A New Extension of the Binomial Error Model for Responses to Items of Varying Difficulty in Educational Testing and Attitude Surveys

We put forward a new item response model which is an extension of the binomial error model first introduced by Keats and Lord. Like the binomial error model, the basic latent variable can be interpreted as a probability of responding in a certain way to an arbitrarily specified item. For a set of dichotomous items, this model gives predictions that are similar to other single parameter IRT models (such as the Rasch model) but has certain advantages in more complex cases. The first is that in specifying a flexible two-parameter Beta distribution for the latent variable, it is easy to formulate models for randomized experiments in which there is no reason to believe that either the latent variable or its distribution vary over randomly composed experimental groups. Second, the elementary response function is such that extensions to more complex cases (e.g., polychotomous responses, unfolding scales) are straightforward. Third, the probability metric of the latent trait allows tractable extensions to cover a wide variety of stochastic response processes.     

Incorporating Inter‐item Correlations in Item Response Data Analysis

This paper concerns with the analysis of item response data, which are usually measured on a rating scale and are therefore ordinal. These study items tended to be highly inter‐correlated. Rasch models, which convert ordinal categorical scales into linear measurements, are widely used in ordinal data analysis. In this paper, we improve the current methodology in order to incorporate inter‐item correlations. We have advocated the latent variable approach for this purpose, in combination with generalized estimating equations to estimate the Rasch model parameters. The data on a study of families of lung cancer patients demonstrate the utility of our methods.     

The cox proportional hazards model with a continuous latent variable measured by multiple binary indicators

This article is motivated by the Women's Health and Aging Study, where information about physical functioning was recorded along with death information in a group of elderly women. The focus is on determining whether having difficulties in daily living tasks is accompanied by a higher mortality rate. To this end, a two-parameter logistic regression model is used for the modeling of binary questionnaire data assuming an underlying continuous latent variable, difficulty in daily living. The Cox model is used for the survival information, and the continuous latent variable is included as an explanatory variable along with other observed variables. Parameters are estimated by maximizing the likelihood for the joint distribution of the items and the time-to-event information. In addition to presenting a new statistical model, this article also illustrates the use of the model in a real data setting and addresses the more practical issues of model building, diagnostics, and parameter interpretation.     

Latent variable models for longitudinal data with multiple continuous outcomes

Multiple outcomes are often used to properly characterize an effect of interest. This paper proposes a latent variable model for the situation where repeated measures over time are obtained on each outcome. These outcomes are assumed to measure an underlying quantity of main interest from different perspectives. We relate the observed outcomes using regression models to a latent variable, which is then modeled as a function of covariates by a separate regression model. Random effects are used to model the correlation due to repeated measures of the observed outcomes and the latent variable. An EM algorithm is developed to obtain maximum likelihood estimates of model parameters. Unit-specific predictions of the latent variables are also calculated. This method is illustrated using data from a national panel study on changes in methadone treatment practices.     

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.     

Differential Item Functioning in the SF-36 Physical Functioning and Mental Health Sub-Scales: A Population-Based Investigation in the Canadian Multicentre Osteoporosis Study

Background

Self-reported health status measures, like the Short Form 36-item Health Survey (SF-36), can provide rich information about the overall health of a population and its components, such as physical, mental, and social health. However, differential item functioning (DIF), which arises when population sub-groups with the same underlying (i.e., latent) level of health have different measured item response probabilities, may compromise the comparability of these measures. The purpose of this study was to test for DIF on the SF-36 physical functioning (PF) and mental health (MH) sub-scale items in a Canadian population-based sample.

Methods

Study data were from the prospective Canadian Multicentre Osteoporosis Study (CaMos), which collected baseline data in 1996–1997. DIF was tested using a multiple indicators multiple causes (MIMIC) method. Confirmatory factor analysis defined the latent variable measurement model for the item responses and latent variable regression with demographic and health status covariates (i.e., sex, age group, body weight, self-perceived general health) produced estimates of the magnitude of DIF effects.

Results

The CaMos cohort consisted of 9423 respondents; 69.4% were female and 51.7% were less than 65 years. Eight of 10 items on the PF sub-scale and four of five items on the MH sub-scale exhibited DIF. Large DIF effects were observed on PF sub-scale items about vigorous and moderate activities, lifting and carrying groceries, walking one block, and bathing or dressing. On the MH sub-scale items, all DIF effects were small or moderate in size.

Conclusions

SF-36 PF and MH sub-scale scores were not comparable across population sub-groups defined by demographic and health status variables due to the effects of DIF, although the magnitude of this bias was not large for most items. We recommend testing and adjusting for DIF to ensure comparability of the SF-36 in population-based investigations.     

The Accuracy of Computerized Adaptive Testing in Heterogeneous Populations: A Mixture Item-Response Theory Analysis

Background

Computerized adaptive testing (CAT) utilizes latent variable measurement model parameters that are typically assumed to be equivalently applicable to all people. Biased latent variable scores may be obtained in samples that are heterogeneous with respect to a specified measurement model. We examined the implications of sample heterogeneity with respect to CAT-predicted patient-reported outcomes (PRO) scores for the measurement of pain.

Methods

A latent variable mixture modeling (LVMM) analysis was conducted using data collected from a heterogeneous sample of people in British Columbia, Canada, who were administered the 36 pain domain items of the CAT-5D-QOL. The fitted LVMM was then used to produce data for a simulation analysis. We evaluated bias by comparing the referent PRO scores of the LVMM with PRO scores predicted by a “conventional” CAT (ignoring heterogeneity) and a LVMM-based “mixture” CAT (accommodating heterogeneity).

Results

The LVMM analysis indicated support for three latent classes with class proportions of 0.25, 0.30 and 0.45, which suggests that the sample was heterogeneous. The simulation analyses revealed differences between the referent PRO scores and the PRO scores produced by the “conventional” CAT. The “mixture” CAT produced PRO scores that were nearly equivalent to the referent scores.

Conclusion

Bias in PRO scores based on latent variable models may result when population heterogeneity is ignored. Improved accuracy could be obtained by using CATs that are parameterized using LVMM.     

Latent class regression on latent factors

In the research of public health, psychology, and social sciences, many research questions investigate the relationship between a categorical outcome variable and continuous predictor variables. The focus of this paper is to develop a model to build this relationship when both the categorical outcome and the predictor variables are latent (i.e. not observable directly). This model extends the latent class regression model so that it can include regression on latent predictors. Maximum likelihood estimation is used and two numerical methods for performing it are described: the Monte Carlo expectation and maximization algorithm and Gaussian quadrature followed by quasi-Newton algorithm. A simulation study is carried out to examine the behavior of the model under different scenarios. A data example involving adolescent health is used for demonstration where the latent classes of eating disorders risk are predicted by the latent factor body satisfaction.     

Modeling longitudinal data with nonignorable dropouts using a latent dropout class model

In longitudinal studies with dropout, pattern-mixture models form an attractive modeling framework to account for nonignorable missing data. However, pattern-mixture models assume that the components of the mixture distribution are entirely determined by the dropout times. That is, two subjects with the same dropout time have the same distribution for their response with probability one. As that is unlikely to be the case, this assumption made lead to classification error. In addition, if there are certain dropout patterns with very few subjects, which often occurs when the number of observation times is relatively large, pattern-specific parameters may be weakly identified or require identifying restrictions. We propose an alternative approach, which is a latent-class model. The dropout time is assumed to be related to the unobserved (latent) class membership, where the number of classes is less than the number of observed patterns; a regression model for the response is specified conditional on the latent variable. This is a type of shared-parameter model, where the shared "parameter" is discrete. Parameter estimates are obtained using the method of maximum likelihood. Averaging the estimates of the conditional parameters over the distribution of the latent variable yields estimates of the marginal regression parameters. The methodology is illustrated using longitudinal data on depression from a study of HIV in women.     

Joint analysis of time-to-event and multiple binary indicators of latent classes

Multiple categorical variables are commonly used in medical and epidemiological research to measure specific aspects of human health and functioning. To analyze such data, models have been developed considering these categorical variables as imperfect indicators of an individual's "true" status of health or functioning. In this article, the latent class regression model is used to model the relationship between covariates, a latent class variable (the unobserved status of health or functioning), and the observed indicators (e.g., variables from a questionnaire). The Cox model is extended to encompass a latent class variable as predictor of time-to-event, while using information about latent class membership available from multiple categorical indicators. The expectation-maximization (EM) algorithm is employed to obtain maximum likelihood estimates, and standard errors are calculated based on the profile likelihood, treating the nonparametric baseline hazard as a nuisance parameter. A sampling-based method for model checking is proposed. It allows for graphical investigation of the assumption of proportional hazards across latent classes. It may also be used for checking other model assumptions, such as no additional effect of the observed indicators given latent class. The usefulness of the model framework and the proposed techniques are illustrated in an analysis of data from the Women's Health and Aging Study concerning the effect of severe mobility disability on time-to-death for elderly women.     

Power and Sample Size Determination in the Rasch Model: Evaluation of the Robustness of a Numerical Method to Non-Normality of the Latent Trait

Patient-reported outcomes (PRO) have gained importance in clinical and epidemiological research and aim at assessing quality of life, anxiety or fatigue for instance. Item Response Theory (IRT) models are increasingly used to validate and analyse PRO. Such models relate observed variables to a latent variable (unobservable variable) which is commonly assumed to be normally distributed. A priori sample size determination is important to obtain adequately powered studies to determine clinically important changes in PRO. In previous developments, the Raschpower method has been proposed for the determination of the power of the test of group effect for the comparison of PRO in cross-sectional studies with an IRT model, the Rasch model. The objective of this work was to evaluate the robustness of this method (which assumes a normal distribution for the latent variable) to violations of distributional assumption. The statistical power of the test of group effect was estimated by the empirical rejection rate in data sets simulated using a non-normally distributed latent variable. It was compared to the power obtained with the Raschpower method. In both cases, the data were analyzed using a latent regression Rasch model including a binary covariate for group effect. For all situations, both methods gave comparable results whatever the deviations from the model assumptions. Given the results, the Raschpower method seems to be robust to the non-normality of the latent trait for determining the power of the test of group effect.     

Biases and Power for Groups Comparison on Subjective Health Measurements

Subjective health measurements are increasingly used in clinical research, particularly for patient groups comparisons. Two main types of analytical strategies can be used for such data: so-called classical test theory (CTT), relying on observed scores and models coming from Item Response Theory (IRT) relying on a response model relating the items responses to a latent parameter, often called latent trait. Whether IRT or CTT would be the most appropriate method to compare two independent groups of patients on a patient reported outcomes measurement remains unknown and was investigated using simulations. For CTT-based analyses, groups comparison was performed using t-test on the scores. For IRT-based analyses, several methods were compared, according to whether the Rasch model was considered with random effects or with fixed effects, and the group effect was included as a covariate or not. Individual latent traits values were estimated using either a deterministic method or by stochastic approaches. Latent traits were then compared with a t-test. Finally, a two-steps method was performed to compare the latent trait distributions, and a Wald test was performed to test the group effect in the Rasch model including group covariates. The only unbiased IRT-based method was the group covariate Wald’s test, performed on the random effects Rasch model. This model displayed the highest observed power, which was similar to the power using the score t-test. These results need to be extended to the case frequently encountered in practice where data are missing and possibly informative.     

A comparison of the multilevel MIMIC model to the multilevel regression and mixed ANOVA model for the estimation and testing of a cross-level interaction effect: A simulation study

When observing data on a patient-reported outcome measure in, for example, clinical trials, the variables observed are often correlated and intended to measure a latent variable. In addition, such data are also often characterized by a hierarchical structure, meaning that the outcome is repeatedly measured within patients. To analyze such data, it is important to use an appropriate statistical model, such as structural equation modeling (SEM). However, researchers may rely on simpler statistical models that are applied to an aggregated data structure. For example, correlated variables are combined into one sum score that approximates a latent variable. This may have implications when, for example, the sum score consists of indicators that relate differently to the latent variable being measured. This study compares three models that can be applied to analyze such data: the multilevel multiple indicators multiple causes (ML-MIMIC) model, a univariate multilevel model, and a mixed analysis of variance (ANOVA) model. The focus is on the estimation of a cross-level interaction effect that presents the difference over time on the patient-reported outcome between two treatment groups. The ML-MIMIC model is an SEM-type model that considers the relationship between the indicators and the latent variable in a multilevel setting, whereas the univariate multilevel and mixed ANOVA model rely on sum scores to approximate the latent variable. In addition, the mixed ANOVA model uses aggregated second-level means as outcome. This study showed that the ML-MIMIC model produced unbiased cross-level interaction effect estimates when the relationships between the indicators and the latent variable being measured varied across indicators. In contrast, under similar conditions, the univariate multilevel and mixed ANOVA model underestimated the cross-level interaction effect.     

Generalized linear latent variable models for repeated measures of spatially correlated multivariate data

Observations of multiple-response variables across space and over time occur often in environmental and ecological studies. Compared to purely spatial models for a single response variable in the exponential family of distributions, fewer statistical tools are available for multiple-response variables that are not necessarily Gaussian. An exception is a common-factor model developed for multivariate spatial data by Wang and Wall (2003, Biostatistics 4, 569-582). The purpose of this article is to extend this multivariate space-only model and develop a flexible class of generalized linear latent variable models for multivariate spatial-temporal data. For statistical inference, maximum likelihood estimates and their standard deviations are obtained using a Monte Carlo EM algorithm. We also use a novel way to automatically adjust the Monte Carlo sample size, which facilitates the convergence of the Monte Carlo EM algorithm. The methodology is illustrated by an ecological study of red pine trees in response to bark beetle challenges in a forest stand of Wisconsin.     

Factor analytic models of clustered multivariate data with informative censoring

This article describes a general class of factor analytic models for the analysis of clustered multivariate data in the presence of informative missingness. We assume that there are distinct sets of cluster-level latent variables related to the primary outcomes and to the censoring process, and we account for dependency between these latent variables through a hierarchical model. A linear model is used to relate covariates and latent variables to the primary outcomes for each subunit. A generalized linear model accounts for covariate and latent variable effects on the probability of censoring for subunits within each cluster. The model accounts for correlation within clusters and within subunits through a flexible factor analytic framework that allows multiple latent variables and covariate effects on the latent variables. The structure of the model facilitates implementation of Markov chain Monte Carlo methods for posterior estimation. Data from a spermatotoxicity study are analyzed to illustrate the proposed approach.     

Ascertainment adjustment in genetic studies of ordinal traits

Most genetic studies recruit high risk families and the discoveries are based on non-random selected groups. We must consider the consequences of this ascertainment process in order to apply the results of genetic research to the general population. In previous reports, we developed a latent variable model to assess the familial aggregation and inheritability of ordinal-scaled diseases, and found a major gene component of alcoholism after applying the model to the data from the Yale family study of comorbidity of alcoholism and anxiety (YFSCAA). In this report, we examine the ascertainment effects on parameter estimates and correct potential bias in the latent variable model. The simulation studies for various ascertainment schemes suggest that our ascertainment adjustment is necessary and effective. We also find that the estimated effects are relatively unbiased for the particular ascertainment scheme used in the YFSCAA, which assures the validity of our earlier conclusion.     

Regression modeling of ordinal data with nonzero baselines

This paper develops regression models for ordinal data with nonzero control response probabilities. The models are especially useful in dose-response studies where the spontaneous or natural response rate is nonnegligible and the dosage is logarithmic. These models generalize Abbott's formula, which has been commonly used to model binary data with nonzero background observations. We describe a biologically plausible latent structure and develop an EM algorithm for fitting the models. The EM algorithm can be implemented using standard software for ordinal regression. A toxicology data set where the proposed model fits the data but a more conventional model fails is used to illustrate the methodology.     

结构方程混合模型在SNP分析中的应用

采用结构方程混合模型(SEMM)对实际SNP数据进行分析,为遗传统计学提供一种新的有效的分析方法。本研究的数据是由GAW17提供的,包含697个个体的22条常染色体的上万个SNP和根据这些SNP所模拟的697个个体的性状特点。随机挑选了1号染色体上的4个SNP和3个定量性状作为研究变量,分别进行潜在类别分析和结构方程混合模型分析。根据4个SNP数据,人群被分为3个潜在类别,概率分别为0.53,0.34,0.13。潜在类别1、2和3中的因子均值Q分别为-4.029、-2.052和0,潜在类别1、2的因子均值均低于3(<0.001)。研究表明:结构方程混合模型(SEMM)综合了结构方程模型和潜在类别模型的思想,形成了自己的优势,可用于处理同时包含分类潜变量和连续潜变量的数据。     

The Depression List for mood evaluation in psychogeriatrics: psychometric analysis and scalability

The Depression List is a Dutch self-report measure of depressive symptoms in persons with dementia. Fifteen items differ to the extent that they provide reliable measurements of latent constructs such as feeling depressed, tired or lonely. Item scalability and construct reliability were studied by administering the questionnaire to 599 consecutive visitors of a psychogeriatric day care department. Using confirmatory factor analysis three competing hypothetical models were tested for model fit. A four-factor model composed of latent constructs for feeling depressed, tired, lonesome and appetite or sleep disturbance provided an adequate fit to the clinical data, which did not apply to a two-factor model (distinguishing a construct of loneliness from a more general mood construct) and a one-factor model, that hypothesized a single general mood construct for all fifteen items. Nine items carried relatively small proportions of error variance. These reliable items were selected for the construction of three scales that satisfied the assumptions of Mokken's criteria for homogeneity. The resulting measures of feeling depressed, tired or lonely showed evidence of convergent validity for a latent affective construct. The affective measures were not associated with severity of cognitive impairment (discriminant validity). The three scales allow a reliable measurement of differences in affective function in cognitively impaired subjects.     

Megavariate data analysis of mass spectrometric proteomics data using latent variable projection method

There are many data mining techniques for processing and general learning of multivariate data. However, we believe the wavelet transformation and latent variable projection method are particularly useful for spectroscopic and chromatographic data. Projection based methods are designed to handle hugely multivariate nature of such data effectively. For the actual analysis of the data we have used latent variable projection methods such as principal component analysis (PCA) and partial least squares projection to latent structures based discriminant analysis (PLS-DA) to analyze the raw data presented to the participants of the First Duke Proteomics Data Mining Conference. PCA was used to solve problem #1 (clustering problem) and the PLS-DA was used to solve problem #2 (classification problem). The idea of internal and external cross-validation was used to validate the model obtained from the classification analysis. The simple two-component PLS-DA model obtained from the analysis performed well. The model has completely separated the two groups from all the data. The same model applied on two-thirds of the data showed good performance by external validation with independent test set of remaining 13 specimens obtained by setting aside the spectra of every third specimen (accuracy of 85%).