states. Am J Ind Med 1998; 33: 247–255.
5. Heineman EF, Olsen JH, Pottern LM, Gomez M, Raffn E, Blair A. Occupational risk factors for multiple myeloma among Danish men. Cancer Causes Control 1992; 3: 555–568.
6. Ji BT, Silverman DT, Stewart PA, et al
. Occupational exposure to pesticides and pancreatic cancer. Am J Ind Med 2001; 39: 92–99.
7. Alguacil J, Kauppinen T, Porta M, et al
. Risk of pancreatic cancer and occupational exposures in Spain. Ann Occup Hyg 2000; 44: 391–403.
8. Benke G, Sim M, Fritschi L, Aldred G, Forbes A, Kauppinen T. Comparison of occupational exposure using three different methods: hygiene panel, job exposure matrix (JEM), and self reports. Appl Occup Environ Hyg 2001; 16: 84–91.
9. Kauppinen T, Partanen T, Degerth R, Ojajarvi A. Pancreatic cancer and occupational exposures. Epidemiology 1995; 6: 498–502.
10. Hammar N, Alfredsson L, Johnson JV. Job strain, social support at work, and incidence of myocardial infarction. Occup Environ Med 1998; 55: 548–553.
11. Johnson JV, Stewart W, Hall EM, Fredlund P, Theorell T. Long-term psychosocial work environment and cardiovascular mortality among Swedish men. Am J Public Health 1996; 86: 324–331.
12. Bouyer J, Hémon D. Comparison of three methods of estimating odds ratios from a job exposure matrix in occupational case-control studies. Am J Epidemiol 1993; 137: 472–481.
13. Bouyer J, Hémon D. Studying the performance of a job exposure matrix. Int J Epidemiol 1993; 22 (suppl 2): S65–S71.
14. Stewart W, Correa-Villasenor A. False positive exposure errors and low exposure prevalence in community-based case-control studies. Appl Occup Environ Hyg 1991; 6: 534–540.
15. Siemiatycki J, Dewar R, Richardson L. Costs and statistical power associated with five methods of collecting occupation exposure information for population-based case-control studies. Am J Epidemiol 1989; 130: 1236–1246.
16. Kauppinen TP, Mutanen PO, Seitsamo JT. Magnitude of misclassification bias when using a job-exposure matrix. Scand J Work Environ Health 1992; 18: 105–112.
17. Björk J, Albin M, Welinder H, et al
. Are occupational, hobby, or lifestyle exposures associated with Philadelphia chromosome-positive chronic myeloid leukemia? Occup Environ Med 2001; 58: 722–727.
18. Bobak M, Leon DA. The effect of air pollution on infant mortality appears specific for respiratory causes in the postneonatal period. Epidemiology 1999; 10: 666–670.
19. Ihrig MM, Shalat SL, Baynes C. A hospital-based case-control study of stillbirths and environmental exposure to arsenic using an atmospheric dispersion model linked to a geographical information system. Epidemiology 1998; 9: 290–294.
20. Nyberg F, Gustavsson P, Jarup L, et al
. Urban air pollution and lung cancer in Stockholm. Epidemiology 2000; 11: 487–495.
21. Biggeri A, Barbone F, Lagazio C, Bovenzi M, Stanta G. Air pollution and lung cancer in Trieste, Italy: spatial analysis of risk as a function of distance from sources. Environ Health Perspect 1996; 104: 750–754.
22. Pershagen G, Rylander E, Norberg S, Eriksson M, Nordvall SL. Air pollution involving nitrogen dioxide exposure and wheezing bronchitis in children. Int J Epidemiol 1995; 24: 1147–1153.
23. Greenland S. Applications of stratified analysis methods. In: Rothman K, Greenland S, eds. Modern Epidemiology. 2nd ed. Philadelphia: Lippincott-Raven, 1998; 459–480.
24. Walter SD. Effects of interaction, confounding and observational error on attributable risk estimation. Am J Epidemiol 1983; 117: 598–604.
25. Hsieh CC, Walter SD. The effect of non-differential exposure misclassification on estimates of the attributable and prevented fraction. Stat Med 1988; 7: 1073–1085.
26. Wacholder S, Benichou J, Heineman EF, Hartge P, Hoover RN. Attributable risk: advantages of a broad definition of exposure. Am J Epidemiol 1994; 140: 303–309.
27. Bruzzi P, Green SB, Byar DP, Brinton LA, Schairer C. Estimating the population attributable risk for multiple risk factors using case-control data. Am J Epidemiol 1985; 122: 904–914.
28. Greenland S. Attributable fractions: bias from broad definition of exposure. Epidemiology 2001; 12: 518–520.
29. Beral V, Chilvers C, Fraser P. On the estimation of relative risk from vital statistical data. J Epidemiol Community Health 1979; 33: 159–162.
30. Rothman K, Greenland S. Case-control studies. In: Rothman K, Greenland S, eds. Modern Epidemiology. 2nd ed. Philadelphia: Lippincott-Raven, 1998; 93–114.
31. Greenland S. Variance estimators for attributable fraction estimates, consistent in both large strata and sparse data. Stat Med 1987; 6: 701–708.
32. Efron B, Tibshirani RJ. An Introduction to the Bootstrap. New York: Chapman and Hall, 1993.
33. Coughlin S, Benichou J, Weed D. Attributable risk estimation in case-control studies. Epidemiol Rev 1994; 16: 51–64.
34. Kuritz SJ, Landis JR. Attributable risk ratio estimation from matched-pairs case-control data. Am J Epidemiol 1987; 125: 324–328.
35. Greenland S, Morgenstern H. Ecological bias, confounding, and effect modification. Int J Epidemiol 1989; 18: 269–274.
36. Greenland S, Robins J. Invited commentary: ecologic studies—biases, misconceptions, and counterexamples. Am J Epidemiol 1994; 139: 747–760.
37. Draper N, Smith H. Applied Regression Analysis. New York: Wiley, 1981.
38. Wacholder S. When measurement errors correlate with truth: surprising effects of nondifferential misclassification. Epidemiology 1995; 6: 157–161.
39. Benichou J. Methods of adjustment for estimating the attributable risk in case-control studies: a review. Stat Med 1991; 10: 1753–1773.
40. Albin M, Björk J, Welinder H, et al
. Acute myeloid leukemia and clonal chromosome aberrations in relation to past exposure to organic solvents. Scand J Work Environ Health 2000; 26: 482–491.
41. Kauppinen T, Toikkanen J, Pukkala E. From cross-tabulations to multipurpose exposure information systems: a new job-exposure matrix. Am J Ind Med 1998; 33: 409–417.
42. Nakayama T. Under-reporting of attributable risk and reporting of the risk ratio in epidemiologic literature. Epidemiology 2000; 11: 366–367.
43. Morgenstern H. Ecologic studies. In: Rothman K, Greenland S, eds. Modern Epidemiology. 2nd ed. Philadelphia: Lippincott-Raven, 1998; 459–480.
44. Cox RD, Hinkley DV. Theoretical Statistics. New York: Chapman and Hall, 1974.
45. Hosmer DW, Lemeshow S. Applied Logistic Regression. New York: Wiley, 1989.