Ignoring Measurement Error Research Articles

How robust are Bayesian posterior inferences based on a Ricker model with regards to measurement errors and prior assumptions about parameters?

We present a Bayesian approach of a Ricker stock-recruitment (S/R) analysis accounting for measurement errors on S/R data. We assess the sensitivity of posterior inferences to (i) the choice of Ricker model parameterizations, with special regards to management-related ones, and (ii) prior parameter distributions. Closed forms for Ricker parameter posterior distributions exist given S/R data known without error. We use this property to develop a procedure based on the Rao–Blackwell formula. This procedure achieves integration of measurement errors by averaging these closed forms over possible S/R data sets sampled from distributions derived from a stochastic model relating field data to the S and R variables. High-quality Bayesian estimates are obtained. The analysis of the influence of different parameterizations and of the priors is made easier. We illustrate our methodological approach by a case study of Atlantic salmon (Salmo salar). Posterior distributions for S and R are computed from a mark–recapture stochastic model. Ignoring measurement errors underestimates parameter uncertainty and overestimates both stock productivity and density dependence. We warn against using management-related parameterizations because it makes the strong prior assumption of long-term sustainability of stocks. Posterior inferences are sensitive to the choice of prior. The use of informative priors as a remedy is discussed.

TheM•‐σ Relation for Supermassive Black Holes

We investigate the differences in the M•-σ relation derived recently by Ferrarese & Merritt and by Gebhardt and collaborators. The shallower slope found by the latter authors (3.75 vs. 4.8) is due partly to the use of a regression algorithm that ignores measurement errors and partly to the value of the velocity dispersion adopted for a single galaxy, the Milky Way. A steeper relation is shown to provide a better fit to black hole masses derived from reverberation mapping studies. Combining the stellar dynamical, gasdynamical, and reverberation mapping mass estimates, we derive a best-fit relation M• = 1.30(±0.36) × 108 M☉(σc/200 km s-1)4.72(±0.36).

ANOCOVA models with measurement errors

This paper deals with analysis of covariance (ANOCOVA) models in a completely randomized design where the explanatory variables are measured with error. The structural models considered encompasses both multiplicative and additive errors in the explanatory variables. Two consistent estimators of the treatment effects are compared in terms of asymptotic relative efficiency. It is shown that the naive estimator which ignores measurement errors is the one that presents the best behavior. A simulation study compares the relative merits of the estimators in finite samples.

Simex approaches to measurement error in roc studies

This paper explores the estimation of the area under the ROC curve when test scores are subject to errors. The naive approach that ignores measurement errors generally yields inconsistent estimates. Finding the asymptotic bias of the naive estimator, Coffin and Sukhatme (1995, 1997) proposed bias-corrected estimators for parametric and nonparametric cases. However, the asymptotic distributions of these estimators have not been developed because of their complexity. We propose several alternative approaches, including the SIMEX procedure of Cook and Stefanski (1994). We also provide the asymptotic distributions of the SIMEX estimators for use in statistical inference. Small simulation studies illustrate that the SIMEX estimators perform reasonably well when compared to the bias-corrected estimators.

The problem of measurement error in modelling the effect of compliance in a randomized trial.

This paper explores the implications of measurement error in the analysis of compliance-response relationships in data from randomized trials. Given that compliance measures are rarely, if ever, error-free indicators of exposure it is argued that both the designs for the collection of compliance data and the statistical models for their resulting analysis should be changed to take the possibility of measurement error into account. An analysis which ignores measurement error in the compliance measurements will provide biased estimates of compliance-response relationships. Provided that one has two or more indicators of compliance for each subject, more appropriate models can be fitted using covariance structure modelling software. If one wishes to explore interactions from repeated measures data on both compliance and response then it is also important that one recognizes that the response measures are also error-prone and that they too are dealt with appropriately.

Estimates of the return to schooling in Sweden from a large sample of twins

A large sample of twins was used to examine whether conventional estimates of the return to schooling in Sweden are biased because ability is omitted from the earnings–schooling relationship. Ignoring measurement error, the results indicate that omitting ability from the earnings–schooling relationship leads to estimates that are positively biased. However, reasonable estimates of the measurement-error-adjusted returns are both above and below the unadjusted estimates, showing that the results depend crucially on a parameter not known at this time. However, an estimate of the reliability ratio was obtained using two measures on educational attainment. With this estimate of the reliability ratio, the measurement-error-adjusted estimate of the return to schooling in the sample of identical twins indicates that there is at most a slight ability bias in the conventional estimates of the return to schooling. The fundamental assumption of this kind of study is that within-pair differences in educational attainment are randomly determined. This assumption was also tested, but no strong evidence to reject it was found.

Estimation in weibull regression model with measurement error

A new estimator is proposed for the Weibull regression model with mea¬surement error. The naive estimator obtained by ignoring measurement error is asymptotically biased. This new estimator is established based on the fact that asymptotically the naive estimator converges to the true parameter times an attenuation factor. A brief review of the bias-corrected estimators literature is presented as well as a considerable simulation comparing the estimators properties.

Correcting surveys for non-response and measurement error using auxiliary information

Surveys often yield estimates that are suspected to be biased, because of either non-response bias and/or measurement error. Numerous methods exist for eliminating non-response bias (e.g., re-weighting the data, individual-level models of selection, or deriving bounds on the true population parameter). But in many settings these methods may not be tractable or helpful, and usually ignore measurement error. I propose a method for dealing with both non-response bias and measurement error simultaneously, when using surveys to estimate rates or proportions. The method exploits `auxiliary information' from analogous surveys, where it is possible to derive the relative amount of non-response bias and measurement error. I demonstrate the method in the case of voter turnout, where a reasonably large body of vote validation studies supply auxiliary information, allowing the components of bias in survey estimates of turnout rates to be isolated. Averaging over the auxiliary information provides bounds on the quantity of interest, yielding an estimate corrected for both non-response bias and measurement error.

Analyzing Real Estate Data Problems Using the Gibbs Sampler

Real estate data are often characterized by data irregularities: missing data, censoring or truncation, measurement error, etc. Practitioners often discard missing‐ or censored‐data cases and ignore measurement error. We argue here that an attractive remedy for these irregularity problems is simulation‐based model fitting using the Gibbs sampler. The style of the paper is primarily pedagogic, employing a simple illustration to convey the essential ideas, unobscured by implementation complications. Focusing on the missing‐data problem, we show dramatic improvement in inference by retaining rather than deleting cases of partially observed data. We also detail Gibbs‐sampler usage for other data problems.

Reconciliation of mineral processing data containing correlated measurement errors

Steady-state values of stream properties (species flowrates) of a mineral processing circuit are usually estimated by averaging values sampled during a given time interval. A major source of data inaccuracy is the integration error, i.e. the error generated by the dynamic variations of the stream properties around their steady-state values. The paper shows that these sampling errors are significantly reduced by appropriate reconciliation methods based on material conservation equations. The optimal error compensation is obtained by minimizing a quadratic function where the weighting factors take into account the integration error covariance from stream to stream and species to species. The method is illustrated for a flotation circuit modelled by a state-space representation. The data variance reduction is compared for different weighting strategies of a quadratic reconciliation criterion. For the studied system, it seems that ignoring measurement error correlation, using a standard weighted least-squares criterion instead of the optimal quadratic function, does not degrade too much the ability of the reconciliation procedure to reduce the data variance.

Bias Analysis and SIMEX Approach in Generalized Linear Mixed Measurement Error Models

We consider generalized linear mixed models (GLMMs) for clustered data when one of the predictors is measured with error. When the measurement error is additive and normally distributed and the error-prone predictor is itself normally distributed, we show that the observed data also follow a GLMM but with a different fixed effects structure from the original model, a different and more complex random effects structure, and restrictions on the parameters. This characterization enables us to compute the biases that result in common GLMMs when one ignores measurement error. For instance, in one common situation the biases in parameter estimates become larger as the number of observations within a cluster increases, both for regression coefficients and for variance components. Parameter estimation is described using the SIMEX method, a relatively new functional method that makes no assumptions about the structure of the unobservable predictors. Simulations and an example illustrate the results.

A Two-Step Approach to Measurement Error in Time-Dependent Covariates in Nonlinear Mixed-Effects Models, With Application to IGF-I Pharmacokinetics

Abstract The usual approach to the analysis of population pharmacokinetic studies is to represent the concentration-time data by a nonlinear mixed-effects model. Primary objectives are to characterize the pattern of drug disposition in the population and to identify individual-specific covariates associated with pharmacokinetic behavior. We consider data from a study of insulin-like growth factor I (IGF-I) administered by intravenous infusion to patients with severe head trauma. Failure to maintain steady-state levels of IGF-I was thought to be related to the temporal pattern of several covariates measured in the study, and an analysis investigating this issue was of interest. Observations on these potentially relevant covariates for each subject were made at time points different from those at which IGF-I concentrations were determined; moreover, the covariates themselves were likely subject to measurement error. The usual approach to time-dependent covariates in population analysis is to invoke a simple interpolation scheme, such as carrying forward the most recent covariate value, ignoring measurement error; however, for these data, the complicated observed covariate pattern makes this approach suspect. A nonlinear mixed-effects model incorporating a model for time-dependent covariates measured with error is used to describe the IGF-I data, and fitting is accomplished by a two-step strategy implemented using standard software. The performance of the method is evaluated via simulation.

A Two-Step Approach to Measurement Error in Time-Dependent Covariates in Nonlinear Mixed-Effects Models, with Application to IGF-I Pharmacokinetics

The usual approach to the analysis of population pharmacokinetic studies is to represent the concentration-time data by a nonlinear mixed-effects model. Primary objectives are to characterize the pattern of drug disposition in the population and to identify individual-specific covariates associated with pharmacokinetic behavior. We consider data from a study of insulin-like growth factor I (IGF-I) administered by intravenous infusion to patients with severe head trauma. Failure to maintain steady-state levels of IGF-I was thought to be related to the temporal pattern of several covariates measured in the study, and an analysis investigating this issue was of interest. Observations on these potentially relevant covariates for each subject were made at time points different from those at which IGF-I concentrations were determined; moreover, the covariates themselves were likely subject to measurement error. The usual approach to time-dependent covariates in population analysis is to invoke a simple interpolation scheme, such as carrying forward the most recent covariate value, ignoring measurement error; however, for these data, the complicated observed covariate pattern makes this approach suspect. A nonlinear mixed-effects model incorporating a model for time-dependent covariates measured with error is used to describe the IGF-I data, and fitting is accomplished by a two-step strategy implemented using standard software. The performance of the method is evaluated via simulation.

Errors-in-variables and the Box-Cox transformation

The Box-Cox family of power transformations is widely used as a means for obtaining an approximate Gaussian distribution. In a regression setting, in which Y (or transformed Y) is regressed on a set of independent variables, the independent variables are usually assumed to be measured exactly, i.e., with no measurement error. In this paper we investigate the effect of measurement error in the independent variables (also known as errors-in-variables) on parameter estimation for the Box and Cox family of transformations (J. Roy. Statist. Soc. Ser. B 26, 1964). Ignoring measurement error in the independent variables causes biased parameter estimation in general. This conclusion is supported by analytical arguments and the results of simulation experiments.

Incorporating measurement error in the estimation of autoregressive models for longitudinal data

Abstract Rosner et al. ( Stat. Med. 4 (1985), 457–467) discussed the use of a first-order autoregressive model for longitudinal data with multiple time-dependent and time-independent covariates. The model assumes that both covariate and outcome variables are measured without error, a condition frequently not satisfied in practice. Here we extend their model to allow for measurement error on both covariate and outcome variables. We obtain maximum likelihood estimates of the model parameters given knowledge of the measurement error variance for both outcome and exposure variables. Our method differs from standard linear model measurement error methods which do not apply to autoregressive models. A simulation study comparing the maximum likelihood estimates (MLEs) with the ordinary least square (OLS) estimates that ignore measurement error shows that the OLS estimates are grossly biased in the presence of substantial measurement error, while the MLEs are essentially unbiased. Furthermore, whereas the coverage probability of the confidence intervals based on the OLS estimates tends to be significantly smaller than the nominal level, the MLEs provide adequate coverage. The relative advantage of MLE versus OLS increases as either the amount of measurement error or the number of visits increases. Finally, we demonstrate that the method corrects for a severe underestimate of the effect of airway responsiveness on pulmonary function decline when both are measured with error.

A Bayesian approach to logistic regression models having measurement error following a mixture distribution.

To estimate the parameters in a logistic regression model when the predictors are subject to random or systematic measurement error, we take a Bayesian approach and average the true logistic probability over the conditional posterior distribution of the true value of the predictor given its observed value. We allow this posterior distribution to consist of a mixture when the measurement error distribution changes form with observed exposure. We apply the method to study the risk of alcohol consumption on breast cancer using the Nurses Health Study data. We estimate measurement error from a small subsample where we compare true with reported consumption. Some of the self-reported non-drinkers truly do not drink. The resulting risk estimates differ sharply from those computed by standard logistic regression that ignores measurement error.

Measurement Error Regression with Unknown Link: Dimension Reduction and Data Visualization

Abstract A general nonlinear regression problem is considered with measurement error in the predictors. We assume that the response is related to an unknown linear combination of a multidimensional predictor through an unknown link function. Instead of observing the predictor, we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance. We identify an important transformation of the surrogate variable. Using this transformed variable, we show that if one proceeds with the usual analysis ignoring measurement error, then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter, up to a constant of proportionality. We derive the asymptotic distribution of the estimates. A simulation study is conducted applying sliced inverse regression in this context.

Efficacy of Repeated Measures in Regression Models with Measurement Error

Ignoring measurement error may cause bias in the estimation of regression parameters. When the true covariates are unobservable, multiple imprecise measurements can be used in the analysis to correct for the associated bias. We suggest a simple estimating procedure that gives consistent estimates of regression parameters by using the repeated measurements with error. The relative Pitman efficiency of our estimator based on models with and without measurement error has been found to be a simple function of the number of replicates and the ratio of intra- to inter-variance of the true covariate. The procedure thus provides a guide for deciding the number of repeated measurements in the design stage. An example from a survey study is presented.

Covariance analysis in generalized linear measurement error models

We summarize some of the recent work on the errors-in-variables problem in generalized linear models. The focus is on covariance analysis, and in particular testing for and estimation of treatment effects. There is a considerable difference between the randomized and non-randomized models when testing for an effect. In randomized studies, simple techniques exist for testing for a treatment effect. In some instances, such as linear and multiplicative regression, simple methods exist for estimating the treatment effect. In other examples such as logistic regression, estimating a treatment effect requires careful attention to measurement error. In non-randomized studies, there is no recourse to understanding and modelling measurement error. In particular ignoring measurement error can lead to the wrong conclusions, for example the true but unobserved data may indicate a positive effect for treatment, while the observed data indicate the opposite. Some of the possible methods are outlined and compared.

The reserve-labor hypothesis, short-run pricing theories, and the employment-output relationship

Short-run price, output, and employment adjustments are studied jointly - in a latent-variable framework - to estimate the effects of unobservable demand and cost conditions. Drawing on the firm-specific human capital and implicit contracts literatures, a microeconomic analysis of firms with hoarded labor guides the econometrics and provides a basis for distinguishing between neoclassical and behavioral pricing. Data from U.S. manufacturing during the 1974-75 recession provide substantial indication of labor hoarding and marginalist pricing, while also suggesting that ‘short-run increasing returns to labor’ are a statistical illusion. Analysis of the same data, ignoring measurement error, suggests very different, and very conventional, inferences.