The use of predicted values for item parameters in item response theory models: an application in intelligence tests

In testing, item response theory models are widely used in order to estimate item parameters and individual abilities. However, even unidimensional models require a considerable sample size so that all parameters can be estimated precisely. The introduction of empirical prior information about candidates and items might reduce the number of candidates needed for parameter estimation. Using data for IQ measurement, this work shows how empirical information about items can be used effectively for item calibration and in adaptive testing. First, we propose multivariate regression trees to predict the item parameters based on a set of covariates related to the item-solving process. Afterwards, we compare the item parameter estimation when tree-fitted values are included in the estimation or when they are ignored. Model estimation is fully Bayesian, and is conducted via Markov chain Monte Carlo methods. The results are two-fold: (a) in item calibration, it is shown that the introduction of prior information is effective with short test lengths and small sample sizes and (b) in adaptive testing, it is demonstrated that the use of the tree-fitted values instead of the estimated parameters leads to a moderate increase in the test length, but provides a considerable saving of resources.     

A multiple group item response theory model with centered skew-normal latent trait distributions under a Bayesian framework

Very often, in psychometric research, as in educational assessment, it is necessary to analyze item response from clustered respondents. The multiple group item response theory (IRT) model proposed by Bock and Zimowski [12] provides a useful framework for analyzing such type of data. In this model, the selected groups of respondents are of specific interest such that group-specific population distributions need to be defined. The usual assumption for parameter estimation in this model, which is that the latent traits are random variables following different symmetric normal distributions, has been questioned in many works found in the IRT literature. Furthermore, when this assumption does not hold, misleading inference can result. In this paper, we consider that the latent traits for each group follow different skew-normal distributions, under the centered parameterization. We named it skew multiple group IRT model. This modeling extends the works of Azevedo et al. [4], Bazán et al. [11] and Bock and Zimowski [12] (concerning the latent trait distribution). Our approach ensures that the model is identifiable. We propose and compare, concerning convergence issues, two Monte Carlo Markov Chain (MCMC) algorithms for parameter estimation. A simulation study was performed in order to evaluate parameter recovery for the proposed model and the selected algorithm concerning convergence issues. Results reveal that the proposed algorithm recovers properly all model parameters. Furthermore, we analyzed a real data set which presents asymmetry concerning the latent traits distribution. The results obtained by using our approach confirmed the presence of negative asymmetry for some latent trait distributions.     

Probability scoring for spelling correction

This paper describes a new program, CORRECT, which takes words rejected by the Unix^® SPELL program, proposes a list of candidate corrections, and sorts them by probability score. The probability scores are the novel contribution of this work. They are based on a noisy channel model. It is assumed that the typist knows what words he or she wants to type but some noise is added on the way to the keyboard (in the form of typos and spelling errors). Using a classic Bayesian argument of the kind that is popular in recognition applications, especially speech recognition (Jelinek, 1985), one can often recover the intended correction,c, from a typo,t, by finding the correctionc that maximizesPr(c) Pr(t/c). The first factor,Pr(c), is a prior model of word probabilities; the second factor,Pr(t/c), is a model of the noisy channel that accounts for spelling transformations on letter sequences (insertions, deletions, substitutions and reversals). Both sets of probabilities were estimated using data collected from the Associated Press (AP) newswire over 1988 and 1989 as a training set. The AP generates about 1 million words and 500 typos per week.In evaluating the program, we found that human judges were extremely reluctant to cast a vote given only the information available to the program, and that they were much more comfortable when they could see a concordance line or two. The second half of this paper discusses some very simple methods of modeling the context usingn-gram statistics. Althoughn-gram methods are much too simple (compared with much more sophisticated methods used in artificial intelligence and natural language processing), we have found that even these very simple methods illustrate some very interesting estimation problems that will almost certainly come up when we consider more sophisticated models of contexts. The problem is how to estimate the probability of a context that we have not seen. We compare several estimation techniques and find that some are useless. Fortunately, we have found that the Good-Turing method provides an estimate of contextual probabilities that produces a significant improvement in program performance. Context is helpful in this application, but only if it is estimated very carefully.At this point, we have a number of different knowledge sources—the prior, the channel and the context—and there will certainly be more in the future. In general, performance will be improved as more and more knowledge sources are added to the system, as long as each additional knowledge source provides some new (independent) information. As we shall see, it is important to think more carefully about combination rules, especially when there are a large number of different knowledge sources.     

ESTIMATING A RATIO OF MEANS FROM BIVARIATE COUNTS WITH APPLICATIONS IN STEREOLOGY

In survey sampling and in stereology, it is often desirable to estimate the ratio of means θ= E(Y)/E(X) from bivariate count data (X, Y) with unknown joint distribution. We review methods that are available for this problem, with particular reference to stereological applications. We also develop new methods based on explicit statistical models for the data, and associated model diagnostics. The methods are tested on a stereological dataset. For point‐count data, binomial regression and bivariate binomial models are generally adequate. Intercept‐count data are often overdispersed relative to Poisson regression models, but adequately fitted by negative binomial regression.     

An elastic net penalized small area model combining unit- and area-level data for regional hypertension prevalence estimation

Hypertension is a highly prevalent cardiovascular disease. It marks a considerable cost factor to many national health systems. Despite its prevalence, regional disease distributions are often unknown and must be estimated from survey data. However, health surveys frequently lack in regional observations due to limited resources. Obtained prevalence estimates suffer from unacceptably large sampling variances and are not reliable. Small area estimation solves this problem by linking auxiliary data from multiple regions in suitable regression models. Typically, either unit- or area-level observations are considered for this purpose. But with respect to hypertension, both levels should be used. Hypertension has characteristic comorbidities and is strongly related to lifestyle features, which are unit-level information. It is also correlated with socioeconomic indicators that are usually measured on the area-level. But the level combination is challenging as it requires multi-level model parameter estimation from small samples. We use a multi-level small area model with level-specific penalization to overcome this issue. Model parameter estimation is performed via stochastic coordinate gradient descent. A jackknife estimator of the mean squared error is presented. The methodology is applied to combine health survey data and administrative records to estimate regional hypertension prevalence in Germany.     

Adaptive Elastic Net for Generalized Methods of Moments

Model selection and estimation are crucial parts of econometrics. This article introduces a new technique that can simultaneously estimate and select the model in generalized method of moments (GMM) context. The GMM is particularly powerful for analyzing complex datasets such as longitudinal and panel data, and it has wide applications in econometrics. This article extends the least squares based adaptive elastic net estimator by Zou and Zhang to nonlinear equation systems with endogenous variables. The extension is not trivial and involves a new proof technique due to estimators’ lack of closed-form solutions. Compared to Bridge-GMM by Caner, we allow for the number of parameters to diverge to infinity as well as collinearity among a large number of variables; also, the redundant parameters are set to zero via a data-dependent technique. This method has the oracle property, meaning that we can estimate nonzero parameters with their standard limit and the redundant parameters are dropped from the equations simultaneously. Numerical examples are used to illustrate the performance of the new method.     

Doubly robust point and variance estimation in the presence of imputed survey data

Imputation is often used in surveys to treat item nonresponse. It is well known that treating the imputed values as observed values may lead to substantial underestimation of the variance of the point estimators. To overcome the problem, a number of variance estimation methods have been proposed in the literature, including resampling methods such as the jackknife and the bootstrap. In this paper, we consider the problem of doubly robust inference in the presence of imputed survey data. In the doubly robust literature, point estimation has been the main focus. In this paper, using the reverse framework for variance estimation, we derive doubly robust linearization variance estimators in the case of deterministic and random regression imputation within imputation classes. Also, we study the properties of several jackknife variance estimators under both negligible and nonnegligible sampling fractions. A limited simulation study investigates the performance of various variance estimators in terms of relative bias and relative stability. Finally, the asymptotic normality of imputed estimators is established for stratified multistage designs under both deterministic and random regression imputation. The Canadian Journal of Statistics 40: 259–281; 2012 © 2012 Statistical Society of Canada     

A Likelihood‐Based Analysis for Relaxing the Exclusion Restriction in Randomized Experiments with Noncompliance

The exclusion restriction is usually assumed for identifying causal effects in true or only natural randomized experiments with noncompliance. It requires that the assignment to treatment does not have a direct causal effect on the outcome. Despite its importance, the restriction can often be unrealistic, especially in situations of natural experiments. It is shown that, without the exclusion restriction, the parametric model is identified if the outcome distributions of various compliance statuses are in the same parametric class and that class is a linearly independent set over the field of real numbers. However, the relaxation of the exclusion restriction yields a parametric model that is characterized by the presence of mixtures of distributions. This scenario complicates the likelihood‐based estimation procedures because it implies more than one maximum likelihood point. A two‐step estimation procedure based on detecting the root that is closest to the method of moments estimate of the parameter vector is then proposed and analyzed in detail, under normally distributed outcomes. An economic example with real data concerning returns to schooling concludes the paper.     

A Random Coefficient Degradation Model With Ramdom Sample Size

In testing product reliability, there is often a critical cutoff level that determines whether a specimen is classified as failed. One consequence is that the number of degradation data collected varies from specimen to specimen. The information of random sample size should be included in the model, and our study shows that it can be influential in estimating model parameters. Two-stage least squares (LS) and maximum modified likelihood (MML) estimation, which both assume fixed sample sizes, are commonly used for estimating parameters in the repeated measurements models typically applied to degradation data. However, the LS estimate is not consistent in the case of random sample sizes. This article derives the likelihood for the random sample size model and suggests using maximum likelihood (ML) for parameter estimation. Our simulation studies show that ML estimates have smaller biases and variances compared to the LS and MML estimates. All estimation methods can be greatly improved if the number of specimens increases from 5 to 10. A data set from a semiconductor application is used to illustrate our methods.     

Bootstrapping in a high dimensional but very low-sample size problem

This article is concerned with testing multiple hypotheses, one for each of a large number of small data sets. Such data are sometimes referred to as high-dimensional, low-sample size data. Our model assumes that each observation within a randomly selected small data set follows a mixture of C shifted and rescaled versions of an arbitrary density f. A novel kernel density estimation scheme, in conjunction with clustering methods, is applied to estimate f. Bayes information criterion and a new criterion weighted mean of within-cluster variances are used to estimate C, which is the number of mixture components or clusters. These results are applied to the multiple testing problem. The null sampling distribution of each test statistic is determined by f, and hence a bootstrap procedure that resamples from an estimate of f is used to approximate this null distribution.     

Rank‐based estimate of four‐parameter logistic model

During drug development, the calculation of inhibitory concentration that results in a response of 50% (IC₅₀) is performed thousands of times every day. The nonlinear model most often used to perform this calculation is a four‐parameter logistic, suitably parameterized to estimate the IC₅₀ directly. When performing these calculations in a high‐throughput mode, each and every curve cannot be studied in detail, and outliers in the responses are a common problem. A robust estimation procedure to perform this calculation is desirable. In this paper, a rank‐based estimate of the four‐parameter logistic model that is analogous to least squares is proposed. The rank‐based estimate is based on the Wilcoxon norm. The robust procedure is illustrated with several examples from the pharmaceutical industry. When no outliers are present in the data, the robust estimate of IC₅₀ is comparable with the least squares estimate, and when outliers are present in the data, the robust estimate is more accurate. A robust goodness‐of‐fit test is also proposed. To investigate the impact of outliers on the traditional and robust estimates, a small simulation study was conducted. Copyright © 2012 John Wiley & Sons, Ltd.     

Bayesian analysis for mixtures of discrete distributions with a non-parametric component

Bayesian finite mixture modelling is a flexible parametric modelling approach for classification and density fitting. Many areas of application require distinguishing a signal from a noise component. In practice, it is often difficult to justify a specific distribution for the signal component; therefore, the signal distribution is usually further modelled via a mixture of distributions. However, modelling the signal as a mixture of distributions is computationally non-trivial due to the difficulties in justifying the exact number of components to be used and due to the label switching problem. This paper proposes the use of a non-parametric distribution to model the signal component. We consider the case of discrete data and show how this new methodology leads to more accurate parameter estimation and smaller false non-discovery rate. Moreover, it does not incur the label switching problem. We show an application of the method to data generated by ChIP-sequencing experiments.     

CONSTRUCTING MORE POWERFUL EXACT TESTS OF EQUIVALENCE FROM BINARY MATCHED PAIRS

In randomized clinical trials, it is often necessary to demonstrate that a new medical treatment does not substantially differ from a standard reference treatment. Formal testing of such ‘equivalence hypotheses’ is typically done by combining two one‐sided tests (TOST). A quite different strand of research has demonstrated that replacing nuisance parameters with a null estimate produces P‐values that are close to exact ( Lloyd 2008a ) and that maximizing over the residual dependence on the nuisance parameter produces P‐values that are exact and optimal within a class ( Röhmel & Mansmann 1999 ; Lloyd 2008a ). The three procedures – TOST, estimation and maximization of a nuisance parameter – can each be expressed as a transformation of an approximate P‐value. In this paper, we point out that TOST‐based P‐values will generally be conservative, even if based on exact and optimal one‐sided tests. This conservatism is avoided by applying the three transforms in a certain order – estimation followed by TOST followed by maximization. We compare this procedure with existing alternatives through a numerical study of binary matched pairs where the two treatments are compared by the difference of response rates. The resulting tests are uniformly more powerful than the considered competitors, although the difference in power can range from very small to moderate.     

Efficiency and Validity Analyses of Two-Stage Estimation Procedures and Derived Testing Procedures in Quantitative Linear Models with AR(1) Errors

Abstract

In a quantitative linear model with errors following a stationary Gaussian, first-order autoregressive or AR(1) process, Generalized Least Squares (GLS) on raw data and Ordinary Least Squares (OLS) on prewhitened data are efficient methods of estimation of the slope parameters when the autocorrelation parameter of the error AR(1) process, ρ, is known. In practice, ρ is generally unknown. In the so-called two-stage estimation procedures, ρ is then estimated first before using the estimate of ρ to transform the data and estimate the slope parameters by OLS on the transformed data. Different estimators of ρ have been considered in previous studies. In this article, we study nine two-stage estimation procedures for their efficiency in estimating the slope parameters. Six of them (i.e., three noniterative, three iterative) are based on three estimators of ρ that have been considered previously. Two more (i.e., one noniterative, one iterative) are based on a new estimator of ρ that we propose: it is provided by the sample autocorrelation coefficient of the OLS residuals at lag 1, denoted r(1). Lastly, REstricted Maximum Likelihood (REML) represents a different type of two-stage estimation procedure whose efficiency has not been compared to the others yet. We also study the validity of the testing procedures derived from GLS and the nine two-stage estimation procedures. Efficiency and validity are analyzed in a Monte Carlo study. Three types of explanatory variable x in a simple quantitative linear model with AR(1) errors are considered in the time domain: Case 1, x is fixed; Case 2, x is purely random; and Case 3, x follows an AR(1) process with the same autocorrelation parameter value as the error AR(1) process. In a preliminary step, the number of inadmissible estimates and the efficiency of the different estimators of ρ are compared empirically, whereas their approximate expected value in finite samples and their asymptotic variance are derived theoretically. Thereafter, the efficiency of the estimation procedures and the validity of the derived testing procedures are discussed in terms of the sample size and the magnitude and sign of ρ. The noniterative two-stage estimation procedure based on the new estimator of ρ is shown to be more efficient for moderate values of ρ at small sample sizes. With the exception of small sample sizes, REML and its derived F-test perform the best overall. The asymptotic equivalence of two-stage estimation procedures, besides REML, is observed empirically. Differences related to the nature, fixed or random (uncorrelated or autocorrelated), of the explanatory variable are also discussed.     

Flexible mixture modeling via the multivariate t distribution with?the?Box-Cox transformation: an?alternative to?the?skew-t distribution

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. Furthermore, normal mixture models are not robust against outliers; they often require extra components for modeling outliers and/or give a poor representation of the data. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling. This class of distributions generalizes the normal distribution with the more heavy-tailed t distribution, and introduces skewness via the Box-Cox transformation. As a result, this provides a unified framework to simultaneously handle outlier identification and data transformation, two interrelated issues. We describe an Expectation-Maximization algorithm for parameter estimation along with transformation selection. We demonstrate the proposed methodology with three real data sets and simulation studies. Compared with a wealth of approaches including the skew-t mixture model, the proposed t mixture model with the Box-Cox transformation performs favorably in terms of accuracy in the assignment of observations, robustness against model misspecification, and selection of the number of components.     

The EM algorithm with gradient function update for discrete mixtures with known (fixed) number of components

The paper is focussing on some recent developments in nonparametric mixture distributions. It discusses nonparametric maximum likelihood estimation of the mixing distribution and will emphasize gradient type results, especially in terms of global results and global convergence of algorithms such as vertex direction or vertex exchange method. However, the NPMLE (or the algorithms constructing it) provides also an estimate of the number of components of the mixing distribution which might be not desirable for theoretical reasons or might be not allowed from the physical interpretation of the mixture model. When the number of components is fixed in advance, the before mentioned algorithms can not be used and globally convergent algorithms do not exist up to now. Instead, the EM algorithm is often used to find maximum likelihood estimates. However, in this case multiple maxima are often occuring. An example from a meta-analyis of vitamin A and childhood mortality is used to illustrate the considerable, inferential importance of identifying the correct global likelihood. To improve the behavior of the EM algorithm we suggest a combination of gradient function steps and EM steps to achieve global convergence leading to the EM algorithm with gradient function update (EMGFU). This algorithms retains the number of components to be exactly k and typically converges to the global maximum. The behavior of the algorithm is highlighted at hand of several examples.     

Generalized Control Charts for Non-Normal Data Using g-and-k Distributions

Statistical control charts are often used in industry to monitor processes in the interests of quality improvement. Such charts assume independence and normality of the control statistic, but these assumptions are often violated in practice. To better capture the true shape of the underlying distribution of the control statistic, we utilize the g-and-k distributions to estimate probability limits, the true ARL, and the error in confidence that arises from incorrectly assuming normality. A sensitivity assessment reveals that the extent of error in confidence associated with control chart decision-making procedures increases more rapidly as the distribution becomes more skewed or as the tails of the distribution become longer than those of the normal distribution. These methods are illustrated using both a frequentist and computational Bayesian approach to estimate the g-and-k parameters in two different practical applications. The Bayesian approach is appealing because it can account for prior knowledge in the estimation procedure and yields posterior distributions of parameters of interest such as control limits.     

SEMI-MECHANISTIC MODELLING IN NONLINEAR REGRESSION: A CASE STUDY

This paper discusses the use of highly parameterized semi‐mechanistic nonlinear models with particular reference to the PARJIB crop response model of Reid (2002) [Yield response to nutrient supply across a wide range of conditions 1. Model derivation. Field Crops Research 77, 161–171]. Compared to empirical linear approaches, such models promise improved generality of application but present considerable challenges for estimation. Some success has been achieved with a fitting approach that uses a Levenberg–Marquardt algorithm starting from initial values determined by a genetic algorithm. Attention must be paid, however, to correlations between parameter estimates and an approach is described to identify these based on large simulated datasets. This work illustrates the value for the scientist in exploring the correlation structure in mechanistic or semi‐mechanistic models. Such information might be used to reappraise the structure of the model itself, especially if the experimental evidence is not strong enough to allow estimation of a parameter free of assumptions about the values of others. Thus statistical modelling and analysis can complement mechanistic studies, making more explicit what is known and what is not known about the processes being modelled and guiding further research.     

An empirical comparison of EM,SEM and MCMC performance for problematic Gaussian mixture likelihoods

We compare EM, SEM, and MCMC algorithms to estimate the parameters of the Gaussian mixture model. We focus on problems in estimation arising from the likelihood function having a sharp ridge or saddle points. We use both synthetic and empirical data with those features. The comparison includes Bayesian approaches with different prior specifications and various procedures to deal with label switching. Although the solutions provided by these stochastic algorithms are more often degenerate, we conclude that SEM and MCMC may display faster convergence and improve the ability to locate the global maximum of the likelihood function.     

Detection of multiple undocumented change-points using adaptive Lasso

The problem of detecting multiple undocumented change-points in a historical temperature sequence with simple linear trend is formulated by a linear model. We apply adaptive least absolute shrinkage and selection operator (Lasso) to estimate the number and locations of change-points. Model selection criteria are used to choose the Lasso smoothing parameter. As adaptive Lasso may overestimate the number of change-points, we perform post-selection on change-points detected by adaptive Lasso using multivariate t simultaneous confidence intervals. Our method is demonstrated on the annual temperature data (year: 1902–2000) from Tuscaloosa, Alabama.