Linear Combination Of Covariates Research Articles

Hypothesis testing of varying coefficients for regional quantiles

Testing the behavior of varying coefficients (VC) over a range of quantiles is important in the field of regression analysis. This study tests whether coefficient functions in varying quantile regression share common structural information across a certain range of quantile levels, even when linear combinations of covariates are unspecified in the null hypothesis. Our approach allows varying the coefficients, β(τ,t), as a function of the quantile level, τ∈Δ, and a random variable, t∈T, where Δ is the quantile region of interest and T is a domain of t. By incorporating an interval of quantiles in the inference, the proposed method can test whether the model can be justified by using a VC quantile model against other important reduced models, such as the linear quantile model, varying coefficient linear model, or linear model with homogeneous errors. We use bivariate B-splines to approximate the varying quantile functions, β(τ,t), and utilize composite quantile regression to estimate the parameters. Furthermore, we develop constrained composite quantile regression to provide a more efficient estimate in case the null hypothesis is not rejected. We show that the proposed test admits normal approximations. Using simulation and real data analysis, we demonstrate the superiority of the proposed test over other tests designed for a finite number of quantile levels.

Weight calibration to improve efficiency for estimating pure risks from the additive hazards model with the nested case-control design.

We study the efficiency of covariate-specific estimates of pure risk (one minus the survival function) when some covariates are only available for case-control samples nested in a cohort. We focus on the semiparametric additive hazards model in which the hazard function equals a baseline hazard plus a linear combination of covariates with either time-varying or time-invariant coefficients. A published approach uses the design-based inclusion probabilities to reweight the nested case-control data. We obtain more efficient estimates of pure risks by calibrating the design weights to data available in the entire cohort, for both time-varying and time-invariant covariate coefficients. We develop explicit variance formulas for the weight-calibrated estimates based on influence functions. Simulations show the improvement in precision by using weight calibration and confirm the consistency of variance estimators and the validity of inference based on asymptotic normality. Examples are provided using data from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial Study (PLCO).

Nonlinear parametric quantile models.

Quantile regression is widely used to estimate conditional quantiles of an outcome variable of interest given covariates. This method can estimate one quantile at a time without imposing any constraints on the quantile process other than the linear combination of covariates and parameters specified by the regression model. While this is a flexible modeling tool, it generally yields erratic estimates of conditional quantiles and regression coefficients. Recently, parametric models for the regression coefficients have been proposed that can help balance bias and sampling variability. So far, however, only models that are linear in the parameters and covariates have been explored. This paper presents the general case of nonlinear parametric quantile models. These can be nonlinear with respect to the parameters, the covariates, or both. Some important features and asymptotic properties of the proposed estimator are described, and its finite-sample behavior is assessed in a simulation study. Nonlinear parametric quantile models are applied to estimate extreme quantiles of longitudinal measures of respiratory mechanics in asthmatic children from an epidemiological study and to evaluate a dose-response relationship in a toxicological laboratory experiment.

Bring More Data!-A Good Advice? Removing Separation in Logistic Regression by Increasing Sample Size.

The parameters of logistic regression models are usually obtained by the method of maximum likelihood (ML). However, in analyses of small data sets or data sets with unbalanced outcomes or exposures, ML parameter estimates may not exist. This situation has been termed ‘separation’ as the two outcome groups are separated by the values of a covariate or a linear combination of covariates. To overcome the problem of non-existing ML parameter estimates, applying Firth’s correction (FC) was proposed. In practice, however, a principal investigator might be advised to ‘bring more data’ in order to solve a separation issue. We illustrate the problem by means of examples from colorectal cancer screening and ornithology. It is unclear if such an increasing sample size (ISS) strategy that keeps sampling new observations until separation is removed improves estimation compared to applying FC to the original data set. We performed an extensive simulation study where the main focus was to estimate the cost-adjusted relative efficiency of ML combined with ISS compared to FC. FC yielded reasonably small root mean squared errors and proved to be the more efficient estimator. Given our findings, we propose not to adapt the sample size when separation is encountered but to use FC as the default method of analysis whenever the number of observations or outcome events is critically low.

Deep learning-based survival prediction of oral cancer patients

The Cox proportional hazards model commonly used to evaluate prognostic variables in survival of cancer patients may be too simplistic to properly predict a cancer patient’s outcome since it assumes that the outcome is a linear combination of covariates. In this retrospective study including 255 patients suitable for analysis who underwent surgical treatment in our department from 2000 to 2017, we applied a deep learning-based survival prediction method in oral squamous cell carcinoma (SCC) patients and validated its performance. Survival prediction using DeepSurv, a deep learning based-survival prediction algorithm, was compared with random survival forest (RSF) and the Cox proportional hazard model (CPH). DeepSurv showed the best performance among the three models, the c-index of the training and testing sets reaching 0.810 and 0.781, respectively, followed by RSF (0.770/0.764), and CPH (0.756/0.694). The performance of DeepSurv steadily improved with added features. Thus, deep learning-based survival prediction may improve prediction accuracy and guide clinicians both in choosing treatment options for better survival and in avoiding unnecessary treatments.

Accounting for non-stationary variance in geostatistical mapping of soil properties

Simple and ordinary kriging assume a constant mean and variance of the soil variable of interest. This assumption is often implausible because the mean and/or variance are linked to terrain attributes, parent material or other soil forming factors. In kriging with external drift (KED) non-stationarity in the mean is accounted for by modelling it as a linear combination of covariates. In this study, we applied an extension of KED that also accounts for non-stationary variance. Similar to the mean, the variance is modelled as a linear combination of covariates. The set of covariates for the mean may differ from the set for the variance. The best combinations of covariates for the mean and variance are selected using Akaike's information criterion. Model parameters of the selected model are then estimated by differential evolution using the Restricted Maximum Likelihood (REML) in the objective function. The methodology was tested in a small area of the Hunter Valley, NSW Australia, where samples from a fine grid with gamma K measurements were treated as measurements of the variable of interest. Terrain attributes were used as covariates. Both a non-stationary variance and a stationary variance model were calibrated. The mean squared prediction errors of the two models were somewhat comparable. However, the uncertainty about the predictions was much better quantified by the non-stationary variance model, as indicated by the mean and median of the standardized squared prediction error and by accuracy plots. We conclude that the non-stationary variance model is more flexible and better suited for uncertainty quantification of a mapped soil property. However, parameter estimation of the non-stationary variance model requires more attention due to possible singularity of the covariance matrix.

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.

Multivariate Regime Switching Model with Flexible Threshold Variable: With an Application to Returns from a Portfolio of U.S. Stocks

This paper proposes a novel multivariate regime switching model that allows the threshold variable to be a linear combination of covariates with unknown coefficients: the model is likely to be more suitable to analyze time series of data in which regimes dynamics are driven by multiple covariates rather than by just one single threshold variable. The paper considers least squares estimation of the model and it proposes a test for the number of regimes based on theoretical results from multivariate statistics. Finite sample results from Monte Carlo analysis strengthen the methodological contribution of the paper. An application to measuring regime-specific cross-sectional dependence in the U.S. stock market illustrates the usefulness of the proposed model for applied work.

Prognostic Value of Podoplanin Expression in Oral Squamous Cell Carcinoma―A Regression Model Auxiliary to UICC Classification

Podoplanin, a type I transmembrane glycoprotein with an effect of platelet aggregation, has been reported to be one of the possible prognostic factors of oral squamous cell carcinoma (OSCC). However, the biological significance of podoplanin is largely unclear. The aim of this study was to develop a practical model for the prediction of prognosis using the grade of podoplanin expression, and also to evaluate the biological function of podoplanin. Eighty-two specimens of patients with previously untreated OSCC, who underwent either biopsy or surgery, were histopathologically and immunohistochemically analyzed. These 82 cases were composed of 66 well-differentiated, 10 moderately differentiated and 6 poorly differentiated OSCC. Podoplanin was successfully immunostained in 78 specimens, and was detected in most cases, but the frequency of positive cells varied. The prognosis of patients with more than 50 % podoplanin-positive tumor cells was significantly poorer than that of the other patients. Multivariate hazards regression analysis suggested that a linear combination of covariates, OSCC patients with more or less than 50 % podoplanin expression, age of more or less than 70 years old, mode of invasion and T3, T4 or T2 versus T1 of the UICC T-stage classification was the most effective model for evaluating the prognosis of OSCC patients. Additionally, podoplanin expression had a significant relationship to UICC clinical stage and the expression of Ki-67. An effective regression model using podoplanin expression was developed for evaluating the prognosis of OSCC and the biological significance of podoplanin was suggested to be associated with the growth and/or progression of OSCC.

P5-13-24: A Predictive Model of Early Systemic Disease Relapse after Standard Adjuvant Therapy for Breast Cancer.

Abstract Background Early relapse after adjuvant therapy for breast cancer is very discouraging and remains a major problem. We sought to identify predictors of early relapse risk and build a predictive model for relapse using prospectively collected data for patients seen at the Markey Cancer Center starting 2007 to date. Methods: Of the 1098 new patients seen, 814 patients had stage I-III disease and were further analyzed for predictors of early relapse risk. Univariate analyses were performed for key variables including patient age, tumor size, grade, estrogen receptor (ER) status, progesterone receptor (PgR) status, and HER2 status. A multivariate Cox regression model was built to identify predictors of systemic relapse and model-building was performed using step-wise model selection to determine candidate models. A risk score was developed based on the linear combination of covariates in the final Cox model. Time-dependent predictive curves, a newly developed statistical methodology, were used to evaluate the predictive accuracy of the proposed risk score. Results: Median patient age was 57 years (Range 25–92) and 88% were white. Forty six (46) % had stage I disease, 36% stage II, and 18% stage III. Median follow up time was 2.3 years. Of this 814 patient cohort, 708 patients had complete baseline covariate data and were used to build the candidate models. The final Cox regression model included 5 covariates that were significantly associated with risk of early relapse: stage III disease (p = 0.0011), grade III (p = 0.0028), PgR-negative status (p = 0.0121), HER2−negative status (p = 0.0305), and node-positive status (p = 0.0360). These five covariates were then used to calculate an early recurrence risk score, which is the weighted average of these risk factors when present, with the weights being the coefficients from the Cox regression model. The 1-year, 2-year and 3-year predictive curves for this risk score decrease considerably, especially for the 2-year and 3-year curves, indicating good predictive accuracy of the risk score. The highest risk score group, which represents 4.8% of the population, has a 1-year, 2-year and 3-year relapse probabilities of 13.0% (95% CI: 4.1%, 27.3%), 39.4 % (95% CI: 20.1%, 58.3%), and 52.3% (95% CI: 28.5%, 71.5%), respectively. In comparison, for the overall population, the corresponding 1-year, 2-year, and 3-year relapse probabilities were only 1.1% (95% CI: 0.5%, 2.1%), 4.2% (95% CI: 2.7%, 6.1%) and 6.2% (95% CI: 4.2%, 8.6%), respectively. Conclusions: The developed risk score based on stage, tumor grade, PgR, HER2, and node status is highly predictive of early relapse in breast cancer patients after standard adjuvant therapy. Our model can be used to identify patients with high risk of early disease relapse who may otherwise benefit from enrollment on novel adjuvant therapeutic trials to improve their outcome. Citation Information: Cancer Res 2011;71(24 Suppl):Abstract nr P5-13-24.

Single-index coefficient models for nonlinear time series

The single-index coefficient model, where the coefficients are functions of an index of a covariate vector, is a powerful tool for modelling nonlinearity in multivariate estimation. By reducing the covariate vector to an index which is usually a linear combination of covariates, the single-index coefficient model overcomes the well-known phenomenon of ‘curse-of-dimensionality’. We estimate the univariate varying coefficients with penalised splines (PS). An iterative data-driven algorithm is developed, adaptively selecting the index. The algorithm is based on the observation that given an estimated index, the varying-coefficient model using PS is essentially a linear ridge regression with spline bases. Our experiments show that the proposed algorithm gives rapid convergence. We also establish large sample properties assuming fixed number of knots. The usual jointly stationary assumption for dependent data is relaxed with weaker size requirements for either φ-mixing or α-mixing. Finally, we present an application to a gross national product data set and a simulated example.

Asymptotic Properties of a Class of Mixture Models for Failure Data: The Interior and Boundary Cases

We analyse an exponential family of distributions which generalises the exponential distribution for censored failure time data, analogous to the way in which the class of generalised linear models generalises the normal distribution. The parameter of the distribution depends on a linear combination of covariates via a possibly nonlinear link function, and we allow another level of heterogeneity: the data may contain "immune" individuals who are not subject to failure. Thus the data is modelled by a mixture of a distribution from the exponential family and a "mass at infinity" representing individuals who never fail. Our results include large sample distributions for parameter estimators and for hypothesis test statistics obtained by maximising the likelihood of a sample. The asymptotic distribution of the likelihood ratio test statistic for the hypothesis that there are no immunes present in the population is shown to be "non-standard"; it is a 50-50 mixture of a chi-squared distribution on 1 degree of freedom and a point mass at 0. Our analysis clearly shows how "negligibility" of individual covariate values and "sufficient followup" conditions are required for the asymptotic properties.