G-computation estimation for causal inference with complex longitudinal data

In a companion paper, [Neugebauer, R., van der Laan, M.J., 2006b. Causal effects in longitudinal studies: definition and maximum likelihood estimation. Comput. Stat. Data. Anal., this issue, doi:10.1016/j.csda.2006.06.013], we provided an overview of causal effect definition with marginal structural models (MSMs) in longitudinal studies. A parametric MSM (PMSM) and a non-parametric MSM (NPMSM) approach were described for the representation of causal effects in pooled or stratified analyses of treatment effects on time-dependent outcomes. Maximum likelihood estimation, also referred to as G-computation estimation, was detailed for these causal effects. In this paper, we develop new algorithms for the implementation of the G-computation estimators of both NPMSM and PMSM causal effects. Current algorithms rely on Monte Carlo simulation of all possible treatment-specific outcomes, also referred to as counterfactuals or potential outcomes. This task becomes computationally impracticable (a) in studies with a continuous treatment, and/or (b) in longitudinal studies with long follow-up with or without time-dependent outcomes. The proposed algorithms address this important computing limitation inherent to G-computation estimation in most longitudinal studies. Finally, practical considerations about the proposed algorithms lead to a further generalization of the definition of NPMSM causal effects in order to allow more reliable applications of these methodologies to a broader range of real-life studies. Results are illustrated with two simulation studies.     

A cross-validation deletion-substitution-addition model selection algorithm: Application to marginal structural models

The cross-validation deletion-substitution-addition (cvDSA) algorithm is based on data-adaptive estimation methodology to select and estimate marginal structural models (MSMs) for point treatment studies as well as models for conditional means where the outcome is continuous or binary. The algorithm builds and selects models based on user-defined criteria for model selection, and utilizes a loss function-based estimation procedure to distinguish between different model fits. In addition, the algorithm selects models based on cross-validation methodology to avoid “over-fitting” data. The cvDSA routine is an R software package available for download. An alternative R-package (DSA) based on the same principles as the cvDSA routine (i.e., cross-validation, loss function), but one that is faster and with additional refinements for selection and estimation of conditional means, is also available for download. Analyses of real and simulated data were conducted to demonstrate the use of these algorithms, and to compare MSMs where the causal effects were assumed (i.e., investigator-defined), with MSMs selected by the cvDSA. The package was used also to select models for the nuisance parameter (treatment) model to estimate the MSM parameters with inverse-probability of treatment weight (IPTW) estimation. Other estimation procedures (i.e., G-computation and double robust IPTW) are available also with the package.     

Comparison of semiparametric maximum likelihood estimation and two-stage semiparametric estimation in copula models

We consider bivariate distributions that are specified in terms of a parametric copula function and nonparametric or semiparametric marginal distributions. The performance of two semiparametric estimation procedures based on censored data is discussed: maximum likelihood (ML) and two-stage pseudolikelihood (PML) estimation. The two-stage procedure involves less computation and it is of interest to see whether it is significantly less efficient than the full maximum likelihood approach. We also consider cases where the copula model is misspecified, in which case PML may be better. Extensive simulation studies demonstrate that in the absence of covariates, two-stage estimation is highly efficient and has significant robustness advantages for estimating marginal distributions. In some settings, involving covariates and a high degree of association between responses, ML is more efficient. For the estimation of association, PML does not offer an advantage.     

Parametric and nonparametric Bayesian model specification: A case study involving models for count data

In this paper we present the results of a simulation study to explore the ability of Bayesian parametric and nonparametric models to provide an adequate fit to count data of the type that would routinely be analyzed parametrically either through fixed-effects or random-effects Poisson models. The context of the study is a randomized controlled trial with two groups (treatment and control). Our nonparametric approach uses several modeling formulations based on Dirichlet process priors. We find that the nonparametric models are able to flexibly adapt to the data, to offer rich posterior inference, and to provide, in a variety of settings, more accurate predictive inference than parametric models.     

Mixed model-based inference in geoadditive hazard regression for interval-censored survival times

Mixed model-based estimation of additive or geoadditive regression models has become popular throughout recent years. It provides a unified and modular framework that facilitates joint estimation of nonparametric covariate effects and the corresponding smoothing parameters. Therefore, extensions of mixed model-based inference to a Cox-type regression model for the hazard rate are considered, allowing for a combination of general censoring schemes for the survival times and a flexible, geoadditive predictor. In particular, the proposed methodology allows for arbitrary combinations of right, left, and interval censoring as well as left truncation. The geoadditive predictor comprises time-varying effects, nonparametric effects of continuous covariates, spatial effects, and potentially a number of extensions such as cluster-specific frailties or interaction surfaces. In addition, all covariates are allowed to be piecewise constant time-varying. Nonlinear and time-varying effects as well as the baseline hazard rate are modeled by penalized splines. Spatial effects can be included based on either Markov random fields or stationary Gaussian random fields. Estimation is based on a reparametrization of the model as a variance component mixed model. The variance parameters, corresponding to inverse smoothing parameters, can then be determined using an approximate marginal likelihood approach. An analysis on childhood mortality in Nigeria serves as an application, where the interval censoring framework additionally allows to deal with the problem of heaped survival times. The effect of ignoring the impact of interval-censored observations is investigated in a simulation study.     

Parametric sensitivity: A case study comparison

This article presents a comparative analysis of three derivative-based parametric sensitivity approaches in multi-response regression estimation: marginal sensitivity, profile-based approach developed by [Sulieman, H., McLellan, P.J., Bacon, D.W., 2004, A Profile-based approach to parametric sensitivity in multiresponse regression models, Computational Statistics & Data Analysis, 45, 721-740] and the commonly used approach of the Fourier Amplitude Sensitivity Test (FAST). We apply the classical formulation of FAST in which Fourier sine coefficients are utilized as sensitivity measures. Contrary to marginal sensitivity, profile-based and FAST approaches provide sensitivity measures that account for model nonlinearity and are pertinent to linear and nonlinear regression models. However, the primary difference between FAST and profile-based sensitivity is that traditional FAST fails to account for parameter dependencies in the model system while these dependencies are considered in the analysis procedure of profile-based sensitivity through the re-estimation of the remaining model parameters conditional on the values of the parameter of interest. An example is discussed to illustrate the comparisons by applying the three sensitivity methods to a model described by set of non-linear differential equations. Some computational aspects are also explored.     

Comparing the estimates of ROC curves by modeling methods

We investigate parameter estimation problems using the ROC curve approach. We compare several parametric, semiparametric, and nonparametric estimates of ROC curves on the assumption that the model is binormal. Our comparison is based on the analysis of numerical examples: we use generalized least square method for parametric estimation, functional modeling for semiparametric, and sdf for nonparametric estimation.     

A new approach to bootstrap inference in functional coefficient models

To infer on functional dependence of regression parameters, a new, factor based bootstrap approach is introduced, that is robust under various forms of heteroskedastic error terms. Modeling the functional coefficient parametrically, the bootstrap approximation of an F-statistic is shown to hold asymptotically. In simulation studies with both parametric and nonparametric functional coefficients, factor based bootstrap inference outperforms the wild bootstrap and pairs bootstrap approach, according to its rejection frequencies under the null hypothesis. Applying the functional coefficient model to a cross sectional investment regression on savings, the saving retention coefficient is found to depend on third variables as the population growth rate and the openness ratio.     

Estimation of characteristics of randomized static models of data (Entropy-robust approach)

A new approach to the definition of static relations between small amounts of input and output data is suggested, which is based on the use of randomized models and the estimation of probability characteristics of their parameters. To work up the procedures of robust parametric and nonparametric estimation, the entropy approach is developed, which uses generalized Boltzmann and Fermi informational entropies.     

Nonparametric Analysis of the Order-Statistic Model in Software Reliability

In the literature on statistical inference in software reliability, the assumptions of parametric models and random sampling of bugs have been pervasive. We argue that both assumptions are problematic, the first because of robustness concerns and the second due to logical and practical difficulties. These considerations motivate the approach taken in this paper. We propose a nonparametric software reliability model based on the order-statistic paradigm. The objective of the work is to estimate, from data on discovery times observed within a type I censoring framework, both the underlying distribution F from which discovery times are generated and N, the unknown number of bugs in the software. The estimates are used to predict the next time to failure. The approach makes use of Bayesian nonparametric inference methods, in particular, the beta-Stacy process. The proposed methodology is illustrated on both real and simulated data     

Hierarchical genetic optimization of a fuzzy logic system for energy flows management in microgrids

Bio-inspired algorithms like Genetic Algorithms and Fuzzy Inference Systems (FIS) are nowadays widely adopted as hybrid techniques in improving goods and services. In this paper we present an interesting application of the fuzzy-GA paradigm to the problem of energy flows management in microgrids, concerning the design, through a data driven synthesis procedure, of an Energy Management System (EMS). The main aim consists in performing decision making for power flow management tasks in the proposed microgrid model, equipped by renewable sources and an energy storage system, aiming to maximize the accounting profit in energy trading with the main-grid. In particular this study focuses on the application of a Hierarchical Genetic Algorithm (HGA) for tuning the Rule Base (RB) of a Fuzzy Inference System (FIS), trying to discover a minimal fuzzy rules set as the core inference engine of an an EMS. The HGA rationale focuses on a particular encoding scheme, based on control genes and parametric genes, applied to the optimization of the FIS parameters, allowing to perform a reduction in the structural complexity of the RB. A performance comparison is performed with a simpler approach based on a classic fuzzy-GA scheme, where both FIS parameters and rule weights are tuned, while the number of fuzzy rules is fixed in advance. Experiments shows how the fuzzy-HGA approach adopted for the synthesis of the proposed controller outperforms the classic fuzzy-GA scheme, increasing the accounting profit by 67% in the considered energy trading problem, yielding at the same time a simpler RB.     

A random effects four-part model, with application to correlated medical costs

In this paper we propose a four-part random effects model, with application to correlated medical cost data. Four joint equations are used to model respectively: (1) the probability of seeking medical treatment, (2) the probability of being hospitalized (conditional on seeking medical treatment), and the actual amount of (3) outpatient and (4) inpatient costs. Our model simultaneously takes account of the inter-temporal (or within-cluster) correlation of each patient and the cross-equation correlation of the four equations, by means of joint linear mixed models and generalized linear mixed models. The estimation is accomplished by the high-order Laplace approximation technique in Raudenbush et al. [Raudenbush, S.W., Yang, M., Yosef, M., 2000. Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation. Journal of Computational and Graphical Statistics 9, 141-157] and Olsen and Schafer [Olsen, M.K., Schafer, J.L., 2001. A two-part random effects model for semicontinuous longitudinal data. Journal of the American Statistical Association 96, 730-745]. Our model is used to analyze monthly medical costs of 1397 chronic heart failure patients from the clinical data repository (CDR) at the University of Virginia.     

A minimum Hellinger distance estimator for stochastic differential equations: An application to statistical inference for continuous time interest rate models

A minimum disparity estimator minimizes a φ-divergence between the marginal density of a parametric model and its non-parametric estimate. This principle is applied to the estimation of stochastic differential equation models, choosing the Hellinger distance as particular φ-divergence. Under an hypothesis of stationarity, the parametric marginal density is provided by solving the Kolmogorov forward equation. A particular emphasis is put on the non-parametric estimation of the sample marginal density which has to take into account sample dependence and kurtosis. A new window size determination is provided. The classical estimator is presented alternatively as a distance minimizer and as a pseudo-likelihood maximizer. The latter presentation opens the way to Bayesian inference. The method is applied to continuous time models of the interest rate. In particular, various models are tested using alternatively tests and their results are discussed.     

非参数回归模型均值函数结构变点的检测与应用

本文将一类系统参数变点检测问题转化为非参数回归模型均值函数结构变点的检测问题.针对当非参数模型均值函数跃度的长期均值为零时,残量累积和(cumulative sum,CUSUM)统计量无效的问题,首先利用均值函数的核估计构造新统计量,给出了原假设和备择假设下统计量的极限分布;进一步构造Bootstrap检验,证明了Bootstrap检验的一致性;最后以模拟结果表明新方法明显优于已有的方法,并应用于两类实际数据分析,说明方法的有效性.     

Orthogonality-projection-based estimation for semi-varying coefficient models with heteroscedastic errors

This paper is concerned with the estimation in semi-varying coefficient models with heteroscedastic errors. An iterated two-stage orthogonality-projection-based estimation is proposed. This method can easily be used to estimate the model parametric and nonparametric parts, as well as the variance function, and in the estimators the parametric part and nonparametric part do not affect each other. Under some mild conditions, the consistency, conditional biases, conditional variances and asymptotic normality of the resulting estimators are studied explicitly. Moreover, some simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by a real data set.     

Simulation-based two-stage estimation for multiple nonparametric regression

To reduce the curse of dimensionality arising from nonparametric estimation procedures for multiple nonparametric regression, in this paper we suggest a simulation-based two-stage estimation. We first introduce a simulation-based method to decompose the multiple nonparametric regression into two parts. The first part can be estimated with the parametric convergence rate and the second part is small enough so that it can be approximated by orthogonal basis functions with a small trade-off parameter. Then the linear combination of the first and second step estimators results in a two-stage estimator for the multiple regression function. Our method does not need any specified structural assumption on the regression function and it is proved that the newly proposed estimation is always consistent even if the trade-off parameter is designed to be small. Thus when the common nonparametric estimator such as local linear smoothing collapses because of the curse of dimensionality, our estimator still works well.     

Testing on the common mean of several normal distributions

Point estimation of the common mean of several normal distributions with unknown and possibly unequal variances has attracted the attention of many researchers over the last five decades. Relatively less attention has been paid to the hypothesis testing problem, presumably due to the complicated sampling distribution(s) of the test statistics(s) involved. Taking advantage of the computational resources available nowadays there has been a renewed interest in this problem, and a few test procedures have been proposed lately including those based on the generalized p-value approach. In this paper we propose three new tests based on the famous Graybill-Deal estimator (GDE) as well as the maximum likelihood estimator (MLE) of the common mean, and these test procedures appear to work as good as (if not better than) the existing test methods. The two tests based on the GDE use respectively a first order unbiased variance estimate proposed by Sinha [Sinha, B.K., 1985. Unbiased estimation of the variance of the Graybill-Deal estimator of the common mean of several normal populations. The Canadian Journal of Statistics 13 (3), 243-247], as well as the little known exact unbiased variance estimator proposed by Nikulin and Voinov [Nikulin, M.S., Voinov, V.G., 1995. On the problem of the means of weighted normal populations. Qüestiió (Quaderns d’Estadistica, Sistemes, Informatica i Investigació Operativa) 19 (1-3), 93-106] (after we’ve fixed a small mistake in the final expression). On the other hand, the MLE, which doesn’t have a closed expression, uses a parametric bootstrap method proposed by Pal, Lim and Ling [Pal, N., Lim, W.K., Ling, C.H., 2007b. A computational approach to statistical inferences. Journal of Applied Probability & Statistics 2 (1), 13-35]. The extensive simulation results presented in this paper complement the recent studies undertaken by Krishnamoorthy and Lu [Krishnamoorthy, K., Lu, Y., 2003. Inferences on the common mean of several normal populations based on the generalized variable method. Biometrics 59, 237-247], and Lin and Lee [Lin, S.H., Lee, J.C., 2005. Generalized inferences on the common mean of several normal populations. Journal of Statistical Planning and Inference 134, 568-582].     

Multivariate Exponential Survival Trees And Their Application to Tooth Prognosis

This paper is concerned with developing rules for assignment of tooth prognosis based on actual tooth loss in the VA Dental Longitudinal Study. It is also of interest to rank the relative importance of various clinical factors for tooth loss. A multivariate survival tree procedure is proposed. The procedure is built on a parametric exponential frailty model, which leads to greater computational efficiency. We adopted the goodness-of-split pruning algorithm of [LeBlanc, M., Crowley, J., 1993. Survival trees by goodness of split. Journal of the American Statistical Association 88, 457-467] to determine the best tree size. In addition, the variable importance method is extended to trees grown by goodness-of-fit using an algorithm similar to the random forest procedure in [Breiman, L., 2001. Random forests. Machine Learning 45, 5-32]. Simulation studies for assessing the proposed tree and variable importance methods are presented. To limit the final number of meaningful prognostic groups, an amalgamation algorithm is employed to merge terminal nodes that are homogeneous in tooth survival. The resulting prognosis rules and variable importance rankings seem to offer simple yet clear and insightful interpretations.     

Unbiased regression trees for longitudinal and clustered data

A new version of the RE–EM regression tree method for longitudinal and clustered data is presented. The RE–EM tree is a methodology that combines the structure of mixed effects models for longitudinal and clustered data with the flexibility of tree-based estimation methods. The RE–EM tree is less sensitive to parametric assumptions and provides improved predictive power compared to linear models with random effects and regression trees without random effects. The previously-suggested methodology used the CART tree algorithm for tree building, and therefore that RE–EM regression tree method inherits the tendency of CART to split on variables with more possible split points at the expense of those with fewer split points. A revised version of the RE–EM regression tree corrects for this bias by using the conditional inference tree as the underlying tree algorithm instead of CART. Simulation studies show that the new version is indeed unbiased, and has several improvements over the original RE–EM regression tree in terms of prediction accuracy and the ability to recover the correct tree structure.