Linear Function Of Covariates Research Articles

Predicting Differences in Model Parameters with Individual Parameter Contribution Regression Using the R Package ipcr

Unmodeled differences between individuals or groups can bias parameter estimates and may lead to false-positive or false-negative findings. Such instances of heterogeneity can often be detected and predicted with additional covariates. However, predicting differences with covariates can be challenging or even infeasible, depending on the modeling framework and type of parameter. Here, we demonstrate how the individual parameter contribution (IPC) regression framework, as implemented in the R package ipcr, can be leveraged to predict differences in any parameter across a wide range of parametric models. First and foremost, IPC regression is an exploratory analysis technique to determine if and how the parameters of a fitted model vary as a linear function of covariates. After introducing the theoretical foundation of IPC regression, we use an empirical data set to demonstrate how parameter differences in a structural equation model can be predicted with the ipcr package. Then, we analyze the performance of IPC regression in comparison to alternative methods for modeling parameter heterogeneity in a Monte Carlo simulation.

Spatial-temporal multivariate semi-Bayesian hierarchical framework for extreme precipitation frequency analysis

We present a semi-Bayesian hierarchical modeling framework for conducting space–time frequency analysis of precipitation extremes over a large domain. In this framework, the data layer, the precipitation extreme – i.e., seasonal maximum precipitation, at each station in each year is modeled using a generalized extreme value (GEV) distribution with temporally varying parameters, which are decomposed as linear functions of covariates. The coefficients of the covariates are estimated via maximum likelihood (ML). In the process layer, the estimated ML coefficients of each of the covariates across the stations are spatially modeled with a Gaussian multivariate process which enables capturing the spatial structure and correlation between the spatial model parameters. Suitable priors are used for the spatial model hyperparameters to complete the Bayesian formulation. Since the Bayesian formulation is only at the second level, our model is semi-Bayesian and thus, the posteriors are conditional posterior distributions. With the conditional posterior distribution of spatial fields of the GEV parameters for each time, conditional posterior distribution of the nonstationary space–time return levels of the precipitation extremes are obtained. We demonstrate this framework by application to summer precipitation extreme at 73 stations covering a large domain of Southwest US consisting of Arizona, New Mexico, Colorado, and Utah. The results from fitting and cross-validation indicate that our model captures the historical variability at the stations very well. Conditional posterior distributions of return levels are simulated on a grid over the domain, which will be of immense utility in management of natural resources and infrastructure.

Sufficient dimension reduction and instrument search for data with nonignorable nonresponse

Consider a response variable subject to nonignorable nonresponse and a fully observed covariate vector. The purpose of our study is threefold. First, we study how to extend nonparametric sufficient dimension reduction to data with nonignorable nonresponse. Second, we utilize sufficient dimension reduction to search an instrument, a linear function of covariates that is related to the response variable but can be excluded from the propensity of nonignorable nonresponse, for the purpose of identifying unknown parameters in a semiparametric propensity and a nonparametric distribution of response variable and covariates. Third, we establish asymptotic results for parameter estimators based on sufficient dimension reduction and instrument search, and investigate the effect on the limiting distribution of parameter estimators due to instrument search. We evaluate the performance of proposed estimators in a Monte Carlo study and illustrate our method in an application to AIDS Clinical Trials Group Protocol 175 data.

Weighted Score test based EWMA control charts for Zero-Inflated Poisson Models

Zero-inflated Poisson models have been widely used to account for excess zero values in count data. Recent literature pointed out that the quality characteristic could be assumed to depend on a linear function of covariates in zero-inflated Poisson models, which is called risk-adjustment. Most existing approaches to monitoring zero-inflated Poisson processes mainly focus on parameter changes based on the observed response variable. The additional covariate information is not generally considered when monitoring the quality characteristic, even though this may cause significant information loss. In this research, we study an upper-sided exponentially weighted moving average control chart using a weighted score test statistic. The proposed control chart can detect the positive shifts in parameters of the Poisson part in zero-inflated Poisson models. Especially, this new method allows improving the capacity of detecting unnatural heterogeneity variability, thanks to the introduced random effect. Moreover, the simulation results via the Monte Carlo methodology are provided, which show that the proposed approach is more efficient than the existing PR-ZIP chart when detecting small to moderate shifts. Finally, the proposed chart is also applied to a practical example to demonstrate its utility.

Truncated location-scale non linear regression models

ABSTRACTWe present a class of truncated non linear regression models for location and scale where the truncated nature of the data is incorporated into the statistical model by assuming that the response variable follows a truncated distribution. The location parameter of the response variable is assumed to be modeled by a continuous non linear function of covariates and unknown parameters. In addition, the proposed model also allows for the scale parameter of the responses to be characterized by a continuous function of the covariates and unknown parameters. Three particular cases of the proposed models are presented by considering the response variable to follow a truncated normal, truncated skew normal, and truncated beta distribution. These truncated non linear regression models are constructed assuming fixed known truncation limits and model parameters are estimated by direct maximization of the log-likelihood using a non linear optimization algorithm. Standardized residuals and diagnostic metrics based on the cases deletion are considered to verify the adequacy of the model and to detect outliers and influential observations. Results based on simulated data are presented to assess the frequentist properties of estimates, and a real data set on soil-water retention from the Buriti Vermelho River Basin database is analyzed using the proposed methodology.

Personalised treatment assignment maximising expected benefit with smooth hinge loss

ABSTRACTIn personalised medicine, the goal is to make a treatment recommendation for each patient with a given set of covariates to maximise the treatment benefit measured by patient's response to the treatment. In application, such a treatment assignment rule is constructed using a sample training data consisting of patients’ responses and covariates. Instead of modelling responses using treatments and covariates, an alternative approach is maximising a response-weighted target function whose value directly reflects the effectiveness of treatment assignments. Since the target function involves a loss function, efforts have been made recently on the choice of the loss function to ensure a computationally feasible and theoretically sound solution. We propose to use a smooth hinge loss function so that the target function is convex and differentiable, which possesses good asymptotic properties and numerical advantages. To further simplify the computation and interpretability, we focus on the rules that are linear functions of covariates and discuss their asymptotic properties. We also examine the performances of our method with simulation studies and real data analysis.

Quantile regression models for current status data

Current status data arise frequently in demography, epidemiology, and econometrics where the exact failure time cannot be determined but is only known to have occurred before or after a known observation time. We propose a quantile regression model to analyze current status data, because it does not require distributional assumptions and the coefficients can be interpreted as direct regression effects on the distribution of failure time in the original time scale. Our model assumes that the conditional quantile of failure time is a linear function of covariates. We assume conditional independence between the failure time and observation time. An M-estimator is developed for parameter estimation which is computed using the concave–convex procedure and its confidence intervals are constructed using a subsampling method. Asymptotic properties for the estimator are derived and proven using modern empirical process theory. The small sample performance of the proposed method is demonstrated via simulation studies. Finally, we apply the proposed method to analyze data from the Mayo Clinic Study of Aging.

Bayesian analysis of joint mean and covariance models for longitudinal data

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. If misspecification occurs, it may lead to inefficient or biased estimators of parameters in the mean. One of the most commonly used methods for handling the covariance matrix is based on simultaneous modeling of the Cholesky decomposition. Therefore, in this paper, we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a fully Bayesian inference for joint mean and covariance models based on this decomposition. A computational efficient Markov chain Monte Carlo method which combines the Gibbs sampler and Metropolis–Hastings algorithm is implemented to simultaneously obtain the Bayesian estimates of unknown parameters, as well as their standard deviation estimates. Finally, several simulation studies and a real example are presented to illustrate the proposed methodology.

Local Empirical Likelihood Inference for Varying-Coefficient Density-Ratio Models Based on Case-Control Data

In this article, we develop a varying-coefficient density-ratio model for case-control studies. The case and control samples come from two different distributions. Under the model assumption, the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function. A special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates. We propose a local empirical likelihood (EL) approach to estimate the nonparametric coefficient functions. Under some regularity assumptions, the proposed estimators are shown to be consistent and asymptotically normally distributed. The sieve empirical likelihood ratio (SELR) test statistic for detecting whether the varying-coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi-squared distribution. We introduce a modified bootstrap procedure to estimate the null distribution of the SELR when sample size is small. We also examine the performance of proposed method for finite sample sizes through simulation studies and illustrate it with a real dataset. Supplementary materials for this article are available online.

A moving average Cholesky factor model in joint mean-covariance modeling for longitudinal data

Modeling the mean and covariance simultaneously is a common strategy to efficiently estimate the mean parameters when applying generalized estimating equation techniques to longitudinal data. In this article, using generalized estimation equation techniques, we propose a new kind of regression models for parameterizing covariance structures. Using a novel Cholesky factor, the entries in this decomposition have moving average and log innovation interpretation and are modeled as linear functions of covariates. The resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed. Simulation studies and a real data analysis show that the proposed approach yields highly efficient estimators for the parameters in the mean, and provides parsimonious estimation for the covariance structure.

On Expressing the Probabilities of Categorical Responses as Linear Functions of Covariates

Logistic regression is usually used to model probabilities of categorical responses as functions of covariates. However, the link connecting the probabilities to the covariates is non-linear. We show in this paper that when the cross-classification of all the covariates and the dependent variable have no empty cells, then the probabilities of responses can be expressed as linear functions of the covariates. We demonstrate this for both the dichotmous and polytomous dependent variables.

Joint Variable Selection of Mean-Covariance Model for Longitudinal Data

In this paper we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a penalized maximum likelihood method for variable selection in joint mean and covariance models based on this decomposition. Under certain regularity conditions, we establish the consistency and asymptotic normality of the penalized maximum likelihood estimators of parameters in the models. Simulation studies are undertaken to assess the finite sample performance of the proposed variable selection procedure.

A moving average Cholesky factor model in covariance modelling for longitudinal data

We propose new regression models for parameterizing covariance structures in longitudinal data analysis. Using a novel Cholesky factor, the entries in this decomposition have a moving average and log-innovation interpretation and are modelled as linear functions of covariates. We propose efficient maximum likelihood estimates for joint mean-covariance analysis based on this decomposition and derive the asymptotic distributions of the coefficient estimates. Furthermore, we study a local search algorithm, computationally more efficient than traditional all subset selection, based on bic for model selection, and show its model selection consistency. Thus, a conjecture of Pan & MacKenzie (2003) is verified. We demonstrate the finite-sample performance of the method via analysis of data on CD4 trajectories and through simulations.

On efficient estimation in additive hazards regression with current status data

The additive hazard regression (AHR) model is known for its convenience in interpretation, as hazard is modeled as a linear function of covariates. One outstanding issue in the application of such a model in the analysis of current status data is that there lacks an efficient and computationally simple approach for parameter estimation. In the current literature, Lin et al.’s (1998) method enjoys the computational ease but it is not semi-parametrically efficient, whereas Martinussen and Scheike’s (2002) method is semi-parametrically efficient but difficult to compute. In this paper, we propose a new estimation approach in the context of Lin et al.’s AHR models where the monitor time process follows a proportional hazard model. We show that not only the proposed estimator achieves semi-parametric information bound, but also its implementation can be done easily using existing statistical software. We evaluate this new method via simulation studies. Also, we illustrate the proposed method through an analysis of renal function recovery data.

Multiple imputation based on restricted mean model for censored data

Most multiple imputation (MI) methods for censored survival data either ignore patient characteristics when imputing a likely event time, or place quite restrictive modeling assumptions on the survival distributions used for imputation. In this research, we propose a robust MI approach that directly imputes restricted lifetimes over the study period based on a model of the mean restricted life as a linear function of covariates. This method has the advantages of retaining patient characteristics when making imputation choices through the restricted mean parameters and does not make assumptions on the shapes of hazards or survival functions. Simulation results show that our method outperforms its closest competitor for modeling restricted mean lifetimes in terms of bias and efficiency in both independent censoring and dependent censoring scenarios. Survival estimates of restricted lifetime model parameters and marginal survival estimates regain much of the precision lost due to censoring. The proposed method is also much less subject to dependent censoring bias captured by covariates in the restricted mean model. This particular feature is observed in a full statistical analysis conducted in the context of the International Breast Cancer Study Group Ludwig Trial V using the proposed methodology.

Accelerated Recurrence Time Models

For the analysis with recurrent events, we propose a generalization of the accelerated failure time model to allow for evolving covariate effects. These so-called accelerated recurrence time models postulate that time to expected recurrence frequency, upon transformation, is a linear function of covariates with frequency-dependent coefficients. This modeling strategy shares the same spirit as quantile regression. An estimation and inference procedure is developed by generalizing the celebrated Powell's (1984, 1986) estimator for censored quantile regression. Consistency and asymptotic normality of the proposed estimator are established. An algorithm is devised to attain good computational efficiency. Simulations demonstrate that this proposal performs well under practical settings. This methodology is illustrated in an application to the well-known bladder cancer study.

Image analysis of the microstructure of magnetorheological elastomers

The results presented refer to the structure analysis of magnetorheological elastomers (MREs), based on ferromagnetic particles in a polyurethane matrix. The influence of the volume fraction of the ferromagnetic particles on their arrangement in relation to the external magnetic field was investigated. The amount of carbonyl iron particles, of the average size 6–9 μm, varied from 1.5 to 33 vol.%. Scanning electron microscopy was used to observe magnetorheological elastomer microstructures. Image analysis has been applied to describe the microstructures. From author’s previous studies it is known that the microstructure anisotropy has significant effect on the properties of magnetorheological elastomers. Two different techniques of image analysis: skeleton by influence zone method and linear covariance function were used to reveal the anisotropy in particle arrangements. These methods allowed quantitative microstructure description. The results show that particle arrangement changes with their volume fraction. The analyses confirm particle chain networks in composites with higher iron content. Differences in particles arrangement for samples prepared under diverse conditions were also clearly shown.

Genetic Evaluation of Mastitis Resistance Using a First-Passage Time Model for Wiener Processes for Analysis of Time to First Treatment

A Wiener process is a Brownian-motion process initiated in a certain state in a state space, and the first passage time is defined as the time of the process to reach a predefined absorbing state where the process stops. Time from 31 d prepartum to first treatment of clinical mastitis (CM) was modeled as first passage times of such Wiener processes. Two processes were used to allow for several risk factors, and for each process, initiation was at some arbitrary time point, in a certain health state with drift toward or away from absorption (disease). The drift parameter of each process was expressed as linear functions of covariates (year of calving and sire). First passage time was de-fined as the time from process initiation until the first health status process reached zero (absorption). The model was fitted to records for 36,178 first-lactation daughters of 245 Norwegian cattle sires using a Bayesian approach and Markov chain Monte Carlo methods. Genetic evaluation of sires was carried out by calculating the posterior probability of no CM (the value of the survival function) by d 331, i.e., 300 d after first calving. Alternatively, sire evaluation was based on the integrated area under the survival curve. These measures were highly correlated (0.999), which indicates a small degree of crossings of the sire-dependent survival curves. Hence, sire-specific hazards were close to proportional, resulting in a higher rank-correlation to sire evaluations from a survival model with proportional hazards than to the results from a multivariate threshold model.

Quasi-likelihood estimation for relative risk regression models

For a prospective randomized clinical trial with two groups, the relative risk can be used as a measure of treatment effect and is directly interpretable as the ratio of success probabilities in the new treatment group versus the placebo group. For a prospective study with many covariates and a binary outcome (success or failure), relative risk regression may be of interest. If we model the log of the success probability as a linear function of covariates, the regression coefficients are log-relative risks. However, using such a log-linear model with a Bernoulli likelihood can lead to convergence problems in the Newton-Raphson algorithm. This is likely to occur when the success probabilities are close to one. A constrained likelihood method proposed by Wacholder (1986, American Journal of Epidemiology 123, 174-184), also has convergence problems. We propose a quasi-likelihood method of moments technique in which we naively assume the Bernoulli outcome is Poisson, with the mean (success probability) following a log-linear model. We use the Poisson maximum likelihood equations to estimate the regression coefficients without constraints. Using method of moment ideas, one can show that the estimates using the Poisson likelihood will be consistent and asymptotically normal. We apply these methods to a double-blinded randomized trial in primary biliary cirrhosis of the liver (Markus et al., 1989, New England Journal of Medicine 320, 1709-1713).

Multiquadric Equations and Optimal Areal Rainfall Estimation

Numerical surface-fitting methods such as kriging, reciprocal distance, and multiquadric equations have often been employed for interpolation of discrete spatial data such as rainfall depth observations. Such methods have been used occasionally for estimation of average areal rainfall depth. When these or similar surface-fitting techniques are used for areal rainfall depth estimation, a weighted average of the discrete rainfall depth observations results. The rainfall depth estimates are unbiased if the numerical surface can fit a nonzero uniform depth exactly everywhere in the region. The unbiased conic multiquadric surface yields optimal gauge weighting coefficients for rainfall data correlated by a linear covariance function. The unbiased conic multiquadric method is a robust estimator of average areal rainfall that does not require extensive rainfall data with which to characterize the stochastic nature of the rainfall process. Unbiased multiquadric analysis appears to reduce the problem of negative g...