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In epidemiology, public health and social science, mediation analysis is often undertaken to investigate the extent to which the effect of a risk factor on an outcome of interest is mediated by other covariates. A pivotal quantity of interest in such an analysis is the mediation proportion. A common method for estimating it, termed the “difference method”, compares estimates from models with and without the hypothesized mediator. However, rigorous methodology for estimation and statistical inference for this quantity has not previously been available. We formulated the problem for the Cox model and generalized linear models, and utilize a data duplication algorithm together with a generalized estimation equations approach for estimating the mediation proportion and its variance. We further considered the assumption that the same link function hold for the marginal and conditional models, a property which we term “g-linkability”. We show that our approach is valid whenever g-linkability holds, exactly or approximately, and present results from an extensive simulation study to explore finite sample properties. The methodology is illustrated by an analysis of pre-menopausal breast cancer incidence in the Nurses’ Health Study. User-friendly publicly available software implementing those methods can be downloaded from the last author’s website (SAS) or from CRAN (R).

Case-control studies are used in epidemiology to try to uncover the causes of diseases, but are a retrospective study design known to suffer from non-participation and recall bias, which may explain their decreased popularity in recent years. Traditional analyses report usually only the odds ratio for given exposures and the binary disease status. Chain event graphs are a graphical representation of a statistical model derived from event trees which have been developed in artificial intelligence and statistics, and only recently introduced to the epidemiology literature. They are a modern Bayesian technique which enable prior knowledge to be incorporated into the data analysis using the agglomerative hierarchical clustering algorithm, used to form a suitable chain event graph. Additionally, they can account for missing data and be used to explore missingness mechanisms. Here we adapt the chain event graph framework to suit scenarios often encountered in case-control studies, to strengthen this study design which is time and financially efficient. We demonstrate eight adaptations to the graphs, which consist of two suitable for full case-control study analysis, four which can be used in interim analyses to explore biases, and two which aim to improve the ease and accuracy of analyses. The adaptations are illustrated with complete, reproducible, fully-interpreted examples, including the event tree and chain event graph. Chain event graphs are used here for the first time to summarise non-participation, data collection techniques, data reliability, and disease severity in case-control studies. We demonstrate how these features of a case-control study can be incorporated into the analysis to provide further insight, which can help to identify potential biases and lead to more accurate study results.

Abstract: We present a general statistical modelling framework for handling multivariate mixed types of outcomes in the context of quantitative genetic analysis. The models are based on the multivariate covariance generalized linear models, where the matrix linear predictor is composed of an identity matrix combined with a relatedness matrix defined by a pedigree, representing the environmental and genetic components, respectively. We also propose a new index of heritability for non-Gaussian data. A case study on house sparrow ( Passer domesticus ) population with continuous, binomial and count outcomes is employed to motivate the new model. Simulation of multivariate marginal models is not trivial, thus we adapt the NORTA (Normal to anything) algorithm for simulation of multivariate covariance generalized linear models in the context of genetic data analysis. A simulation study is presented to assess the asymptotic properties of the estimating function estimators for the correlation between outcomes and the new heritability index parameters. The data set and R code are available in the supplementary material.

The determinant of the covariance matrix for high-dimensional data plays an important role in statistical inference and decision. It has many real applications including statistical tests and information theory. Due to the statistical and computational challenges with high dimensionality, little work has been proposed in the literature for estimating the determinant of high-dimensional covariance matrix. In this paper, we estimate the determinant of the covariance matrix using some recent proposals for estimating high-dimensional covariance matrix. Specifically, we consider a total of eight covariance matrix estimation methods for comparison. Through extensive simulation studies, we explore and summarize some interesting comparison results among all compared methods. We also provide practical guidelines based on the sample size, the dimension, and the correlation of the data set for estimating the determinant of high-dimensional covariance matrix. Finally, from a perspective of the loss function, the comparison study in this paper may also serve as a proxy to assess the performance of the covariance matrix estimation.

Linear mixed-effects models are linear models with several variance components. Models with a single random-effects factor have two variance components: the random-effects variance, i. e., the inter-subject variance, and the residual error variance, i. e., the intra-subject variance. In many applications, it is practice to report variance components as coefficients of variation. The intra- and inter-subject coefficients of variation are the square roots of the corresponding variances divided by the mean. This article proposes methods for computing confidence intervals for intra- and inter-subject coefficients of variation using generalized pivotal quantities. The methods are illustrated through two examples. In the first example, precision is assessed within and between runs in a bioanalytical method validation. In the second example, variation is estimated within and between main plots in an agricultural split-plot experiment. Coverage of generalized confidence intervals is investigated through simulation and shown to be close to the nominal value.

The identification of genetic variants that are associated with disease risk is an important goal of genetic association studies. Standard approaches perform univariate analysis where each genetic variant, usually Single Nucleotide Polymorphisms (SNPs), is tested for association with disease status. Though many genetic variants have been identified and validated so far using this univariate approach, for most complex diseases a large part of their genetic component is still unknown, the so called missing heritability. We propose a Kernel-based measure of variable importance (KVI) that provides the contribution of a SNP, or a group of SNPs, to the joint genetic effect of a set of genetic variants. KVI can be used for ranking genetic markers individually, sets of markers that form blocks of linkage disequilibrium or sets of genetic variants that lie in a gene or a genetic pathway. We prove that, unlike the univariate analysis, KVI captures the relationship with other genetic variants in the analysis, even when measured at the individual level for each genetic variable separately. This is specially relevant and powerful for detecting genetic interactions. We illustrate the results with data from an Alzheimer’s disease study and show through simulations that the rankings based on KVI improve those rankings based on two measures of importance provided by the Random Forest. We also prove with a simulation study that KVI is very powerful for detecting genetic interactions.

Suppose we observe n$n$ independent and identically distributed observations of a finite dimensional bounded random variable. This article is concerned with the construction of an efficient targeted minimum loss-based estimator (TMLE) of a pathwise differentiable target parameter of the data distribution based on a realistic statistical model. The only smoothness condition we will enforce on the statistical model is that the nuisance parameters of the data distribution that are needed to evaluate the canonical gradient of the pathwise derivative of the target parameter are multivariate real valued cadlag functions (right-continuous and left-hand limits, (G. Neuhaus. On weak convergence of stochastic processes with multidimensional time parameter. Ann Stat 1971;42:1285–1295.) and have a finite supremum and (sectional) variation norm. Each nuisance parameter is defined as a minimizer of the expectation of a loss function over over all functions it its parameter space. For each nuisance parameter, we propose a new minimum loss based estimator that minimizes the loss-specific empirical risk over the functions in its parameter space under the additional constraint that the variation norm of the function is bounded by a set constant. The constant is selected with cross-validation. We show such an MLE can be represented as the minimizer of the empirical risk over linear combinations of indicator basis functions under the constraint that the sum of the absolute value of the coefficients is bounded by the constant: i.e., the variation norm corresponds with this L1$L_1$-norm of the vector of coefficients. We will refer to this estimator as the highly adaptive Lasso (HAL)-estimator. We prove that for all models the HAL-estimator converges to the true nuisance parameter value at a rate that is faster than n−1/4$n^{-1/4}$ w.r.t. square-root of the loss-based dissimilarity. We also show that if this HAL-estimator is included in the library of an ensemble super-learner, then the super-learner will at minimal achieve the rate of convergence of the HAL, but, by previous results, it will actually be asymptotically equivalent with the oracle (i.e., in some sense best) estimator in the library. Subsequently, we establish that a one-step TMLE using such a super-learner as initial estimator for each of the nuisance parameters is asymptotically efficient at any data generating distribution in the model, under weak structural conditions on the target parameter mapping and model and a strong positivity assumption (e.g., the canonical gradient is uniformly bounded). We demonstrate our general theorem by constructing such a one-step TMLE of the average causal effect in a nonparametric model, and establishing that it is asymptotically efficient.