Odds Ratio

BackgroundIn epidemiological studies researchers use logistic regression as an analytical tool to study the association of a binary outcome to a set of possible exposures.MethodsUsing a simulation study we illustrate how the analytically derived bias of odds ratios modelling in logistic regression varies as a function of the sample size.ResultsLogistic regression overestimates odds ratios in studies with small to moderate samples size. The small sample size induced bias is a systematic one, bias away from null. Regression coefficient estimates shifts away from zero, odds ratios from one.ConclusionIf several small studies are pooled without consideration of the bias introduced by the inherent mathematical properties of the logistic regression model, researchers may be mislead to erroneous interpretation of the results.

BackgroundPropensity score (PS) methods are increasingly used, even when sample sizes are small or treatments are seldom used. However, the relative performance of the two mainly recommended PS methods, namely PS-matching or inverse probability of treatment weighting (IPTW), have not been studied in the context of small sample sizes.MethodsWe conducted a series of Monte Carlo simulations to evaluate the influence of sample size, prevalence of treatment exposure, and strength of the association between the variables and the outcome and/or the treatment exposure, on the performance of these two methods.ResultsDecreasing the sample size from 1,000 to 40 subjects did not substantially alter the Type I error rate, and led to relative biases below 10%. The IPTW method performed better than the PS-matching down to 60 subjects. When N was set at 40, the PS matching estimators were either similarly or even less biased than the IPTW estimators. Including variables unrelated to the exposure but related to the outcome in the PS model decreased the bias and the variance as compared to models omitting such variables. Excluding the true confounder from the PS model resulted, whatever the method used, in a significantly biased estimation of treatment effect. These results were illustrated in a real dataset.ConclusionEven in case of small study samples or low prevalence of treatment, PS-matching and IPTW can yield correct estimations of treatment effect unless the true confounders and the variables related only to the outcome are not included in the PS model.

Comparative trials that report binary outcome data are commonly pooled in systematic reviews and meta-analyses. This type of data can be presented as a series of 2-by-2 tables. The pooled odds ratio is often presented as the outcome of primary interest in the resulting meta-analysis. We examine the use of 7 models for random-effects meta-analyses that have been proposed for this purpose. The first of these models is the conventional one that uses normal within-study approximations and a 2-stage approach. The other models are generalised linear mixed models that perform the analysis in 1 stage and have the potential to provide more accurate inference. We explore the implications of using these 7 models in the context of a Cochrane Review, and we also perform a simulation study. We conclude that generalised linear mixed models can result in better statistical inference than the conventional 2-stage approach but also that this type of model presents issues and difficulties. These challenges include more demanding numerical methods and determining the best way to model study specific baseline risks. One possible approach for analysts is to specify a primary model prior to performing the systematic review but also to present the results using other models in a sensitivity analysis. Only one of the models that we investigate is found to perform poorly so that any of the other models could be considered for either the primary or the sensitivity analysis.

It is well-established that the odds ratio (OR) is non-collapsible and that adjustment of the OR for a risk factor can produce a change in the main effect estimate even when there is no confounding. In contrast, the risk ratio (RR) is collapsible, whereas the rate ratio is non-collapsible, but usually to a lesser extent than is the case with ORs. However, almost all discussions of non-collapsibility have been conducted in the context of effect measures in cohort studies. In this paper, I consider non-collapsibility in case-control studies where the OR inherits the collapsibility or non-collapsibility of the ratio effect measure that the study was designed to estimate. This is most clearly illustrated in the situation of a closed cohort with a fixed follow-up time. In this situation, choosing controls by cumulative incidence sampling estimates the OR in the corresponding cohort study, and therefore the resulting case-control OR is non-collapsible. Choosing controls by case-cohort sampling estimates the cohort study RR, and the resulting case-control OR is collapsible. Choosing controls by density sampling (perhaps the most common approach) estimates the cohort study rate ratio, and the resulting case-control OR is non-collapsible, but to a lesser extent than for cumulative incidence sampling. Moreover, if the disease under study is rare (in the cohort study), all three methods produce case-control ORs that are approximately collapsible.