Flexible TwoPhase studies for rare exposures: Feasibility, planning and efficiency issues of a new variant
 Pascal Wild^{1_48}Email author,
 Nadine Andrieu^{2_48, 3_48, 4_48},
 Alisa M Goldstein^{5_48} and
 Walter Schill^{6_48}
DOI: 10.1186/1742557354
© Wild et al. 2008
Received: 28 May 2008
Accepted: 01 October 2008
Published: 01 October 2008
Abstract
The twophase design consists of an initial (Phase One) study with known disease status and inexpensive covariate information. Within this initial study one selects a subsample on which to collect detailed covariate data. Twophase studies have been shown to be efficient compared to standard casecontrol designs. However, potential problems arise if one cannot assure minimum sample sizes in the rarest categories or if recontact of subjects is difficult.
In the case of a rare exposure with an inexpensive proxy, the authors propose the flexible twophase design for which there is a single time of contact, at which a decision about full covariate ascertainment is made based on the proxy. Subjects are screened until the desired numbers of cases and controls have been selected for full data collection. Strategies for optimizing the cost/efficiency of this design and corresponding software are presented. The design is applied to two examples from occupational and genetic epidemiology. By ensuring minimum numbers for the rarest diseasecovariate combination(s), we obtain considerable efficiency gains over standard twophase studies with an improved practical feasibility.
The flexible twophase design may be the design of choice in the case of well targeted studies of the effect of rare exposures with an inexpensive proxy.
Introduction
For rare exposures, the power of epidemiological studies depends mainly on the rarest diseaseexposure combinations. For example, in populationbased casecontrol studies the limiting factor is frequently the number of exposed cases and/or controls. One approach that may substantially increase power for these types of studies is the twophase study design.
The twophase design [1–4] consists of an initial (Phase One) large study with known disease status and easily collectible or inexpensive covariate information. Within this initial study one selects a subsample on which to collect detailed covariate data (Phase Two). In Phase Two, one may deliberately oversample the subjects with the rarest exposuredisease combinations based on the available Phase One information, consequently increasing power. Appropriate statistical methods [5] correct for the biased sampling by incorporating the statistical distribution of the available information among cases and controls from Phase One. The data collection of Phase Two usually proceeds in one of two ways. The first approach includes recontacting selected study subjects from Phase One to obtain detailed covariate information. However, with secondary data collection, potential problems may arise if recontacting subjects is difficult, if cases have died, or if response rates are low. Alternatively, one may collect full raw data at first contact for all participants and process only selected subjects. An example would be a molecular or genetic epidemiologic study in which biological specimens were obtained for all cases and controls but only a subsample were genotyped (see [6] for another example). This may, however, be considered wasteful since only a fraction of the collected data is used.
As an alternative, we propose a new variant of the twophase design called the flexible twophase design, for which there is a single time of contact. Phase One data are collected for all subjects and Phase Two subjects are selected for immediate complete data collection based on their basic Phase One information. The key principle of this new variant is to fix a priori stratumwise numbers of cases and controls for full data collection and recruit Phase One subjects until the required numbers of subjects in each stratum are reached.
We describe the proposed study design and its implementation in terms of power, cost/efficiency considerations and statistical analysis. We illustrate its applicability using two examples from occupational and molecular/genetic epidemiology.
Steps in the planning and the analysis of flexible twophase studies
We start by defining several key variables and then describe the proposed setup for the study design. First, define Z, a discrete proxy variable for the exposure(s) of interest (X). Z needs to be collected and available at Phase One. Then, compute the power for several design options within the flexible twophase design (see below). Based on these computations, select the design option which produces the best compromise between power and feasibility in terms of subject availability, cost and other studyspecific criteria that will permit achievement of the study aims.
The four major steps for the setup of a study with the proposed design are as follows:
Design setup
1. Identify a stratification variable Z which is an easily available proxy of the exposure(s) of interest X. The number of strata (J) will equal the number of response choices for Z.
2. For each stratum, fix the number of cases and controls (n_{ij}), based on study power and cost considerations, for whom the exposure of interest X and covariates will be assessed. From n_{ij}, compute their expected distributions according to X and the numbers of cases and controls who will need to be screened at Phase One.
Data collection
3. Screen subjects for Z and keep cases and controls for full data collection (i.e. the variable(s) of interest X and potential confounders) until the numbers of cases and controls fixed in step 2 are reached.
4. Within each stratum j, count the number of cases and controls that were screened in Phase One at Step 3.
Computation of expected numbers and power
As mentioned above, the expected Phase One numbers depend on the fixed stratumspecific Phase Two numbers. They also depend on the study hypotheses including exposure prevalences and odds ratios. Other assumptions, common to all types of twophase studies, quantify how well the Phase One strata predict the exposure of interest (sensitivity and specificity of proxy Z). The formulas for expected Phase One and Phase Two numbers are given in Appendix 1. From these numbers, one can compute, using specific variance computations given in Schill and Drescher [5], the expected asymptotic variance and the statistical power. A corresponding STATA (StataCorp, College Station Texas) program for data analysis and power computations is included as an online addon to this paper.
Planning options
A critical issue is how to optimize, in terms of cost and power, the fixed stratumwise numbers of cases and controls with full data collection. This complex problem has been addressed in different contexts [7–10]. However, one can formulate a general heuristic rule, which has worked well in our applications using Maximum Likelihood as the analysis method. Specifically, choose the numbers of cases and controls for full data collection so that, within both controls and cases, the overall expected Phase Two exposure proportions are as equally distributed as possible. For rare exposures, this means choosing cases and controls to oversample the rarest exposure categories among both groups.
Statistical analysis
The collected data can be analyzed using any twophase analysis software. As the second phase sample is a biased sample of the original population, a combined analysis of the Phase One and the Phase Two data relies on weighting of the Phase Two data by the inverse sampling fractions. The two main methods for analysis are maximum likelihood (ML) and weighted likelihood (WL) which differ in the weights used; the more efficient ML estimate iteratively adjusts these weights using the estimated disease model. As such software is not readily available, we included our STATAbased twophase analysis program "blogit_2P.ado" [see additional file 1]. The software takes as input the disease indicator, the stratum indicator, the Phase One frequencies, the Phase Two frequencies and the independent variables. A help file accessible from within STATA "blogit_2P.hlp" [see additional file 2] is also included as well as an illustrative example [see additional files 3 and 4]. In this paper, we use the ML approach.
Examples
To demonstrate the potential efficiency of the flexible twophase approach, we present two examples from occupational and molecular/genetic epidemiology. In the first example, we detail the computations for a given design; in the second, we perform a full search for optimal designs for given scenarios.
Example 1: Metalworking fluids and bladder cancer
A number of populationbased casecontrol studies have found an association between bladder cancer and metalworking fluids (MWF) exposure (see Calvert [11] for a review). However, because of the low prevalence of the exposure, the numbers of exposed cases and controls in each study were too small to produce a stable estimate of the association. We use a flexible twophase study to illustrate the efficiency gain over a standard casecontrol study, considering as a proxy of MWF exposure "having worked in the metal industry". In practice, when contacting cases and controls, for instance in a telephone interview, one of the first questions to the volunteers would be: "Have you ever worked in the metal industry?". Based on the answer to this question the subject would then be included (or not) in Phase Two; that is, the interview would be continued to assess a detailed work history and confounder information.
Scenario for Example 1
Variables and parameters characterizing the setup  Values of parameters and variables 

Stratification/Proxy Z (with J strata)  Past work in metal industry 
No: Z = 1  
Yes: Z = 2  
Phase One prevalence among controls (τ^{0} _{j})  Z = 1: τ^{0} _{1} = 80% 
Z = 2: τ^{0} _{2} = 20%*  
Risk factor X (with K outcomes)  Exposure to MWF 
No: X = 1  
Yes: X = 2  
Disease Model (Odds Ratios) (ψ_{k})  ψ_{1} = 1: baseline risk 
ψ_{2} = 2^{#}  
Phase Two prevalence of X among controls by stratum (π^{0} _{jk})  Z = 1: π^{0} _{11} = 97.5%, π^{0} _{12} = 2.5%^{&} Z = 2: π^{0} _{21} = 75%, π^{0} _{22} = 25%^{@} 
Study design
1. Stratify subjects by Z (Table 1, Line 1).
Design of the flexible twophase study for Example 1
Disease Status (D)  Metalworkers Z (τ ^{ i } _{ j })  Fixed number of subjects to be included in Phase Two (n _{ ij })  Expected Phase One numbers of subjects to be screened (N _{ ij })  Expected Proportion of MWF exposure within strata § X(π ^{ i } _{ jk })  Expected distribution of subjects by MWF in Phase Two 

N_{0} = Max(160/20%, 40/80%) = 800  
Control  No (80%*)  40  800*80% = 640  No (97.5%*)  40*97.5% = 39 
Yes (2.5%*)  40*2.5% = 1  
Yes (20%*)  160  800*20% = 160  No (75%*)  160*75% = 120  
Yes (25%*)  160*25% = 40  
N_{1} = Max(85/23.4%, 20/76.6%) = 364  
Case  No (76.6%#)  20  364*76.6% = 278.8  No (95.1%#)  20*95.1% = 19.02 
Yes (4.9%#)  20*4.9% = 0.98  
Yes (23.4%#)  85  364*23.4% = 85  No (60%#)  85*60% = 51  
Yes (40%#)  85*40% = 34 
Planned data collection
3. Screen cases and controls until the required numbers in each stratum are reached and assess the detailed exposure to MWF and potential confounders in this sample of 305 subjects.
4. Record the number of subjects screened in order to reach the required sample size. At the planning stage, these numbers are not yet available, but expected numbers can be computed. Assuming 20% metalworkers in the general population, we would expect to screen 800 controls (N_{0}) to obtain 160 metalworkers (20% × 800 = 160). Therefore, the number of nonmetal worker controls that would have been screened (N_{00}) is expected to be 640 (800–160) of which 40 are included in Phase Two for detailed exposure assessment. For the corresponding computations for cases, see Table 2 and Appendix 2.
We note that oversampling the metalworkers has achieved our aim of increased numbers of MWF exposed cases and controls. Among the 200 controls, 41 are exposed (20.5% versus 7% in Phase One) and among the 105 cases, 35 are exposed (33.3% versus 13% in Phase One) (Table 2, Column 6 and Footnote §.
Example 2: Detection of geneenvironment interaction
Molecular/genetic epidemiology studies identify genes involved in disease risk, estimate the strength of the diseasegene association and investigate modifier factors that may interact with the susceptibility genes. The study of interactions between genes and "environmental" factors is often challenging because of the rarity of having both factors, i.e., being exposed to the environmental factor of interest and carrying a deleterious allele.
We present a search for an optimized flexible TwoPhase design, in this setting, assuming that an inexpensive proxy of the deleterious allele (e.g., family history of disease) is available.
The scenarios
Scenarios for Example 2
Variables and parameters required for setup  Formulas and values of parameters 

Stratification/Proxy Z (with J strata)  Environmental exposure E and Gene proxy S_{G} 
J = 4  
Z = 1: E^{} S_{G} ^{}, Z = 2: E^{} S_{G} ^{+}, Z = 3: E^{+} S_{G} ^{}, Z = 4: E^{+} S_{G} ^{+}  
Phase One prevalence among controls (τ^{0} _{j}):  τ^{0} _{1} = Pr(E^{})Pr(S_{G} ^{}) = (1  P_{E})[(1Se)P_{G}+Sp(1P_{G})] 
P_{E} = 20%  τ^{0} _{2} = Pr(E^{})Pr(S_{G} ^{+}) = (1  P_{E})[SeP_{G}+(1Sp).(1P_{G})] 
P_{G} = 1%  τ^{0} _{3} = Pr(E^{+})Pr(S_{G} ^{}) = P_{E}[(1Se)P_{G}+Sp(1P_{G})] 
τ^{0} _{4} = Pr(E^{+})Pr(S_{G} ^{+}) = P_{E}[SeP_{G}+(1Sp).(1P_{G})]  
Risk factor X (with K outcomes)  Exposure to E and exposure to G: K = 4 
X = 1: E^{} G^{}, X = 2: E^{} G^{+}, X = 3: E^{+} G^{}, X = 4: E^{+} G^{+}  
Disease Model (Odds Ratios ψ_{k})  ψ_{1} = 1, ψ_{2} = 3, ψ_{3} = 2, ψ_{4} = ψ_{2} × ψ_{3} × OR_{I} = 30 
Phase Two prevalence of X among controls by stratum (π^{0} _{jk})  Z = 1: π^{0} _{11} = (1 P_{E})Sp(1P_{G})/Pr(S_{G} ^{}), 
π^{0} _{12} = 1  π^{0} _{11}, π^{0} _{13} = π^{0} _{14} = 0  
Z = 2: π^{0} _{21} = (1  P_{E})(1  Sp)(1P_{G})/Pr(S_{G} ^{+}),  
π^{0} _{22} = 1  π^{0} _{21}, π^{0} _{23} = π^{0} _{24} = 0  
Z = 3: π^{0} _{31} = π^{0} _{32} = 0, π^{0} _{33} = P_{E} Sp(1P_{G})/Pr(S_{G} ^{}),  
π^{0} _{34} = 1  π^{0} _{33}  
Z = 4: π^{0} _{41} = = π^{0} _{42} = 0, π^{0} _{43} = P_{E} (1  Sp)(1P_{G})/Pr(S_{G} ^{+}),  
π^{0} _{44} = 1  π^{0} _{43} 
We further assume that the proxy of the susceptibility gene (SG) and the environmental exposure (E) are available at Phase One for an unlimited number of controls. However, we restrict the number of cases available in Phase One to a maximum of 2000 cases. We further assume that capacities for genotyping restrict the total number of subjects (cases + controls) that can be included in Phase Two to a maximum of 1200 subjects. We assume that the cost of genotyping is 20 times the cost of screening. Such a cost ratio would arise if, for example, a SNP array costs $100 and 15 minutes for a trained interviewer screening a subject for E and SG costs $5. We repeat the design search for each combination of sensitivity (Se) and specificity (Sp) of 0.6, 0.7, 0.8, and 0.9.
Planning the design
The aim of the flexible twophase approach is to choose subjects for genotyping to optimize the study power for given costs. This is achieved by oversampling subjects with positive gene proxy and environmental exposure.
In practice, such oversampling could be done during case/control recruitment using a short interview that allows assessment of the environmental exposure and the gene proxy (e.g., a family history of disease) and getting a blood/buccal sample (for genotyping) only for the subjects sampled for Phase Two based on the results of this first interview.
Step 1: The stratification is by gene surrogate and environmental exposure (Table 3, line 1).
Step 2 entails choosing the stratumwise numbers of cases and controls to be included in Phase Two. We use our general heuristic rule with respect to E and fix at 50% the target numbers of E+ and E to be included in Phase Two among cases and controls. The amount by which we oversample SG+ will be considered through use of two additional parameters, the proportion of controls ρ0 with SG+ and the proportion ρ1 of cases with SG+. For example, if we selected 800 controls and 400 cases with proportions ρ0 = 80% and ρ1 = 60%, this would correspond to 800*50%*80% = 320 E+ SG+ controls, 400*50%*60% = 120 E+ SG+ cases, 800*50%*20% = 80 E+ SG controls and so on.
Comparing designs
We now consider a series of design options for this example for which we compare power and cost. To meet the constraints on availability and capacity fixed above, the designs considered have numbers of cases ranging from 100 to 600 and numbers of controls from 400 to 1100 in steps of 100, with a maximum of 1200 subjects to be included in Phase Two. For each of these combinations, ρ0 and ρ1 are varied from 40% to 90%. This corresponds to several hundred possible designs for each combination of sensitivity and specificity of SG.
Designs with maximal power of detecting the interaction, according to sensitivity and specificity
Genesurrogate  Flexible twophase design options  Expected Phase One counts  Power#  Cost*  

Spec  Sens  n_{0}  n_{1}  ρ_{0}†  ρ_{1}‡  N_{0}  N_{1}  
70%  80%  800  400  90%  90%  5902  1373  83%  1564 
70%  90%  800  400  90%  90%  5882  1325  87%  1560 
80%  60%  800  400  90%  90%  8824  1988  87%  1741 
80%  70%  800  400  90%  90%  8780  1889  91%  1733 
80%  80%  800  400  90%  90%  8738  1800  94%  1727 
80%  90%  800  400  90%  90%  8696  1718  96%  1720 
90%  60%  900  300  90%  90%  19286  2000  98%  2264 
90%  70%  900  300  90%  90%  19104  2000  99%  2255 
90%  80%  900  300  90%  90%  18925  1960  99.6%  2244 
90%  90%  900  300  90%  90%  18750  1835  99.8%  2229 
Designs with minimum cost among designs with 80% power of detecting the interaction
Genesurrogate  Flexible twophase design options  Expected Phase One counts  Power#  Cost*  

Spec  Sens  n_{0}  n_{1}  ρ_{0}†  ρ_{1}‡  N_{0}  N_{1}  
70%  80%  700  500  90%  80%  5163  1525  81%  1534 
70%  90%  600  500  90%  80%  4412  1472  80%  1394 
80%  60%  700  300  90%  90%  7721  1491  80%  1461 
80%  70%  600  300  90%  80%  6585  1259  80%  1292 
80%  80%  500  300  90%  80%  5461  1200  80%  1133 
80%  90%  400  400  90%  80%  4348  1528  81%  1094 
90%  60%  400  400  70%  50%  6667  1683  81%  1217 
90%  70%  500  300  50%  50%  5896  1169  80%  1153 
90%  80%  500  300  40%  60%  4673  1307  80%  1099 
90%  90%  500  300  40%  50%  4630  1019  82%  1082 
Note that the better the proxy, the more effective the flexible twophase approach. For example, for a gene proxy with 70% specificity and 80% sensitivity, the most cost effective design costs 1534 units whereas the most cost effective design for a gene proxy with 90% specificity and 90% sensitivity costs 1082 units.
Comparison with standard casecontrol studies
For the scenario considered, the most powerful standard casecontrol study with 1200 genotyped subjects would include 300 cases and 900 controls with an expected var(β_{I}) = 0.96, corresponding to a statistical power of 37%. Achieving 80% power would require var(β_{I}) = 0.33. Thus, for a standard casecontrol study to attain 80% power, it would require genotyping of 870 cases (i.e. 300 × 0.96/0.33) and 2610 controls (i.e. 900 × 0.96/0.33), totaling a cost of 3480 units. This compares to 1534 units in the most costeffective flexible twophase design assuming 70% specificity and 80% sensitivity.
Comparison with balanced twophase studies
A second comparison of interest would be a comparison with balanced twophase studies, the design that is generally recommended in papers on twophase studies (see [1, 2, 12]). As mentioned in the introduction, these studies start from a fixed Phase One sample and draw equal numbers in each stratum for Phase Two data collection. In order to be comparable to our flexible design, we considered a design in which 8000 controls and 2000 cases were assessed in Phase One and 800 controls and 400 cases included in Phase Two. As the design is balanced, we selected equal numbers, i.e., 200 controls and 100 cases from each stratum defined by SG × E.
This balanced TwoPhase design is always less efficient than the Flexible TwoPhase design although more efficient than the standard casecontrol design. For instance, in the preceding example with 70% specificity and 80% sensitivity, the expected variance is var(β_{I}) = 0.47, corresponding to a statistical power of 65%. The corresponding cost is 1200+(10000:20) = 1700 units.
Discussion
Twophase studies are efficient compared to standard casecontrol designs. The variant design presented in this paper improves on some aspects of standard twophase studies. Specifically, with respect to data collection there is only one time of contact. At a time when studies are struggling with decreasing response rates, collection of all necessary data at a single time of contact may result in improved overall participation rates. Moreover, for rare exposures, minimum numbers of exposed subjects can be guaranteed in this design, thus increasing the power, even compared with standard balanced TwoPhase designs. The disadvantage of the flexible twophase design compared to other designs, including standard twophase, is the additional complexity in design planning. Another possible disadvantage is that the categories that are relatively easy to fill will be filled quickly during recruitment, while the hardtofill categories will take longer to reach their sampling targets. This can produce complex relationships between covariates and recruitment times. This could be alleviated by the randomized recruitment approach proposed by Weinberg and Sandler [13] in which the most common Phase One category would be included in Phase Two with a given probability, chosen so that all categories are filled in at about the same time.
In the examples presented, we focused on rare exposures for which one could identify inexpensive proxies. Using our proposed heuristic rule, this allows oversampling the rare exposure and thus increasing power. This approach is efficient provided the analysis method used is maximum likelihood, thus, implicitly assuming nondifferential misclassification, i.e., that the proxy is not a confounder. In practical terms, this means that the disease risk, given exposure, is the same in all strata. If the disease risk varies across strata, the effect of exposure may have to be assessed separately in each stratum resulting in reduced power to detect the effect of exposure in the underrepresented strata.
One major consideration for the flexible twophase design is the availability of an adequate proxy for Phase One screening. The proxy must be easily obtained on all screened subjects but must also have high sensitivity and specificity. For a study focused on occupational exposures, as in example 1, a question about working in the industry of interest is easily collected and should yield a reasonable proxy for exposure. This binary stratification for the proxy may be extended to increase sensitivity and specificity. For example, one could ask about duration of work in a particular industry, thereby obtaining a proxy of the actual cumulative dose. Similarly, a positive family history was previously shown [14] to be a good proxy for a rare gene with a strong effect. However, as the effect of the allele decreases and its frequency increases (as would be the situation for a lowrisk gene) the sensitivity and specificity for family history decreases. In such situations, an alternative proxy for G may need to be considered, such as age at diagnosis, or a quick inexpensive physiologic test during the inperson interview at Phase One. Of course, the more information obtained at Phase One, the more expensive Phase One becomes.
We acknowledge that a geneenvironment interaction odds ratio of 5 may be rather extreme for most diseases, particularly given some recent findings, as in [15]. We are currently working on a more topicoriented comparison of different study designs for detecting geneenvironment interactions using a wider range of scenarios and including the Flexible TwoPhase design and caseonly design (under the assumption of independence of Genetic and Environmental factors in the population).
In the present paper, we focused on the estimation of a single oddsratio. However, doseresponse estimation is possible, as long as detailed data are available at Phase Two. Similarly, it is possible to adjust for confounders as long as the relevant data are available in Phase Two. However, since the flexible twophase design is mostly targeted on predefined hypotheses, especially if one oversamples some strata, there may be limited power to test other hypotheses or perform exploratory analyses. For example, exposure to some aromatic amines increases risk for bladder cancer, but this exposure is rare in the metal industry. Thus, the design we considered would have low power for detecting this risk. Many epidemiologic studies are exploratory in that they assess the effects of a large spectrum of factors without focusing on predefined hypotheses. The Flexible TwoPhase design is not adapted to this situation and focuses necessarily on a restricted number of explicitly stated hypotheses. We are, however, convinced that in many circumstances, only studies with predefined hypotheses will allow progress in understanding disease etiology.
Conclusion
In conclusion, the flexible twophase design expands the advantages of twophase designs to substantially increase power for studies of rare diseaseexposure combinations. The flexible twophase design may be the design of choice in well targeted studies of the effect of rare exposures for which inexpensive proxies are available.
Abbreviations
 MWF:

metal working fluid
 SG:

the surrogate of the gene G considered as a risk factor.
Declarations
Acknowledgements
This work was funded by the National Cancer Institute, NIH (Intramural Research Program to A.G.) and the Deutsche Forschungsgemeinschaft (PI 345/1–2 to W.S.)
Authors’ Affiliations
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