Improving the efficiency of nested case-control studies of interaction by selecting controls using counter matching on exposure

John B Cologne1, Gerald B Sharp2, Kazuo Neriishi3, Pia K Verkasalo5, Charles E Land6 and Kei Nakachi4

Departments of 1 Statistics, 2 Epidemiology, 3 Clinical Studies, 4 Radiobiology/Molecular Epidemiology, Radiation Effects Research Foundation, Hiroshima, Japan
5 Unit of Environmental Epidemiology, National Public Health Institute, Finland
6 Radiation Epidemiology Branch, National Cancer Institute, US

Correspondence: John B Cologne, Department of Statistics, Radiation Effects Research Foundation, 5-2 Hijiyama Park, Minami-ku, Hiroshima 732-0815, Japan. E-mail: cologne@rerf.or.jp


    Abstract
 Top
 Abstract
 Subjects and Methods
 Results
 Conclusions and Discussion
 References
 
Background Studies of the effect of exposure to a risk factor measured in an entire cohort may be augmented by nested case-control subsets to investigate confounding or effect modification by additional factors not practically assessed on all cohort members. We compared three control-selection strategies—matching on exposure, counter matching on exposure, and random sampling—to determine which was most efficient in a situation where exposure is a known, continuous variable and high doses are rare.

Methods We estimated the power to detect interaction using four control-to-case ratios (1:1, 2:1, 4:1, and 8:1) in a planned case-control study of the joint effect of atomic bomb radiation exposure and serum oestradiol levels on breast cancer. Radiation dose is measured in the entire cohort, but because neither serum oestradiol level nor the true degree of interaction was known, we simulated values of oestradiol and hypothetical levels of oestradiol–radiation interaction.

Results Compared with random sampling, power to detect interaction was similarly higher with either matching or counter matching with two or more controls.

Conclusions Because counter matching is generally at least as efficient as random sampling, whereas matching on exposure can result in loss of efficiency and precludes estimation of exposure risk, we recommend counter matching for selecting controls in nested case-control studies of the joint effects of multiple risk factors when one is previously measured in the full cohort.


Keywords Nested case-control studies, probability sample, matching, counter matching, breast cancer, radiation effects

Accepted 17 December 2003

In cohort studies aimed at investigating the effects of already measured risk factors—such as radiation exposure in the survivors of the atomic bombings of Hiroshima and Nagasaki, Japan—nested case-control studies may be conducted to analyse the effects of additional factors that cannot be assessed practically in the entire cohort. The purpose might be to study the effects of potential confounders or effect-modifying factors on the exposure risk. If the exposure is rare or has a skewed distribution, ignoring it in selecting controls can lead to a loss of statistical efficiency, so exposure-based methods of control selection might be considered.

As an example, although radiation and oestradiol are both strong risk factors for breast cancer, only radiation dose is known for atomic bomb survivors. The relative risk of early-onset breast cancer (diagnosis under age 35) for 1 Sv of radiation is 14 among women irradiated by the atomic bombs before age 20; the overall relative risk of breast cancer ranges from 2.3 to 3.4 among all women exposed under the age of 40.1 Key, Verkasalo, and Banks showed that serum oestradiol is positively associated with risk of breast cancer, with risks being about twice as high for postmenopausal women with high, as opposed to low, serum oestradiol concentrations.2 Little is known, however, about oestradiol levels and the risk of pre-menopausal breast cancer. Furthermore, the joint effect of radiation and oestradiol has not been studied, although Land et al.3 demonstrated interactions between radiation and breast cancer risk factors that may be related to constitutional hormone levels. At the Radiation Effects Research Foundation (RERF), we are conducting a study of radiation and oestradiol as joint risk factors for pre-menopausal breast cancer using stored sera obtained from atomic bomb survivors who participated in biennial clinical examinations conducted for RERF's Adult Health Study. Oestradiol, which is expensive to assay and requires sera, for which supplies are limited, will be measured in all cases but only a subset of controls. This raises the issue of how best to select controls to provide maximum statistical efficiency.

Selecting controls by individually matching them to cases on radiation exposure can improve statistical efficiency for testing interaction with another factor.4 If there is evidence of interaction or possible confounding in the case-control sample, a logical next step in the analysis would be to examine how exposure risk estimates vary with the level of the other factor. Matching on exposure allows studying the effect of con-founder/effect-modifier per se but precludes studying its effect on the exposure risk without additional information, such as comes from the cohort.3 An alternative to matching is weighted sampling of controls using counter-matching, where controls are selected to fill exposure strata not occupied by the case.5–8 Counter matching also allows estimation of the exposure risk, and the efficiency for studying both confounding and effect modification can be improved relative to random sampling of controls.9 Furthermore, counter matching allows the investigator to fix the number of controls in advance and is easily implemented with prospective, risk-set based selection. Many of the references on these designs provide justification and intuitive explanation as to why exposure-based sampling is efficient.

Counter matching has been shown to generate better efficiency for testing interaction than matching over a wide range of exposure risks and degrees of correlation between exposure and another risk factor when both are dichotomous,8 but comparisons have not been made for continuous risk factors, such as radiation dose and oestradiol. The proposed breast cancer study provides a basis for making that comparison in the case of a rare exposure that may interact positively with an additional factor, the situation in which matching achieves the greatest gain in efficiency.4 Our objective was to assess the extent to which matching and counter matching impact statistical power for detecting interaction relative to random control selection.


    Subjects and Methods
 Top
 Abstract
 Subjects and Methods
 Results
 Conclusions and Discussion
 References
 
Study design
We are conducting a nested case-control study of pre-menopausal breast cancer, radiation exposure, and serum oestradiol levels using all currently available cases.10 Radiation doses are known, but oestradiol is to be measured in the case-control subset using stored sera. Day of menstrual cycle at the time of serum collection is not known; therefore it was decided to use the average of two measurements on serum specimens collected on different occasions. There were 80 cases of pre-menopausal breast cancer and 5644 cancer-free women with stored serum. Radiation doses were calculated according to Dosimetry System 1986 (DS86).11 Radiation doses to the breast were calculated in Sievert (Sv), combining gamma and neutron components with a relative weight of 10 for neutrons.

To assess the power of the study for detecting interaction, we simulated oestradiol levels and their interaction with radiation. We then calculated the resulting power using one, two, four, or eight controls selected from case risk-sets using three approaches: (1) random sampling, (2) matching as closely as possible on radiation exposure, or (3) counter matching on radiation exposure.

Selecting controls by random sampling is simple. Within risk sets defined by age, date, and availability of stored serum, controls are selected at random, without regard for exposure status. Matching on radiation exposure is also straightforward; within each risk set, the potential controls whose exposure values are closest to the case's exposure are selected. Note, however, that matching on exposure in addition to matching on other factors can be more complicated.12 If there are more tied values among the potential controls than the number needed, the controls are selected at random from among the tied subjects.

With counter matching, exposure strata are defined based on the number of controls to be selected per case and on the distribution of exposure among the cases. In each risk set one control is selected from each of the exposure strata not occupied by the case. In the study described here, there were many people with dose zero; we therefore defined the lowest exposure category to be zero. The other categories were determined by the appropriate percentiles of the distribution of exposure values among the exposed cases. For example, with eight controls per case there were nine exposure strata: zero plus eighths defined by octiles of the case exposure values (cutpoints: 0.14, 0.45, 0.66, 0.79, 1.15, 1.50, and 2.04 Sv; Supplementary Material: Distribution of Radiation Doses). With four controls per case there were five strata: zero plus fourths defined by quartiles of the case exposures (cutpoints: 0.45, 0.79, and 1.50 Sv; Figure 1).



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Figure 1 Illustration of control-selection strategies with four controls per case. The case exposure value (x) in this illustration is close to zero and in the lowest of the fourths derived from quartiles among non-zero-dose cases. With random sampling, controls would be selected randomly without regard to their dose. With matching, the four controls with doses closest to that of the case would be selected. With counter matching, one control would be selected randomly from each of the non-case strata (the remaining three exposure fourths and the zero-dose stratum)

 
Simulation of serum oestradiol levels and interaction with radiation
Oestradiol values were computer generated to have overall mean and variance equal to those from a previous study performed in the same cohort.13 Because there was no information about day of menstrual cycle on which serum was collected, we accounted for day-to-day variation by mimicking random sampling from the menstrual cycle using data for British women from Verkasalo and her associates.14 First, a random integer representing day of cycle was obtained using a uniform random number generator. Then, a random log10-oestradiol value was obtained by generating a random Normal variable with mean and variance equal to those for the same cycle day among the British study group. Finally, the resulting values were adjusted to have overall (not day-specific) mean and variance equal to those found in the previous atomic bomb survivor study. This adjustment was made separately for cases and controls. The simulated values were standardized to produce a log odds ratio of {varphi}A = 0.359, the result obtained with an average log10-oestradiol difference of 7% between cases and controls in the previous study, where controls were matched to cases on radiation dose. This corresponds to odds of disease of 1.43 for a 1.41-fold higher oestradiol level. For each subject we used the average of two simulated oestradiol values from arbitrary cycle days to reduce the error due to lack of knowledge of cycle day on which the serum was collected.

Because radiation dose and case status were already known in the cohort, we simulated the interaction by adjusting the mean of the Normal distribution from which case log10-oestradiol values were generated so that the average case-control mean difference in log10-oestradiol level increased linearly with radiation dose, while log10-oestradiol values among the controls were generated independently of radiation dose. The dose-dependent case log10-oestradiol means were calculated to produce log odds ratios (OR) according to the following model:

so that the log OR is {varphi}A at a dose of 0.75 Sv, close to the average dose among the cases. Three levels of interaction were selected to produce low, moderate, and high power of detection in the full cohort. These were simulated by setting {gamma} so that, with a doubling of dose to 1.5 Sv, the relative changes in the log OR for log10-oestradiol with interaction were 0.5 (low degree of interaction), 0.75 (moderate degree of interaction), or 1.0 (high degree of interaction); Figure 2. The low degree of interaction results in a log OR of 0.5{varphi} (OR = 1.20) at 0 Sv and a log OR of 1.5{varphi} (OR = 1.71) at twice the median dose (1.5 Sv). The moderate degree of interaction results in corresponding log OR of 0.25{varphi} and 1.75{varphi} (OR = 1.09 and 1.87). The high degree of interaction results in corresponding log OR of 0 and 2.0{varphi} (OR = 1.00 and 2.05); it represents extreme effect modification in that it results in no oestradiol effect (OR = 1.0) among women not exposed to radiation.



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Figure 2 The three levels of hypothetical oestradiol-radiation interaction used in the simulations. Plots show the odds ratio for oestradiol as a function of radiation dose. The overall odds ratio (1.43) for the case-control difference of 7% in log oestradiol value was set to occur at a dose of 0.75 Sv, near the median dose of the cases, corresponding to the result of the previous study

 
We simulated 500 random case-control study outcomes for each of the 12 configurations defined by the number of controls per case selected (1, 2, 4, or 8) and the degree of interaction assumed (low, moderate, or high). All simulated data were generated using S-plus (MathSoft Inc., Seattle, Washington).

Statistical analysis of simulated data
We analysed the counter-matched design according to Langholz and Borgan,5 using conditional logistic regression with sampling weights as offsets.15 (An offset is added to the logistic regression model by entering it as a covariate with coefficient fixed at 1.) Counter-matched sampling weights were calculated separately for each risk set (i, i = 1,...,I), and exposure stratum (j, j = 1,...,J) as:

Weights are calculated in the same way for both cases and controls (Figure 1).

We studied effect modification via statistical interaction by fitting the model:

where f(d, e) is the log odds of breast cancer, ß is the log OR for a unit (1 Sv) difference in radiation dose (d), {theta} is the log OR for a unit difference in log10-oestradiol (e), and {gamma} is the interaction parameter. Interaction was tested by the {chi}2 approximation to the likelihood ratio test of the null hypothesis {gamma} = 0, which was rejected if the test statistic exceeded 3.84. Note that this is equivalent to failure of the likelihood-based CI to include the null value {gamma} = 0. We calculated power as the proportion of 500 simulations in which the null hypothesis of no interaction was rejected. Because the power of a cohort based analysis—if oestradiol levels were known for all cohort members—would increase with increasing degree of interaction, we compared the power of each case-control design to that of the entire cohort. For each level of interaction we simulated 1000 cohort oestradiol values.

Counter-matched case-control samples were analysed using conditional logistic regression with weights as described above. Matched and randomly selected case-control samples were analysed using conditional logistic regression without weights. Cohort data were analysed using unconditional logistic regression. All analyses were conducted using Epicure (Hirosoft Inc., Seattle, Washington).


    Results
 Top
 Abstract
 Subjects and Methods
 Results
 Conclusions and Discussion
 References
 
Figure 3 shows the distribution of randomly generated pre-menopausal oestradiol values for 80 cases and 160 controls from one set of 500 simulations assuming no interaction with radiation. There was substantial variation in simulated oestradiol levels within day and due to day of cycle. Oestradiol levels were 16% higher on average (log10-oestradiol values were 7% higher) in cases than in controls.



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Figure 3 Boxplots illustrating computer-generated oestradiol values by day of cycle among pre-menopausal breast cancer (a) cases and (b) selected controls. Data are from 500 simulations with 80 cases and 2 controls per case. Day of cycle is defined with 0 being 28 days prior to the start of the subsequent cycle (i.e. day 28 is the start of the next cycle). Case values are 16% higher on average than control values. The boxplots show the median (white dot), inter-quartile range (filled bar), and the most extreme observations (brackets)

 
As shown in Table 1, compared with random selection of controls, the counter-matched and matched strategies increased power to detect interaction for each level of interaction when at least two controls per case were selected. All three methods demonstrated similar levels of power relative to the full cohort (slightly less than 40%) with only one control per case. Relative differences in power between the three strategies were similar regardless of the level of interaction. In Figure 4 we summarize the power of each design using curves fit to the average power (averaged over levels of interaction). Random control selection resulted in power that ranged from 50% of the cohort level with two controls per case to slightly more than 80% with eight controls per case. On the other hand, the exposure-based sampling strategies achieved levels of power ranging from almost 80% with two controls per case to 100% (maximum) power with eight controls per case. For random sampling to achieve an acceptable level of power equivalent to that of the exposure-based control-selection designs required about twice as many controls, indicating that the exposure-based designs were about twice as efficient as random sampling for detecting interaction.


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Table 1 Estimated power for detecting interaction relative to the full cohort for three control-selection strategies

 


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Figure 4 Power of the control-sampling strategies for detecting interaction relative to the power of the full cohort. Points are values from Table 1, jittered slightly on the abscissa to reduce overlap. Lines are the results of fitting cubic power curves for purposes of smoothing the means (averaged over level of interaction)

 
Repeated runs with counter matching and four controls per case with the moderate degree of interaction resulted in absolute power estimates of 0.466, 0.516, 0.518, 0.474, and 0.530 (average: 0.501, standard deviation 0.029). Because the variance of a binomial variable is a maximum when the proportion is 0.5, the other simulation results would be equally, or more, precise. The differences between exposure-related sampling designs and random sampling were much larger than this repeated simulation variability. However, differences between power for counter matching and matching were small and on the order of the simulation precision.


    Conclusions and Discussion
 Top
 Abstract
 Subjects and Methods
 Results
 Conclusions and Discussion
 References
 
Case-control studies of interaction between multiple risk factors generally have low statistical power,16 so designs that are suitably sized for studying main effects may be inadequate for studying effect modification.17 The two exposure-related control-selection strategies studied here (matching and counter matching) resulted in similar gains in efficiency for testing interaction when two or more controls were selected per case. We conclude that, while more than 10 controls per case would be needed to achieve greater than 90% of the maximum, cohort level of power with random control selection, only 5–6 controls—about half as many as with random sampling—would be needed using either of the exposure-based control-selection designs. In practice, there could be missing specimens or refusal of informed consent, so using more than 5–6 controls per case might be considered based on the specific application. We do not recommend the use of only one control per case for studies of interaction.

The risk factors in our investigation were assumed to have positive interaction and one was a rare exposure already measured in the full cohort. This is the situation where matching performs best; in more general situations—i.e., when the two risk factors do not interact positively or when the matching factor is not rare—matching can lead to a loss of efficiency.4 Counter matching generally improves statistical efficiency for studying interaction9 and, unlike matching, further allows studying the exposure risk with adjustment for the other factor measured only in the case-control sample. Counter matching using sampling within risk-set strata is no more difficult to perform than matching on exposure, which can be complicated when additional risk-set matching factors are involved,11 and both strategies require the use of con-ditional logistic regression. Counter matching additionally requires sampling weights, which are calculated from the numbers of cohort subjects in each risk-set exposure stratum in the cohort. Being able to examine the adjusted exposure risk would usually outweigh the extra effort involved in calculating the weights. We conclude that counter matching, and not matching, should generally be used to increase efficiency if a nested case-control study of joint effects is planned when one risk factor is known in the cohort.

When calculating the power of a study involving sampling from a cohort, two issues deserve consideration: power of the full cohort and power of the study design. There is little point in selecting a subset to investigate interaction if even the cohort is too small or the effect too weak to provide sufficient statistical power. If the cohort has sufficient power, then the question becomes what type and size of design will provide the greatest possible efficiency within the limitations of financial cost, time, biological specimen availability, and other considerations. Breslow and Day point out that some sampling of cohort risk sets can generally be performed with little loss of efficiency.18 We have not considered the trade-off between cost and benefit here (see, for example, Reilly19), but in designing studies investigators must decide how much of the cohort power they are willing to sacrifice to achieve the necessary logistical savings. We have not investigated all possible designs, but for two approaches to nested case-control selection with fixed risk-set size, we have demonstrated that using counter matching can allow the researcher to achieve the same level of efficiency using about half as many controls as would be needed if controls were selected randomly.

The nested case-control design allows repeated selection of subjects in different risk sets; even cases can serve as controls in risk sets prior to their disease onset. Thus, there can be greater efficiency (in terms of number of subjects) depending on how many subjects are selected repeatedly by chance. In our application, the number of potential controls was large com-pared with the number of risk sets, so the probability of repeated selection was small. The total number of subjects needed for a study will depend on this ratio as well as on the random draw of subjects. With matching, the number of potential controls at rare levels of exposure is limited and may lead to repeated selection. However, when counter matching is used, because dose strata are defined by quantiles, repeated selection is not likely to occur except for very large control:case ratios.

In studies with exposure known in the entire cohort, there is additional information on exposure risk in the non-selected subjects. Two-stage designs can improve efficiency.20–22 Langholz and Goldstein23 proposed a likelihood for analysing the case-control data only using a proportional odds model with multi-stage sampling. Land and others3 proposed a method for incorporating the cohort risk estimate into the analysis of case-control subsets matched on exposure using more general risk models. There are also alternatives to the nested case-control design with counter matching. Borgan et al. addressed exposure-stratified selection in the case-cohort design.24 Randomized recruitment as an alternative to counter matching can also result in efficiency gains.25 Much remains to be done to synthesize the various designs and methods of analysis, but that is beyond the scope of the present work.

Because the present investigation was based on our interest in effect modification, we had to speculate as to what form it might take in order to simulate study power. Huang et al. reported that risk of breast cancer for medical irradiation to the chest in pre-/peri-menopausal women tended to be associated with oestrogen-receptor negative tumours,26 suggesting that mechanisms other than those dependent on hormonal exposure may be involved. Radiation might cause additional genetic alterations that result in more rapid progression of breast cancer associated with oestrogen receptor negative phenotype. Oestrogen receptor negative breast cancer cells have been reported to be relatively resistant to IL-6 induced apoptosis,27 so they may be more proliferative. If radiation induced alterations in signal transduction systems that were independent of the oestrogen receptor signalling system, then the joint effect of radiation and oestradiol could be multiplicative. If such alterations were dependent on the oestrogen receptor signalling system, then the joint effect could be greater than multiplicative. On the other hand, radiation exposure may lead to early onset of menopause,28 which could indirectly reduce the risk of breast cancer by decreasing the duration of exposure to constitutional estrogens. Therefore, in studying the joint effects of radiation and oestradiol, it is important to have sufficient power to detect or rule out interaction on the multiplicative scale to facilitate the planning of in-depth mechanistic studies.

Because these possibilities for interaction are mostly speculative, there was, in the present study, no basis to assume any particular type of effect modification between radiation and oestradiol. We therefore studied several arbitrary degrees of statistical interaction using a log-linear model. Effect modification can take other forms, including interaction on an additive scale. Such statistical interactions have been defined as effect-measure modification as distinguished from true effect modification, or biological interaction,29,30 which implies that the joint effect of multiple risk factors exceeds the sum of their individual risks. In the analysis of data from a nested case-control study, one should consider alternatives to the standard log-linear logistic-regression model for the joint effect of multiple risk factors, such as additive or mixture models.31

In summary, we have demonstrated that matching and counter-matching on a known, continuous exposure variable provide equal gains in statistical power in a nested case-control study of risk-factor interaction with a control:case ratio of at least 2:1. However, matching on exposure prevents studying the effect of exposure after adjusting for one or more other risk factors which might confound or modify the exposure risk, study aspects that counter matching addresses with greater efficiency than random sampling. We conclude that counter matching is superior to both matching on exposure and random control selection for nested case-control studies of effect modification when there is a known exposure.


KEY MESSAGES

  • Selecting controls with consideration of a known, rare exposure can improve efficiency over random control selection in nested case-control studies of interaction.
  • Counter matching produces the same gain in efficiency as matching for such studies but affords greater flexibility.
  • Implementation of counter matching and estimation of power of a nested case-control study are illustrated using a case study of breast cancer and two risk factors, one of which is known in the cohort.

 


    Acknowledgments
 
Special thanks go to Professor Bryan Langholz for advice and instruction regarding the method of counter matching. The study would not have been possible without the skilful technical support of Ms Sachiyo Funamoto. This publication was supported in part by research protocol 6–02 of the Radiation Effects Research Foundation (RERF), Hiroshima and Nagasaki, Japan. RERF is a private, non-profit foundation funded by the Japanese Ministry of Health, Labour and Welfare and the US Department of Energy, the latter through the National Academy of Sciences. The authors also acknowledge the support of Grants-in-Aid Nos. 14580356 and 14031227 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and grant number NCI-4893–8–001 from the US National Institutes of Health.


    References
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 Subjects and Methods
 Results
 Conclusions and Discussion
 References
 
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