Case-Base Study Assuming Gene-Environment Independence and Hardy-Weinberg Equilibrium

Let , 1 and 2 represent the number of the variant allele(s) a subject carries. We define two dummy variables: and , with , if , and 0 if otherwise; , if , and 0 if otherwise. Let be the exposure status of a subject which can be in any measurement scale: binary, ordinal or continuous. Let represent the disease status of a subject, with for diseased and for non-diseased. We assume that the disease risk in the study population follows a logistic model:(1)

The is the baseline disease odds in the population; is the odds ratio of disease for those with vs. those with ; and , the odds ratio for those with vs. those with . The is the odds ratio associated with the environmental exposure. The and are the odds ratios associated with gene-environment interactions.

In a case-base study, researchers implement two sampling schemes: the ‘case’ and the ‘control’ sampling schemes [9]-[13]. The case sampling scheme targets the diseased subjects. Let indicate that a diseased subject is recruited in the case sample of a case-base study; , otherwise. In such a case sampling scheme:(2)where is a constant between 0 and 1. Note that the sampling probability depends only on , not on and . In this sampling scheme, the diseased subjects have a constant non-zero probability of being recruited [], whereas the non-diseased subjects have a probability of zero of being recruited [].

The control sampling scheme targets the entire population (the study base) without regard to disease status. Let indicate that a subject is recruited in the control sample; , otherwise. Such a control sampling scheme is noted as:(3)where is a constant between 0 and 1. Note that this sampling scheme essentially is a random sampling of the study population at large; it depends on neither , nor .

The two sampling schemes are assumed to be independent of each other. A diseased subject can be recruited in a case-base study simultaneously in the case sample and in the control sample. The probability of a diseased subject entering the case-base study as a duplicate sample () is:(4)which depends on neither nor . The event of indicates that a subject is recruited in a case-base study through case sampling, control sampling or both. Let be the probability that a diseased subject recruited in a case-base study can be found in the control sample, i.e.:(5)

Again, this depends on neither nor .

The maximum likelihood estimate of and its variance (see Chui and Lee [13]) are:(6)and(7)where is the total number of distinct diseased subject recruited in the case-base study, and is the number of diseased subjects recruited in the control sample. Chui and Lee [13] showed that the disease risk in a case-base sample also follows a logistic model:(8)where

As in Lee et al. [8], we assume that among the non-diseased subjects in the study population, the gene () is independent of environmental exposure () [the first equality in the following Equation (9)] and in the Hardy-Weinberg equilibrium (the second equality), i.e.:(9)where is the allele frequency and , the log allele frequency odds, among the non-diseased subjects in the study population. Combining Equations (8) and (9), the likelihood function for a case-base study under the assumptions of gene-environment independence and Hardy-Weinberg equilibrium is found to be (Exhibit S1):(10)

Model (10) above has exactly the same form of the likelihood function of a 1∶5 matched case-control study. [The denominator in Model (10) has a total of six terms, corresponding to one ‘case’ and five ‘controls’.] Therefore, one can adopt the ‘counterfactual approach’ [8] to fit Model (10). To be precise, one first creates a recruitment indicator, , for each and every distinct subject actually recruited in the case-base study. Next, one creates a total of five counterfactual subjects (all of them with ) to each recruited subject; the exposure status () of the five counterfactual subjects is deliberately set to be exactly the same as the recruited subject to whom they are matched, but the disease status () and gene () are different. (The five subjects represent the five different ways of making []-different counterfactuals.) Treating as the outcome variable, one then performs a conditional logistic regression analysis (using existing statistical software, such as SAS) with the following regression equation: based on the above created 1 (factual): 5 (counterfactuals) matched data. The results are the conditional maximum likelihood estimates of the total 7 parameters in Model (10), together with their variance-covariance matrix, , a matrix.

Among the parameter estimates obtained from a fitting of Model (10) to data, is an estimate for log allele frequency odds in Equation (9), and are estimates for log genetic odds ratios in Model (1), is an estimate for log environmental odds ratio in Model (1), and and are estimates associated with gene-environment interactions in Model (1). The estimate from Model (10) is to be further combined with the estimate from Equation (6) to provide an estimate for the log baseline disease odds in Model (1):

(11).

Because is independent of [13], the variance of is:(12)

Now, we can estimate the absolute and relative risks. An estimate of the (absolute) disease risk for subjects in the study population with (,,) is:(13)

where is a gene-environment profile vector and is a vector of parameter estimates. Because is independent of all the other parameters [13], the variance of the estimate (in logit scale) is:(14)where is readily available by simply deleting the first row and the first column of . [ from Model (10) plays no role in the disease risk estimation.] Let the reference group be those subjects in the study population with (,,), and let be the gene-environment profile vector for them. An estimate of the relative risk is therefore:(15)

Using the delta method, the variance of the estimate (in log scale) is:(16)where

Monte-Carlo Simulations

For simplicity, we assume a binary exposure E () and a biallelic gene with genotype G (). For the non-diseased subjects in the study population, we assume gene-environment independence and Hardy-Weinberg equilibrium, with the exposure prevalence (for ) and the allele frequency of the variant allele both being set at 0.5. The disease probabilities of subjects in the study population are assumed to follow the logistic model [Model (1)]. We assume an autosomal recessive gene with a genetic odds ratio of 2.0 [ and ], an environmental odds ratio of 2.5 [] and a gene-environmental interaction odds ratio of 2.0 [ and ]. The disease prevalence in the study population is set at 0.1. Therefore, the six disease risks are: , , , , , and , respectively. And the five relative risks are (with as the reference level) , , , , and , respectively. A case-base study is conducted in the study population (population size: 100,000) with a case sampling probability () of 0.05 and a control sampling probability () of 0.005. Under such a sampling scheme, the case-base study is expected to recruit a total of 500 distinct diseased and 500 distinct non-diseased subjects. Exhibit S2 presents the SAS code for simulating data.

Using the formulas derived by Cheng and Lin [6], we compare the relative efficiencies of the estimates in the conditional logistic regression [with the assumptions of gene-environment independence and Hardy-Weinberg equilibrium, Model (10)] relative to the corresponding estimates in Chui and Lee's [13] unconditional logistic regression [without the dual assumptions, Model (8)]. [Relative efficiency of A method relative to B method is defined as the ratio of the variance of B estimator and that of A estimator. A larger-than-one relative efficiency implies a better statistical performance (larger power and better precision, etc) for A method as compared to B method.]

In regard to the estimates for relative risks and absolute risks, we perform a total of 10,000 simulations to compare the performances of the two approaches (the conditional logistic regression with the dual assumptions vs. the unconditional logistic regression without the dual assumptions). The means of the estimates for relative risks (in log scale) and absolute risks (in logit scale) are calculated. The variance of an estimate is calculated as the sample variance of the estimates. We also calculate the coverage probabilities and the average lengths of the 95% confidence intervals for the estimates. [The coverage probability of a 95% confidence interval for a parameter estimate is the probability that the interval covers the true value of the parameter in a repeated sampling (simulation) experiment.]