A latent variable regression model for asymmetric bivariate ordered categorical data

In many areas of medical research, especially in studies that involve paired organs, a bivariate ordered categorical response should be analyzed. Using a bivariate continuous distribution as the latent variable is an interesting strategy for analyzing these data sets. In this context, the bivariate standard normal distribution, which leads to the bivariate cumulative probit regression model, is the most common choice. In this paper, we introduce another latent variable regression model for modeling bivariate ordered categorical responses. This model may be an appropriate alternative for the bivariate cumulative probit regression model, when postulating a symmetric form for marginal or joint distribution of response data does not appear to be a valid assumption. We also develop the necessary numerical procedure to obtain the maximum likelihood estimates of the model parameters. To illustrate the proposed model, we analyze data from an epidemiologic study to identify some of the most important risk indicators of periodontal disease among students 15-19 years in Tehran, Iran.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

Variable selection via penalized minimum φ-divergence estimation in logistic regression

We propose penalized minimum φ-divergence estimator for parameter estimation and variable selection in logistic regression. Using an appropriate penalty function, we show that penalized φ-divergence estimator has oracle property. With probability tending to 1, penalized φ-divergence estimator identifies the true model and estimates nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. The advantage of penalized φ-divergence estimator is that it produces estimates of nonzero parameters efficiently than penalized maximum likelihood estimator when sample size is small and is equivalent to it for large one. Numerical simulations confirm our findings.     

Model Selection,Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.     

Truncated location-scale non linear regression models

We 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.     

Modeling Longitudinal Obesity Data with Intermittent Missingness Using a New Latent Variable Model

We propose a latent variable model for informative missingness in longitudinal studies which is an extension of latent dropout class model. In our model, the value of the latent variable is affected by the missingness pattern and it is also used as a covariate in modeling the longitudinal response. So the latent variable links the longitudinal response and the missingness process. In our model, the latent variable is continuous instead of categorical and we assume that it is from a normal distribution. The EM algorithm is used to obtain the estimates of the parameter we are interested in and Gauss–Hermite quadrature is used to approximate the integration of the latent variable. The standard errors of the parameter estimates can be obtained from the bootstrap method or from the inverse of the Fisher information matrix of the final marginal likelihood. Comparisons are made to the mixed model and complete-case analysis in terms of a clinical trial dataset, which is Weight Gain Prevention among Women (WGPW) study. We use the generalized Pearson residuals to assess the fit of the proposed latent variable model.     

A note on millers's empirical weights for heteroscedastic linear regression

For heteroscedastic simple linear regression when the variances are proportional to a power of the mean of the response variable, Miller (1986) recommends the following procedure: do a weighted least squares regression with the weights (empirical weights) estimated by the inverse of the appropriate power of the response variable. The practical appeal of this approach is its simplicity.

In this article some of the consequences of this simple procedure are considered. Specifically, the effect of this procedure on the bias of the point estimators of the regression coefficients and on the coverage probabilities of their corresponding confidence intervals is examined. It is found that the performance of the process of employing empirical weights in a weighted least squares regression depends on : (1) the particular regression parameter (slope or intercept) of interest, (2) the appropriate power of the mean of the response variable involved, and (3) the amount of variation in the data about the true regression line.     

A class of finite mixture of quantile regressions with its applications

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.     

Properties of Added Variable Plots in Cox's Regression Model

The added variable plot is useful for examining the effect of a covariate in regression models. The plot provides information regarding the inclusion of a covariate, and is useful in identifying influential observations on the parameter estimates. Hall et al. (1996) proposed a plot for Cox's proportional hazards model derived by regarding the Cox model as a generalized linear model. This paper proves and discusses properties of this plot. These properties make the plot a valuable tool in model evaluation. Quantities considered include parameter estimates, residuals, leverage, case influence measures and correspondence to previously proposed residuals and diagnostics.     

Variable selection in heteroscedastic single-index quantile regression

We propose a new algorithm for simultaneous variable selection and parameter estimation for the single-index quantile regression (SIQR) model . The proposed algorithm, which is non iterative , consists of two steps. Step 1 performs an initial variable selection method. Step 2 uses the results of Step 1 to obtain better estimation of the conditional quantiles and , using them, to perform simultaneous variable selection and estimation of the parametric component of the SIQR model. It is shown that the initial variable selection method consistently estimates the relevant variables , and the estimated parametric component derived in Step 2 satisfies the oracle property.     

Non-iterative Estimation and Variable Selection in the Single-index Quantile Regression Model

A new estimation procedure is proposed for the single-index quantile regression model. Compared to existing work, this approach is non-iterative and hence, computationally efficient. The proposed method not only estimates the index parameter and the link function but also selects variables simultaneously. The performance of the variable selection is enhanced by a fully adaptive penalty function motivated by the sliced inverse regression technique. Finite sample performance is studied through a simulation study that compares the proposed method with existing work under several criteria. A data analysis is given that highlights the usefulness of the proposed methodology.     

Reducing bias in parameter estimates from stepwise regression in proportional hazards regression with right-censored data

When variable selection with stepwise regression and model fitting are conducted on the same data set, competition for inclusion in the model induces a selection bias in coefficient estimators away from zero. In proportional hazards regression with right-censored data, selection bias inflates the absolute value of parameter estimate of selected parameters, while the omission of other variables may shrink coefficients toward zero. This paper explores the extent of the bias in parameter estimates from stepwise proportional hazards regression and proposes a bootstrap method, similar to those proposed by Miller (Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, 2002) for linear regression, to correct for selection bias. We also use bootstrap methods to estimate the standard error of the adjusted estimators. Simulation results show that substantial biases could be present in uncorrected stepwise estimators and, for binary covariates, could exceed 250% of the true parameter value. The simulations also show that the conditional mean of the proposed bootstrap bias-corrected parameter estimator, given that a variable is selected, is moved closer to the unconditional mean of the standard partial likelihood estimator in the chosen model, and to the population value of the parameter. We also explore the effect of the adjustment on estimates of log relative risk, given the values of the covariates in a selected model. The proposed method is illustrated with data sets in primary biliary cirrhosis and in multiple myeloma from the Eastern Cooperative Oncology Group.     

Bayesian composite quantile regression

One advantage of quantile regression, relative to the ordinary least-square (OLS) regression, is that the quantile regression estimates are more robust against outliers and non-normal errors in the response measurements. However, the relative efficiency of the quantile regression estimator with respect to the OLS estimator can be arbitrarily small. To overcome this problem, composite quantile regression methods have been proposed in the literature which are resistant to heavy-tailed errors or outliers in the response and at the same time are more efficient than the traditional single quantile-based quantile regression method. This paper studies the composite quantile regression from a Bayesian perspective. The advantage of the Bayesian hierarchical framework is that the weight of each component in the composite model can be treated as open parameter and automatically estimated through Markov chain Monte Carlo sampling procedure. Moreover, the lasso regularization can be naturally incorporated into the model to perform variable selection. The performance of the proposed method over the single quantile-based method was demonstrated via extensive simulations and real data analysis.     

Bayesian variable selection and estimation in maximum entropy quantile regression

Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis.     

Ordinal ridge regression with categorical predictors

In multi-category response models, categories are often ordered. In the case of ordinal response models, the usual likelihood approach becomes unstable with ill-conditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraint on the order of intercept values is not met during the iterative procedure. Proportional odds models (POMs) are most commonly used for ordinal responses. In this paper, penalized likelihood with quadratic penalty is used to address these issues with a special focus on POMs. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized-likelihood function penalizes the parameter estimates or differences between the parameter estimates according to the type of predictors. Mean-squared error for parameter estimates, deviance of fitted probabilities and prediction error for ridge regression are compared with usual likelihood estimates in a simulation study and an application.     

Variable selection in joint location and scale models of the skew-normal distribution

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the data set under consideration involves asymmetric outcomes. Variable selection is an important issue in all regression analyses, and in this paper, we investigate the simultaneously variable selection in joint location and scale models of the skew-normal distribution. We propose a unified penalized likelihood method which can simultaneously select significant variables in the location and scale models. Furthermore, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the location and scale models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies and a real example are used to illustrate the proposed methodologies.     

On the asymptotic properties of parameter estimates in a regression model with non-normally distributed errors

As pointed out in a recent paper by Amirkhalkhali and Rao (1986) (henceforth referred to as A&R), the usual assumption of normality for the error terms of a regression model isoften untenable. However, when this assumption is dropped, it may be difficult to characterize parameter estimates for the model. For example, A&R (p. 189) state that “if the regression errors are non-normal, we are not even sure of their [e.g., the generalized least squares parameter estimates1] asymptotic properties.” A partial answer, however, is given by Spall and Wall (1984), which presents an asymptotic distribution theory for Kalman filter estimates for cases where the random terms of the state space model are not necessarily Gaussian. Certain of these asymptotic distribution results are also discussed in Spall (1985) in the context of model validation (diagnostic checking)     

On estimation and diagnostics analysis in log-generalized gamma regression model for interval-censored data

The interval-censored survival data appear very frequently, where the event of interest is not observed exactly but it is only known to occur within some time interval. In this paper, we propose a location-scale regression model based on the log-generalized gamma distribution for modelling interval-censored data. We shall be concerned only with parametric forms. The proposed model for interval-censored data represents a parametric family of models that has, as special submodels, other regression models which are broadly used in lifetime data analysis. Assuming interval-censored data, we consider a frequentist analysis, a Jackknife estimator and a non-parametric bootstrap for the model parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some techniques to perform global influence.     

The log-odd log-logistic Weibull regression model: modelling,estimation, influence diagnostics and residual analysis

In applications of survival analysis, the failure rate function may frequently present a unimodal shape. In such cases, the log-normal and log-logistic distributions are used. In this paper, we shall be concerned only with parametric forms, so a location-scale regression model based on the odd log-logistic Weibull distribution is proposed for modelling data with a decreasing, increasing, unimodal and bathtub failure rate function as an alternative to the log-Weibull regression model. For censored data, we consider a classic method to estimate the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals is determined and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the new regression model applied to censored data. We analyse a real data set using the log-odd log-logistic Weibull regression model.