IJE Advance Access originally published online on March 23, 2006
International Journal of Epidemiology 2006 35(4):1067-1073; doi:10.1093/ije/dyl048
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Published by Oxford University Press
Methodology |
Unconditional analyses can increase efficiencyin assessing gene–environment interaction of the case-combined-control design
1 Genetic Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, NIH, DHHS, Bethesda, MD 20892, USA
2 Inserm, IC10213, Service de Biostatistiques, Institut Curie, 26 rue d'Ulm, 75248 Paris Cedex 5, France
3 Inserm, EMI00-06, Service de Biostatistiques, Institut Curie, 26 rue d'Ulm, 75248 Paris Cedex 5, France
* Corresponding author. Genetic Epidemiology Branch/NCI/NIH/DHHS, Executive Plaza South, Room 7004, 6120 Executive Blvd, MSC 7236, Bethesda, MD 20892-7236, USA. E-mail: goldstea{at}exchange.nih.gov
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Abstract |
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Background A design combining both related and unrelated controls, named the case-combined-control design, was recently proposed to increase the power for detecting gene–environment (GxE) interaction. Under a conditional analytic approach, the case-combined-control design appeared to be more efficient and feasible than a classical case–control study for detecting interaction involving rare events.
Methods We now propose an unconditional analytic strategy to further increase the power for detecting gene–environment (GxE) interactions. This strategy allows the estimation of GxE interaction and exposure (E) main effects under certain assumptions (e.g. no correlation in E between siblings and the same exposure frequency in both control groups). Only the genetic (G) main effect cannot be estimated because it is biased.
Results Using simulations, we show that unconditional logistic regression analysis is often more efficient than conditional analysis for detecting GxE interaction, particularly for a rare gene and strong effects. The unconditional analysis is also at least as efficient as the conditional analysis when the gene is common and the main and joint effects of E and G are small.
Conclusions Under the required assumptions, the unconditional analysis retains more information than does the conditional analysis for which only discordant case–control pairs are informative leading to more precise estimates of the odds ratios.
Keywords GxE interaction, unconditional logistic regression analysis, conditional analysis, sibling controls, population-based controls
Accepted 2 March 2006
The desire to examine gene–environment (GxE) interactions continues to increase, particularly, as molecular genetic technology improves and genotyping costs decrease. However, most study designs appear to be inefficient for detecting interaction involving rare event(s), particularly for moderate values of the GxE interaction effect. We recently proposed a design using both related and unrelated controls (simultaneously), named the case-combined-control design, to increase the power for detecting GxE interaction when the involved factors are rare without increasing dramatically the number of study subjects required. This design permitted the estimation of both GxE interaction and main effects.1 For this design to be valid, a number of assumptions were required including no population stratification bias, no difference in the distribution of variables of interest between cases who have sibling controls vs those cases without such sibling controls, and exchangeability of covariates of interest in cases and sibling controls.
For ease of computation, the proposed analysis was a conditional analysis with each matched set comprising a case, an unrelated control, and an unaffected sibling of the case (for the cases with an available sibling control). In addition to the assumptions listed above, this analysis approach required homogeneity between the odds ratios of the variables involved in the GxE interactions using either of the two types of controls. This conditional analysis strategy produced unbiased main effect estimates for the environmental exposure E, negligible bias for the interaction effect I, and minimal bias for the genetic effect G (Pascal Wild, InRS, France, personal communication). The case-combined-control design appeared to be more efficient and feasible than a classical case–control study for detecting interaction involving rare events. The number of available sibling controls per case and the frequencies of the risk factors were the most important parameters for determining relative efficiency (RE). The case-combined-control design was, however, less efficient for common genes with moderate effects.1
We now propose an unconditional analytic strategy to further increase the power for detecting GxE interactions. Since unconditional analysis uses information from all subjects, an increase in statistical power is expected. This strategy allows estimation of the GxE interaction effect and the E main effect under certain assumptions. Only the G main effect cannot be estimated from the unconditional analysis because it is biased. We present the assumptions and requirements needed for the unconditional analysis to be valid for GxE interaction estimation and illustrate situations when this analytic strategy leads to improved efficiency for detecting GxE interaction relative to a conditional analytic approach.
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Methods |
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The population for the case-combined-control design consists of cases and two types of controls, unrelated controls and sibling controls. The parameters for modelling an interaction between a genetic factor G and an environmental exposure E are defined in Table 1. G and E are assumed to be independent events. Limited examinations have suggested that
50% of cases may have appropriate sibling controls2,3; thus, we use this observation to represent the average number of available controls per case (defined as F). Thus, in most of our evaluations, F = 0.5, meaning that approximately half the cases have a sibling control. Table 2 shows the subgroups of the population at different risks of disease when there is a GxE interaction. For illustrative purposes, we used an autosomal dominant inheritance model. The results showed similar patterns for an autosomal recessive model (data not shown). We calculated the expected distributions of the environmental exposure and genetic susceptibility in cases, matched unrelated, and matched related controls according to Table 2 and as previously described.1
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Simulation studies
Random numbers were generated to determine the number of controls each case had for each of the studies (i.e. one unrelated control plus approximately F having one related control for the case-combined-control study; 1 unrelated plus approximately F having a second unrelated control for the classical case–control study).
When E and G were relatively common (e.g. both >0.05), we simulated 2500 data sets with 1000 cases: 1000 matched unrelated controls: 500 matched sibling controls with F = 0.5. When E and G were relatively rare (e.g. either <0.05) (or very rare; e.g. both 
0.01), we simulated 1000 case–control studies with 5000 (or 10 000) cases: 5000 (or 10 000) unrelated controls: 2500 (or 5000) sibling controls. In addition, a second set of 1500 or 7500 (or 15 000) unrelated controls was matched to the cases to conduct a classical case-unrelated-control study. All subjects were simulated using random numbers generated by the SAS function RANUNI (SAS, version 8, Cary, NC) to assign each of the cases and controls to the different possible E and G categories.
General assumptions/requirements
For purposes of presentation, we make several assumptions about the study population. We assume that there is no population stratification bias. Second, we assume that baseline disease risks do not differ in the study population. That is, we assume that there are no other factors in addition to G and E that differentially influence risk of disease. To examine the impact of this assumption, however, we examine the effect of a family-specific variable denoted by H on the estimates of the interaction effect (GxE) and the relative efficiencies. H is defined as follows. ORH for a given family was randomly determined from a normal distribution with mean of log(2) and variance of 0.5. H was defined such that it was independent of E and G and had no confounding effect; the frequency of H was set to 0.1. Thus, H is used to generate families with different baseline risks due to factors other than G or E. We further assume that there is no difference in the distribution of variables of interest between cases who have sibling controls vs those cases without such sibling controls and that there is exchangeability of covariates of interest in cases and sibling controls, i.e. the covariate distribution does not depend on calendar time, birth order, or geographic location.4,5 Finally, we assume homogeneity between the odds ratios of the variables involved in the GxE interactions using either of the two types of controls.1
Additional assumptions for unmatched strategy
We assume that there is no correlation in E between siblings leading to the equality between the prevalence of E among unrelateds [P(E)unr] and among relateds [P(E)rel], i.e. P(E)unr = P(E)rel. Thus, the E main effect odds ratio across control groups is the same, i.e. 
. This is equivalent to equality in the interaction effect across control groups, i.e. 
. Further descriptions of these assumptions are presented in the Appendix.
Analysis approach
To assess the proposed analytic strategy, we compared the case-combined-control design with a classical case-unrelated-control study. The parameter of interest is the interaction odds ratio (RI) defined on a multiplicative scale. We defined the RE of the case-combined-control study compared with a classical case–control study, as the ratio of the variances of ßI, i.e. the variance of ßI of the classical case–control study divided by the variance of ßI of the case-combined-control study. We used the same case:control ratio, i.e. number of cases/number of controls, in the two designs. We compared the RE for the unconditional analyses RE(U) with the RE for the conditional analyses RE(C). We denote the ratio of relative efficiencies RE(U)/RE(c), U/C. Thus, when U/C > 1, the unconditional analysis is more powerful than the corresponding conditional analysis; when U/C < 1, the unconditional analysis is less powerful.
For the matched and unmatched strategies, each simulated case–control study was analysed with conditional and unconditional logistic regression, respectively, using the program STATA6 with a binary variable for E and a binary variable for G (based on the genotypes and inheritance model).1
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Results |
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We compare RE for the unmatched vs the matched strategies for specific situations to illustrate the potential improvement in efficiency from the unconditional analysis approach. Figure 1 presents RE(U) and RE(C) for different frequencies of G for a dominant gene with RG = 3, RE = 2, RI = 5, P(E) = 0.2, and F = 0.5. The results show a dramatic effect of P(G) on RE for both analyses. RE(U) decreases from 1.82 at P(G) = 0.001 to 1.12 when P(G) = 0.2. The slope for RE(C) is less steep as RE(C) decreases from 1.26 to 1.08 when P(G) increases from 0.001 to 0.2. Figure 1 also illustrates a more pronounced improvement in power for the unconditional analysis relative to the conditional analysis when P(G) is rare. For example, U/C = 1.44 when P(G) = 0.001 [and 1.36 when P(G) = 0.01] vs 1.12 when P(G) = 0.1 [and 1.04 when P(G) = 0.2].
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The gain and/or change in RE(U) and U/C is insignificant for common genes and/or moderate effect estimates. For example, when P(G) = 0.2 and RI = RG = RE = 1.5, RE(U) = 1.03 and U/C = 1.00. In addition, for moderate effect estimates when P(G)
0.1, there is little change in RE as P(E) changes and U/C approximates one (i.e. 1.00–1.02) (data not shown).
Table 3 compares RE(U) and RE(C) for different RG, RE, and RI effects when P(G) = 0.01, P(E) = 0.2, and F = 0.5. The results show that RG and RI have the strongest effect on RE(U) and that RE(U) increases substantially as either RG or RI increases. An increase in RE leads to a minor increase in RE(U). In contrast, RE and RI have almost no effect on RE(C); only an increase in RG leads to any observable increase in RE(C). Thus, U/C ranges from 1.03 for moderate effects (RI = RG = RE = 1.5) to 2.13 for very strong effects (RG = 10, RI = RE = 5). Indeed, there is generally 50% improvement in power for the unconditional analysis compared with the conditional analysis with high or very high estimates of RG, RI or even RE. Also, as previously mentioned, when P(G) is common, there is little increase in U/C as RE, RG, or RI increase from moderate (1.5) to high (5.0) values (data not shown).
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Table 3 also presents the 95% confidence intervals (95% CIs) for RI. The lower and upper bounds for the unconditional analysis are contained within the bounds for the conditional analysis illustrating the improved efficiency of the unconditional analysis. This comparison also serves as a check on the validity of the required assumptions for the unconditional analysis. For example, when there is a correlation in E between siblings (ORec
1), thus violating one of the required assumptions, the confidence bounds for RI from the unconditional analysis is no longer contained within the bounds from the conditional analysis. To illustrate this situation, consider a model with P(G) = 0.01, P(E) = 0.2, RE = 1.5, RG = 3, and RI = 5, and a moderate correlation in E between siblings, i.e. ORec = 2. Under this scenario, the estimates of P(E) among the related and unrelated controls differ by 7.7% and yield biased estimates of RE, RI, and RG for the unconditional analysis. This produces 95% CIs of RI that are no longer nested (i.e. 95% CI from conditional analysis: 2.89–9.41; 95% CI from unconditional analysis: 2.55–6.54). In addition, as ORec increases, the bias in RE and RI increase (data not shown). Finally, Table 3 shows that, as expected, as U/C increases, the CI for the unconditional analysis becomes narrower relative to the CI for the conditional analysis.
Incorporation of family-specific baseline risks (represented by H) had no effect on RI for either the conditional or unconditional analysis (data not shown). This finding was as expected since H was not associated with either G or E. If there had been a correlation with E, for example, the results would have been comparable with that observed when ORec 1 since P(E) would not have been equal in the two control groups. Specifically, the unconditional analysis would not be valid and estimates of RE, RI, and RG would all be biased.
Figure 2 presents RE(U) and RE(C) for different values of F for a rare [P(G) = 0.01] and a common [P(G) = 0.2] dominant gene. F varies from 0 to 200% resulting in 0–2 sibling controls per case. All other parameters are fixed with P(E) = 0.2, RE = 2, RG = 3, RI = 5. The results show a similar effect of F on RE for both analyses. When P(G) = 0.01, there is a substantial increase in RE from F = 0.25 to F = 1.0 followed by a plateau. The magnitude of the RE is much greater for RE(U) compared with RE(C) and U/C varies from 1.3 to 1.5 with a mean improvement for the unconditional analysis of 40%. In contrast, when the gene is common [P(G) = 0.2], there is a negligible change in RE as F increases and there is essentially no difference between the unconditional and conditional analyses, i.e. U/C varies from 1.0 to 1.04.
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Discussion |
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We have shown that unconditional logistic regression analysis of data from a case-combined-control study for detecting GxE interaction is often more efficient than a conditional analysis, particularly for a rare gene and strong effects. The unconditional analysis is also at least as efficient as the conditional analysis when the gene is common and the main and joint effects of E and G are small. Naturally, under the required assumptions, the unconditional analysis retains more data information than does the conditional analysis for which only discordant case–control pairs are informative leading to more precise estimates of the odds ratios.7
The gain in efficiency for detecting a GxE interaction using unconditional analysis may have a non-negligible effect on the feasibility of a study, i.e. on the required sample size. We previously illustrated1 several feasible scenarios with 80% power involving a rare gene that required 1000 cases: 1500 controls when a conditional analysis was used. Performing an unconditional analysis at the same power (80%) would only require 750 cases, leading to a total decrease in study subjects of 625 (250 cases, 250 unrelated, and 125 related controls). Obviously, for situations that require many more subjects (e.g. >20 000 cases and 30 000 controls), even the substantial increase in efficiency associated with the unconditional analysis will not lead to reasonable sample sizes.
As we previously discussed,1 the major assumptions (e.g. no population stratification bias for the unrelated controls and no difference in the distribution of variables of interest between cases who have sibling controls vs those cases without such sibling controls plus exchangeability of covariates of interest in cases and sibling controls4,5,8–10) required for the case-combined-control study to produce valid estimates of main and/or interaction effects are not testable before the data have been collected. If these major assumptions are not met, then alternative analytic strategies will be required. However, if these assumptions are met, then conducting conditional and/or unconditional analyses will depend on the goals of the study and a second set of assumptions. Specifically, for conditional analyses to estimate both main and interaction effects, we require that there is homogeneity between the odds ratios of the variables involved in the GxE interactions using either of the two types of controls. Additionally, for the unconditional analyses to be valid there must be no correlation in E between siblings and the same frequency of E in the two control groups. Although P(E)rel = P(E)unr leads to and as long as G and E are independent and the other required assumptions are met, in real settings, these relationships may be more complicated. Specifically, because of confounding, effect modification, etc., the equality of the frequency of E in the two control groups may not necessarily translate to equivalency of the effect estimates. Therefore, the necessary assumptions for the validity of the unconditional analysis should also include the equality . For purposes of this study, we implicitly assumed that there was no correlation within families owing to shared, but unmeasured, factors. Examination of H with different risks across families showed that if H was not correlated with E or G, it had no effect on RI or its variance and, thus, no effect on the RE. However, if H was correlated with E, it would be comparable with having a correlation in E between siblings with different values across families leading to different frequencies of E between control groups and, therefore, invalid estimates for the E main effect as well as the GxE interaction effect under an unconditional analysis.
Under the assumptions listed above and detailed in the Methods section, unconditional analysis of data from the case-combined-control study yields unbiased estimates of the E main effect and GxE interaction effect. The conditional analysis, however, can serve as a check on the validity of the unconditional analysis strategy by evaluation of the CIs for the GxE interaction effect estimate RI. Specifically, if the unmatched approach is valid, given the increased efficiency for an unconditional vs a conditional analysis, the lower and upper bounds for the RI CI from the unconditional analysis should be contained within the CI for the conditional analysis. If the CI bounds for the unconditional analysis are not within the bounds for the conditional analysis, then one may expect that one or more of the required assumptions is not valid. However, there may be situations when the deviation from the required assumptions (e.g. existence of a correlation in E between siblings) may be small such that unconditional analysis could still be conducted without much bias. Further study is required to determine how robust RE and RI are to small or moderate deviation from these assumptions. Moreover, it may be helpful to consider the types of scenarios when the required assumptions would be likely to apply. Depending on the disease and exposures of interest, external data may be available to provide a priori information about, for example, the chances of a correlation in exposure E between siblings. If such a correlation is known or strongly suspected, unconditional analysis would not be appropriate and a different analytic strategy would be required.
If the necessary assumptions for the conditional or unconditional analysis of the case-combined-control design are not met, then one may choose an alternative analytical strategy such as polytomous regression or incorporation of weights to account for the differential ascertainment of the two control groups. Another possible approach would be to combine the conditional and unconditional logistic regressions in a single analysis. That is, one could use a conditional likelihood for the sibling controls and a full likelihood for the unrelated controls and maximize the product of the likelihoods. Limited simulations using such an approach suggest that this analytical strategy is substantially less efficient than the unconditional analysis but appears to be generally as efficient as the conditional analysis (data not shown). An advantage of this approach, however, is that it appears to produce minimally biased estimates of RG and RE even when there is a correlation in E between siblings. Since this approach may require fewer assumptions than either unconditional or conditional analysis strategies, the efficiency reductions might be offset by improved estimation capabilities. Further evaluation of this strategy and other analytical approaches are planned. If one is interested in estimating the main effect of G, then unconditional analysis cannot be conducted. Also, since validity of the unconditional analyses requires the same frequency for at least one variable in the interaction for both control groups, it follows that unconditional analysis cannot be used to estimate GxG interaction effects. The unconditional analysis approach is limited to GxE interactions or ExE interactions so long as at least one of the exposure variables is not correlated among siblings and has the same frequency in the two control groups.
The case-combined-control design was developed to increase efficiency for GxE interaction detection by simultaneously using two types of controls. Other designs such as the case-control-family design11 and the triads (affected offspring plus two parents) and unrelated subjects design12,13 with simultaneous use of multiple control groups have also been proposed. As molecular genetic data become more easily collectable and less expensive, there will be increased opportunities for these types of hybrid designs to increase power for the detection of genetic or environmental main effects and interaction effects.
These results show that unconditional analysis of the case-combined-control design for estimating GxE interaction is unbiased under certain conditions and may produce a substantial increase in power. However, necessary assumptions for such an analysis may not be met for all variables of interest. Future studies will examine whether the case-combined-control design would still offer advantages over a single case-unrelated-control or case-related-control study design in these situations.
Appendix
The above table shows the E and G distributions for cases, related, and unrelated controls. If one is interested in using the case-combined-control design approach to estimate main and interaction effects, then the approach requires homogeneity between the odds ratios of the variables involved in the GxE interactions using either of the two types of controls.1 That is, the ORs for G and E for the two control groups are required to be equal. For this assumption to be valid, a matched analytic strategy is required since siblings will have a greater frequency of G (and also E if there is a positive correlation in E between cases and their sibling controls) compared with unrelated controls.
However, if one is mainly interested in estimating the GxE interaction effect, then only the equality is required where rel denotes related and unr, unrelated controls. Thus, from the above table
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For this equality to hold, we assume that G and E are independent. In addition, since G is correlated in siblings, we also require no correlation in E between siblings, which means that P(E)rel = P(E)unr. Given these requirements, from the above table, we have i/c = k/e and j/d = l/f. Thus, in addition to and an unconditional analysis may be performed in the case-combined-control design to estimate the GxE interaction effect. Both the GxE interaction effect and E main effect estimates are unbiased. If, however, there is a correlation in E between siblings and/or P(E)rel P(E)unr, then i/c k/e and j/d l/f. And it follows that 
and 
. Under such a scenario, unconditional analysis of case-combined-control data would not be valid.
KEY MESSAGES
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Acknowledgments |
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This project was supported in part by INSERM, the Foundation Philippe INC, the Association pour la Recherche contre le Cancer, and the University of Paris Sud. This work was supported in part by the Intramural Research Program of the National Institutes of Health, National Cancer Institute, Division of Cancer Epidemiology and Genetics. This work was initiated while Nadine Andrieu was a guest researcher at the Genetic Epidemiology Branch of the National Cancer Institute. The authors thank Angela Fahey, IMS for her assistance with the simulations, Dr John Thompson (University of Leicester, UK) for analytic support, and the reviewers for helpful comments.
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