Covariate Measurement Error in Quadratic Regression

We consider quadratic regression models where the explanatory variable is measured with error. The effect of classical measurement error is to flatten the curvature of the estimated function. The effect on the observed turning point depends on the location of the true turning point relative to the population mean of the true predictor. Two methods for adjusting parameter estimates for the measurement error are compared. First, two versions of regression calibration estimation are considered. This approximates the model between the observed variables using the moments of the true explanatory variable given its surrogate measurement. For certain models an expanded regression calibration approximation is exact. The second approach uses moment-based methods which require no assumptions about the distribution of the covariates measured with error. The estimates are compared in a simulation study, and used to examine the sensitivity to measurement error in models relating income inequality to the level of economic development. The simulations indicate that the expanded regression calibration estimator dominates the other estimators when its distributional assumptions are satisfied. When they fail, a small-sample modification of the method-of-moments estimator performs best. Both estimators are sensitive to misspecification of the measurement error model.     

The presence of an error term does not preclude causal inference in regression: a comment on Krause (2012)

A recent article by Krause (Qual Quant, doi:10.1007/s11135-012-9712-5, Krause (2012)) maintains that: (1) it is untenable to characterize the error term in multiple regression as simply an extraneous random influence on the outcome variable, because any amount of error implies the possibility of one or more omitted, relevant explanatory variables; and (2) the only way to guarantee the prevention of omitted variable bias and thereby justify causal interpretations of estimated coefficients is to construct fully specified models that completely eliminate the error term. The present commentary argues that such an extreme position is impractical and unnecessary, given the availability of specialized techniques for dealing with the primary statistical consequence of omitted variables, namely endogeneity, or the existence of correlations between included explanatory variables and the error term. In particular, the current article discusses the method of instrumental variable estimation, which can resolve the endogeneity problem in causal models where one or more relevant explanatory variables are excluded, thus allowing for accurate estimation of effects. An overview of recent methodological resources and software for conducting instrumental variables estimation is provided, with the aim of helping to place this crucial technique squarely in the statistical toolkit of applied researchers.     

Adjusting for differential misclassification in multilevel models: the relationship between child exposure to smoke and cognitive development

Misclassification is found in many of the variables used in social sciences and, in practice, tends to be ignored in statistical analyses, and this can lead to biased results. This paper shows how to correct for differential misclassification in multilevel models and illustrates the extent to which this changes fixed and random parameter estimates. Reliability studies on self-reported behaviour of pregnant women suggest that there may be differential misclassification related to smoking and, thus, to child exposure to smoke. Models are applied to the Millennium Cohort Study data. The response variable is the child cognitive development assessed by the British Ability Scales at 3 years of age and explanatory variables are child exposure to smoke and family income. The proposed method allows a correction for misclassification when the specificity and sensitivity are known, and the assessment of potential biases occurring in the multilevel model parameter estimates if a validation data sample is not available, which is often the case.     

Consistent estimation of a regression with errors in the variables

Summary As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.     

Health Effects of Air Pollution: A Statistical Review

We critically review and compare epidemiological designs and statistical approaches to estimate associations between air pollution and health. More specifically, we aim to address the following questions:

• 1 Which epidemiological designs and statistical methods are available to estimate associations between air pollution and health?
• 2 What are the recent methodological advances in the estimation of the health effects of air pollution in time series studies?
• 3 What are the the main methodological challenges and future research opportunities relevant to regulatory policy?

In question 1, we identify strengths and limitations of time series, cohort, case‐crossover and panel sampling designs. In question 2, we focus on time series studies and we review statistical methods for: 1) combining information across multiple locations to estimate overall air pollution effects; 2) estimating the health effects of air pollution taking into account of model uncertainties; 3) investigating the consequences of exposure measurement error in the estimation of the health effects of air pollution; and 4) estimating air pollution‐health exposure‐response curves. Here, we also discuss the extent to which these statistical contributions have addressed key substantive questions. In question 3, within a set of policy‐relevant‐questions, we identify research opportunities and point out current data limitations.     

Measurement error in causal analysis of panel data: Attenuated versus inflated relationships

The presence of random measurement error is commonly thought to cause attenuation of statistical relationships. While this is an unquestionable truth in bivariate analysis, it cannot be generalized to the multivariate case without qualification. This paper shows that measurement error may give rise to overestimates of parameters in causal analysis whenever there is more than one independent variable and the independent variables are correlated. If the independent variables are not measured with the same amount of reliability, there may also be considerable error in estimates of the relative magnitude of their impact. Both problems are particularly serious when the amount of measurement error is large relative to some of the causal effects such as in panel analysis with lagged dependent variables.     

Adjusting for measurement error in the value added model: evidence from Portugal

The multilevel value added approach to measuring school effectiveness is now widely used. We propose a method to adjust for measurement error to investigate the extent to which this changes school effect estimates. It is applied to longitudinal data collected in the region of Cova da Beira (NUT III) for 1st, 3rd, 5th, 7th and 8th grades. Three different variance component models are considered, depending on the predictor variables included. Assuming measurement error occurs in explanatory and/or response variables, corrections are made for different values of the coefficient of reliability. Moreover, models are fitted under the assumption of either independent or correlated measurement errors.     

自相关检验方法的检验功效研究

研究目标:探究DW检验和LM检验的检验功效及其渐近性。研究方法:运用蒙特卡罗模拟实验方法结合相关影响因素对两种检验方法进行分析与比较。影响因素包括样本容量、解释变量的随机性及自相关性、随机误差项的自相关程度以及分布形态。研究发现:DW和LM检验功效与样本容量和随机误差项的自相关程度正相关,与解释变量的自相关程度负相关;解释变量的随机性对DW和LM检验功效无显著影响;误差项的几种常见分布形态的变化对DW和LM检验功效的影响可以忽略;在误差项存在一阶自相关的情况下,DW检验效果优于LM检验效果。研究创新:以DW检验和LM检验的假设条件为出发点,探究比较不同条件下自相关检验方法的检验功效。研究价值:在实证研究背景下为有效地选择自相关检验方法提供借鉴参考。     

An analysis of some short-term health effects of air pollution in the Washington,D.C. metropolitan area

This study assesses some of the short-term health effects of air pollution in Washington, D.C. Specifically, regression models are formulated to explain health-care visits to a group practice medical care plan. Primary interest is focused on the effects of mobile-source air pollutants, particularly photo-chemical oxidants. Meteorological conditions, as well as other variables thought to influence the consumption of medical services, are included in the models as explanatory variables. The study found only a small effect of air pollution levels on the health-care visits to the group practice.     

Stratified partial likelihood estimation

When multiple durations are generated by a single unit, they may be related in a way that is not fully captured by the regressors. The omitted unit-specific variables might vary over the durations. They might also be correlated with the variables in the regression component. The authors propose an estimator that responds to these concerns and develop a specification test for detecting unobserved unit-specific effects. Data from Malaysia reveal that concentration of child mortality in some families is imperfectly explained by observed explanatory variables, and that failure to control for unobserved heterogeneity seriously biases the parameter estimates.     

The occurrence of outliers in the explanatory variable considered in an errors-in-variables framework

Summary The problem of estimating the slope of a linear relationship between two jointly normally distributed random variables is considered when outliers may occur in the explanatory variable. It will be studied as a special case of an errors-in-variables problem where the explanatory variable is measured which a nonnormally distributed error. In this more general model and under certain conditions a consistent estimator can be given with a normal limiting distribution. Applications to cases of outliers in the explanatory variable will be presented.     

MSE-improvement of the least squares estimator by dropping variables

It is well known that dropping variables in regression analysis decreases the variance of the least squares (LS) estimator of the remaining parameters. However, after elimination estimates of these parameters are biased, if the full model is correct. In his recent paper, Boscher (1991) showed that the LS-estimator in the special case of a mean shift model (cf. Cook and Weisberg, 1982) which assumes no “outliers” can be considered in the framework of a linear regression model where some variables are deleted. He derived conditions under which this estimator outperforms the LS-estimator of the full model in terms of the mean squared error (MSE)-matrix criterion. We demonstrate that this approach can be extended to the general set-up of dropping variables. Necessary and sufficient conditions for the MSE-matrix superiority of the LS-estimator in the reduced model over that in the full model are derived. We also provide a uniformly most powerful F-statistic for testing the MSE-improvement.     

Semiparametric deconvolution with unknown error variance

Deconvolution is a useful statistical technique for recovering an unknown density in the presence of measurement error. Typically, the method hinges on stringent assumptions about the nature of the measurement error, more specifically, that the distribution is entirely known. We relax this assumption in the context of a regression error component model and develop an estimator for the unknown density. We show semi-uniform consistency of the estimator and provide an application to the stochastic frontier model.     

Modelling response error in school effectiveness research

Statistical modelling of school effectiveness in educational research is considered. Variance component models are generally accepted for the analysis of such studies. A shortcoming is that outcome variables are still treated as measured without an error. Unreliable variables produce biases in the estimates of the other model parameters. The variability of the relationships across schools and the effects of schools on students' outcomes differ substantially when taking the measurement error in the dependent variables of the variance component models into account. The random effects model can be extended to handle measurement error using a response model, leading to a random effects item response theory model. This extended random effects model is in particular suitable when subjects are measured repeatedly on the same outcome at several points in time.     

A new measurement procedure for attitudinal research. Analysis of its psychometric and informational properties

Since measurement errors have strong effects in all relationships (statistical or otherwise) studied, there is an increasing interest in the data quality, which is the major justification for this research. This paper aims to present a new measurement procedure, the letter scale, which avoids many of the problems connected with the response modalities traditionally used in attitudinal research, especially the ordinal categorical scales. This paper analyzes the error composition of the scores obtained with this new measurement procedure. The validity of the procedure is also analyzed and the observed variance is assessed to determine which part of the observed variance is “valid”, which part is random error (attenuating relationships) and which is correlated error (magnifying relationships). Structural equation models will be used to provide estimates of the measurement quality: (i) Reliability, (ii) Construct validity, method effect and residual variance. In addition, this letter scale is evaluated under another different perspective, Information Theory measures are also used to assess the amount of information transmitted. The relative merits of this new measurement procedure as opposed to other common response modalities will be discussed in both cases.     

A data-driven approach to forecasting ground-level ozone concentration

The ability to forecast the concentration of air pollutants in an urban region is crucial for decision-makers wishing to reduce the impact of pollution on public health through active measures (e.g. temporary traffic closures). In this study, we present a machine learning approach applied to forecasts of the day-ahead maximum value of ozone concentration for several geographical locations in southern Switzerland. Due to the low density of measurement stations and to the complex orography of the use-case terrain, we adopted feature selection methods instead of explicitly restricting relevant features to a neighborhood of the prediction sites, as common in spatio-temporal forecasting methods. We then used Shapley values to assess the explainability of the learned models in terms of feature importance and feature interactions in relation to ozone predictions. Our analysis suggests that the trained models effectively learned explanatory cross-dependencies among atmospheric variables. Finally, we show how weighting observations helps to increase the accuracy of the forecasts for specific ranges of ozone’s daily peak values.     

OR vice: Forecasting and classification techniques for narcotics interdiction

Counternarcotics interdiction efforts have traditionally relied on historically determined sorting criteria or “best guess” to find and classify suspected smuggling traffic. We present a more quantitative approach which incorporates customized database applications, graphics software and statistical modeling techniques to develop forecasting and classification models. Preliminary results show that statistical methodology can improve interdiction rates and reduce forecast error. The idea of predictive modeling is thus gaining support in the counterdrug community. The problem is divided into sea, air and land forecasting, only part of which will be addressed here. The maritime problem is solved using multiple regression in lieu of multivariate time series. This model predicts illegal boat counts by behavior and geographic region. We developed support software to present the forecasts and to automate the process of performing periodic model updates. During the period, the model was in use at. Coast Guard Headquarters. Because of deterrence provided by improved intervention, the vessel seizure rate declined from 1 every 36 hours to 1 every 6 months. Due in part to the success of the sea model, the maritime movement of marijuana has ceased to be a major threat. The air problem is more complex, and required us to locally design data collection and display software. Intelligence analysts are using a customized relational database application with a map overlay to perform visual pattern recognition of smuggling routes. We are solving the modeling portion of the air problem using multiple regression for regional forecasts of traffic density, and discriminant analysis to develop tactical models that classify “good guys” and “bad guys”. The air models are still under development, but we discuss some modeling considerations and preliminary results. The land problem is even more difficult, and data collection is still in progress.     

Estimating panel data models in the presence of endogeneity and selection

We consider estimation of panel data models with sample selection when the equation of interest contains endogenous explanatory variables as well as unobserved heterogeneity. Assuming that appropriate instruments are available, we propose several tests for selection bias and two estimation procedures that correct for selection in the presence of endogenous regressors. The tests are based on the fixed effects two-stage least squares estimator, thereby permitting arbitrary correlation between unobserved heterogeneity and explanatory variables. The first correction procedure is parametric and is valid under the assumption that the errors in the selection equation are normally distributed. The second procedure estimates the model parameters semiparametrically using series estimators. In the proposed testing and correction procedures, the error terms may be heterogeneously distributed and serially dependent in both selection and primary equations. Because these methods allow for a rather flexible structure of the error variance and do not impose any nonstandard assumptions on the conditional distributions of explanatory variables, they provide a useful alternative to the existing approaches presented in the literature.     

Statistical inference on regression with spatial dependence

Central limit theorems are developed for instrumental variables estimates of linear and semiparametric partly linear regression models for spatial data. General forms of spatial dependence and heterogeneity in explanatory variables and unobservable disturbances are permitted. We discuss estimation of the variance matrix, including estimates that are robust to disturbance heteroscedasticity and/or dependence. A Monte Carlo study of finite-sample performance is included. In an empirical example, the estimates and robust and non-robust standard errors are computed from Indian regional data, following tests for spatial correlation in disturbances, and nonparametric regression fitting. Some final comments discuss modifications and extensions.     

Making measurement errors and interpreting path coefficients: a practical perspective

The purpose of this analysis is to provide a practical approach to the assessment of reliability. In particular, we examine the impact of random measurement error upon the magnitude and interpretation of standardized regression coefficients (or path coefficients) and the specification of regression models. With the proper research the relationship between measured and true values can be inferred by using path coefficients. Such inferences allow assessments of the specification of statistical models. Several examples illustrate how researchers can be misled without knowledge of the impact of measurement error.