Incident + prevalent case control analysis
We propose and study a fully efficient method to estimate associations of an exposure with disease incidence when both, incident cases and prevalent cases, i.e. individuals who were diagnosed with the disease at some prior time point and are alive at the time of sampling, are included in a case-control study. We extend the exponential tilting model for the relationship between exposure and case status to accommodate two case groups, and correct for the survival bias in the prevalent cases through a tilting term that depends on the parametric distribution of the backward time, i.e. the time from disease diagnosis to study enrollment. We construct an empirical likelihood that also incorporates the observed backward times for prevalent cases, obtain efficient estimates of odds ratio parameters that relate exposure to disease incidence and propose a likelihood ratio test for model parameters that has a standard chi-squared distribution. We quantify the changes in efficiency of association parameters when incident cases are supplemented with, or replaced by, prevalent cases in simulations. We illustrate our methods by estimating associations of single nucleotide polymorphisms (SNPs) with breast cancer incidence in a sample of controls and incident and prevalent cases from the U.S. Radiologic Technologists Health Study.READ FULL TEXT VIEW PDF
Incident + prevalent case control analysis
We propose and study a fully efficient method to estimate associations of an exposure with disease incidence when both, incident cases and prevalent cases, i.e. individuals who were diagnosed with the disease at some prior time point and are alive at the time of sampling, are included in a case-control study.
We extend the exponential tilting model for the relationship between exposure and case status to accommodate two case groups, and correct for the survival bias in the prevalent cases through a tilting term that depends on the parametric distribution of the backward time, i.e. the time from disease diagnosis to study enrollment. We construct an empirical likelihood that also incorporates the observed backward times for prevalent cases, obtain efficient estimates of odds ratio parameters that relate exposure to disease incidence and propose a likelihood ratio test for model parameters that has a standard chi-squared distribution. We quantify the changes in efficiency of association parameters when incident cases are supplemented with, or replaced by, prevalent cases in simulations. We illustrate our methods by estimating associations of single nucleotide polymorphisms (SNPs) with breast cancer incidence in a sample of controls and incident and prevalent cases from the U.S. Radiologic Technologists Health Study.
Keywords: Outcome dependent sampling, survival bias, empirical likelihood, exponential tilting model, density ratio model, length biased sampling.
Case-control studies that compare the frequency of exposures in incident cases to that in healthy individuals to assess associations with risk of disease incidence are popular for rare outcomes, as they are more economical than prospective cohorts. However, like all observational studies, case-control studies are also vulnerable to biases that result in distorted estimates of exposures’ associations with disease risk. One of several possible biases , sometimes called survival bias, occurs when prevalent cases, i.e. individuals who were diagnosed with the disease at some prior time point and are alive at the time of sampling for the case-control study, are used in addition to, or instead of, individuals newly diagnosed with disease, namely incident cases. If the exposure also impacts survival after disease onset, the estimated association of an exposure with disease incidence over- or underestimates the true association. This is a particularly serious problem for diseases, or outcomes, that are rapidly fatal, as survivors may comprise a very special subgroup of cases.
Many epidemiologic textbooks (e.g. Schlesselman, 1982, p. 133) point out that simply including prevalent cases in case-control studies of rare diseases without any adjustment for the survival bias leads to biased estimates of incidence odds ratios. While several authors have proposed approaches to correct for survival bias in the analysis of cohorts comprised of prevalent cases that are then followed to some failure event of interest (e.g. death) (e.g. Cheng and Huang, 2014), only one approach has been proposed to explicitly correct for survival bias when prevalent cases are compared with controls. Begg and Gray (1987) subtracted a bias term estimated from a survival model for the backward time from the log odds ratio estimates obtained from a standard logistic model fit to controls and prevalent cases. A statistical approach to allow incorporating information from prevalent cases in addition to incident cases is thus needed to enhance inference based on case-control data for rare disease like cancer, where prevalent cases become more readily available due to improvements in treatment.
Our work was motived by a case-control study conducted within the U.S. Radiologic Technologists Study (USRTS) to assess the associations of single nucleotide polymorphisms (SNPs) with risk of female breast cancer (Bhatti et al., 2008)
. The USRTS, initiated in 1982 by the National Cancer Institute and other institutions to study radiation-related health effects from low-dose occupational radiation exposure, enrolled 146,022 radiologic technologists at baseline. Information on participants’ characteristics, exposures and prior health outcomes was collected via several surveys conducted between 1984 and 2014, and blood sample collection for molecular studies began in 1999. As the number of incident breast cancer cases with blood samples available for genetic analysis was limited, we developed methods that allow one to also include information on prevalent cases, i.e. women whose breast cancers were diagnosed prior to blood sample collection, to obtain unbiased estimates of odds ratios for the associations of SNPs with breast cancer incidence.
Our work is based on the well known result on the equivalence between the logistic regression model for prospectively collected data and the exponential tilting, or density ratio model, for retrospectively collected data(Qin, 1998)
. To accommodate data from incident cases, prevalent cases and controls, we discuss a three-sample exponential tilting density ratio model. For prevalent cases, in addition to covariate information, we observe their backward time, i.e. the time between disease diagnosis and sampling. We model the backward time distribution based on a parametric model for the survival time conditional on surviving to time of sampling (Section 2). In Section 3 we derive a semi-parametric likelihood that combines information from controls, incident and prevalent cases. We estimate log odds ratios for the associations between disease incidence and exposures, and parameters in the model for the backward time using empirical likelihood techniques, and derive the asymptotic properties of the estimates. In Section 4, we assess the performance of the method in simulations and study efficiency of the estimates when prevalent cases are used in addition to, or instead of, incident cases in a study under various scenarios. We illustrate the methods with data from the motivating study on the association of breast cancer risk and SNPs among women sampled from the USRTS (Section 5), before closing with a discussion (Section 6).
Let denote the disease indicator, with for individuals newly diagnosed with disease (incident cases) and for those without (controls), and
is a vector of covariates. We assume that the association betweenand in the population is captured by the prospective logistic model
where denotes an intercept term, and the log odds ratio for the association of with
, the parameter of interest. In the general population, the marginal probability of disease iswhere is the density of , that is unspecified.
In a case-control study, independent samples of fixed sizes and are drawn from controls () and cases (), respectively, and then information on the exposure is obtained. Due to the retrospective sampling, only the conditional densities and are observed. Using Bayes’ rule, the prospective model in (1), and letting ,
Model (2) is called a two-sample exponential tilting model or density ratio model.
and the corresponding standard errors.Qin (1998) profiled out the baseline distribution in equation (2) and derived a constrained empirical likelihood to estimate and the nuisance parameter . We adapt this profile likelihood method in the next section to incorporate information on prevalent cases.
We now assume that in addition to incident cases, on whom exposure information is ascertained at time of diagnosis, we also have information on exposures from prevalent cases, i.e. individuals who developed disease previously and are alive at the time of sample selection for the case-control study. To formalize the notion of a prevalent case, let denote the (unobserved) survival time from disease diagnosis to death, with a survival function , and let denote the backward time, defined as the time between disease diagnosis and sampling. We only observe prevalent cases who are alive at the time of sampling, i.e. if . The sampling scheme for incident cases, prevalent cases and controls is depicted in Figure 1. If is related to survival, that is simply combining prevalent with incident cases and fitting model (1) to the data will lead to biased estimates of . In what follows we assume that belongs to a known parametric family indexed by parameters and use the notation .
Using information on the covariates and the observed backward time, , we now derive the likelihood contribution for the prevalent cases and extend the exponential tilting model in (2) to obtain unbiased estimates of
. The joint distribution ofand in the population of prevalent cases alive at the time of sampling is
Assuming that the disease incidence is stationary over time, based on standard renewal process theory, we assume that has a uniform density on , i.e. for and zero otherwise, for some known value . Then using equation (2
) and Bayes’ theorem, the density of the covariatesfor prevalent cases is
where and }.
The density of the covariates for the prevalent cases, in (4) can thus also be expressed in terms of and a parametric tilting term that, in addition to and an intercept, depends on the survival distribution . Notice that when does not depend on , the tilting term in (4) depends on only through , i.e. is the same as for the incident cases, but with a different intercept. To derive the conditional density of in (3), we use Bayes’ theorem and the fact that is independent of both, and , to obtain
For , . Any parametric survival model can be used to model in equation (5) for the backward time . We assume with hazard where , the baseline hazard function, is modeled as a constant, Weibull or a piecewise-constant hazard.
Let denote the covariates for the controls, the covariates for the incident cases and and the covariates and backward times for the prevalent cases, where . Using the exponential tilting models in equations (2) and (4), and the distribution for the backward time in (5), the likelihood for the controls and the two case groups is
Similar to Qin (1998), we estimate empirically under the following constraints that ensure that are, in fact, distributions: (1) , (2) , (3) . These constraints are accommodated via Lagrange multipliers in the log-likelihood. After maximizing the log-likelihood for subject to constraints (see Appendix 1), and letting , , the profile log-likelihood for the remaining parameters is
We refer to the above likelihood as the IP-case-control likelihood.
Denote the maximum likelihood estimator of in (7) by , and the true value by . To derive the large-sample properties of , we first define a matrix that is important in the asymptotic behavior of .
For ease of exposition, let , and . Let denote the expectation with respect to , the expectation with respect to , for any vector , and denote the differentiation operator with respect to a generic parameter .
We assume the following regularity conditions hold:as for . For each , both and have continuous first derivatives with respect to in a neighborhood of . and is finite and positive definite.
Condition (C1) requires that the sample sizes of the observed controls, incident cases and prevalent cases grow at the same rate. Note that and
Conditions (C2) and (C3) together guarantee that the matrix , where
is well-defined. Condition (C3) ensures that all the moments in the definition of the’s are finite.
Assume regularity conditions (C1-C3) hold and that the matrix is positive definite. Then, as ,
in distribution, , with defined in Supplementary Section 1;
the likelihood ratio in distribution, where is the dimension of ;
the likelihood ratio for any sub-vector of in distribution, where is the dimension of .
The proof of the Theorem is given in Appendix 2. Statement (3) implies that the standard asympotics also hold for the likelihood ratio test restricted to the parameters of the model, which is of primary interest in many practical settings.
We assessed the proposed model and estimating procedure in samples of realistic size, and characterized efficiency of estimates of the log odds ratio parameters when prevalent cases are used in addition to, or instead of, incident cases in a case-control study.
We generated data directly from the exponential tilting models (2) and (4). For the controls, we simulated normally distributed covariates , where for , and , , with or . For incident cases, we generated values , where , or . To simulate exposure data for prevalent cases, we first generated a large set of values , , where . For each we computed a weight , which is proportional to the tilt in equation (4), and then drew a sample of size with replacement, where each was sampled with probability . As , the resulting sample has density as in equation (4).
For the survival distribution , we assumed a proportional hazards model with a Weibull baseline hazard where and are the shape and scale parameters, respectively, leading to , where are the parameters associated with covariate vectors , with , or . Then, whereand scale parameter one, which can be evaluated using standard statistical software.
In all simulations the backward times for the prevalent cases were generated letting to obtain a closed-form expression for in (5). To obtain backward times, we first generated and then computed .
Estimates (Est) of the parameters, empirical standard deviations (SD) of the estimates and standard deviations (SD) were based on K=1000 replications for each parameter setting. SD where is the estimate of in Theorem 1 scaled by the sample size. We estimate this quantity as , where is the numerical estimate of the Hessian, and
is obtained by computing the empirical variance of the scores for controls, incident and prevalent cases separately, and scaling each of the variance estimates by the respective sample size, to account for the retrospective sampling.
We first examined the efficiency of the log odds ratio estimates , and the estimates () of the parameters in the survival sub-model when or prevalent cases were added to a study with controls and incident cases.
Results for and , both with , and presented in Table 1 show that the estimates of all parameters were virtually unbiased, and the asymptotic and empirical standard deviation estimates agreed well. As increased from 500 to 1000, the SD for ’s decreased at the rate of , from 0.105 to 0.070 for and from 0.100 to 0.070 for .
Efficiency results in terms of the ratio of the variance of estimated using only the original incident cases and controls, compared to the variance of when prevalent cases were added, for all combinations of , and are shown in Figure 2. As the number
of prevalent cases increased from 0 to 1000, efficiency gains under the null hypothesis,, were modest and did not depend on the values of or (Figure 2a); the standard deviations (SD) for ’s decreased slightly from 0.073 for to 0.069 for (Table 1a).
Efficiency gains were somewhat more noticeable for and were greatest when had the same magnitude and signs as , and and were correlated (Table 1b). For example, for , , and , the ratio of the variance of based on incident cases and controls alone was three times larger compared to the variance after adding prevalent cases (Figure 2b, and Supplementary Table S6). Additional results are given in Supplementary Tables S1-S5.
|(a)||= 0||= 0||= 1||= 1||= 1||= -1|
|= 500, = 500|
|= 500, = 1000|
|(b)||= 1||= -1||= 1||= 1||= 1||= -1|
|= 500, = 500|
|= 500, = 1000|
We next examined the efficiency of estimates when the total number of cases was fixed at , but the proportion of prevalent cases increased, or . The number of controls was .
Results in Figure 3 show that replacing incident with prevalent cases resulted in an appreciable loss of efficiency of estimates in most settings. This was especially apparent when , where all the ratios var()/var() were below one. However, similar to the results in Section 4.2, when and had the same sign and magnitude, there was a gain in efficiency of when prevalent, instead of incident, cases were used. For example, for and with , the SD for ’s decreased from 0.107 to 0.091 as the proportion of prevalent cases increased from 0 to 100%, resulting in a 28% efficiency gain, as measured by the ratio of the corresponding variances (Supplementary Table S9 and additional results in Supplementary Tables S7-S13).
When designing a study, an investigator may have the choice of including additional incident or additional prevalent cases, possibly associated with different costs. We thus further investigated the difference in efficiency of when adding either incident or prevalent cases to a study comprised of a “base sample” of controls and incident cases. We first added from 20 to 1000 incident cases in increments of 20 to the base sample, and estimated var. Then, for each value of var, we found the number of prevalent cases that, if added to the controls and incident cases, resulted in . We also increased in increments of 20.
Figure 4 shows the relationship between additional incident and additional prevalent cases for all scenarios considered in Section 4.2. For most settings, using prevalent cases led to less efficient estimates of , indicated by lines above the 45 (gray dot-dashed) line, that corresponds to equal variance for the same number of added incident or prevalent cases. This loss of efficiency was particularly apparent when , where even for , approximately prevalent cases yielded the same variance of as 200 additional incident cases. However, when and had the same sign and magnitude, a prevalent case provided more information than an additional incident case, as indicated by the lines below the 45 line. For example, for and , using prevalent cases resulted in the same variance of as adding 400 incident cases to the base study sample.
As our method requires specifying a parametric survival distribution to model the backward time, we examined the robustness of the method to misspecification of .
First, we studied the estimates of when the backwards time was generated using that had a proportional hazards form with a Weibull baseline and was a function of three covariates , but we omitted in fitting the model. When was uncorrelated with and , all parameter estimates were unbiased and there was virtually no loss of efficiency (data not shown). When , there was appreciable bias in the parameter estimates of , but no noticeable bias in . For example, when and prevalent cases were added to incident cases and controls, the log odds ratio estimates were and . However, and instead of (Supplementary Table S14). There was no noticeable impact of the survival model misspecification on the efficiency of estimates of .
Assuming a proportional hazards model for , we also assessed the robustness of our method to misspecification of the baseline hazard of . We simulated the backward time for prevalent cases with a Weibull baseline hazard with shape and scale or a piecewise-constant baseline hazard with = 0.025, = 0.1, = 0.25 with breakpoints at = 0, = 10, = 30, , and fit the IP-case-control likelihood (7) using either a Weibull or piecewise-constant baseline hazard. We generated the data with or and or with sample sizes . Estimates of were virtually unbiased, despite misspecification of the baseline hazard, as were estimates for the (Supplementary Table S15). For this simulation the data were generated prospectively, as described in Supplementary Section 4.4.1.
We now illustrate our model using data from the study that motivated this work, a case-control study conducted within the United States Radiologic Technologists Study to assess associations of single nucleotide polymorphisms (SNPs) in candidate genes with risk of female breast cancer (Bhatti et al., 2008). This study used information from the first two surveys, conducted between 1984-1989, 1993-1998. Incident cases were women who answered both surveys and were diagnosed with a primary breast cancer between the two surveys. Eligible controls were women who at the age of diagnosis of a case were were breast cancer free. Controls were frequency matched to cases by year of birth in five year strata. Prevalent cases were women who answered only one of the two surveys and who reported a prior breast cancer diagnosis. Their backward time was defined as the difference between the year of the survey and the year of their diagnosis. All breast cancer diagnoses were confirmed based on pathology or medical records.
The covariates used in our analysis were age at diagnosis for cases or age at selection for controls (in categories: ); the year when the woman started working as a radiation technologist (1 if 1955, 0 if ); smoking status (1 if former/current, 0 if never); history of breast cancer among first degree relatives (yes/no); BMI (kg/m) during their 20’s (in categories: , (20-25], ); BMI during 20’s among women diagnosed after 50 years of age (coded as BMI during 20’s for women diagnosed at , and 0 otherwise, to capture the age dependent effect of BMI on breast cancer risk); history of heart disease (yes/no); alcohol consumption (1 if drinks/week, 0 otherwise); and genotype for three SNPs: rs2981582 (1 if TC/TT, 0 if CC); rs889312 (1 if CA/CC, 0 if AA); and rs13281615 (1 if GG/GA, 0 if AA). We restricted our analysis to women with complete covariate information, and used data from 663 controls, 345 incident cases, and 213 prevalent cases.
The demographic characteristics differed between controls, incident and prevalent cases (Supplementary Table S16). The prevalent cases were older than incident cases and controls, more likely to have started work as a radiation technologist before 1955, more likely to be current smokers, and to have a first-degree relative with breast cancer.
We compared log odds ratio estimates from the following models: (A) Incident model: standard logistic regression model fit to incident cases only and controls; (B) Naïve model: standard logistic regression fit to controls and incident plus prevalent cases combined without accounting for survival bias in the prevalent cases; (C) IP-case-control: IP-case-control likelihood (7) fit to incident cases, controls, and prevalent cases accounting for survival bias.
The covariates in the logistic models were rs2981582, rs889312, rs13281615, age at diagnosis or selection, year first worked, family history, BMI in 20’s, BMI in 20’s (50+), and alcohol consumption. The survival sub-model in (C) was a Cox proportional hazards model with a Weibull baseline hazard, with the same covariates as in the logistic sub-model plus smoking status and history of heart disease. The support of the backward time was 0 to , where
was chosen to be larger than the maximum backward time (35 years) among the prevalent cases. We computed jackknife standard errors (SEs) and also used them to compute 95% confidence intervals (CIs), assuming normality of the log odds ratio (log(OR)) or log hazard ratio (log(HR)) estimates.
Results of the analysis are summarized in Table 2. For model (A), the following covariates were significantly associated with breast cancer incidence (95% CIs are given in parentheses): SNP rs13281615, log(OR) = 0.40 (0.12, 0.69), year first worked, log(OR) = -0.84 (-1.23, -0.45), family history of breast cancer, log(OR) = 0.54 (0.25, 0.83), and BMI in ones 20’s, log(OR) = -0.34 (-0.60, -0.08). For model (B), the significant covariates were: SNP rs13281615, log(OR) = 0.34 (0.10, 0.58), family history of breast cancer, log(OR) = 0.57 (0.32, 0.83), and BMI in ones 20’s, log(OR) = -0.37 (-0.59, -0.14). Nearly all estimates from model B were attenuated compared to those from model A.
For model (C), the following covariates were associated with breast cancer incidence: SNP rs13281615, log(OR) = 0.32 (0.05, 0.58), year first worked, log(OR) = -0.34 (-0.65, -0.03), family history of breast cancer, log(OR) = 0.53 (0.27, 0.79), and BMI in ones 20’s, log(OR) = -0.34 (-0.57, -0.11). The log(OR) estimates in the IP-case-control model for rs981782, rs889312 and BMI in 20’s were close to those estimated from the incident model, with smaller standard errors. The log(OR) estimates for age at diagnosis and year first worked were somewhat lower than the estimates of model (A) (Table 2). However, those two variables were the ones significantly associated with the backward time, with log(HR) = 0.48 (0.28, 0.68) for age at diagnosis, log(HR) = -1.48 (-2.03, -0.93) for year first worked. Not surprisingly, the baseline hazard increased with increasing backward time.
Based on the likelihood ratio test, using an asymptotic cutoff value, the IP-case control model with the three SNPs in the logistic and the survival models fit the data significantly better than a model without the SNPs (p = 0.033).
|Incident (A)||Naïve (B)||IP-case-control (C)|
|log(OR) (SE)||log(OR) (SE)||log(OR) (SE)|
|rs2981582||0.170 (0.144)||0.122 (0.123)||0.137 (0.134)|
|rs889312||0.228 (0.137)||0.188 (0.118)||0.237 (0.127)|
|rs13281615||0.404 (0.147)||0.341 (0.124)||0.317 (0.134)|
|Age at diagnosis/selection||0.133 (0.073)||-0.056 (0.060)||0.059 (0.063)|
|Year first worked||-0.838 (0.198)||0.022 (0.151)||-0.341 (0.159)|
|Family history||0.542 (0.148)||0.574 (0.128)||0.527 (0.133)|
|BMI in 20s||-0.341 (0.132)||-0.366 (0.113)||-0.342 (0.118)|
|BMI in 20s (50+)||0.226 (0.213)||0.221 (0.184)||0.162 (0.199)|
|7+ alcoholic drinks/week||0.131 (0.203)||0.109 (0.173)||-0.004 (0.196)|
|Age at diagnosis/selection||0.483 (0.103)|
|Year first worked||-1.482 (0.280)|
|Ever smoker||-0.136 (0.165)|
|Family history||-0.188 (0.153)|
|BMI in 20s||0.094 (0.158)|
|BMI in 20s (50+)||-0.240 (0.314)|
|History of heart disease||0.021 (0.406)|
|7+ alcoholic drinks/week||-0.417 (0.350)|
|, , Est (SE)||1.581 (0.317), 11.147 (2.134)|
rs2981582: 1 if TC/TT, 0 if CC; rs889312 SNP: 1 if CA/CC, 0 if AA; rs13281615 SNP: 1 if GA/GG, 0 if AA; age at diagnosis/selection: coded with a trend based on categories ; year first worked: 1 if 1955, 0 otherwise; BMI (kg/m) in 20s: coded with a trend based on categories ; BMI in 20s (50+): coded as BMI in 20s among subjects with age at diagnosis of years, 0 otherwise; ever smoker: 1 if current or former smoker, 0 otherwise; 7+ alcoholic drinks/week: 1 if drinks per week, 0 otherwise.
and are Weibull baseline hazard shape and scale parameters, respectively.
The distribution of exposures among prevalent cases, individuals who have a prior disease diagnosis and are alive at the time of sampling for a case-control study, differs from that among incident cases when the exposures are associated with survival after disease onset. Thus naïvely combining prevalent cases with incident cases in the analysis of case-control data without accounting for their survival bias leads to biased estimates of log odds ratios for association (Begg and Gray, 1987).
In this paper we propose a semi-parametric model to incorporate data on covariates and the observed backward time from prevalent cases, to obtain unbiased estimates of exposure-disease association. We propose a three-group exponential tilting, or density ratio, model to accommodate two case groups and one control group, tht we assume is an appropriate comparison group for the incident cases. We provide a semi-parametric method for estimation based on empirical likelihood (Qin and Lawless, 1994; Qin, 1998).
Many authors dealt with the issue of length-bias when estimating survival parameters based on a prevalent cohort (e.g. Cook and Bergeron, 2011; Huang and Qin, 2012; Zhu et al., 2017). However, very few publications use prevalent cases when samples are ascertained cross-sectionally. Without using any information on follow-up, Chan (2013) estimated the impact of a covariate on the survival distribution in a log-linear model by showing that the covariate sampling distribution of prevalent cases compared to incident cases could be expressed using an exponential tilting model. To our knowledge only Begg and Gray (1987) addressed adjusting for survival bias when comparing prevalent cases to controls to estimate incidence odds ratios, again, not using any follow-up information. They modeled the backward time distribution based on an accelerated failure time model for survival and estimated the parameters using quasi-likelihood techniques. Incidence log odds ratio parameters were then estimated by subtracting a bias term from the log odds ratio estimates obtained from a standard logistic model fit to controls and prevalent cases.
In contrast to the approach by Begg and Gray (1987), we propose a semi-parametric likelihood that yields root consistent and fully efficient estimates of the incident log odds parameters. We show that the corresponding likelihood ratio statistic has a standard asymptotic chi-square distribution, which makes the test easy to use and therefore relevant for practical applications. Based on simulations, the efficiency gains or losses when prevalent cases are added to, or used instead of, incident cases depend on the ratio of the incident to prevalent cases, and the correlation structure among the covariates in the incidence and survival sub-models. Surprisingly, in some settings, prevalent cases were more informative than incident cases, which warrants further investigation in future work.
A limitation of our approach is that the model for the backward time is fully parametric. However, based on simulations, the estimates of the log odds ratios were not affected by reasonable misspecification of the model for the backward time. Our method is thus very appealing in settings where there is little concern about recall bias for the main exposure and the number of available incident cases is limited.
Web Appendices referenced in Sections 3, 4.2, 4.3, 4.5, 5, and Appendix 2, include proofs of equations (17) and (20), calculation of in equation (17), and additional simulation results including misspecified models, are available with this paper at the Biometrics website on Wiley Online Library.
The authors thank Michele Doody at the National Cancer Institute for providing the data, Jerry Reid at the American Registry of Radiologic Technologists, Diane Kampa and Allison Iwan at the University of Minnesota, Jeremy Miller and Laura Bowen at Information Management Services, and the radiologic technologists who participated. This work utilized the computational resources of the NIH HPC Biowulf cluster (http://hpc.nih.gov).
Following Qin (1998), we estimate , , empirically under the following constraints: (1) , (2) , (3) , by accommodating them in the log-likelihood using Lagrange multipliers, , . The ’s and ’s are explicitly computed by maximizing the constrained log-likelihood: