Quantitative ambiguities in matched versus unmatched analyses of the 2x2 table for a case-control study

When controls are individually matched to the cases in a case-control study, the subsequent data can be analysed in a matched or unmatched format. If done with careful attention to clinical variables that can produce important bias or confounding, the matching would have a scientific basis that warrants preservation of a matched analysis. If done, however, in the more common manner, as an act of demographic convenience, the matching is not based on a cogent 'correlation'; and an unmatched analysis may be preferred because it uses all the data and it is easier to understand. Regardless of the merits of the arguments, investigators can choose (and can often justify) either a matched or unmatched analysis. If the matched table is structured in the customary format of (a b/c d), the results for the odds ratio and chi-square test in the matched and unmatched analyses will be relatively similar if ad congruent to bc, but be strikingly disparate if ad is substantially higher or lower than bc. The same distinction can be noted by comparing a (or any value observed in the four cells) with the corresponding value that would be expected for that cell as calculated from the marginal totals. If the observed and expected values sharply disagree, the values of the odds ratio and chi-square will sharply disagree in the matched and unmatched formats. To avoid invidious choices when disparate results emerge from the matched and unmatched methods, investigators can routinely apply and routinely report what is found with both methods. Readers can then see both sets of results and can take their choice.     

A comparison of different matching designs in case-control studies: An empirical example using continuous exposures,continuous confounders and incidence of myocardial infarction

The paper presents a case-control study involving a disease, exposures and several continuous confounders. The relative efficiency and validity of a fully matched design is compared with random sampling of controls. We test a viable option of a partially matched design when inability to match all study subjects on all confounders occurs. The degree of bias in the odds ratios introduced by the different designs and by the different analytic models is assessed in comparison with the estimates obtained from a total cohort, from which both cases and controls were selected. Matched designs and analytic strategies are also evaluated in terms of the variances of the odds ratios. The results indicate that matching on continuous variables may lead to a more precise estimate of odds ratio than statistical control of confounding in unmatched designs. Partial selection of controls by matching may be a useful strategy when complete matching cannot be achieved; in practice, partial matching achieves most of the benefits of full matching.     

Polychotomous logistic regression methods for matched case-control studies with multiple case or control groups

Two statistical methods, a polychotomous and pairwise approach, are presented to derive estimates of the relative odds in a matched case-control design when multiple case or control groups are used. Test statistics are derived to determine if the relative odds between groups are different. The polychotomous method is limited to case-control sets, i.e., where data are available on all members of a matched set. In contrast, the pairwise method makes use of data from both complete and incomplete sets. Nonetheless, efficiency calculations show that the polychotomous logistic regression model is more efficient even when 40 per cent of the case-control sets are incomplete. An example using a single dichotomous variable is provided.     

Dose-response in case-control studies.

The evidence provided by a case-control study on the association between a disease and some factor is strengthened if the extent of exposure to the factor is categorised into several groups or measured on a continuous scale. Then dose-response relationships can be estimated. The methods available are illustrated by application to data on lung cancer and chrysotile asbestos exposure from Quebec in which there were three matched controls for each case. Regression-type models were fitted assuming that the relative risk of lung cancer was linearly related to an exposure measure; a covariate, smoking, was also included in the analysis. The data were first analysed ignoring the matching and secondly taking account of the matching. The methodology for the latter analysis has only recently been developed; formerly, matched studies were of necessity analysed as unmatched. Although, in this particular example, the unmatched and matched analyses gave similar results, this is not always the case and it is argued that, now that the methodology is available, matched case-control studies should be analysed taking proper account of the matching.     

Combining matched and unmatched control groups in case-control studies

Multiple control groups in case-control studies are used to control for different sources of confounding. For example, cases can be contrasted with matched controls to adjust for multiple genetic or unknown lifestyle factors and simultaneously contrasted with an unmatched population-based control group. Inclusion of different control groups for a single exposure analysis yields several estimates of the odds ratio, all using only part of the data. Here the authors introduce an easy way to combine odds ratios from several case-control analyses with the same cases. The approach is based upon methods used for meta-analysis but takes into account the fact that the same cases are used and that the estimated odds ratios are therefore correlated. Two ways of estimating this correlation are discussed: sandwich methodology and the bootstrap. Confidence intervals for the pooled estimates and a test for checking whether the odds ratios in the separate case-control studies differ significantly are derived. The performance of the method is studied by simulation and by applying the methods to a large study on risk factors for thrombosis, the MEGA Study (1999-2004), wherein cases with first venous thrombosis were included with a matched control group of partners and an unmatched population-based control group.     

Breaking the matches in a paired t-test for community interventions when the number of pairs is small

There is considerable interest in community interventions for health promotion, where the community is the experimental unit. Because such interventions are expensive, the number of experimental units (communities) is usually small. Because of the small number of communities involved, investigators often match treatment and control communities on demographic variables before randomization to minimize the possibility of a bad split. Unfortunately, matching has been shown to decrease the power of the design when the number of pairs is small, unless the matching variable is very highly correlated with the outcome variable (in this case, with change in the health behaviour). We used computer simulation to examine the performance of an approach in which we matched communities but performed an unmatched analysis. If the appropriate matching variables are unknown, and there are fewer than ten pairs, an unmatched design and analysis has the most power. If, however, one prefers a matched design, then for N > 10, power can be increased by performing an unmatched analysis of the matched data. We also discuss a variant of this procedure, in which an unmatched analysis is performed only if the matching ‘did not work’.     

Analysis of matched case-control data with incomplete strata: applying longitudinal approaches

BACKGROUND: Matched case-control data have a structure that is similar to longitudinal data with correlated outcomes, except for a retrospective sampling scheme. In conditional logistic regression analysis, sets that are incomplete due to missing covariates and sets with identical values of the covariates do not contribute to the estimation; both situations may cause a loss in efficiency. These problems are more severe when sample sizes are small. We evaluated retrospective models for longitudinal data as alternatives in analyzing matched case-control data. METHODS: We conducted simulations to compare the properties of matched case-control data analyses using conditional likelihood and a commonly used longitudinal approach generalized estimating equation (GEE). We simulated scenarios for one-to-one and one-to-two matching designs, each with various sizes of matching strata, with complete and incomplete strata, and with dichotomous and normal exposures. RESULTS AND CONCLUSIONS: The simulations show that the estimates by conditional likelihood and GEE methods are consistent, and a proper coverage was reached for both binary and continuous exposures. The estimates produced by conditional likelihood have greater standard errors than those obtained by GEE. These relative efficiency losses are more substantial when data contain incomplete matched sets and when the data have small sizes of matching strata; these can be improved by including more controls in the strata. These losses of efficiency also increase as the magnitude of the association increases.     

Hepatitis B vaccination and first central nervous system demyelinating events: reanalysis of a case-control study using the self-controlled case series method

The hypothesis that hepatitis B vaccination is a risk factor for multiple sclerosis has been discussed at length. The data from an earlier case-control study were reanalyzed using the self-controlled case series method. Using the matched cases from the case-control study, we found a relative incidence of 1.68, 95% CI (0.77-3.68) for the 0-60-day post-vaccination risk period; this compares to an odds ratio of 1.8, 95% CI (0.7-4.6). When an additional 53 unmatched cases not used in the case-control study were included, the relative incidence was 1.35, 95% CI (0.66-2.79). Our results throw further light on the methodological aspects of the case series method. We recommend that, when case-control studies of vaccination and adverse events are planned, case series analyses based on the cases are also undertaken when appropriate.     

Pooled exposure assessment for matched case-control studies

Exposure assessment using biologic specimens is important for epidemiology but may become impracticable if assays are expensive, specimen volumes are marginally adequate, or analyte levels fall below the limit of detection. Pooled exposure assessment can provide an effective remedy for these problems in unmatched case-control studies. We extend pooled exposure strategies to handle specimens collected in a matched case-control study. We show that if a logistic model applies to individuals, then a logistic model also applies to an analysis using pooled exposures. Consequently, the individual-level odds ratio can be estimated while conserving both cost and specimen. We discuss appropriate pooling strategies for a single exposure, with adjustment for multiple, possibly continuous, covariates (confounders) and assessment of effect modification by a categorical variable. We assess the performance of the approach via simulations and conclude that pooled strategies can markedly improve efficiency for matched as well as unmatched case-control studies.     

Potential gain in efficiency and power to detect gene-environment interactions by matching in case-control studies

Background: There is growing interest in interactions between genetic and environmental risk factors of disease, but adequate power to detect such interactions in epidemiologic studies is of concern. The aim of this paper is to quantify the effect of matching on the efficiency of estimation and power to detect gene-environment interactions in case-control studies. Methods: Starting from an empirical example in cancer epidemiology, we simulated frequency matched and unmatched case-control studies for a wide range of assumptions regarding the prevalence and the effects of an environmental and a genetic factor on disease risk as well as the quality and quantity of the interaction between these factors. Simulated studies were analyzed with multivariable logistic regression. Results: Matching increased the efficiency and power in most scenarios. The gain was most pronounced in scenarios assuming a low prevalence of the environmental exposure. In such scenarios, equivalent power was only obtained with more than twice as many unmatched than matched controls. Conclusions: Frequency matching for known environmental risk factors with a low prevalence in the population may increase the efficiency of estimation and power of case-control studies to detect gene-environment interactions considerably. Investigators should weigh the gain in efficiency and power against known potential disadvantages of matching.     

Degree of matching and gain in power and efficiency in case-control studies

Frequency matching can be used to increase the precision and power of case-control studies. Unmatched and frequency-matched designs are only two distinct possibilities of control selection in a continuum regarding the frequency of the matching factor in controls. We assessed the power and efficiency of case-control studies under a variety of assumptions regarding the prevalence and the effects of the matching factor and the exposure of interest as well as their association in the population. For each set of parameters, we simulated 10,000 case-control studies varying the degree of matching, that is, the proportion of the matching factor in selected controls over a wide range including the proportion in cases (matched design) and the population (unmatched design) as two special options. Traditional frequency matching increased the precision and power in most scenarios, but most of the gain was often achieved by incomplete (less than perfect) matching. Even greater gains were sometimes observed by increasing the prevalence of the matching factor in controls above the one in cases. In the scenarios assessed, perfect matching was neither necessary nor the optimum degree of matching in many circumstances. It might be worthwhile to evaluate the optimum degree of matching for specific settings in the design of case-control studies.     

Application of odds ratio regression models for assessing familial aggregation from case-control studies

A regression model for estimating covariate effects on odds ratios to test for familial aggregation of common disease in first-degree relatives of cases and controls is presented and illustrated by using family data from a study of chronic obstructive pulmonary disease. These estimators are in essence an extension of the Mantel-Haenszel estimator of odds ratio but do not require the assumption of independence among relatives. A robust test statistic for possible effects of covariates such as the matching variables for cases and controls on odds ratio is also presented. In data on 156 adult first-degree relatives of 28 cases with demonstrated airway obstruction and 28 controls matched for age, sex, race, and hospital status, there appeared to be a difference in the odds ratio among families of black and white case-control pairs. However, the small sample size available prevents conclusive interpretation of this observation.     

Methodological issues in case-control studies: II. Test statistics as measures of efficiency

The ratio of test statistics has been used to compare the efficiency of matched and unmatched designs, and stratified and pooled analyses for case-control studies. The index has been computed for a wide range of population conditions and it is concluded that (a) pooled analysis is always more efficient than stratified analysis when such pooling leads to a valid estimate of the relative risk, (b) the loss of efficiency by matching in the classic overmatching situation can be substantial, (c) in the confounding situation either design may be more efficient but generally the difference is small. These results add support to the contention that matching is rarely justified in case-control studies.     

Effect heterogeneity by a matching covariate in matched case-control studies: a method for graphs-based representation

The authors describe a method for assessing and characterizing effect heterogeneity related to a matching covariate in case-control studies, using an example from veterinary medicine. Data are from a case-control study conducted in Texas during 1997-1998 of 498 pairs of horses with colic and their controls. Horses were matched by veterinarian and by month of examination. The number of matched pairs of cases and controls varied by veterinarian. The authors demonstrate that there is effect heterogeneity related to this characteristic (i.e., cluster size of veterinarians) for the association of colic with certain covariates, using a moving average approach to conditional logistic regression and graphs-based methods. The method described in this report can be applied to examining effect heterogeneity (or effect modification) by any ordered categorical or continuous covariates for which cases have been matched with controls. The method described enables one to understand the pattern of variation across ordered categorical or continuous matching covariates and allows for any shape for this pattern. This method applies to effect modification when causality might be reasonably assumed.     

Log-linear models for the analysis of matched cohort studies

The application of conditional logistic regression to the analysis of matched case-control studies has now become quite customary. In addition, it is well known that software designed to fit linear logistic and log-linear models can be used in these analyses. The application of conditional logistic regression to cohort designs is described, and an approach is developed that adapts the linear logistic and log-linear models for the analysis of prospectively collected data. Specific situations discussed include matched pairs, 2:1 matching, and studies in which some subjects are pair matched and others matched 2:1. The methods are illustrated with numeric examples.     

Multivariate analysis for matched case-control studies

A multivariate method based on the linear logistic model is presented for the analysis of case-control studies with pairwise matching. This technique enables one to investigate the effect of several variables simultaneously in the analysis while allowing for the matched design. The odds ratio is used as the basic measure of risk. One is able to control for variables which are not matching variables while investigating the odds ratio for a particular factor, and to estimate the change in the odds ratio as the level of one or more interval variables changes. The computing methods used for obtaining maximum conditional likelihood estimates of the parameters of interest are modifications of standard programs for logit regression.     

Random-digit dialing for control selection in childhood cancer studies: the geographic proximity and demographics within matched sets.

The purpose of this analysis was to evaluate the degree of matching in 95 individually matched pairs from a case-control study of childhood leukemia that used random-digit dialing to select control subjects. Both geographic proximity (of each case subject to his or her matched control subject) and differences in socioeconomic status were evaluated. The median distance between matched pairs was 3.2 km. There were no significant differences in distance between matched pairs by urban/rural status and geographic location. For studies of childhood cancer drawn from pediatric referral centers, random-digit dialing appears to provide a suitable control group.     

Assessing the adequacy of the logistic regression model for matched case-control studies

We apply results for the detection of outliers in logistic regression to the analysis of matched pairs data and illustrate the techniques with data from a matched pairs study of the relationship between iron-binding proteins and mortality in the Solomon Islands. Although the study includes 90 matched pairs, the conclusions change substantially with deletion of one or two pairs. Thus, even with reasonably large data sets, single observations may have a great impact on the results of an analysis. The identification of influential observations should constitute part of the analysis of matched case-control data.     

A method for combining matched and unmatched binary data. Application to randomized, controlled trials of photocoagulation in the treatment of diabetic retinopathy

A method for combining matched and unmatched data is described and was applied to the results of randomized, controlled trials of photocoagulation in the treatment of diabetic retinopathy. A pooled estimate from the matched and unmatched studies was obtained by adaptation of the Mantel-Haenszel method, where the strata were unmatched studies and matched pairs within studies. A test of significance was based on the Mantel-Haenszel chi-square statistic, the latter also being used to calculate test-based confidence intervals. A test of homogeneity was performed by combining Mantel-Haenszel chi-square statistics from the matched and unmatched studies. By these methods, the combined estimate of the risk of deterioration of visual acuity for those receiving photocoagulation (relative to a risk of unity for those not receiving photocoagulation) was 0.37 (95 per cent confidence interval 0.29-0.46). The chi-square statistic (1 df) for an effect of treatment was 87.75 (p less than 0.0001). The chi-square statistic for homogeneity of relative risk among studies was 11.20 (4 df, p less than 0.05). However, this result was influenced disproportionately by one small matched study.