Correcting data for measurement error in generalized linear models

This paper discusses a general strategy for reducing measurement-error-induced bias in statistical models. It is assumed that the measurement error is unbiased with a known variance although no other distributional assumptions on the measurement-error are employed,

Using a preliminary fit of the model to the observed data, a transformation of the variable measured with error is estimated. The transformation is constructed so that the estimates obtained by refitting the model to the ‘corrected’ data have smaller bias,

Whereas the general strategy can be applied in a number of settings, this paper focuses on the problem of covariate measurement error in generalized linear models, Two estimators are derived and their effectiveness at reducing bias is demonstrated in a Monte Carlo study.     

Nonlinear models of disability and age applied to census data

It is usually considered that the proportion of handicapped people grows with age. Namely, the older the man/woman, the more the level of disability he/she suffers. However, empirical evidence shows that this assessment is not always true, or at least, it is not true in the Spanish population. The study tries to assess the impact of age on disability in Spain. Each gender has been treated separately because it can be shown that men and women have their own pattern of behaviour. Three different methods of estimation have been used to check the link between those variables. The results seem to support the idea that the relationship among age and the intensity of disability is not always direct. One of the concluding remarks in this analysis is that the method of estimation has a great incidence in the final results, especially in central ages between 20 and 80 years old.     

Some observations on semi-markov models for partially censored data

Cause-specific hazard functions are employed to analyze a semi-Markov model which could be used to describe data arising from clinical trials or certain types of observational studies. The use of these hazard functions to fit a set of data arising from N possibly incomplete case histories is shown to have several notable advantages over the approach adopted by Lagakos, Sommer, and Zelen (1978).     

Some recent advances in measurement error models and methods

Summary A measurement error model is a regression model with (substantial) measurement errors in the variables. Disregarding these measurement errors in estimating the regression parameters results in asymptotically biased estimators. Several methods have been proposed to eliminate, or at least to reduce, this bias, and the relative efficiency and robustness of these methods have been compared. The paper gives an account of these endeavors. In another context, when data are of a categorical nature, classification errors play a similar role as measurement errors in continuous data. The paper also reviews some recent advances in this field. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the frame of the Sonderforschungsbereich SFB 386. We thank two anonymous referees for their helpful comments.     

Analysis of inaccurate data with mixture measurement error models

Measurement error, the difference between a measured (observed) value of quantity and its true value, is perceived as a possible source of estimation bias in many surveys. To correct for such bias, a validation sample can be used in addition to the original sample for adjustment of measurement error. Depending on the type of validation sample, we can either use the internal calibration approach or the external calibration approach. Motivated by Korean Longitudinal Study of Aging (KLoSA), we propose a novel application of fractional imputation to correct for measurement error in the analysis of survey data. The proposed method is to create imputed values of the unobserved true variables, which are mis-measured in the main study, by using validation subsample. Furthermore, the proposed method can be directly applicable when the measurement error model is a mixture distribution. Variance estimation using Taylor linearization is developed. Results from a limited simulation study are also presented.     

Estimating partially linear panel data models with one-way error components

We consider the problem of estimating a partially linear panel data model whenthe error follows an one-way error components structure. We propose a feasiblesemiparametric generalized least squares (GLS) type estimator for estimating the coefficient of the linear component and show that it is asymptotically more efficient than a semiparametric ordinary least squares (OLS) type estimator. We also discussed the case when the regressor of the parametric component is correlated with the error, and propose an instrumental variable GLS-type semiparametric estimator.     

Some asymptotic results for semiparametric nonlinear mixed-effects models with incomplete data

In modeling complex longitudinal data, semiparametric nonlinear mixed-effects (SNLME) models are very flexible and useful. Covariates are often introduced in the models to partially explain the inter-individual variations. In practice, data are often incomplete in the sense that there are often measurement errors and missing data in longitudinal studies. The likelihood method is a standard approach for inference for these models but it can be computationally very challenging, so computationally efficient approximate methods are quite valuable. However, the performance of these approximate methods is often based on limited simulation studies, and theoretical results are unavailable for many approximate methods. In this article, we consider a computationally efficient approximate method for a class of SNLME models with incomplete data and investigate its theoretical properties. We show that the estimates based on the approximate method are consistent and asymptotically normally distributed.     

Some simple log-linear models for the analysis of ordinal response data

A family of log-linear models are proposed to describe contingency tables in which one variable can be considered as the response to the remaining. The proposed models take into account the ordering nature of the response categories and have structure similar to that employed in polynomial regression. Stochastic ordering of the response distributions under the proposed models is discussed and the model-reduction techniques are developed. The proposed models are applied to two data sets previously analysed in the literature.     

Semiparametric methods for survival data with measurement error under additive hazards cure rate models

Lifetime Data Analysis - It is well established that measurement error has drastically negative impact on data analysis. It can not only bias parameter estimates but may also cause loss of power...     

Estimation of regression parameters in generalized linear models for cluster correlated data with measurement error

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.     

Skew-probit measurement error models

In this paper we extend the structural probit measurement error model by considering the unobserved covariate to follow a skew-normal distribution. The new model is termed the structural skew-normal probit model. As in the normal case, the likelihood function is obtained analytically, and can be maximized by using existing statistical software. A Bayesian approach using Markov chain Monte Carlo techniques for generating from the posterior distributions is also developed. A simulation study demonstrates the usefulness of the approach in avoiding attenuation which arises with the naive procedure. Moreover, a comparison of predicted and true success probabilities indicates that it seems to be more efficient to use the skew probit model when the distribution of the covariate (predictor) is skew. An application to a real data set is also provided.     

Doubly robust estimation of partially linear models for longitudinal data with dropouts and measurement error in covariates

In longitudinal studies, missing responses and mismeasured covariates are commonly seen due to the data collection process. Without cautiousness in data analysis, inferences from the standard statistical approaches may lead to wrong conclusions. In order to improve the estimation for longitudinal data analysis, a doubly robust estimation method for partially linear models, which can simultaneously account for the missing responses and mismeasured covariates, is proposed. Imprecisions of covariates are corrected by taking advantage of the independence between replicate measurement errors, and missing responses are handled by the doubly robust estimation under the mechanism of missing at random. The asymptotic properties of the proposed estimators are established under regularity conditions, and simulation studies demonstrate desired properties. Finally, the proposed method is applied to data from the Lifestyle Education for Activity and Nutrition study.     

Extended complex error correction models for seasonal cointegration

In this paper, we extend the complex error correction model (ECM) of [Cubadda, G. (2001). Complex reduced rank models for seasonally cointegrated time series. Oxford Bulletin of Economics and Statistics, 63, 497–511] to models with two types of deterministic terms: (i) restricted seasonal dummies and constant; (ii) restricted seasonal dummies and unrestricted constant. These types of deterministic terms are most frequently adopted in the analysis of seasonal cointegration by many practitioners and researchers, because the other type–where all seasonal dummies and constant terms are unrestricted–may yield oscillating trends. We obtain the limiting distribution of the likelihood ratio (LR) test for the seasonal cointegrating (CI) rank in the extended models. We also provide asymptotic and finite critical values for the test.     

Model bias effects in small area coverage error estimation

For estimation of population totals, dual system estimation (d.s.e.) is often used. Such a procedure is known to suffer from bias under certain conditions. In the following, a simple model is proposed that combines three conditions under which bias of the DSE can result. The conditions relate to response correlation, classification and matching error. The resulting bias is termed model bias. The effects of model bias and synthetic bias in a small area estimation application are illustrated. The illustration uses simulated population data     

Dynamic models for spatiotemporal data

We propose a model for non-stationary spatiotemporal data. To account for spatial variability, we model the mean function at each time period as a locally weighted mixture of linear regressions. To incorporate temporal variation, we allow the regression coefficients to change through time. The model is cast in a Gaussian state space framework, which allows us to include temporal components such as trends, seasonal effects and autoregressions, and permits a fast implementation and full probabilistic inference for the parameters, interpolations and forecasts. To illustrate the model, we apply it to two large environmental data sets: tropical rainfall levels and Atlantic Ocean temperatures.     

Statistical models for e-learning data

In this paper we analyse a real e-learning dataset derived from the e-learning platform of the University of Pavia. The dataset concerns an online learning environment with in-depth teaching materials. The main focus of this paper is to supply a measure of the relative importance of the exercises (test) at the end of each training unit; to build predictive models of student’s performance and finally to personalize the e-learning platform. The methodology employed is based on nonparametric statistical methods for kernel density estimation and generalized linear models and generalized additive models for predictive purposes.     

Regression models for cyclic data

Summary. Geologists take slices through rock samples at a series of different orientations. Each slice is examined for a phenomenon which may occur in one of two states labelled clockwise and anticlockwise. The probability of an occurrence being in the clockwise state is dependent on the orientation. Motivated by these data, two models are presented that relate the probability of an event to orientation. Each model has two parameters, one identifying the orientation corresponding to the peak probability, and the other controlling the rate of change of probability with orientation. One model is a logistic model, whereas the other involves a power of the sine function. For the given data neither model consistently outperforms the other.     

Multilevel models for longitudinal data

Summary.  Repeated measures and repeated events data have a hierarchical structure which can be analysed by using multilevel models. A growth curve model is an example of a multilevel random-coefficients model, whereas a discrete time event history model for recurrent events can be fitted as a multilevel logistic regression model. The paper describes extensions to the basic growth curve model to handle auto-correlated residuals, multiple-indicator latent variables and correlated growth processes, and event history models for correlated event processes. The multilevel approach to the analysis of repeated measures data is contrasted with structural equation modelling. The methods are illustrated in analyses of children's growth, changes in social and political attitudes, and the interrelationship between partnership transitions and childbearing.