Nonparametric Prediction in Measurement Error Models

Conformal normal curvature and detection of masked observations in multivariate null intercept measurement error models

Measurement errors occur very commonly in practice. After fitting the model, influence diagnostics is an important step in statistical data analysis. The most frequently used diagnostic method for measurement error models is the local influence. However, this methodology may fail to detect masked influential observations. To overcome this limitation, we propose the use of the conformal normal curvature with the forward search algorithm. The results are presented through easy to interpret plots considering different perturbation schemes. The proposed methodology is illustrated with three real data sets and one simulated data set, two of which have been previously analyzed in the literature. The third data set deals with the stability of the hygroscopic solid dosage in pharmaceutical processes to ensure the maintenance of product safety quality. In this application, the analytical mass balance is subject to measurement errors, which require attention in the modeling process and diagnostic analysis.

MALMEM: model averaging in linear measurement error models

We develop model averaging estimation in the linear regression model where some covariates are subject to measurement error. The absence of the true covariates in this framework makes the calculation of the standard residual-based loss function impossible. We take advantage of the explicit form of the parameter estimators and construct a weight choice criterion. It is asymptotically equivalent to the unknown model average estimator minimizing the loss function. When the true model is not included in the set of candidate models, the method achieves optimality in terms of minimizing the relative loss, whereas, when the true model is included, the method estimates the model parameter with root n rate. Simulation results in comparison with existing Bayesian information criterion and Akaike information criterion model selection and model averaging methods strongly favour our model averaging method. The method is applied to a study on health.

Calibrate Variations in Biomarker Measures for Improving Prediction with Time-to-event Outcomes

Novel biologic markers have been used to predict clinical outcomes of many diseases. One specific feature of biomarkers is that they often are measured with variations due to factors such as sample preparation and specific laboratory process. Statistical methods have been proposed to characterize the effects of underlying error-free quantity in association with an outcome, yet the impact of measurement errors in terms of prediction has not been well studied. We focus in this manuscript on using biomarkers for predicting an individual's future risk for survival outcome. In the setting where replicates of error-prone biomarkers are available in a 'training' population and risk projection is applied to individuals in a 'prediction' population, we propose two-step measurement-error-corrected estimators of absolute risks. We conducted numerical studies to evaluate the predictive performance of the proposed and routine approaches under various assumptions about the measurement error distributions to pinpoint situations when correction of measurement errors might be necessary. We studied the asymptotic properties of the proposed estimators. We applied the estimators to a liver cancer biomarker study to predict risk of liver cancer incidence using age and a novel biomarker, α-Fetoprotein.

Linear Model Selection when Covariates Contain Errors

Prediction precision is arguably the most relevant criterion of a model in practice and is often a sought after property. A common difficulty with covariates measured with errors is the impossibility of performing prediction evaluation on the data even if a model is completely given without any unknown parameters. We bypass this inherent difficulty by using special properties on moment relations in linear regression models with measurement errors. The end product is a model selection procedure that achieves the same optimality properties that are achieved in classical linear regression models without covariate measurement error. Asymptotically, the procedure selects the model with the minimum prediction error in general, and selects the smallest correct model if the regression relation is indeed linear. Our model selection procedure is useful in prediction when future covariates without measurement error become available, e.g., due to improved technology or better management and design of data collection procedures.

Strong Relationships in Acid-Base Chemistry - Modeling Protons Based on Predictable Concentrations of Strong Ions, Total Weak Acid Concentrations, and pCO2

Understanding acid-base regulation is often reduced to pigeonholing clinical states into categories of disorders based on arterial blood sampling. An earlier ambition to quantitatively explain disorders by measuring production and elimination of acid has not become standard clinical practice. Seeking back to classical physical chemistry we propose that in any compartment, the requirement of electroneutrality leads to a strong relationship between charged moieties. This relationship is derived in the form of a general equation stating charge balance, making it possible to calculate [H+] and pH based on all other charged moieties. Therefore, to validate this construct we investigated a large number of blood samples from intensive care patients, where both data and pathology is plentiful, by comparing the measured pH to the modeled pH. We were able to predict both the mean pattern and the individual fluctuation in pH based on all other measured charges with a correlation of approximately 90% in individual patient series. However, there was a shift in pH so that fitted pH in general is overestimated (95% confidence interval -0.072-0.210) and we examine some explanations for this shift. Having confirmed the relationship between charged species we then examine some of the classical and recent literature concerning the importance of charge balance. We conclude that focusing on the charges which are predictable such as strong ions and total concentrations of weak acids leads to new insights with important implications for medicine and physiology. Importantly this construct should pave the way for quantitative acid-base models looking into the underlying mechanisms of disorders rather than just classifying them.

A calibration hierarchy for risk models was defined: from utopia to empirical data

OBJECTIVE

Calibrated risk models are vital for valid decision support. We define four levels of calibration and describe implications for model development and external validation of predictions.

STUDY DESIGN AND SETTING

We present results based on simulated data sets.

RESULTS

A common definition of calibration is "having an event rate of R% among patients with a predicted risk of R%," which we refer to as "moderate calibration." Weaker forms of calibration only require the average predicted risk (mean calibration) or the average prediction effects (weak calibration) to be correct. "Strong calibration" requires that the event rate equals the predicted risk for every covariate pattern. This implies that the model is fully correct for the validation setting. We argue that this is unrealistic: the model type may be incorrect, the linear predictor is only asymptotically unbiased, and all nonlinear and interaction effects should be correctly modeled. In addition, we prove that moderate calibration guarantees nonharmful decision making. Finally, results indicate that a flexible assessment of calibration in small validation data sets is problematic.

CONCLUSION

Strong calibration is desirable for individualized decision support but unrealistic and counter productive by stimulating the development of overly complex models. Model development and external validation should focus on moderate calibration.

The impact of covariate measurement error on risk prediction

In the development of risk prediction models, predictors are often measured with error. In this paper, we investigate the impact of covariate measurement error on risk prediction. We compare the prediction performance using a costly variable measured without error, along with error-free covariates, to that of a model based on an inexpensive surrogate along with the error-free covariates. We consider continuous error-prone covariates with homoscedastic and heteroscedastic errors, and also a discrete misclassified covariate. Prediction performance is evaluated by the area under the receiver operating characteristic curve (AUC), the Brier score (BS), and the ratio of the observed to the expected number of events (calibration). In an extensive numerical study, we show that (i) the prediction model with the error-prone covariate is very well calibrated, even when it is mis-specified; (ii) using the error-prone covariate instead of the true covariate can reduce the AUC and increase the BS dramatically; (iii) adding an auxiliary variable, which is correlated with the error-prone covariate but conditionally independent of the outcome given all covariates in the true model, can improve the AUC and BS substantially. We conclude that reducing measurement error in covariates will improve the ensuing risk prediction, unless the association between the error-free and error-prone covariates is very high. Finally, we demonstrate how a validation study can be used to assess the effect of mismeasured covariates on risk prediction. These concepts are illustrated in a breast cancer risk prediction model developed in the Nurses' Health Study.

Varying Coefficient Models for Sparse Noise-contaminated Longitudinal Data

In this paper we propose a varying coefficient model for highly sparse longitudinal data that allows for error-prone time-dependent variables and time-invariant covariates. We develop a new estimation procedure, based on covariance representation techniques, that enables effective borrowing of information across all subjects in sparse and irregular longitudinal data observed with measurement error, a challenge in which there is no adequate solution currently. More specifically, sparsity is addressed via a functional analysis approach that considers the observed longitudinal data as noise contaminated realizations of a random process that produces smooth trajectories. This approach allows for estimation based on pooled data, borrowing strength from all subjects, in targeting the mean functions and auto- and cross-covariances to overcome sparse noisy designs. The resulting estimators are shown to be uniformly consistent. Consistent prediction for the response trajectories are also obtained via conditional expectation under Gaussian assumptions. Asymptotic distribution of the predicted response trajectories are derived, allowing for construction of asymptotic pointwise confidence bands. Efficacy of the proposed method is investigated in simulation studies and compared to the commonly used local polynomial smoothing method. The proposed method is illustrated with a sparse longitudinal data set, examining the age-varying relationship between calcium absorption and dietary calcium. Prediction of individual calcium absorption curves as a function of age are also examined.

Deconvolution Estimation in Measurement Error Models: The R Package decon

Data from many scientific areas often come with measurement error. Density or distribution function estimation from contaminated data and nonparametric regression with errors-in-variables are two important topics in measurement error models. In this paper, we present a new software package decon for R, which contains a collection of functions that use the deconvolution kernel methods to deal with the measurement error problems. The functions allow the errors to be either homoscedastic or heteroscedastic. To make the deconvolution estimators computationally more efficient in R, we adapt the fast Fourier transform algorithm for density estimation with error-free data to the deconvolution kernel estimation. We discuss the practical selection of the smoothing parameter in deconvolution methods and illustrate the use of the package through both simulated and real examples.

Impact of climate change on ambient ozone level and mortality in southeastern United States

There is a growing interest in quantifying the health impacts of climate change. This paper examines the risks of future ozone levels on non-accidental mortality across 19 urban communities in Southeastern United States. We present a modeling framework that integrates data from climate model outputs, historical meteorology and ozone observations, and a health surveillance database. We first modeled present-day relationships between observed maximum daily 8-hour average ozone concentrations and meteorology measured during the year 2000. Future ozone concentrations for the period 2041 to 2050 were then projected using calibrated climate model output data from the North American Regional Climate Change Assessment Program. Daily community-level mortality counts for the period 1987 to 2000 were obtained from the National Mortality, Morbidity and Air Pollution Study. Controlling for temperature, dew-point temperature, and seasonality, relative risks associated with short-term exposure to ambient ozone during the summer months were estimated using a multi-site time series design. We estimated an increase of 0.43 ppb (95% PI: 0.14-0.75) in average ozone concentration during the 2040's compared to 2000 due to climate change alone. This corresponds to a 0.01% increase in mortality rate and 45.2 (95% PI: 3.26-87.1) premature deaths in the study communities attributable to the increase in future ozone level.