Bias in Association Studies Resulting from Gene-Environment Interactions and Competing Risks

Alexis Elbaz and Annick Alpérovitch

From the Institut National de la Santé et de la Recherche Médicale (INSERM), Unit 360, Paris, France.


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
The etiology, particularly the genetic basis, of multifactorial late-onset diseases is the subject of many genetic epidemiologic studies. The authors' aim in this paper was to investigate the circumstances under which competing risks can lead to bias in studies of genetic susceptibility to late-onset diseases. The authors used a model built in an epidemiologic framework to show that when a genetic risk factor and an environmental risk factor interact to cause a frequent competing risk of death, the measure of the association between the disease under investigation and the genetic risk factor will be biased if the environmental risk factor is also associated with the latter disease and is omitted from the analysis. This is an example of confounding bias, and it is the consequence of an association between the genetic risk factor and the environmental risk factor that appears over time. Numerical examples show that under certain conditions this bias can be substantial. The authors present several patterns of association in favor of such a bias. Because competing risks of death are likely to be present in older subjects, researchers studying the etiology of late-onset diseases should be aware of the possibility of this bias.

association; bias (epidemiology); competing risks; interaction; observational study

Abbreviations: RH, relative hazard


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
Many diseases are considered multifactorial, i.e., caused by multiple environmental and genetic factors. Available data and pathophysiologic knowledge suggest that gene-environment interactions are not just a theoretical concept, and methodological papers have addressed the issue of an interaction between genetic and environmental risk factors (1GoGo–3Go).

Prospective and retrospective observational studies (also known as "association studies" in the genetic literature) are a popular approach to investigating the relation between genetic factors and diseases, as well as gene-environment interactions (2Go, 4Go). Association studies have been carried out to explore the genetic basis of several late-onset diseases (e.g., Alzheimer's disease, Parkinson's disease, stroke), sometimes producing conflicting or unexpected findings. A possible explanation for these discrepancies may be competing risks of death.

The present work is an extension of previous work to genetic epidemiologic studies (5Go, 6Go). It was motivated by a previous study on stroke (7Go) and by an increasing interest in gene-environment interactions. Our aim was to investigate circumstances under which competing risks can lead to bias in studies of genetic susceptibility to late-onset diseases. In this paper, we present an epidemiologic model and give numerical examples to provide a sense of the magnitude of the biases.


    METHODS
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
The model
We consider a closed cohort of n subjects of the same age with a follow-up starting at time t0. Subjects may suffer from diseases D1 and D2, with respective instantaneous hazards {lambda}1(t) and {lambda}2(t). We suppose that D1 is a constantly and immediately lethal condition, whereas D2 does not increase the risk of death. Subjects with D2 are still at risk of D1, whereas those with D1 are not at risk of D2. Thus, D1 is a competing risk for D2; {lambda}2(t) is the hazard of D2 in the presence of a competing risk of death (D1) (8Go, 9Go).

We consider two dichotomous factors, one environmental (E) and one genetic (G). The subscripts E and G denote the presence of these factors, and and denote their absence. We assume that E and G are independent at time t0, with initial frequencies PE(t0) and PG(t0). At t0, there are nE and nG subjects exposed to only one of the risk factors, nGE subjects exposed to both risk factors, and n subjects exposed to none. Let {lambda}ij(t) denote the instantaneous hazard of Di (i = 1, 2) in the jth exposure stratum (j = , G, E, GE).

Under a proportional hazards model, RHij = {lambda}ij(t)/{lambda}i(t) is the relative hazard (RH) of Di for the jth category. A demonstration based on an additive model is presented in appendix 1. When E has not been measured or is unknown, the crude relative hazard of D2 associated with G may vary over time and is



(1)
where PE/G(t), PE/(t), P/G(t), and P/(t) are conditional frequencies of the environmental factor given the genetic factor at time t and Sj(t) is the probability of surviving without D1 and D2 from time t0 to time t in the jth stratum, given that the person was disease-free and alive at t0 (appendix 2).

Conditions under which no confounding is present
Our aim is to determine when the crude relative hazard of D2 associated with G remains constant over time and equal to the relative hazard of D2 associated with G among subjects unexposed to E (i.e., the relation between G and D2 is not confounded by E). We assume that E and G do not interact on a multiplicative scale for D2. Then, RH2G(t) = RH2G for all t is equivalent to (RH2E - 1)(PE/G(t) - PE/(t)) = 0. This expression is true for all t if 1) E has no effect among persons unexposed to G (RH2E = 1) or 2) E and G remain independent in the population at each time point (PE/G(t) = PE/(t)). Using the formulas for PE/G(t) and PE/(t) given in appendix 2, the latter expression is equivalent to nnGES(t)SGE(t) - nGnESG(t)SE(t) = 0. Because E and G are initially independent, the odds ratio relating E and G is 1 (i.e., nnGE = nGnE), thus leading to

(2)
Expression 2 is true for all t if we have independence of the two effects on an additive scale for D1 and D2, that is,


(10Go).

We first consider the particular situation in which RH2E = 1 or RH2G = 1. In this case, and because there is no departure from multiplicative joint effects of E and G for D2, the expression {lambda}2GE(t) = {lambda}2G(t) + {lambda}2E(t) - {lambda}2(t) is true (11Go). Therefore, for expression 2 to be true, we only need to have independence of the two effects on an additive scale for D1; if so, RH2G(t) will remain constant and will be an unbiased estimate of RH2G for all t.

More generally (RH2E != 1 and RH2G != 1), because we postulated that there was no departure from multiplicative joint effects of E and G for D2, the expression {lambda}2GE(t) = {lambda}2G(t) + {lambda}2E(t) - {lambda}2(t) will not be true: RH2G(t) will not remain constant and equal to RH2G for all t, even if E and G have additive joint effects for D1. This bias is the consequence of omitting a risk factor associated with the disease. If incidences follow a proportional hazards model, omitting a variable associated with the disease leads to a model which is misspecified and not proportional (12Go). However, our numerical examples show that even if RH2G(t) does not remain strictly constant over time, this bias is small for usual values of disease cumulative risks (<=20 percent) (appendix 3).

Thus, because this bias related to misspecification of the model is small and can usually be considered negligible, we conclude that RH2G(t) is a correct measure of the association between D2 and G, when E and G do not interact on an additive scale for D1.

Conditions under which confounding is present
We now focus on studying the bias in estimating RH2G(t) when there is an interaction between E and G on an additive scale in causing D1. To be consistent with the proportional hazards model, we write

so RHiGE is constant, with RHiGE = RHiG + RHiE + xi - 1, where xi is an interaction parameter measuring the departure from additive joint effects (11Go). In terms of survival probabilities, we can write

where

and

After replacing SGE(t) in equation 1, we can show that if RH2E is greater than 1, RH2G(t) is a decreasing function of {Omega}(t): For positive values of {Omega}(t), RH2G(t) underestimates RH2G, while for negative values, RH2G(t) overestimates RH2G. When RH2E is less than 1, opposite results are found.

We first consider the particular situation in which G and D2 are not associated (RH2G = 1). The relation {lambda}2GE(t) = {lambda}2G(t) + {lambda}2E(t) - {lambda}2(t) is true (so x2 = {Omega}2(t) = 0 and {Omega}(t) = {Omega}1(t)). The bias in estimating the relation between G and D2 depends exclusively on the interaction on an additive scale between E and G for D1.

When there is an association between G and D2 (RH2G != 1), the bias in estimating the relation between G and D2 depends on {Omega}1(t) and {Omega}2(t). The term {Omega}2(t) represents the bias due to the misspecification of the model mentioned above; if E and G do not interact for D2 on a multiplicative scale, {Omega}2(t) is small (appendix 3). In addition, when the incidence of D2 is small compared with that of D1, {Omega}2(t) is negligible compared with {Omega}1(t). Thus, as illustrated below (see "Numerical examples" section), the values of {Omega}(t) and {Omega}1(t) are close, and the bias mainly depends on the departure from additive effects of E and G for D1.

The interaction on an additive scale between E and G for D1 results in an association between both factors appearing over time. The difference in the frequency of G among subjects at risk exposed and unexposed to E is proportional to exp(-{Omega}(t)) - 1. Thus, if there is no interaction, E and G will remain independent, whereas in the case of positive interaction, the frequency of G among subjects at risk who are exposed to E will be lower than its frequency among subjects at risk who are unexposed to E(PG/E(t) < PG/(t)). Opposite results are true in the case of negative interaction (PG/E(t) > PG/(t)). The crude RH2G(t) is a biased measure because of the correlation that appears between E and G over time. A positive interaction between E and G results in an increasing depletion of individuals with both risk factors from the population. The decrease in the frequency of G will be more pronounced among persons exposed to E. Thus, if E is associated with D2, the frequency of G will be lower among those who are more likely to suffer from D2, leading to underestimation of the association between G and D2.

We present this model using relative hazards because they are the most popular measure of association. Similar results are obtained using an additive model (appendix 1). The main difference is that the bias related to the misspecification of the model does not occur. In an additive model, the bias in estimating the relation between G and D2 depends exclusively on the departure from additive joint effects for D1.


    NUMERICAL EXAMPLES
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
We estimated the relation between D2 and G for different values of the model parameters, when mortality from D1 is not uncommon and D2 incidence is lower than D1 mortality. We assumed that D1 mortality and D2 incidence follow a Weibull model in each exposure category (13Go) and that the shape parameters are equal to {gamma} for any ij, so that the proportional hazards hypothesis holds. We selected ischemic heart disease as D1. We used age-specific mortality rates (14Go) and a likelihood maximization procedure for nonlinear models (Proc NLIN in SAS) (15Go) to estimate parameters {gamma} and {lambda}0,1 for D1, with the following assumptions: PE(t0) = PG(t0) = 0.3; E and G are initially independent; RH1G = RH1E = 2.00; and there is no interaction on a multiplicative scale. The remaining parameters ({lambda}0,1j) were deduced from these estimates. The estimated values (0,1 = 9.28 x 10-3; = 7.638) resulted in good agreement between observed and calculated mortality rates (data not shown). Sensitivity analysis showed that deviations from our assumptions had little effect on parameter estimates. Finally, we assumed that the D2 instantaneous hazard was 10 times lower than the D1 mortality rate in each exposure group. The values of RH2G(t) were calculated using equation 1 for different values of the following parameters: the cumulative risk R1(t) of suffering from D1 before time t, RH2E, PE(t0), PG(t0), and the interaction between G and E for D1 on an additive scale (x1). Tables 1Go3 give numerical illustrations of the bias in the estimation of the association between G and D2, when E and G interact positively (tables 1 and 2) or negatively (table 3) for D1 and when G is (table 2) or is not (tables 1 and 3) a risk factor for D2.


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TABLE 1. Crude relative hazard of D2 associated with G(RH2G(t)) for different values of the cumulative risk of D1 up to time t (R1(t)), of the positive interaction between E and G for D1 (x1), and of RH2E, when RH2G = 1*

 

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TABLE 2. Crude relative hazard of D2 associated with G (RH2G(t)) for different values of the cumulative risk of D1 up to time t (R1(t)), of the positive interaction between E and G for D1 (x1), and of RH2E, when RH2G = 2*

 

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TABLE 3. Crude relative hazard of D2 associated with G(RH2G(t)) for different values of the cumulative risk of D1 up to time t (R1(t)), of the negative interaction between E and G for D1 (x1), and of RH2E, when RH2G = 1*

 
When R1(t), RH2E, and |x1| increase, RH2G(t) underestimates (positive interactions) or overestimates (negative interactions) RH2G. There is no important bias when R1(t) is small. It becomes nonnegligible when x1 >= 5 and R1(t) is greater than 5 percent. For weak or moderate associations between E and D2 (RH2E <= 2), the bias only appears for strong interactions and high risks of D1. When RH2G = 1, there is no bias if x1 = 0. When x1 = 1 (no interaction on a multiplicative scale), the bias is small, even for high R1(t) values. Thus, although strictly speaking the bias occurs in the case of departures from additive joint effects, it only becomes substantial for departures from multiplicative joint effects.

Figure 1 shows the influence of the initial frequencies of E and G. For a given PG(t0), the bias increases with increasing PE(t0) up to a certain value and then decreases more slowly. For a given PE(t0), higher frequencies of G are associated with smaller biases. This is more pronounced when E is moderately frequent. For extreme frequencies of E, variations in the frequency of G have little impact on the magnitude of the bias.



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FIGURE 1. Value of the crude relative hazard (RH) of D2 associated with G(RH2G(t)) for different values of the initial frequencies of G(PG(t0)) and E (PE(t0))(R1(t) = 0.1, x1 = 10, RH2E = 4, and RH2G = 1).

 

    DISCUSSION
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
In this paper, we have shown that gene-environment interactions can bias the measure of the association between a disease and a genetic factor, when a frequent competing risk of death is involved and an environmental risk factor is omitted. If a genetic factor and an environmental factor interact on an additive scale in causing a disease that results in increased mortality, an association between these two factors will appear in the population over time. There is potential for confounding in the analysis of the relation between the genetic factor and another disease, if the environmental risk factor is also associated with this disease and is not taken into account in the analysis. In the case of rare competing risks, environmental risk factors with weak effects, and small interactions, the bias is likely to be small. We studied this bias in the context of cohort studies; because the odds ratio is an estimate of the relative hazard even in the presence of competing risks (16GoGo–18Go), our findings also apply to case-control studies.

Slud and Byar (5Go) previously identified this bias. They showed mathematically that omitting a risk factor can result in erroneous conclusions about the effect of another risk factor when dependent causes of death are present. Schatzkin and Slud (6Go) gave two numerical examples based on pulmonary cancer, coronary artery disease, and cholesterol level to offer an explanation for a negative relation between cholesterol level and pulmonary cancer.

Our model was built in the context of an association study, with the genetic factor being the one under investigation. It can be extended to interactions between two environmental factors, between two genetic factors, or between a genetic factor and an environmental factor, the latter being the one under investigation. For instance, in the Framingham Study, a surprising inverse relation was found between fat intake and stroke risk (19Go). Since myocardial infarction occurs at a younger mean age than stroke and has a higher incidence and mortality, and because both diseases share several environmental risk factors, stroke is a good example of a disease for which this bias may be involved. Other possible examples are vascular dementia (for the same reasons as stroke) and Parkinson's disease, because environmental exposures (pesticides, smoking) have been linked to Parkinson's disease and cancer (20Go), and it has been hypothesized that these exposures may interact with detoxifying enzymes in causing cancer (21Go). We considered a general model involving two diseases, with one of them acting as a competing risk for the other. A model involving a single disease, with the disease competing against itself and causing selective attrition of subjects exposed to E and G, could also be considered.

Association studies of late-onset diseases have been carried out in different populations, often with conflicting results. Our findings may contribute to an understanding of discrepancies, since the populations studied may have been different with respect to age, prevalence of risk factors, or incidence of competing risks of death.

This model may help to explain why relative hazards or odds ratios vary with age. For instance, the relative hazard of Alzheimer's disease associated with the {varepsilon}4 allele of the apolipoprotein E gene decreases with age (22GoGo–24Go). Similarly, the strength of associations between family history and neurodegenerative diseases (25Go, 26Go) and between cardiovascular disease risk factors and stroke (27Go) decreases with age.

If a genetic factor increases the risk of a lethal disease, its frequency should decrease with increasing age. For instance, decreasing frequencies with age of the {varepsilon}4 allele of the apolipoprotein E gene (28GoGoGo–31Go), or the alanine/valine polymorphism in the methylenetetrahydrofolate reductase gene (32Go)—two polymorphisms thought to be associated with myocardial infarction—have been reported. This emphasizes the need to take age into account in association studies of late-onset diseases (33Go). Our model provides further information. If a gene-environment interaction results in an increased risk of death from a disease, the age-related decrease in the frequency of the genetic factor will depend on the environmental factor. Some examples suggest that this may occur. For instance, it was reported that the decrease with age of the frequency of the adenine allele of the G/A-455 polymorphism in the fibrinogen ß-subunit gene was more pronounced among smokers than among nonsmokers (34Go). Similarly, the frequency of the insertion allele of a polymorphism in the NOS2A gene has been shown to decrease with age only among hypertensive persons (35Go).

Case-only designs have been proposed for investigation of gene-environment interactions (1Go). The main assumption underlying this approach is that genetic and environmental factors are independent in the population from which cases originated. Some situations in which genetic and environmental factors may be associated have been discussed (1Go, 36Go). We provide another example of how an association between these factors may arise, even if they were initially independent. Investigators using this design, particularly for late-onset conditions, should be aware of this possibility and should be able to test the independence assumption.

When do we suspect that this bias is involved? Ideally, the analysis of the relation between the data on environmental factors that have been collected and the data on the genetic factor may detect an association, leading to the appropriate adjustment. Furthermore, if the relation between the genetic and environmental factors is modified by age, this may provide evidence in favor of this bias.

It is difficult to think of a study in which data on every risk factor have been collected. It is therefore important to assess whether age modifies the association between the disease and the genetic factor. If the relative hazard associated with the polymorphism increases with age, this unusual pattern may suggest this kind of bias, particularly if a protective exposure later becomes harmful. If the relative hazard decreases with increasing age, two situations should be distinguished: 1) if its initial value is 1 (or less) and the genetic factor becomes protective as age increases, again this kind of bias may be involved; and 2) if its initial value is greater than 1, there is no easy way to identify the bias, unless the polymorphism appears to be protective in the oldest subjects. If the relative hazard remains constant over time, this kind of bias is unlikely. Finally, even if some patterns of association are in favor of this bias, it is essential to consider alternative explanations, particularly biologic explanations (e.g., age-related etiologic heterogeneity with a stronger genetic contribution for early-onset cases).

Our numerical examples had limitations. The exposure status of a subject did not vary with time; this assumption is questionable for many nongenetic risk factors (e.g., smoking, hypertension). More complex models including time-dependent covariates could be considered. We specified a simple relation between the disease under investigation and a competing risk; more complex multistate models could be elaborated. A Weibull model was used for our numerical examples because good agreement between observed rates and calculated rates was noted, but other incidence models could be used.

In conclusion, we have shown that failure to take an environmental factor into account may lead to erroneous conclusions about the relation between a genetic factor and a disease, when gene-environment interactions and competing risks are involved. The bias can sometimes be substantial, and we suggest some patterns of association in favor of this bias. These findings emphasize the need for application of rigorous epidemiologic principles in association studies. Researchers conducting studies on the etiology of late-onset diseases should be aware of this bias.


    APPENDIX 1
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
Under an additive model, the measure of association is the excess risk {Delta}ij. The question is whether the crude excess risk due to G is equal to the excess risk due to G among persons unexposed to E. By assuming that E and G do not interact on an additive scale for D2, we can show that {Delta}2G = {Delta}2G is equivalent to {Delta}2E(PE/G(t) - PE/(t)) = 0 (similar to expression 2).


    APPENDIX 2
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
If the probability of suffering from D1 and D2 in a short time interval is negligible, the total hazard in the jth exposure category is {lambda}j(t) = {lambda}1j(t) + {lambda}2j(t) 8Go, 13Go. The probability of surviving without D1 and D2 from time t0 to time t in the jth stratum is

The proportion of subjects who are free of D1 and D2 at time t in the jth exposure group is

The frequency of E conditional on being free of D1 and D2 at time t and exposed to G is PE/G(t) = nGESGE(t)/(nGSG(t) + nGESGE(t)), and the frequency of E conditional on being free of D1 and D2 at time t and unexposed to G is PE/(t) = nESE(t)/(nS(t) + nESE(t)). Similar expressions can be derived for P/G(t) and P/(t).

The total hazard of D2 is

i.e., the sum of hazards in each exposure stratum weighted by the frequency of subjects at risk at time t in each stratum. This expression is equivalent to {lambda}2(t) = P(t){lambda}2(t) + PG(t){lambda}2G(t), where {lambda}2G(t) = {lambda}2GE(t)PE/G(t) + {lambda}2G(t)P/G(t) and {lambda}2(t) = {lambda}2E(t)PE/(t) + {lambda}2(t)P/(t). The crude relative hazard (RH) of D2 associated with G is

Dividing the numerator and the denominator by {lambda}2(t) yields

and replacing the conditional frequencies leads to expression 1.


    APPENDIX 3
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 
We consider a cohort of subjects at risk of a single disease D (i.e., no competing risks are present). We suppose that G and E do not interact on a multiplicative scale for D. Appendix table 1 shows the values of the crude relative hazard (RH) of D associated with G according to the risk of D and RHG, when RHE != 1. As expected, when RHG = 1, RHG(t) is constant and equal to RHG. Otherwise, RHG(t) underestimates RHG. However, the bias is small unless the risk of D is very high (>=20 percent). The bias related to the misspecification of the model when a factor is omitted appears to be negligible for usual disease risks in our example.


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APPENDIX TABLE 4. Crude relative hazard of D associated with G (RH2G(t)) when competing risks are not involved and there is no departure from multiplicative joint effects*

 

    ACKNOWLEDGMENTS
 
The authors acknowledge the use of Canadian mortality and incidence data obtained at http://www.hc-sc.gc.ca/hpb/lcdc/webmap/.

The authors are grateful to Drs. Philippe Amouyel, Jean Bouyer, Catherine Com-Nougué, Joseph Lellouch, Susan Slager, and Laurence Tiret for helpful discussions and suggestions.


    NOTES
 
Correspondence to Dr. Alexis Elbaz, INSERM, Unité 360, Hôpital de la Salpêtrière, 47 Boulevard de l'Hôpital, 75651 Paris, Cedex 13, France (e-mail: alexis.elbaz{at}chups.jussieu.fr).


    REFERENCES
 TOP
 ABSTRACT
 INTRODUCTION
 METHODS
 NUMERICAL EXAMPLES
 DISCUSSION
 APPENDIX 1
 APPENDIX 2
 APPENDIX 3
 REFERENCES
 

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Received for publication March 1, 2001. Accepted for publication July 19, 2001.





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