Modeling within-subject dependencies in ordinal paired comparison data

This paper presents two probabilistic models based on the logistic and the normal distribution for the analysis of dependencies in individual paired comparison judgments. It is argued that a core assumption of latent class choice models, independence of individual decisions, may not be well-suited for the analysis of paired comparison data. Instead, the analysis and interpretation of paired comparison data may be much simplified by allowing for within-person dependencies that result from repeated evaluations of the same options in different pairs. Moreover, by relating dependencies among the individual-level responses to (in)consistencies in the judgmental process, we show that the proposed graded paired comparison models reduce to ranking models under certain conditions. Three applications are presented to illustrate the approach.This research was partially supported by NSF grant SBR-9409531. The authors are grateful to the reviewers, Alan Agresti and Herbert Hoijtink for their helpful comments on this research.     

Generalized multilevel structural equation modeling

A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.gllamm can be downloaded from http://www.gllamm.org. The paper was written while Sophia Rabe-Hesketh was employed at and Anders Skrondal was visiting the Department of Biostatistics and Computing, Institute of Psychiatry, King's College London.     

The use of limit theorems in paired and triple comparison model building

The psychological framework proposed by Thompson and Singh (1967) is extended to obtain a statistical interpretation of the stimulus-response phenomenon permitting ties when comparing two stimuli. Using this interpretation, the generalized Bradley-Terry model (Rao and Kupper, 1967) for paired comparisons and the  and  for triple comparisons are derived as the appropriate asymptotic models for experiments permitting tied observations.     

Estimating Complex Measurement and Growth Models Using the R Package PLmixed

Measurement models, such as factor analysis and item response theory models, are commonly implemented within educational, psychological, and behavioral science research to mitigate the negative effects of measurement error. These models can be formulated as an extension of generalized linear mixed models within a unifying framework that encompasses various kinds of multilevel models and longitudinal models, such as partially nonlinear latent growth models. We introduce the R package PLmixed, which implements profile maximum likelihood estimation to estimate complex measurement and growth models that can be formulated within the general modeling framework using the existing R package lme4 and function optim. We demonstrate the use of PLmixed through two examples before concluding with a brief overview of other possible models.     

Remarks on the identifiability of thurstonian paired comparison models under multiple judgment

It is well-known that the representations of the Thurstonian models for difference judgment data are not unique. It has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the Thurstonian ranking models. In this paper, we examine the equivalence relations between Thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models.The author is indebted to Ulf Böckenholt and Albert Maydeu-Olivares for their significant comments and suggestions which led to considerable improvement in this article.     

Analyzing temporal patterns of infant sleep and negative affective behavior: a comparison between different statistical models

Objective

Variability in infant sleep and negative affective behavior (NAB) is a developmental phenomenon that has long been of interest to researchers and clinicians. However, analyses and delineation of such temporal patterns were often limited to basic statistical approaches, which may prevent adequate identification of meaningful variation within these patterns. Modern statistical procedures such as additive models may detect specific patterns of temporal variation in infant behavior more effectively.

Method

Hundred and twenty-one mothers were asked to record different behaviors of their 4–44 weeks old healthy infants by diaries for three days consecutively. Circadian patterns as well as individual trajectories and day-to-day variability of infant sleep and NAB were modeled with generalized linear models (GLMs) including a linear and quadratic polynomial for time, a GLM with a polynomial of the 8th order, a GLM with a harmonic function, a generalized linear mixed model (GLMM) with a polynomial of the 8th order, a generalized additive model, and a generalized additive mixed model (GAMM).

Results

The semi-parametric model GAMM was found to fit the data of infant sleep better than any other parametric model used. GLMM with a polynomial of the 8th order and GAMM modeled temporal patterns of infant NAB equally well, although the GLMM exhibited a slightly better model fit while GAMM was easier to interpret. Besides the well-known evening clustering in infant NAB we found a significant second peak in NAB around midday that was not affected by the constant decline in the amounts of NAB across the 3-day study period.

Conclusion

Using advanced statistical procedures (GAMM and GLMM) even small variations and phenomena in infant behavior can be reliably detected. Future studies investigating variability and temporal patterns in infant variables may benefit from these statistical approaches.     

Additive and multiplicative models for gamma distributed random variables,and their application as psychometric models for response times

A class of models for gamma distributed random variables is presented. These models are shown to be more flexible than the classical linear models with respect to the structure that can be imposed on the expected value. In particular, both additive, multiplicative, and combined additive-multiplicative models can be formulated. As a special case, a class of psychometric models for reaction times is presented, together with their psychological interpretation. By means of a comparison with existing models, this class of models is shown to offer some possibilities that are not available in existing methods. Parameter estimation by means of maximum likelihood (ML) is shown to have some attractive properties, since the models belong to the exponential family. Then, the results of a simulation study of the bias in the ML estimates are presented. Finally, the application of these models is illustrated by an analysis of the data from a mental rotation experiment. This analysis is preceded by an evaluation of the appropriateness of the gamma distribution for these data.     

Regularized Finite Mixture Models for Probability Trajectories

Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse. This work was supported in part by NIAAA grants R37 AA07065 and R01 AA12217 to RAZ.     

A model for repeat paired comparison preference tests

A model is proposed that describes subject behavior on repeat paired comparison preference tests. The model extends prior work in this area in that it explicitly allows for abstentions and permits the derivation of individual true scores for discrimination ability as well as conditional estimates of proportionate preference. With these results, the properties of a paired comparison test can be thoroughly explored. An empirical example is presented, and test design issues are considered. In particular, repeat paired comparison preference tests are shown to be inherently less efficient discrimination tests than are pick 1 of 2 tests.     

A generalized,likelihood-free method for posterior estimation

Recent advancements in Bayesian modeling have allowed for likelihood-free posterior estimation. Such estimation techniques are crucial to the understanding of simulation-based models, whose likelihood functions may be difficult or even impossible to derive. However, current approaches are limited by their dependence on sufficient statistics and/or tolerance thresholds. In this article, we provide a new approach that requires no summary statistics, error terms, or thresholds and is generalizable to all models in psychology that can be simulated. We use our algorithm to fit a variety of cognitive models with known likelihood functions to ensure the accuracy of our approach. We then apply our method to two real-world examples to illustrate the types of complex problems our method solves. In the first example, we fit an error-correcting criterion model of signal detection, whose criterion dynamically adjusts after every trial. We then fit two models of choice response time to experimental data: the linear ballistic accumulator model, which has a known likelihood, and the leaky competing accumulator model, whose likelihood is intractable. The estimated posterior distributions of the two models allow for direct parameter interpretation and model comparison by means of conventional Bayesian statistics—a feat that was not previously possible.     

Analyzing preventive trials with generalized additive models

Described a new class of nonparametric regression procedures called generalized additive models (Hastie and Tibshirani, 1991) for assessing intervention effects in mental health preventive field trials. Such models are often better than analysis of covariance models for examining intervention effects adjusted for one or more baseline characteristics and for assessing potential interactions between the intervention and baseline characteristics. Because of these advantages, generalized additive models are important tools analysts should consider in evaluating preventive field trials. We apply generalized additive models as well as more standard linear models to data from a preventive trial aimed at improving mental health and school performance outcomes through a universal intervention in first and second grades. Practical guidance is given to researchers regarding when generalized additive models would be beneficial.     

Linear structural equations with latent variables

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.     

Improved Regression Calibration

The likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form, which makes maximum likelihood estimation taxing. A popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation. We propose an improved regression calibration approach, a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood. It is both consistent and computationally efficient, and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood. Simulations suggest that improved regression calibration, which is easy to implement in standard software, works well in a range of situations.     

A general approach to categorical data analysis with missing data,using generalized linear models with composite links

A general approach for analyzing categorical data when there are missing data is described and illustrated. The method is based on generalized linear models with composite links. The approach can be used (among other applications) to fill in contingency tables with supplementary margins, fit loglinear models when data are missing, fit latent class models (without or with missing data on observed variables), fit models with fused cells (including many models from genetics), and to fill in tables or fit models to data when variables are more finely categorized for some cases than others. Both Newton-like and EM methods are easy to implement for parameter estimation.The author thanks the editor, the reviewers, Laurie Hopp Rindskopf, and Clifford Clogg for comments and suggestions that substantially improved the paper.     

Inference in regression models of heavily skewed alcohol use data: a comparison of ordinary least squares, generalized linear models, and bootstrap resampling.

Analysis of alcohol use data and other low base rate risk behaviors using ordinary least squares regression models can be problematic. This article presents 2 alternative statistical approaches, generalized linear models and bootstrapping, that may be more appropriate for such data. First, the basic theory behind the approaches is presented. Then, using a data set of alcohol use behaviors and consequences, results based on these approaches are contrasted with the results from ordinary least squares regression. The less traditional approaches consistently demonstrated better fit with model assumptions, as demonstrated by graphical analysis of residuals, and identified more significant variables potentially resulting in theoretically different interpretations of the models of alcohol use. In conclusion, these models show significant promise for furthering the understanding of alcohol-related behaviors.     

A general diagnostic model applied to language testing data

Probabilistic models with one or more latent variables are designed to report on a corresponding number of skills or cognitive attributes. Multidimensional skill profiles offer additional information beyond what a single test score can provide, if the reported skills can be identified and distinguished reliably. Many recent approaches to skill profile models are limited to dichotomous data and have made use of computationally intensive estimation methods such as Markov chain Monte Carlo, since standard maximum likelihood (ML) estimation techniques were deemed infeasible. This paper presents a general diagnostic model (GDM) that can be estimated with standard ML techniques and applies to polytomous response variables as well as to skills with two or more proficiency levels. The paper uses one member of a larger class of diagnostic models, a compensatory diagnostic model for dichotomous and partial credit data. Many well‐known models, such as univariate and multivariate versions of the Rasch model and the two‐parameter logistic item response theory model, the generalized partial credit model, as well as a variety of skill profile models, are special cases of this GDM. In addition to an introduction to this model, the paper presents a parameter recovery study using simulated data and an application to real data from the field test for TOEFL^® Internet‐based testing.     

On the importance of distinguishing between within- and between-subject effects in intransitive intertemporal choice

Most probabilistic paired comparison models treat inconsistent choices as caused by independent and random errors in the pairwise judgments. In this paper, we argue that this assumption is too restrictive for the analysis of paired comparison data obtained from multiple judges when transitivity violations are systematic. We present a new framework that contains the random error assumption as a special case but also allows for systematic changes in an option's utility assessments over the pairwise comparisons. Accounting for both between- and within-judge sources of variability, we demonstrate in an application on intertemporal choice that the proposed framework can capture systematic transitivity violations as well as individual taste differences.     

解释性项目反应理论模型：理论与应用

解释性项目反应理论模型(Explanatory Item Response Theory Models, EIRTM)是指基于广义线性混合模型和非线性混合模型构建的项目反应理论(Item Response Theory, IRT)模型。EIRTM能在IRT模型的基础上直接加入预测变量, 从而解决各类测量问题。首先介绍EIRTM的相关概念和参数估计方法, 然后展示如何使用EIRTM处理题目位置效应、测验模式效应、题目功能差异、局部被试依赖和局部题目依赖, 接着提供实例对EIRTM的使用进行说明, 最后对EIRTM的不足之处和应用前景进行讨论。     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.