Marginal regression models with a time to event outcome and discrete multiple source predictors

In health services research, researchers often use clustered data to estimate the independent association between individual outcomes and cluster-level covariates after adjusting for individual-level characteristics. Marginal generalized linear models estimated using generalized estimating equation (GEE) methods or hierarchical (or multilevel) regression models can be used when there is a single source of clustering (e.g., patients nested within hospitals). Hierarchical regression models can also be used when there are multiple sources of clustering (e.g., patients nested within surgeons who in turn are nested within hospitals). Methods for estimating marginal regression models are less well-developed when there are multiple sources of non-nested clustering (e.g., patients are clustered both within hospitals and within in neighborhoods, but neither neighborhoods or hospitals are nested in the other). Miglioretti and Heagerty developed a GEE-type variance estimator for use when fitting marginal generalized linear models to non-nested multilevel data. We propose a variance estimator for a marginal Cox regression model fit to non-nested multilevel data that combined their approach with Lin and Wei's robust variance estimator for the Cox model. We evaluated the performance of the proposed variance estimator using an extensive set of Monte Carlo simulations. We illustrated the use of the variance estimator in a case study consisting of patients hospitalized with an acute myocardial infarction who were clustered within hospitals and who were also clustered in neighborhoods. In summary, a variance estimator motivated by that proposed by Miglioretti and Heagerty can be used with marginal Cox regression models fit to non-nested multilevel data.

BackgroundLongitudinal ordinal data are commonly analyzed using a marginal proportional odds model for relating ordinal outcomes to covariates in the biomedical and health sciences. The generalized estimating equation (GEE) consistently estimates the regression parameters of marginal models even if the working covariance structure is misspecified. For small-sample longitudinal binary data, recent studies have shown that the bias of regression parameters may result from the GEE and have addressed the issue by applying Firth’s adjustment for the likelihood score equation to the GEE as if generalized estimating functions were likelihood score functions. In this manuscript, for the proportional odds model for longitudinal ordinal data, the small-sample properties of the GEE were investigated, and a bias-reduced GEE (BR-GEE) was derived.MethodsBy applying the adjusted function originally derived for the likelihood score function of the proportional odds model to the GEE, we produced the BR-GEE. We investigated the small-sample properties of both GEE and BR-GEE through simulation and applied them to a clinical study dataset.ResultsIn simulation studies, the BR-GEE had a bias closer to zero, smaller root mean square error than the GEE with coverage probability of confidence interval near or above the nominal level. The simulation also showed that BR-GEE maintained a type I error rate near or below the nominal level.ConclusionsFor the analysis of longitudinal ordinal data involving a small number of subjects, the BR-GEE is advantageous for obtaining estimates of the regression parameters of marginal proportional odds models.

In this chapter we consider dependent data but move from the linear models of Chap.8 to general regression models. As in Chap.6, we consider generalized linear models (GLMs) and, more briefly, nonlinear models. We first give an outline of this chapter. In Sect.9.2 we describe three motivating datasets to which we return throughout the chapter. The GLMs discussed in Sect.6.3 can be extended to incorporate dependences in observations on the same unit; as with the linear model, an obvious way to carry out modeling in this case is to introduce unit-specific random effects. Within a GLM a natural approach is for these random effects to be included on the linear predictor scale. The resultant conditional models are known as generalized linear mixed models (GLMMs), and these are introduced in Sect.9.3. In Sects.9.4 and9.5 we describe likelihood and conditional likelihood methods of estimation, with Sect.9.6 devoted to a Bayesian treatment. Section9.7 illustrates some of the flexibility of GLMMs by describing and applying a particular model for spatial dependence. An alternative random effects specification, based on conjugacy, is described in Sect.9.8. An important approach to the modeling and analysis of dependent data that is philosophically different from the random effects formulation is via marginal models and generalized estimating equations (GEE), and these are the subject of Sect.9.9. In Sect.9.10, a second GEE approach is described in which the estimating equations for the mean are supplemented with a second set for the variances/covariances. For GLMMs, extra care must be taken with parameter interpretation, and Sect.9.11 discusses this issue, emphasizing how interpretation differs between conditional and marginal models. In PartII of the book, which focused on independent data, Chap.7 was devoted to models for binary data. For dependent data, models binary data are less well developed, and so we do not devote a complete chapter to their description. However, Sect.9.12 introduces the modeling of dependent binary data, and, subsequently, Sects.9.13 and 9.14 describe conditional (mixed) and marginal models for binary data. Section 9.15 considers how nonlinear models, as defined in Sect.6.10, can be extended to the dependent data case. For such models, many applications concentrate on inference for units, and so the introduction of random effects is again suggested. We refer to the resultant class of models as nonlinear mixed models (NLMMs). Section9.16 considers issues related to the parameterization of the nonlinear model. Inference for nonlinear mixed models via likelihood and Bayes approaches is covered in Sects.9.17 and9.18, while GEE is briefly considered in Sect.9.19. The assessment of assumptions for general regression models is described in Sect.9.20, with concluding comments contained in Sect.9.21. Additional references appear in Sect.9.22.

본 연구는 비연속 분할함수 성장혼합모형의 접근법에 따라 계층 간 원격 결과변인 추정치에 어떠한 차이가 있는지를 확인하는 데 목적이 있다. 이를 위하여 본 연구는 3단계 ML접근법과 3단계 BCH 접근법을 적용하여 모의실험 연구를 수행하였다. 모의실험에 계층 분류정도(1.3, 2.6), 계층 혼합비율(90:10, 70:30, 50:50), 표본크기(200, 500, 1000, 2000)의 조건을 설정하였다. 분석결과, 3단계 ML접근법과 3단계 BCH접근법의 원격 결과변인에서 양호도의 차이는 미미한 것으로 나타났지만, 소표본일 때 3단계 BCH접근법이 3단계 ML접근법에 비해 상대적으로 효율성이 높은 것으로 나타났다. 본 연구는 원격 결과변인의 보조변수 접근법에 기존에 활용되지 않았던 비연속 분할함수 성장혼합모형을 적용하고, 다양한 모의실험 조건을 설정하여 모의실험 연구를 수행하였다는 점에서 의의가 있다. 본 연구가 연구자들에게 실질적인 모형 활용 방안을 제시할 수 있을 것으로 기대하며, 연구의 제한점을 바탕으로 후속연구를 제언하였다.The purpose of this study was to compare the estimates of distal outcome variable in discontinuous growth mixture model. For this purpose, this study carried out a simulation study using a three-step maximum likelihood (ML) approach and a three-step BCH approach. The manipulated conditions were the class separation (1.3, 2.6), the mixing proportions (90:10, 70:30, 50:50), and the sample size (200, 500, 1000, 2000). The differences between ML and a three-step BCH approaches were not substantial, however, the three-step BCH approach was relatively more efficient than the three-step ML approach in the conditions of relatively small sample sizes. In particular, the bias of standard error was smaller in the conditions of the relatively less mixing proportions when three-step BCH approach was used. Limitations of the current study and suggestions for future study were provided.

The construction of rank hierarchies based on agonistic interactions between two individuals (“dyads”) is an important component in the characterization of the social structure of groups. To this end, winner‐loser matrices are typically created, which collapse the outcome of dyadic interactions over time, resulting in the loss of all information contained in the temporal domain. Methods that track changes in the outcome of dyadic interactions (such as “Elo‐ratings”) are receiving increasing interest. Critically, individual ratings are not just based on the succession of wins and losses, but depend on the values of start ratings and a shift coefficient. Recent studies improved existing methods by introducing a point estimation of these auxiliary parameters on the basis of a maximum likelihood (ML) approach. For a sound assessment of the rank hierarchies generated this way, we argue that measures of uncertainty of the estimates, as well as a quantification of the robustness of the methods, are also needed. We introduce a Bayesian inference (BI) approach using “partial pooling”, which rests on the assumption that all start ratings are samples from the same distribution. We compare the outcome of the ML approach to that of the BI approach using real‐world data. In addition, we simulate different scenarios to explore in which way the Elo‐rating responds to social events (such as rank changes), and low numbers of observations. Estimates of the start ratings based on “partial pooling” are more robust than those based on ML, also in scenarios where some individuals have only few observations. Our simulations show that assumed rank differences may fall well within the “uncertain” range, and that low sampling density, unbalanced designs, and coalitionary leaps involving several individuals within the hierarchy may yield unreliable results. Our results support the view that Elo‐rating can be a powerful tool in the analysis of social behaviour, when the data meet certain criteria. Assessing the uncertainty greatly aids in the interpretation of results. We advocate running simulation approaches to test how well Elo‐ratings reflect the (simulated) true structure and how sensitive the rating is to true changes in the hierarchy.

A comparative study of short cut procedures for parameter estimation in differential equations

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

Uncertainty assessment of flow and contaminant transport in the vadose zone entails probability density functions (PDFs) of soil hydraulic parameters. An unconventional maximum likelihood (ML) approach was used in this study to estimate the PDFs of water retention parameters (e.g., van Genuchten α and m) for a situation common in field‐scale modeling where core samples are sparse and prior PDFs of the parameters are unknown. In this situation, the unconventional ML approach approximates the PDFs as multivariate Gaussian. This study developed a method of estimating the mean and covariance of the multivariate Gaussian PDF based on the results of least square methods that can be easily obtained in practice. The developed method was applied to and evaluated through numerical simulation of unsaturated flow and tracer transport at the proposed Yucca Mountain geologic repository. Another focus of this study was to investigate the effect of uncertainty in the water retention parameters on predictive uncertainty. By comparing the predictive uncertainty before and after incorporating random water retention parameters, it was found that the random water retention parameters had limited effects on the mean predictions of the state variables including percolation flux and tracer travel time from the potential repository to the water table. Incorporating the uncertainty in the water retention parameters, however, significantly increased the magnitude and spatial extent of predictive uncertainty of the state variables. In particular, incorporating the random water retention parameters significantly changed the 5th and 95th percentiles of the tracer travel time by tens of thousands of years.

A fundamental issue in knowledge-aided space-time adaptive processing (KA-STAP) is to determine the degree of accuracy of the a~priori knowledge and the optimal emphasis that should be placed on it. In KA-STAP, the a priori knowledge consists usually of an initial guess of the clutter covariance matrix. This can be obtained either by previous radar probings or by a map-based study. We consider a linear combination of the a~priori clutter covariance matrix with the sample covariance matrix obtained from secondary data, and derive an optimal weighting factor on the a priori knowledge by a maximum likelihood (ML) approach. The performance of the ML approach for KA-STAP is evaluated based on numerically simulated data.

Co-clustering addresses the problem of simultaneous clustering of both dimensions of a data matrix. When dealing with high dimensional sparse data, co-clustering turns out to be more beneficial than one-sided clustering even if one is interested in clustering along one dimension only. Aside from being high dimensional and sparse, some datasets, such as document-term matrices, exhibit directional characteristics, and the $$L_2$$ normalization of such data, so that it lies on the surface of a unit hypersphere, is useful. Popular co-clustering assumptions such as Gaussian or Multinomial are inadequate for this type of data. In this paper, we extend the scope of co-clustering to directional data. We present Diagonal Block Mixture of Von Mises–Fisher distributions (dbmovMFs), a co-clustering model which is well suited for directional data lying on a unit hypersphere. By setting the estimate of the model parameters under the maximum likelihood (ML) and classification ML approaches, we develop a class of EM algorithms for estimating dbmovMFs from data. Extensive experiments, on several real-world datasets, confirm the advantage of our approach and demonstrate the effectiveness of our algorithms.

Building forecasting models for tropical cyclone intensity is one of the most challenging area in tropical cyclone research. Most, if not all, of the existing models have been built using variants of Maximum Likelihood (ML) approach. The need to partition data into two sets for model development is seen to be one of the drawbacks of ML approach in the face of limited available data. This paper proposes a way to build forecasting model using a number of model selection criteria which take the penalized-likelihood approach, namely MML, MDL, CAICF, SRM. These criteria claim to have the mechanism to balance between model complexity and goodness of fit. The models selected are then compared with the benchmark models being used in operation.

In longitudinal studies, missing data are ubiquitous. This article made a comparison of three model-based techniques for handling different types of missing data (i.e., missing at random (MAR)-based maximum likelihood (ML) approach, missing not at random (MNAR)-based Diggle–Kenward (DK) selection model and MNAR-based pattern mixture (PM) model) in longitudinal studies through a Monte Carlo simulation study. Two influential factors were considered: the dropout rates (5%, 10%, 20%, and 40%) and the sample sizes (100, 300, 500, and 1000) under MAR and MNAR missingness mechanisms respectively. The results indicated that the model selection was a crucial issue when researchers were dealing with missing data in longitudinal studies because under MNAR mechanism, DK method outperformed MAR-based ML approach, but PM method performed worse than MAR-based method did. The differences of the parameter estimation among three methods became more significant as the sample size and the dropout rate increased.

The present paper deals with the comparison of the performance of different estimation methods for regression models with correlated binary responses. Throughout, we consider probit models where an underlying latent continous random variable crosses a threshold. The error variables in the unobservable latent model are assumed to be normally distributed. The estimation procedures considered are (1) marginal maximum likelihood estimation using Gauss-Hermite quadrature, (2) generalized estimation equations (GEE) techniques with an extension to estimate tetrachoric correlations in a second step, and, (3) the MECOSA approach proposed by Schepers, Arminger and Kusters (1991) using hierarchical mean and covariance structure models. We present the results of a simulation study designed to evaluate the small sample properties of the different estimators and to make some comparisons with respect to technical aspects of the estimation procedures and to bias and mean squared error of the estimators. The results show that the calculation of the ML estimator requires the most computing time, followed by the MECOSA estimator. For small and moderate sample sizes the calculation of the MECOSA estimator is problematic because of problems of convergence as well as a tendency of underestimating the variances. In large samples with moderate or high correlations of the errors in the latent model, the MECOSA estimators are not as efficient as ML or GEE estimators. The higher the `true' value of an equicorrelation structure in the latent model and the larger the sample sizes are, the more is the efficiency gain of the ML estimator compared to the GEE and MECOSA estimators. Using the GEE approach, the ML estimates of tetrachoric correlations calculated in a second step are biased to a smaller extent than using the MECOSA approach.

Family-based association analysis unconditional on parental genotypes models the effects of observed genotypes. This approach has been shown to have greater power than conditional methods. In this chapter, we review popular association analysis methods accounting for familial correlations: the marginal model using generalized estimating equations (GEE), the mixed model with a polygenic random component, and genome-wide association analyses. The marginal approach does not explicitly model familial correlations but uses the information to improve the efficiency of parameter estimates. This model, using GEE, is useful when the correlation structure is not of interest; the correlations are treated as nuisance parameters. In the mixed model, familial correlations are modeled as random effects, e.g., the polygenic inheritance model accounts for correlations originating from shared genomic components within a family. These unconditional methods provide a flexible modeling framework for general pedigree data to accommodate traits with various distributions and many types of covariate effects. Genome-wide association studies usually test more than 10,000 SNPs and thus traditional statistical methods accounting for the familial correlations often suffer from a computational burden. Multiple approaches that have been recently proposed to avoid this computational issue are reviewed. The single-marker analysis procedures are demonstrated using the R package gee and the ASSOC program in the S.A.G.E. package, including how to prepare input data, conduct the analysis, and interpret the output. ASSOC allows models to include random components of additional familial correlations that may be not sufficiently explained by a polygenic effect and addresses nonnormality of response variables by transformation methods. With its ease of use, ASSOC provides a useful tool for association analysis of large pedigree data.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The estimation of the symbol rate of a linearly modulated signal is addressed, with special focus on low signal-to-noise ratio (SNR) scenarios. This problem finds application in automatic modulation classification and signal monitoring. A maximum-likelihood (ML) approach is adopted to derive practical estimators, exploiting information on the cyclostationarity of the modulated signal as well as knowledge of the received signaling pulse shape. The structure of the ML estimator suggests a two-step estimation procedure, whereby an initial coarse search determines first a neighborhood from which a subsequent fine search yields the final symbol rate estimate. Links between the ML approach and previous results from the literature in symbol rate estimation are established as well. The proposed scheme is applicable even for small excess bandwidths, at the cost of a higher complexity with respect to simpler estimators known to fail under such conditions. </para>