American Journal of Epidemiology Advance Access originally published online on May 22, 2008
American Journal of Epidemiology 2008 168(2):197-201; doi:10.1093/aje/kwn130
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PRACTICE OF EPIDEMIOLOGY |
Population Stratification Bias in the Case-Only Study for Gene-Environment Interactions
From the Research Center for Genes, Environment, and Human Health and the Graduate Institute of Epidemiology, College of Public Health, National Taiwan University, Taipei, Taiwan, Republic of China
Correspondence to Dr. Wen-Chung Lee, Room 536, 17 Xuzhou Road, Taipei 100, Taiwan, Republic of China (e-mail: wenchung{at}ntu.edu.tw).
Received for publication October 15, 2007. Accepted for publication April 15, 2008.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONPOPULATION STRATIFICATION BIAS...MAGNITUDE OF BIASDISCUSSIONReferences |
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The case-only study is a convenient approach and provides increased statistical efficiency in detecting gene-environment interactions. The validity of a case-only study hinges on one well-recognized assumption: The susceptibility genotypes and the environmental exposures of interest are independent in the population. Otherwise, the study will be biased. The authors show that hidden stratification in the study population could also ruin a case-only study. They derive the formulas for population stratification bias. The bias involves three terms: 1) the coefficient of variation of the exposure prevalence odds, 2) the coefficient of variation of the genotype frequency odds, and 3) the correlation coefficient between the exposure prevalence odds and the genotype frequency odds. The authors perform simulation to investigate the magnitude of bias over a wide range of realistic scenarios. It is found that the estimated interaction effect is frequently biased by more than 5%. For a rarer gene and a rarer exposure, the bias becomes even larger (>30%). Because of the potentially large bias, researchers conducting case-only studies should use the boundary formula presented in this paper to make more prudent interpretations of their results, or they should use stratified analysis or a modeling approach to adjust for population stratification bias in their studies.
bias (epidemiology); data interpretation, statistical; environment; epidemiologic methods; genetics
Abbreviations: CIR, confounding interaction ratio; CV, coefficient of variation; RR, relative risk
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONPOPULATION STRATIFICATION BIAS...MAGNITUDE OF BIASDISCUSSIONReferences |
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The interplay between susceptibility genotypes and environmental exposures on the risks of human diseases has important public health implications (1–3). The traditional epidemiologic designs, for example, the case-control study and the cohort study, can be applied to detect gene-environment interactions, but they require large sample sizes to reach enough power (4, 5). The "control-free" case-only study, on the other hand, is a convenient approach and provides increased statistical efficiency in detecting interactions (6–9).
The validity of a case-only study hinges on one well-recognized assumption: The susceptibility genotypes and the environmental exposures of interest are independent in the population (6, 10, 11). The assumption is reasonable, because genes are constitutional and often will not predict a subject's exposure profile that is acquired environmentally through life. On the other hand, counterexamples are not difficult to find. For example, a person with a strong family history of a particular disease is more likely to carry the risk genotype and at the same time may be more likely (because of the awareness of the disease) to avoid risk exposure, thus violating the gene-environment independence assumption.
However, the nonindependence of the susceptibility genotype that a person inherits and the environmental exposure that he or she will have is not the only condition that can invalidate a case-only study. In this paper, we show that hidden stratification in the study population could easily ruin a case-only study as well, even though within each and every one of the population strata, there is perfect gene-environment independence.
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POPULATION STRATIFICATION BIAS IN THE CASE-ONLY STUDY |
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TOPABSTRACTINTRODUCTIONPOPULATION STRATIFICATION BIAS...MAGNITUDE OF BIASDISCUSSIONReferences |
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Let E(
) indicate the presence(absence) of the environmental exposure of interest, and G(
), the presence(absence) of the genotype of interest. In a population composed of J strata (j = 1,...,J), we define pj to be the prevalence of E (
, the exposure prevalence odds); qj as the frequency of G (
, the genotype frequency odds); bj as the background disease risk; and mj as the total number of subjects in the jth stratum. In a comparison of risks with the (, )-subjects, we assume that the relative risks (RR) of disease are RREG for the (E, G)-subjects, RRE for the (E, )-subjects, and RRG for the (, G)-subjects. Note that we assume the RREG, RRE, and RRG to be constant across the strata. In other words, the stratification in the population only segregates people. If sufficiently long, the segregation could result in disparate exposure prevalences, genotype frequencies, and background disease risks in various population strata. However, it could not by itself modify the magnitudes of effects of exposures and/or genes in causing human diseases.
The gene-environment interaction effect (RRINT) on a multiplicative scale is 
. There is no gene-environment interaction if RRINT=1, and interaction exists when RRINT>1 (synergistic interaction) or RRINT<1 (antagonistic interaction). The RRINT is estimable by use of a case-only study. If a study collects each and every diseased subject (the cases) in the population, the case numbers would be 
for the (E, G)-subjects, 
for the (E, )-subjects, 
for the (, G)-subjects, and 
for the (, )-subjects (the environmental exposure and genotype are assumed to be independent within each and every stratum). Thus, the gene-environment interaction effect can be estimated as
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We use the confounding interaction ratio (CIR) to quantify the degree of population stratification bias of RR
in the case-only study. The CIR is defined as the ratio of the crude interaction effect and the true (or adjusted) interaction effect: 
. We found out, interestingly, that the mathematical formula for the CIR for the interaction effect is very similar to the confounding risk ratio (CRR) for the main genetic effect in our previous study (12).
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and 
denote the means; 
and 
, the standard deviations; CVE and CVG, the coefficients of variation of the exposure prevalence odds and the genotype frequency odds, respectively; and rEG is the correlation coefficient between the exposure prevalence odds and the genotype frequency odds. Note that 
is the "weight" used in the above summation. It is clear from the formula above for CIR that there would be no population stratification bias (CIR=1) if, across the strata, 1) the exposure prevalence odds and the genotype frequency odds are uncorrelated, 2) there is no variation in the exposure prevalence odds, or 3) there is no variation in the genotype frequency odds. Also, one notices that overestimation (CIR>1) would occur when the prevalence odds of exposure and the frequency odds of genotype are positively correlated, and underestimation (CIR<1) would occur when these are negatively correlated. Higher relative bias (CIR away from 1) exists with higher variation of the prevalence odds of exposure or the frequency odds of genotype.
Researchers often do not have such detailed knowledge of the study population, however. They may instead have an educated guess as to the magnitudes of the "variations." Let 
G (G
1) denote the ratio of the largest and the smallest genotype frequency odds, and E (E1), the ratio of the largest and the smallest exposure prevalence odds, among all the strata in the population. Assuming that G and E are known to researchers, it is easy to show (similar to the derivation of the bounds for the confounding risk ratio) (13) that such limited information is enough to set an upper (U) and a lower (L) bounds of the CIR:
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Note, again, that there is no bias (U=L=1) when there is either no variation in the genotype frequency odds (G=1) or no variation in the exposure prevalence odds (E=1). |
MAGNITUDE OF BIAS |
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TOPABSTRACTINTRODUCTIONPOPULATION STRATIFICATION BIAS...MAGNITUDE OF BIASDISCUSSIONReferences |
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We investigate the CIR in a case-only study over some realistic ranges of genotype frequency, exposure prevalence, and background disease risk. We follow the procedure of Wacholder et al. (14) (with some modifications) by assuming that, in a population, there are eight strata of equal size but possibly with different values of genotype frequency, exposure prevalence, and background disease risk. The ranges of the genotype frequencies and the exposure prevalence considered are 0.01–0.30, 0.10–0.40, or 0.30–0.60. Note that the variations in terms of the difference between the largest and the smallest values are
0.3 for all the situations considered, but that the variations in terms of odds ratios of the largest and the smallest values are extremely disparate: 42.43 (for 0.01–0.30), 6.00 (for 0.10–0.40), and 3.50 (for 0.30–0.60). The risk ratio ranges of the background disease considered are 1.0–1.5 or 1.0–3.0. Within the indicated ranges above, the genotype frequency and the exposure prevalence are spaced to be equidistant on a logistic scale (for eight values), while the background disease risk ratio is spaced to be equidistant on a logarithmic scale (also for eight values). In the simulation, we fix the background disease risk ratio for the eight strata, and in each round of the simulation, we randomly permute the eight values for the genotype frequencies and the eight values for the exposure prevalences to the eight strata. The simulation was done for a total of 100,000 times. Using the equations in the previous section, we calculate the CIRs for all the simulations and show the 25th, 50th, and 75th percentiles in table 1. The same table also presents the upper and the lower bounds of CIR over these ranges.
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From table 1, we see that the estimated interaction effect in a case-only study is frequently biased by more than 5 percent (CIR
1.05 or 
0.95), a nonnegligible amount of bias. For a rarer gene and/or a rarer exposure, the bias becomes larger. For example, with a genotype frequency range of 0.01–0.30 and an exposure prevalence range of 0.01–0.30, the bias is usually over 30 percent. The theoretical analysis of the bounds of CIR reveals that up to 1,000 percent bias is possible. On the other hand, the magnitude in the variation in background disease risks has no influence on the magnitude of bias. |
DISCUSSION |
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The impacts of population stratification bias on the "main genetic effect" in a case-control study have been extensively studied in the literature, both theoretically and empirically (12–19). It is of interest to compare the results of the "main-effect/case-control" in those studies and the "interaction/case-only" in the present study. On the surface, the biases in both situations appear similar, viewed through their respective mathematical formulas (13). The variation in the genotype frequency odds is involved in both situations. The variation in the exposure prevalence odds is involved in the present interaction/case-only situation but not in the main-effect/case-control situation, and the variation in the background disease risks is to the contrary. However, the magnitudes of the biases in the two situations are drastically different: Wacholder et al. (14) reported that the main-effect/case-control bias is generally no more than 1 percent, whereas our study showed that the interaction/case-only bias is frequently over 5 percent and could be alarmingly larger. The reason for this disparity lies in that "minor" difference in formulas: the variation in "background disease risks" (main effect/case-control) versus the variation in "exposure prevalence odds" (interaction/case-only). It turns out that the measurement scale of an "odds" acts as a bias magnifier: What appears to be a nothing-out-of-the-ordinary variation in exposure prevalence from 0.1 to 0.4 can easily translate into a sixfold variation when measured in prevalence odds.
Wang et al. (20) studied the impacts of population stratification bias on gene-environment interaction in a case-control study, through computer simulations. They concluded that the interaction/case-control bias is small to nonexistent. This is in sharp contrast to our interaction/case-only situation, where the bias is real and isn't to be taken lightly. A possible explanation for this disparity is that the exposed and the unexposed subgroups of a case-control study suffer from a similar population stratification bias. Therefore, the bias in a gene-environment interaction and the ratio of the main genetic effects between the exposed and the unexposed cancel each other out. By contrast, our interaction/case-only situation does not have two subgroups to cancel each other out.
In this paper, we have shown that, in addition to the bias due to gene-environment nonindependence, the bias due to population stratification is also a major threat to the validity of a case-only study. One may argue that the latter is but a special case for the former, because population stratification by itself also creates gene-environment nonindependence at the population level. However, we believe that distinguishing these two types of biases is important. For the "ordinary" gene-environment nonindependence bias, the researchers should try to evaluate on the basis of their subject-matter knowledge whether the gene under study will influence the exposure status, from the point of view of an individual subject in the population. For the population stratification bias, the attention should, instead, be paid to the study population itself. If the researchers have some rough guess about the ratio of the largest and the smallest genotype frequency odds (G) and that of the exposure prevalence odds (E), they can use the boundary formula in this paper to set the limits of the population stratification bias. This would help for a more prudent interpretation of the results. To really achieve control of the two types of biases in a case-only study, a stratified analysis (11) or a modeling approach (21, 22) should be used, adjusting for any confounding factor causing the gene-environment nonindependence in the population, as well as any variable, such as race, ethnicity, nationality, ancestry, and birthplace, that helps to delineate population strata.
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ACKNOWLEDGMENTS |
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This paper is partly supported by funding from the National Science Council, Taiwan, Republic of China (grants NSC 95-2314-B-002-242, NSC 95-3114-P-002-005-Y, and NSC 96-2314-B-002-143).
Conflict of interest: none declared.
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