Volume 9, Issue 2 -

In this article, we present a sensitivity analysis for drawing inferences about parameters that are not estimable from observed data without additional assumptions. We present the methodology using two different examples: a causal parameter that is not identifiable due to violations of the randomization assumption, and a parameter that is not estimable in the nonparametric model due to measurement error. Existing methods for tackling these problems assume a parametric model for the type of violation to the identifiability assumption and require the development of new estimators and inference for every new model. The method we present can be used in conjunction with any existing asymptotically linear estimator of an observed data parameter that approximates the unidentifiable full data parameter and does not require the study of additional models.

Assessing the causal effect of an exposure often involves the definition of counterfactual outcomes in a hypothetical world in which the stochastic nature of the exposure is modified. Although stochastic interventions are a powerful tool to measure the causal effect of a realistic intervention that intends to alter the population distribution of an exposure, their importance to answer questions about plausible policy interventions has been obscured by the generalized use of deterministic interventions. In this article, we follow the approach described in Díaz and van der Laan (2012) to define and estimate the effect of an intervention that is expected to cause a truncation in the population distribution of the exposure. The observed data parameter that identifies the causal parameter of interest is established, as well as its efficient influence function under the non-parametric model. Inverse probability of treatment weighted (IPTW), augmented IPTW and targeted minimum loss-based estimators (TMLE) are proposed, their consistency and efficiency properties are determined. An extension to longitudinal data structures is presented and its use is demonstrated with a real data example.

Point estimation is particularly important in predicting weight loss in individuals or small groups. In this analysis, a new health response function is based on a model of human response over time to estimate long-term health outcomes from a change point in short-term linear regression. This important estimation capability is addressed for small groups and single-subject designs in pilot studies for clinical trials, medical and therapeutic clinical practice. These estimations are based on a change point given by parameters derived from short-term participant data in ordinary least squares (OLS) regression. The development of the change point in initial OLS data and the point estimations are given in a new semiparametric ratio estimator (SPRE) model. The new response function is taken as a ratio of two-parameter Weibull distributions times a prior outcome value that steps estimated outcomes forward in time, where the shape and scale parameters are estimated at the change point. The Weibull distributions used in this ratio are derived from a Kelvin model in mechanics taken here to represent human beings. A distinct feature of the SPRE model in this article is that initial treatment response for a small group or a single subject is reflected in long-term response to treatment. This model is applied to weight loss in obesity in a secondary analysis of data from a classic weight loss study, which has been selected due to the dramatic increase in obesity in the United States over the past 20 years. A very small relative error of estimated to test data is shown for obesity treatment with the weight loss medication phentermine or placebo for the test dataset. An application of SPRE in clinical medicine or occupational therapy is to estimate long-term weight loss for a single subject or a small group near the beginning of treatment.

A method to produce exact simultaneous confidence bands for the empirical cumulative distribution function that was first described by Owen, and subsequently corrected by Jager and Wellner, is the starting point for deriving exact nonparametric confidence bands for the survivor function of any positive random variable. We invert a nonparametric likelihood test of uniformity, constructed from the Kaplan–Meier estimator of the survivor function, to obtain simultaneous lower and upper bands for the function of interest with specified global confidence level. The method involves calculating a null distribution and associated critical value for each observed sample configuration. However, Noe recursions and the Van Wijngaarden–Decker–Brent root-finding algorithm provide the necessary tools for efficient computation of these exact bounds. Various aspects of the effect of right censoring on these exact bands are investigated, using as illustrations two observational studies of survival experience among non-Hodgkin’s lymphoma patients and a much larger group of subjects with advanced lung cancer enrolled in trials within the North Central Cancer Treatment Group. Monte Carlo simulations confirm the merits of the proposed method of deriving simultaneous interval estimates of the survivor function across the entire range of the observed sample. This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. It was begun while the author was visiting the Department of Statistics, University of Auckland, and completed during a subsequent sojourn at the Medical Research Council Biostatistics Unit in Cambridge. The support of both institutions, in addition to that of NSERC and the University of Waterloo, is greatly appreciated.

Clustered interval-censored failure time data are commonly encountered in many medical settings. In such situations, one issue that often arises in practice is that the cluster size is related to the risk for the outcome of interest. It is well-known that ignoring the informativeness of the cluster size can result in biased parameter estimates. In this article, we consider regression analysis of clustered interval-censored data with informative cluster size with the focus on semiparametric methods. For the problem, two approaches are presented and investigated. One is a within-cluster resampling procedure and the other is a weighted estimating equation approach. Unlike previously published methods, the new approaches take into account cluster sizes and heterogeneous correlation structures without imposing strong parametric assumptions. A simulation experiment is carried out to evaluate the performance of the proposed approaches and indicates that they perform well for practical situations. The approaches are applied to a lymphatic filariasis study that motivated this study.

Propensity score (PS) matching is widely used for studying treatment effects in observational studies. This article proposes the method of matching weights (MWs) as an analog to one-to-one pair matching without replacement on the PS with a caliper. Compared with pair matching, the proposed method offers more efficient estimation, more accurate variance calculation, better balance, and simpler asymptotic analysis. A statistical test for the misspecification of the PS model is proposed for balance checking purposes. An augmented version of the MW estimator is developed that has the double robust property, that is, the estimator is consistent, if either the outcome model or the PS model is correct. The proposed method is studied in simulations and illustrated through a real data example.

Estimating the direct effect of a treatment on an outcome is often the focus of epidemiological and clinical research, when the treatment has more than one specified pathway to the defined outcome. Even if the total effect is unconfounded, the direct effect is not identified when unmeasured variables affect the intermediate and outcome variables. Therefore, bounds on direct effects have been presented via linear programming under two common definitions of direct effects: controlled and natural. Here, we propose bounds on natural direct effects without using linear programming, because such bounds on controlled direct effects have already been proposed. To derive narrow bounds, we introduce two monotonicity assumptions that are weaker than those in previous studies and another monotonicity assumption. Furthermore, we do not assume that an outcome variable is binary, whereas previous studies have made that assumption. An additional advantage of our bounds is that the bounding formulas are extremely simple. The proposed bounds are illustrated using a randomized trial for coronary heart disease.

The risk ratio effect measure is often the main parameter of interest in epidemiologic studies with a binary outcome. In this paper, the author presents a simple and efficient two-stage approach to estimate the risk ratios directly, which does not directly rely on consistency for an estimate of the baseline risk. This latter property is a key advantage of the approach over existing methods, because, unlike these other methods, the proposed approach obviates the need to restrict the predicted risk probabilities to fall below one, in order to recover efficient inferences about risk ratios. An additional appeal of the approach is that it is easy to implement. Finally, when the primary interest is in the effect of a specific binary exposure, a simple doubly robust closed-form estimator is derived, for the multiplicative effect of the exposure. Specifically, we show how one can adjust for confounding by incorporating a working regression model for the propensity score so that the correct inferences about the multiplicative effect of the exposure are recovered if either this model is correct or a working model for the association between confounders and outcome risk is correct, but both do not necessarily hold.

: In this article, we consider the problem of comparing the distribution of observations in a planar region to a pre-specified null distribution. Our motivation is a surveillance setting where we map locations of incident disease, aiming to monitor these data over time, to locate potential areas of high/low incidence so as to direct public health actions. We propose a non-parametric approach to distance-based disease risk mapping inspired by tomographic imaging. We consider several one-dimensional projections via the observed distribution of distances to a chosen fixed point; we then compare this distribution to that expected under the null and average these comparisons across projections to compute a relative-risk-like score at each point in the region. The null distribution can be established from historical data. Scores are displayed on the map using a color scale. In addition, we give a detailed description of the method along with some desirable theoretical properties. To further assess the performance of this method, we compare it to the widely used log ratio of kernel density estimates. As a performance metric, we evaluate the accuracy to locate simulated spatial clusters superimposed on a uniform distribution in the unit disk. Results suggest that both methods can adequately locate this increased risk but each relies on an appropriate choice of parameters. Our proposed method, distance-based mapping (DBM), can also generalize to arbitrary metric spaces and/or high-dimensional data.

: Statistical analysis of longitudinal outcomes is often complicated by the absence of observable values in patients who die prior to their scheduled measurement. In such cases, the longitudinal data are said to be “truncated by death” to emphasize that the longitudinal measurements are not simply missing, but are undefined after death. Recently, the truncation by death problem has been investigated using the framework of principal stratification to define the target estimand as the survivor average causal effect (SACE), which in the context of a two-group randomized clinical trial is the mean difference in the longitudinal outcome between the treatment and control groups for the principal stratum of always-survivors. The SACE is not identified without untestable assumptions. These assumptions have often been formulated in terms of a monotonicity constraint requiring that the treatment does not reduce survival in any patient, in conjunction with assumed values for mean differences in the longitudinal outcome between certain principal strata. In this paper, we introduce an alternative estimand, the balanced-SACE, which is defined as the average causal effect on the longitudinal outcome in a particular subset of the always-survivors that is balanced with respect to the potential survival times under the treatment and control. We propose a simple estimator of the balanced-SACE that compares the longitudinal outcomes between equivalent fractions of the longest surviving patients between the treatment and control groups and does not require a monotonicity assumption. We provide expressions for the large sample bias of the estimator, along with sensitivity analyses and strategies to minimize this bias. We consider statistical inference under a bootstrap resampling procedure.