On the simultaneous effects of model misspecification and errors in variables

Model misspecification and noisy covariate measurements are two common sources of inference bias. There is considerable literature on the consequences of each problem in isolation. In this paper, however, the author investigates their combined effects. He shows that in the context of linear models, the large‐sample error in estimating the regression function may be partitioned in two terms quantifying the impact of these sources of bias. This decomposition reveals trade‐offs between the two biases in question in a number of scenarios. After presenting a finite‐sample version of the decomposition, the author studies the relative impacts of model misspecification, covariate imprecision, and sampling variability, with reference to the detectability of the model misspecification via diagnostic plots.     

Effect of Berkson measurement error on parameter estimates in Cox regression models

We study the effect of additive and multiplicative Berkson measurement error in Cox proportional hazard model. By plotting the true and the observed survivor function and the true and the observed hazard function dependent on the exposure one can get ideas about the effect of this type of error on the estimation of the slope parameter corresponding to the variable measured with error. As an example, we analyze the measurement error in the situation of the German Uranium Miners Cohort Study both with graphical methods and with a simulation study. We do not see a substantial bias in the presence of small measurement error and in the rare disease case. Even the effect of a Berkson measurement error with high variance, which is not unrealistic in our example, is a negligible attenuation of the observed effect. However, this effect is more pronounced for multiplicative measurement error.     

Microeconometric models and anonymized micro data

Summary The paper first provides a short review of the most common microeconometric models including logit, probit, discrete choice, duration models, models for count data and Tobit-type models. In the second part we consider the situation that the micro data have undergone some anonymization procedure which has become an important issue since otherwise confidentiality would not be guaranteed. We shortly describe the most important approaches for data protection which also can be seen as creating errors of measurement by purpose. We also consider the possibility of correcting the estimation procedure while taking into account the anonymization procedure. We illustrate this for the case of binary data which are anonymized by ‘post-randomization’ and which are used in a probit model. We show the effect of ‘naive’ estimation, i. e. when disregarding the anonymization procedure. We also show that a ‘corrected’ estimate is available which is satisfactory in statistical terms. This is also true if parameters of the anonymization procedure have to be estimated, too. Research in this paper is related to the project “Faktische Anonymisierung wirtschaftsstatistischer Einzeldaten” financed by German Ministry of Research and Technology.     

Prediction and forecasting in linear models with measurement error

We consider measurement error models within the time series unobserved component framework. A variable of interest is observed with some measurement error and modelled as an unobserved component. The forecast and the prediction of this variable given the observed values is given by the Kalman filter and smoother along with their conditional variances. By expressing the forecasts and predictions as weighted averages of the observed values, we investigate the effect of estimation error in the measurement and observation noise variances. We also develop corrected standard errors for prediction and forecasting accounting for the fact that the measurement and observation error variances are estimated by the same sample that is used for forecasting and prediction purposes. We apply the theory to the Yellowstone grizzly bears and US index of production datasets.     

The effects of model mis-specifications in linear measurement error models

In the literature, there are many results on the consequences of mis-specified models for linear models with error in the response only, see, e.g., Seber(1977). There are also discussions of estimation for the model writh errors both in the response and in the predictor variables (called measurement error models; see, e.g., Fuller(1987)). In this paper, we consider the problem of model mis-specification for measurement error models. Only a few special cases have been tackled in the past (Edland, 1996; Carroll and Ruppert, 1996 and Lakshminarayanan Amp; Gunst, 1984); we deal with the situation here in some generality. Results have been obtained as follows: (a) When a model is under-fitted, the estimate of the variance of the measurement error will be asymptotically biased, as will the regression coefficients, and the asymptotic biases in the estimates of the regression coefficients will always exist for under-fitted models. Even orthogonality of the variables in the model will not make the biases vanish. (b)For over-fitting, the estimates of the variances of measurement errors and of the regression coefficients are asymptotically unbiased. However, the variance of the estimated regression coefficients will increase. Over-fitting will cause larger changes in the variances of the estimated parameters in measurement error models than in no measurement error models.     

Locally efficient semiparametric estimators for a class of Poisson models with measurement error

The presence of measurement error may cause bias in parameter estimation and can lead to incorrect conclusions in data analyses. Despite a large body of literature on general measurement error problems, relatively few works exist to handle Poisson models. In this article we thoroughly study Poisson models with errors in covariates and propose consistent and locally efficient semiparametric estimators. We assess the finite sample performance of the estimators through extensive simulation studies and illustrate the proposed methodologies by analyzing data from the Stroke Recovery in Underserved Populations Study. The Canadian Journal of Statistics 47: 157–181; 2019 © 2019 Statistical Society of Canada     

A Bayesian ordinal logistic regression model to correct for interobserver measurement error in a geographical oral health study

Summary.  We present an approach for correcting for interobserver measurement error in an ordinal logistic regression model taking into account also the variability of the estimated correction terms. The different scoring behaviour of the 16 examiners complicated the identification of a geographical trend in a recent study on caries experience in Flemish children (Belgium) who were 7 years old. Since the measurement error is on the response the factor 'examiner' could be included in the regression model to correct for its confounding effect. However, controlling for examiner largely removed the geographical east–west trend. Instead, we suggest a (Bayesian) ordinal logistic model which corrects for the scoring error (compared with a gold standard) using a calibration data set. The marginal posterior distribution of the regression parameters of interest is obtained by integrating out the correction terms pertaining to the calibration data set. This is done by processing two Markov chains sequentially, whereby one Markov chain samples the correction terms. The sampled correction term is imputed in the Markov chain pertaining to the regression parameters. The model was fitted to the oral health data of the Signal–Tandmobiel^® study. A WinBUGS program was written to perform the analysis.     

Ordered response models

Summary We discuss regression models for ordered responses, such as ratings of bonds, schooling attainment, or measures of subjective well-being. Commonly used models in this context are the ordered logit and ordered probit regression models. They are based on an underlying latent model with single index function and constant thresholds. We argue that these approaches are overly restrictive and preclude a flexible estimation of the effect of regressors on the discrete outcome probabilities. For example, the signs of the marginal probability effects can only change once when moving from the smallest category to the largest one. We then discuss several alternative models that overcome these limitations. An application illustrates the benefit of these alternatives. We are grateful to an anonymous referee for valuable comments.     

Assessing the impact of measurement error in modeling change in the absence of auxiliary data

Measurement error is well known to cause bias in estimated regression coefficients and a loss of power for detecting associations. Methods commonly used to correct for bias often require auxiliary data. We develop a solution for investigating associations between the change in an imprecisely measured outcome and precisely measured predictors, adjusting for the baseline value of the outcome when auxiliary data are not available. We require the specification of ranges for the reliability or the measurement error variance. The solution allows one to investigate the associations for change and to assess the impact of the measurement error.     

Stochastic approximation methods in linear regression models (with consideration of errors in the regressors) 1

Usually, the dependence in stationary processes is described by a set of coefficients. In this paper, a measure of dependence is proposed which can be used instead of the autocorrelation function, and another measure for the dependence between two processes instead of cross-correlation function and coherence coefficients. In the end, an improvement of extrapolation of a process is investigated which is caused by the knowledge of another related process.