We propose a two-step procedure to estimate structural equation models (SEMs). In a first step, the latent variable is replaced by its conditional expectation given the observed data. This conditional expectation is estimated using a James–Stein type shrinkage estimator. The second step consists of regressing the dependent variables on this shrinkage estimator. In addition to linear SEMs, we also derive shrinkage estimators to estimate polynomials. We empirically demonstrate the feasibility of the proposed method via simulation and contrast the proposed estimator with ML and MIIV estimators under a limited number of simulation scenarios. We illustrate the method on a case study.

When latent variables are used as outcomes in regression analysis, a common approach that is used to solve the ignored measurement error issue is to take a multilevel perspective on item response modeling (IRT). Although recent computational advancement allows efficient and accurate estimation of multilevel IRT models, we argue that a two-stage divide-and-conquer strategy still has its unique advantages. Within the two-stage framework, three methods that take into account heteroscedastic measurement errors of the dependent variable in stage II analysis are introduced; they are the closed-form marginal MLE, the expectation maximization algorithm, and the moment estimation method. They are compared to the naïve two-stage estimation and the one-stage MCMC estimation. A simulation study is conducted to compare the five methods in terms of model parameter recovery and their standard error estimation. The pros and cons of each method are also discussed to provide guidelines for practitioners. Finally, a real data example is given to illustrate the applications of various methods using the National Educational Longitudinal Survey data (NELS 88).

An asymptotic expression for the reliability of a linearly equated test is developed using normal theory. The reliability is expressed as the product of two terms, the reliability of the test before equating, and an adjustment term. This adjustment term is a function of the sample sizes used to estimate the linear equating transformation. The results of a simulation study indicate close agreement between the theoretical and simulated reliability values for samples greater than 200. Findings demonstrate that samples as small as 300 can be used in linear equating without an appreciable decrease in reliability.

This article considers the application of the simulation-extrapolation (SIMEX) method for measurement error correction when the error variance is a function of the latent variable being measured. Heteroskedasticity of this form arises in educational and psychological applications with ability estimates from item response theory models. We conclude that there is no simple solution for applying SIMEX that generally will yield consistent estimators in this setting. However, we demonstrate that several approximate SIMEX methods can provide useful estimators, leading to recommendations for analysts dealing with this form of error in settings where SIMEX may be the most practical option.

The achievement level is a variable measured with error, that can be estimated by means of the Rasch model. Teacher grades also measure the achievement level but they are expressed on a different scale. This paper proposes a method for combining these two scores to obtain a synthetic measure of the achievement level based on the theory developed for regression with covariate measurement error. In particular, the focus is on ordinal scaled grades, using the SIMEX method for measurement error correction. The result is a measure comparable across subjects with smaller measurement error variance. An empirical application illustrates the method.

This article discusses alternative procedures to the standard F-test for ANCOVA in case the covariate is measured with error. Both a functional and a structural relationship approach are described. Examples of both types of analysis are given for the simple two-group design. Several cases are discussed and special attention is given to issues of model identifiability. An approximate statistical test based on the functional relationship approach is described. On the basis of Monte Carlo simulation results it is concluded that this testing procedure is to be preferred to the conventional F-test of the ANCOVA null hypothesis. It is shown how the standard null hypothesis may be tested in a structural relationship approach. It is concluded that some knowledge of the reliability of the covariate is necessary in order to obtain meaningful results.

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.

Covariate-adjusted treatment effects are commonly estimated in non-randomized studies. It has been shown that measurement error in covariates can bias treatment effect estimates when not appropriately accounted for. So far, these delineations primarily assumed a true data generating model that included just one single covariate. It is, however, more plausible that the true model consists of more than one covariate. We evaluate when a further covariate may reduce bias due to measurement error in another covariate and in which cases it is not recommended to include a further covariate. We analytically derive the amount of bias related to the fallible covariate’s reliability and systematically disentangle bias compensation and amplification due to an additional covariate. With a fallible covariate, it is not always beneficial to include an additional covariate for adjustment, as the additional covariate can extensively increase the bias. The mechanisms for an increased bias due to an additional covariate can be complex, even in a simple setting of just two covariates. A high reliability of the fallible covariate or a high correlation between the covariates cannot in general prevent from substantial bias. We show distorting effects of a fallible covariate in an empirical example and discuss adjustment for latent covariates as a possible solution.

Item calibration is an essential issue in modern item response theory based psychological or educational testing. Due to the popularity of computerized adaptive testing, methods to efficiently calibrate new items have become more important than that in the time when paper and pencil test administration is the norm. There are many calibration processes being proposed and discussed from both theoretical and practical perspectives. Among them, the online calibration may be one of the most cost effective processes. In this paper, under a variable length computerized adaptive testing scenario, we integrate the methods of adaptive design, sequential estimation, and measurement error models to solve online item calibration problems. The proposed sequential estimate of item parameters is shown to be strongly consistent and asymptotically normally distributed with a prechosen accuracy. Numerical results show that the proposed method is very promising in terms of both estimation accuracy and efficiency. The results of using calibrated items to estimate the latent trait levels are also reported.

This paper proposes a structural analysis for generalized linear models when some explanatory variables are measured with error and the measurement error variance is a function of the true variables. The focus is on latent variables investigated on the basis of questionnaires and estimated using item response theory models. Latent variable estimates are then treated as observed measures of the true variables. This leads to a two-stage estimation procedure which constitutes an alternative to a joint model for the outcome variable and the responses given to the questionnaire. Simulation studies explore the effect of ignoring the true error structure and the performance of the proposed method. Two illustrative examples concern achievement data of university students. Particular attention is given to the Rasch model.

The ability of bisection procedures to specify the form of the psychophysical scale depends upon the precision of the technique. It is demonstrated that the precision of bisection techniques is a function of the stimulus interval bisected. Consequently, the choice of stimuli in a bisection experiment may predispose the ability of the experiment to distinguish between alternative psychophysical scales. The testing of interval scale properties of derived scales and the assessment of context effects in bisection experiments was also discussed.

Asymptotic expansions of the maximum likelihood estimator (MLE) and weighted likelihood estimator (WLE) of an examinee’s ability are derived while item parameter estimators are treated as covariates measured with error. The asymptotic formulae present the amount of bias of the ability estimators due to the uncertainty of item parameter estimators. A numerical example is presented to illustrate how to apply the formulae to evaluate the impact of uncertainty about item parameters on ability estimation and the appropriateness of estimating ability using the regular MLE or WLE method.

This paper is concerned with combining observed scores from sections of tests. It is demonstrated that in the presence of population information a linear combination of true scores can be estimated more efficiently than by the same linear combination of the observed scores. Three criteria for optimality are discussed, but they yield the same solution which can be described and motivated as a multivariate shrinkage estimator.

In survey research it is not uncommon to ask questions of the following type: “How many times did you undertake action a in reference period T of length τ?.” The relationship is established between τ and the correlation of the number of reported actions with some background variable. To this end it is assumed that the process of actions satisfies a renewal model with individual heterogeneity. Also a model has to be formulated for possible recall effects. Applications are given in the field of medical consumption.

A type of data layout that may be considered as an extension of the two-way random effects analysis of variance is characterized and modeled based on group invariance. The data layout seems to be suitable for several scenarios in psychometrics, including the one in which multiple measurements are taken on each of a set of variables, and the measurements can be divided into exchangeable subsets. The algebraic structure of the model is studied, which leads to results that are applicable to such problems as estimating correlation matrix corrected for attenuation and testing symmetry hypotheses.

In psychometrics, the canonical use of conditional likelihoods is for the Rasch model in measurement. Whilst not disputing the utility of conditional likelihoods in measurement, we examine a broader class of problems in psychometrics that can be addressed via conditional likelihoods. Specifically, we consider cluster-level endogeneity where the standard assumption that observed explanatory variables are independent from latent variables is violated. Here, “cluster” refers to the entity characterized by latent variables or random effects, such as individuals in measurement models or schools in multilevel models and “unit” refers to the elementary entity such as an item in measurement. Cluster-level endogeneity problems can arise in a number of settings, including unobserved confounding of causal effects, measurement error, retrospective sampling, informative cluster sizes, missing data, and heteroskedasticity. Severely inconsistent estimation can result if these challenges are ignored.

Household survey estimates of retirement income suffer from substantial underreporting which biases downward measures of elderly financial well-being. Using data from both the 2016 Current Population Survey Annual Social and Economic Supplement (CPS ASEC) and the Health and Retirement Study (HRS), matched with administrative records, we examine to what extent underreporting of retirement income affects key statistics: elderly reliance on social security benefits and poverty. We find that retirement income is underreported in both the CPS ASEC and the HRS. Consequently, the relative importance of social security income remains overstated – 53 percent of elderly beneficiaries in the CPS ASEC and 49 percent in the HRS rely on social security for the majority of their incomes compared to 42 percent in the administrative data. The elderly poverty rate is also overstated – 8.8 percent in the CPS ASEC and 7.4 percent in the HRS compared to 6.4 percent in the administrative data.