Identifiability of causal effects for binary variables with baseline data missing due to death

We discuss identifiability and estimation of causal effects of a treatment in subgroups defined by a covariate that is sometimes missing due to death, which is different from a problem with outcomes censored by death. Frangakis et al. (2007, Biometrics 63, 641-662) proposed an approach for estimating the causal effects under a strong monotonicity (SM) assumption. In this article, we focus on identifiability of the joint distribution of the covariate, treatment and potential outcomes, show sufficient conditions for identifiability, and relax the SM assumption to monotonicity (M) and no-interaction (NI) assumptions. We derive expectation-maximization algorithms for finding the maximum likelihood estimates of parameters of the joint distribution under different assumptions. Further we remove the M and NI assumptions, and prove that signs of the causal effects of a treatment in the subgroups are identifiable, which means that their bounds do not cover zero. We perform simulations and a sensitivity analysis to evaluate our approaches. Finally, we apply the approaches to the National Study on the Costs and Outcomes of Trauma Centers data, which are also analyzed by Frangakis et al. (2007) and Xie and Murphy (2007, Biometrics 63, 655-658).     

Estimation and interpretation of incidence rate difference for recurrent events when the estimation model is misspecified

Recurrent events data are common in experimental and observational studies. It is often of interest to estimate the effect of an intervention on the incidence rate of the recurrent events. The incidence rate difference is a useful measure of intervention effect. A weighted least squares estimator of the incidence rate difference for recurrent events was recently proposed for an additive rate model in which both the baseline incidence rate and the covariate effects were constant over time. In this article, we relax this model assumption and examine the properties of the estimator under the additive and multiplicative rate models assumption in which the baseline incidence rate and covariate effects may vary over time. We show analytically and numerically that the estimator gives an appropriate summary measure of the time‐varying covariate effects. In particular, when the underlying covariate effects are additive and time‐varying, the estimator consistently estimates the weighted average of the covariate effects over time. When the underlying covariate effects are multiplicative and time‐varying, and if there is only one binary covariate indicating the intervention status, the estimator consistently estimates the weighted average of the underlying incidence rate difference between the intervention and control groups over time. We illustrate the method with data from a randomized vaccine trial.     

Permutation Testing for Treatment–Covariate Interactions and Subgroup Identification

We consider the problem of using permutation-based methods to test for treatment–covariate interactions from randomized clinical trial data. Testing for interactions is common in the field of personalized medicine, as subgroups with enhanced treatment effects arise when treatment-by-covariate interactions exist. Asymptotic tests can often be performed for simple models, but in many cases, more complex methods are used to identify subgroups, and non-standard test statistics proposed, and asymptotic results may be difficult to obtain. In such cases, it is natural to consider permutation-based tests, which shuffle selected parts of the data in order to remove one or more associations of interest; however, in the case of interactions, it is generally not possible to remove only the associations of interest by simple permutations of the data. We propose a number of alternative permutation-based methods, designed to remove only the associations of interest, but preserving other associations. These methods estimate the interaction term in a model, then create data that “looks like” the original data except that the interaction term has been permuted. The proposed methods are shown to outperform traditional permutation methods in a simulation study. In addition, the proposed methods are illustrated using data from a randomized clinical trial of patients with hypertension.     

Bias and Sensitivity Analysis When Estimating Treatment Effects from the Cox Model with Omitted Covariates

Summary

Omission of relevant covariates can lead to bias when estimating treatment or exposure effects from survival data in both randomized controlled trials and observational studies. This paper presents a general approach to assessing bias when covariates are omitted from the Cox model. The proposed method is applicable to both randomized and non‐randomized studies. We distinguish between the effects of three possible sources of bias: omission of a balanced covariate, data censoring and unmeasured confounding. Asymptotic formulae for determining the bias are derived from the large sample properties of the maximum likelihood estimator. A simulation study is used to demonstrate the validity of the bias formulae and to characterize the influence of the different sources of bias. It is shown that the bias converges to fixed limits as the effect of the omitted covariate increases, irrespective of the degree of confounding. The bias formulae are used as the basis for developing a new method of sensitivity analysis to assess the impact of omitted covariates on estimates of treatment or exposure effects. In simulation studies, the proposed method gave unbiased treatment estimates and confidence intervals with good coverage when the true sensitivity parameters were known. We describe application of the method to a randomized controlled trial and a non‐randomized study.     

Estimating predictors for long- or short-term survivors

Suppose that the response variable in a well-executed clinical or observational study to evaluate a treatment is the time to a certain event, and a set of baseline covariates or predictors was collected for each study patient. Furthermore, suppose that a significant number of study patients had nontrivial, long-term adverse effects from the treatment. A commonly posed question is how to use these covariates from the study to identify future patients who would (or would not) benefit from the treatment. In this article, we present "point" and "interval" estimates for the set of covariate or predictor vectors associated with a specific patient survival status, e.g., long- (or short-) term survival, in the presence of censoring. These estimates can be easily displayed on a two-dimensional plane, even for the case with high-dimensional covariate vectors. These simple numerical and graphical procedures provide useful information for patient management and/or the design of future studies, which are key issues in pharmacogenomics with genetic markers. The new proposal is illustrated with a data set from a cancer study for treating multiple myeloma.     

Semiparametric models for missing covariate and response data in regression models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.     

Smooth individual level covariates adjustment in disease mapping

Spatial models for disease mapping should ideally account for covariates measured both at individual and area levels. The newly available “indiCAR” model fits the popular conditional autoregresssive (CAR) model by accommodating both individual and group level covariates while adjusting for spatial correlation in the disease rates. This algorithm has been shown to be effective but assumes log‐linear associations between individual level covariates and outcome. In many studies, the relationship between individual level covariates and the outcome may be non‐log‐linear, and methods to track such nonlinearity between individual level covariate and outcome in spatial regression modeling are not well developed. In this paper, we propose a new algorithm, smooth‐indiCAR, to fit an extension to the popular conditional autoregresssive model that can accommodate both linear and nonlinear individual level covariate effects while adjusting for group level covariates and spatial correlation in the disease rates. In this formulation, the effect of a continuous individual level covariate is accommodated via penalized splines. We describe a two‐step estimation procedure to obtain reliable estimates of individual and group level covariate effects where both individual and group level covariate effects are estimated separately. This distributed computing framework enhances its application in the Big Data domain with a large number of individual/group level covariates. We evaluate the performance of smooth‐indiCAR through simulation. Our results indicate that the smooth‐indiCAR method provides reliable estimates of all regression and random effect parameters. We illustrate our proposed methodology with an analysis of data on neutropenia admissions in New South Wales (NSW), Australia.     

Graphical exploration of covariate effects on survival data through nonparametric quantile curves

Kaplan-Meier curves provide an effective means of presenting the distributional pattern in a sample of survival data. However, in order to assess the effect of a covariate, a standard scatterplot is often difficult to interpret because of the presence of censored observations. Several authors have proposed a running median as an effective way of indicating the effect of a covariate. This article proposes a form of kernel estimation, employing double smoothing, that can be applied in a simple and efficient manner to construct an estimator of a percentile of the survival distribution as a function of one or two covariates. Permutations and bootstrap samples can be used to construct reference bands that help identify whether particular features of the estimates indicate real features of the underlying curve or whether this may be due simply to random variation. The techniques are illustrated on data from a study of kidney transplant patients.     

Indirect genetic effect model using feeding behaviour traits to define the degree of interaction between mates: an implementation in pigs growth rate

An alternative implementation of the animal model including indirect genetic effect (IGE) is presented considering pair-mate-specific interaction degrees to improve the performance of the model. Data consisted of average daily gain (ADG) records from 663 pigs kept in groups of 10 to 14 mates during the fattening period. Three types of models were used to fit ADG data: (i) animal model (AM); (ii) AM with classical IGE (AM-IGE); and (iii) AM fitting IGE with a specific degree of interaction between each pair of mates (AM-IGE_i). Several feeding behavior phenotypes were used to define the pair-mate-specific degree of interaction in AM-IGE_i: feeding rate (g/min), feeding frequency (min/day), the time between consecutive visits to the feeder (min/day), occupation time (min/day) and an index considering all these variables. All models included systematic effects batch, initial age (covariate), final age (covariate), number of pigs per pen (covariate), plus the random effect of the pen. Estimated posterior mean (posterior SD) of heritability was 0.47 (0.15) using AM. Including social genetic effects in the model, total heritable variance expressed as a proportion of total phenotypic variance (T²) was 0.54 (0.29) using AM-IGE, whereas it ranged from 0.51 to 0.55 (0.12 to 0.14) with AM-IGE_i, depending on the behavior trait used to define social interactions. These results confirm the contribution of IGEs to the total heritable variation of ADG. Moreover, important differences between models were observed in EBV rankings. The percentage of coincidence of top 10% animals between AM and AM-IGE_i ranged from 0.44 to 0.89 and from 0.41to 0.68 between AM-IGE and AM-IGE_i. Based on the goodness of fit and predictive ability, social models are preferred for the genetic evaluation of ADG. Among models including IGEs, when the pair-specific degree of interaction was defined using feeding behavior phenotypes we obtained an increase in the accuracy of genetic parameters estimates, the better goodness of fit and higher predictive ability. We conclude that feeding behavior variables can be used to measure the interaction between pen mates and to improve the performance of models including IGEs.     

An indicator quantitatively comparing two treatment effect sizes on responder and non-responder groups – exponential of estimated interaction parameter

 Sometimes a specific treatment is effective in one subgroup but not in another. An indicator allowing quantitative comparison of treatment effect in two subgroups would be useful in clinical medicine. We have developed such an indicator. It is obtained by calculations using Cox’s proportional hazard or logistic model with therapy, subgroup, and confounding explanatory variables. The parameter of the interaction between therapy and subgroup can be estimated and tested statistically. The exponential value of the interaction parameter is what we tentatively call the “hazard ratio ratio”, meaning the ratio between the treatment effects in two subgroups. The 95% confidence interval of the indicator can also be calculated. As a numerical example, the hazard ratio between the survival times of postoperative gastric cancer patients treated by adjuvant immunochemotherapy and patients without adjuvant immunochemotherapy in a subgroup with high serum glycosidically bound sialic acid (SA) level was lower than that in a low-SA subgroup using an estimate for hazard ratio ratio of less than 0.5 with statistical significance. We propose this indicator be used as a “responder/non-responder ratio” of therapy effect. Received: 11 April 1995 / Accepted: 5 September 1995     

Dealing with limited overlap in estimation of average treatment effects

Estimation of average treatment effects under unconfounded orignorable treatment assignment is often hampered by lack ofoverlap in the covariate distributions between treatment groups.This lack of overlap can lead to imprecise estimates, and canmake commonly used estimators sensitive to the choice of specification.In such cases researchers have often used ad hoc methods fortrimming the sample. We develop a systematic approach to addressinglack of overlap. We characterize optimal subsamples for whichthe average treatment effect can be estimated most precisely.Under some conditions, the optimal selection rules depend solelyon the propensity score. For a wide range of distributions,a good approximation to the optimal rule is provided by thesimple rule of thumb to discard all units with estimated propensityscores outside the range [0.1,0.9].     

Simple methods for the estimation and sensitivity analysis of principal strata effects using marginal structural models: Application to a bone fracture prevention trial

In randomized clinical trials, it is often of interest to estimate the effect of treatment on quality of life (QOL), in addition to those on the event itself. When an event occurs in some patients prior to QOL score assessment, investigators may compare QOL scores between patient subgroups defined by the event after randomization. However, owing to postrandomization selection bias, this analysis can mislead investigators about treatment efficacy and result in paradoxical findings. The recent Japanese Osteoporosis Intervention Trial (JOINT‐02), which compared the benefits of a combination therapy for fracture prevention with those of a monotherapy, exemplifies the case in point; the average QOL score was higher in the combination therapy arm for the unfractured subgroup but was lower for the fractured subgroup. To address this issue, principal strata effects (PSEs), which are treatment effects estimated within subgroups of individuals stratified by potential intermediate variable, have been discussed in the literature. In this paper, we describe a simple procedure for estimating the PSEs using marginal structural models. This procedure utilizes SAS code for the estimation. In addition, we present a simple sensitivity analysis method for examining the resulting estimates. The analyses of JOINT‐02 data using these methods revealed that QOL scores were higher in the combination therapy arm than in the monotherapy arm for both subgroups.     

Patient-specific dose finding based on bivariate outcomes and covariates

SUMMARY: A Bayesian sequential dose-finding procedure based on bivariate (efficacy, toxicity) outcomes that accounts for patient covariates and dose-covariate interactions is presented. Historical data are used to obtain an informative prior on covariate main effects, with uninformative priors assumed for all dose effect parameters. Elicited limits on the probabilities of efficacy and toxicity for each of a representative set of covariate vectors are used to construct bounding functions that determine the acceptability of each dose for each patient. Elicited outcome probability pairs that are equally desirable for a reference patient are used to define two different posterior criteria, either of which may be used to select an optimal covariate-specific dose for each patient. Because the dose selection criteria are covariate specific, different patients may receive different doses at the same point in the trial, and the set of eligible patients may change adaptively during the trial. The method is illustrated by a dose-finding trial in acute leukemia, including a simulation study.     

Change-plane analysis for subgroup detection with a continuous treatment

Detecting and characterizing subgroups with differential effects of a binary treatment has been widely studied and led to improvements in patient outcomes and population risk management. Under the setting of a continuous treatment, however, such investigations remain scarce. We propose a semiparametric change-plane model and consequently a doubly robust test statistic for assessing the existence of two subgroups with differential treatment effects under a continuous treatment. The proposed testing procedure is valid when either the baseline function for the covariate effects or the generalized propensity score function for the continuous treatment is correctly specified. The asymptotic distributions of the test statistic under the null and local alternative hypotheses are established. When the null hypothesis of no subgroup is rejected, the change-plane parameters that define the subgroups can be estimated. This paper provides a unified framework of the change-plane method to handle various types of outcomes, including the exponential family of distributions and time-to-event outcomes. Additional extensions with nonparametric estimation approaches are also provided. We evaluate the performance of our proposed methods through extensive simulation studies under various scenarios. An application to the Health Effects of Arsenic Longitudinal Study with a continuous environmental exposure of arsenic is presented.     

A new nonparametric approach for baseline covariate adjustment for two-group comparative studies

SUMMARY: We consider two-armed clinical trials in which the response and/or the covariates are observed on either a binary, ordinal, or continuous scale. A new general nonparametric (NP) approach for covariate adjustment is presented using the notion of a relative effect to describe treatment effects. The relative effect is defined by the probability of observing a higher response in the experimental than in the control arm. The notion is invariant under monotone transformations of the data and is therefore especially suitable for ordinal data. For a normal or binary distributed response the relative effect is the transformed effect size or the difference of response probability, respectively. An unbiased and consistent NP estimator for the relative effect is presented. Further, we suggest a NP procedure for correcting the relative effect for covariate imbalance and random covariate imbalance, yielding a consistent estimator for the adjusted relative effect. Asymptotic theory has been developed to derive test statistics and confidence intervals. The test statistic is based on the joint behavior of the estimated relative effect for the response and the covariates. It is shown that the test statistic can be used to evaluate the treatment effect in the presence of (random) covariate imbalance. Approximations for small sample sizes are considered as well. The sampling behavior of the estimator of the adjusted relative effect is examined. We also compare the probability of a type I error and the power of our approach to standard covariate adjustment methods by means of a simulation study. Finally, our approach is illustrated on three studies involving ordinal responses and covariates.     

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

For randomized clinical trials where the endpoint of interest is a time-to-event subject to censoring, estimating the treatment effect has mostly focused on the hazard ratio from the Cox proportional hazards model. Since the model’s proportional hazards assumption is not always satisfied, a useful alternative, the so-called additive hazards model, may instead be used to estimate a treatment effect on the difference of hazard functions. Still, the hazards difference may be difficult to grasp intuitively, particularly in a clinical setting of, e.g., patient counseling, or resource planning. In this paper, we study the quantiles of a covariate’s conditional survival function in the additive hazards model. Specifically, we estimate the residual time quantiles, i.e., the quantiles of survival times remaining at a given time t, conditional on the survival times greater than t, for a specific covariate in the additive hazards model. We use the estimates to translate the hazards difference into the difference in residual time quantiles, which allows a more direct clinical interpretation. We determine the asymptotic properties, assess the performance via Monte-Carlo simulations, and demonstrate the use of residual time quantiles in two real randomized clinical trials.     

Bayesian Space-Time Patterns and Climatic Determinants of Bovine Anaplasmosis

The space-time pattern and environmental drivers (land cover, climate) of bovine anaplasmosis in the Midwestern state of Kansas was retrospectively evaluated using Bayesian hierarchical spatio-temporal models and publicly available, remotely-sensed environmental covariate information. Cases of bovine anaplasmosis positively diagnosed at Kansas State Veterinary Diagnostic Laboratory (n = 478) between years 2005–2013 were used to construct the models, which included random effects for space, time and space-time interaction effects with defined priors, and fixed-effect covariates selected a priori using an univariate screening procedure. The Bayesian posterior median and 95% credible intervals for the space-time interaction term in the best-fitting covariate model indicated a steady progression of bovine anaplasmosis over time and geographic area in the state. Posterior median estimates and 95% credible intervals derived for covariates in the final covariate model indicated land surface temperature (minimum), relative humidity and diurnal temperature range to be important risk factors for bovine anaplasmosis in the study. The model performance measured using the Area Under the Curve (AUC) value indicated a good performance for the covariate model (> 0.7). The relevance of climatological factors for bovine anaplasmosis is discussed.     

Assessing the Effect of the Best Treatment. A Non-parametric Inference Problem

For patients of a certain type, a number of treatments are available. The effect of each such treatment is assumed to be described by a shift model; it is, however, admitted that there may be an interaction between patient and treatment, meaning in particular that the treatment which is best for one patient is not necessarily best for another. The problem is the following: if each patient is given the treatment which is optimal for that particular patient, will this produce a significant effect and, if so, how large is the effect?     

Analysis of multi-type recurrent events in longitudinal studies; application to a skin cancer prevention trial

We consider the statistical modeling and analysis of replicated multi-type point process data with covariates. Such data arise when heterogeneous subjects experience repeated events or failures which may be of several distinct types. The underlying processes are modeled as nonhomogeneous mixed Poisson processes with random (subject) and fixed (covariate) effects. The method of maximum likelihood is used to obtain estimates and standard errors of the failure rate parameters and regression coefficients. Score tests and likelihood ratio statistics are used for covariate selection. A graphical test of goodness of fit of the selected model is based on generalized residuals. Measures for determining the influence of an individual observation on the estimated regression coefficients and on the score test statistic are developed. An application is described to a large ongoing randomized controlled clinical trial for the efficacy of nutritional supplements of selenium for the prevention of two types of skin cancer.     

Using aggregate data to estimate the standard error of a treatment–covariate interaction in an individual patient data meta‐analysis

Subgroup analyses are important to medical research because they shed light on the heterogeneity of treatment effectts. A treatment–covariate interaction in an individual patient data (IPD) meta‐analysis is the most reliable means to estimate how a subgroup factor modifies a treatment's effectiveness. However, owing to the challenges in collecting participant data, an approach based on aggregate data might be the only option. In these circumstances, it would be useful to assess the relative efficiency and power loss of a subgroup analysis without patient‐level data. We present methods that use aggregate data to estimate the standard error of an IPD meta‐analysis’ treatment–covariate interaction for regression models of a continuous or dichotomous patient outcome. Numerical studies indicate that the estimators have good accuracy. An application to a previously published meta‐regression illustrates the practical utility of the methodology.