Logistic Models

In diverse fields of empirical research-including many in the biological sciences-attempts are made to decompose the effect of an exposure on an outcome into its effects via a number of different pathways. For example, we may wish to separate the effect of heavy alcohol consumption on systolic blood pressure (SBP) into effects via body mass index (BMI), via gamma-glutamyl transpeptidase (GGT), and via other pathways. Much progress has been made, mainly due to contributions from the field of causal inference, in understanding the precise nature of statistical estimands that capture such intuitive effects, the assumptions under which they can be identified, and statistical methods for doing so. These contributions have focused almost entirely on settings with a single mediator, or a set of mediators considered en bloc; in many applications, however, researchers attempt a much more ambitious decomposition into numerous path-specific effects through many mediators. In this article, we give counterfactual definitions of such path-specific estimands in settings with multiple mediators, when earlier mediators may affect later ones, showing that there are many ways in which decomposition can be done. We discuss the strong assumptions under which the effects are identified, suggesting a sensitivity analysis approach when a particular subset of the assumptions cannot be justified. These ideas are illustrated using data on alcohol consumption, SBP, BMI, and GGT from the Izhevsk Family Study. We aim to bridge the gap from "single mediator theory" to "multiple mediator practice," highlighting the ambitious nature of this endeavor and giving practical suggestions on how to proceed.

Recent successful discoveries of potentially causal single nucleotide polymorphisms (SNPs) for complex diseases hold great promise, and commercialization of genomics in personalized medicine has already begun. The hope is that genetic testing will benefit patients and their families, and encourage positive lifestyle changes and guide clinical decisions. However, for many complex diseases, it is arguable whether the era of genomics in personalized medicine is here yet. We focus on the clinical validity of genetic testing with an emphasis on two popular statistical methods for evaluating markers. The two methods, logistic regression and receiver operating characteristic (ROC) curve analysis, are applied to our age-related macular degeneration dataset. By using an additive model of the CFH, LOC387715, and C2 variants, the odds ratios are 2.9, 3.4, and 0.4, with p-values of 10(-13), 10(-13), and 10(-3), respectively. The area under the ROC curve (AUC) is 0.79, but assuming prevalences of 15%, 5.5%, and 1.5% (which are realistic for age groups 80 y, 65 y, and 40 y and older, respectively), only 30%, 12%, and 3% of the group classified as high risk are cases. Additionally, we present examples for four other diseases for which strongly associated variants have been discovered. In type 2 diabetes, our classification model of 12 SNPs has an AUC of only 0.64, and two SNPs achieve an AUC of only 0.56 for prostate cancer. Nine SNPs were not sufficient to improve the discrimination power over that of nongenetic predictors for risk of cardiovascular events. Finally, in Crohn's disease, a model of five SNPs, one with a quite low odds ratio of 0.26, has an AUC of only 0.66. Our analyses and examples show that strong association, although very valuable for establishing etiological hypotheses, does not guarantee effective discrimination between cases and controls. The scientific community should be cautious to avoid overstating the value of association findings in terms of personalized medicine before their time.

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.

AimsMultinomial logistic regression models allow one to predict the risk of a categorical outcome with > 2 categories. When developing such a model, researchers should ensure the number of participants (n) is appropriate relative to the number of events (Ek) and the number of predictor parameters (pk) for each category k. We propose three criteria to determine the minimum n required in light of existing criteria developed for binary outcomes.Proposed criteriaThe first criterion aims to minimise the model overfitting. The second aims to minimise the difference between the observed and adjusted R2 Nagelkerke. The third criterion aims to ensure the overall risk is estimated precisely. For criterion (i), we show the sample size must be based on the anticipated Cox-snell R2 of distinct 'one-to-one' logistic regression models corresponding to the sub-models of the multinomial logistic regression, rather than on the overall Cox-snell R2 of the multinomial logistic regression.Evaluation of criteriaWe tested the performance of the proposed criteria (i) through a simulation study and found that it resulted in the desired level of overfitting. Criterion (ii) and (iii) were natural extensions from previously proposed criteria for binary outcomes and did not require evaluation through simulation.SummaryWe illustrated how to implement the sample size criteria through a worked example considering the development of a multinomial risk prediction model for tumour type when presented with an ovarian mass. Code is provided for the simulation and worked example. We will embed our proposed criteria within the pmsampsize R library and Stata modules.