Markov chain Monte Carlo (MCMC) estimation strategies represent a powerful approach to estimation in psychometric models. Popular MCMC samplers and their alignment with Bayesian approaches to modeling are discussed. Key historical and current developments of MCMC are surveyed, emphasizing how MCMC allows the researcher to overcome the limitations of other estimation paradigms, facilitates the estimation of models that might otherwise be intractable, and frees the researcher from certain possible misconceptions about the models.

The last decade has seen an explosion in the use of Markov chain Monte Carlo (MCMC) techniques in fitting statistical psychometric models. In this time, MCMC has been put to advantageous use in estimating existing models and, more importantly, supporting the development of new models that are otherwise computationally intractable. This paper surveys the use of MCMC in modern psychometric models, namely, models that employ (a) probabilistic reasoning in the form of statistical models to facilitate inference from observations of behaviors made by subjects to more broadly conceived statements about the subjects and/or the domain and (b) latent variables to model the presence of measurement error. Additional modeling archetypes, including hierarchical and mixture modeling, are noted where they intersect or overlay with the psychometric modeling paradigm of interest.

Psychometric models are typically organized in terms of assumptions about the latent and observable variables. Factor analysis (FA; Bollen [

This list is far from exhaustive and in later sections we will discuss these and other psychometric models, some of which can be viewed as extensions or combinations of those already mentioned. It is however important to recognize that these latent variable modeling traditions evolved somewhat independently from one another, yielding a current state in which a repository of fairly mature models possess only at best partially overlapping foci, literatures, notational schemes, and—of interest in this work—paradigmatic estimation routines and strategies. To illustrate, FA and SEM have historically been employed to model relationships among constructs and typically do not involve inferences at the subject level. Estimation typically involves maximum likelihood (ML) or least squares (LS) using first- and second-order moments from sample data, with an emphasis on the estimation of structural parameters, that is, parameters for the conditional distributions of observed scores given latent variables (e.g., factor loadings) but not on the values of the latent variables (factors) for individual subjects (Bollen, [

This paper is organized as follows. A brief description of the Bayesian approach to psychometric modeling is advanced, followed by an overview of the most popular MCMC samplers for psychometric models, where the emphasis is placed on how the elements of the latter align with the features and challenges of estimating posterior distributions in the former. Next, an overview of the key historical developments and current work on MCMC for psychometric modeling is presented, emphasizing how MCMC overcomes the limitations of other estimation paradigms, facilitates the estimation of models that would be otherwise intractable, and frees the researcher from possible misconceptions about the models. A discussion concludes the paper.

Because MCMC procedures yield empirical approximations to probability distributions, it fits naturally with the Bayesian approach to statistical analysis in which unknown parameters are treated as random and represented with probability distributions (Gelman et al. [

To formulate a Bayesian psychometric model, let

To conduct Bayesian inference, we specify prior distributions for the unknown parameters. An assumption of exchangeability (Lindley and Smith [

Turning to

Following the conditional independence assumptions inherent in the above treatment, the joint probability distribution for all the entities in the model can be expressed as

Model estimation comes to estimating the posterior distribution. Analytical solutions, though ideal, are often impractical or impossible due to the necessity to evaluate the high-dimensional integrals to obtain the marginal distribution in the denominator in (

To construct a Markov chain, initial values for all parameters must be specified. Subsequent values for the parameters are repeatedly drawn creating a sequence that constitutes the chain. Given certain general conditions hold (e.g., Roberts [

In the following sections, several of the most popular MCMC routines are described. Emphasis is placed on the connection between the features of the MCMC routines that align with features of Bayesian approaches to psychometric modeling.

Let

Let

Initialize the parameters by assigning values for

For

Repeat step 2 for some large number

The conditional independence assumptions greatly reduce the set of parameters that need to be conditioned on in each of distributions in step 2. For example, in drawing a value for a subject’s value of

In complex models, it may be the case that while full conditional distributions may be constructed, they are too complex to sample from. More complex sampling schemes, such as the Metropolis-Hastings and Metropolis samplers, described in this and the following section, are required.

To simplify notation, let

Initialize the parameters by assigning a value for

Draw a

Accept

Repeat steps 2 and 3 for some large number

The

In Metropolis sampling (Metropolis et al. [

It is easily seen that the Metropolis sampler is a special case of the Metropolis-Hastings sampler. It is somewhat less obvious that the Gibbs sampler may be viewed as a special case of the Metropolis sampler, namely, where the proposal distribution for each parameter is the full conditional distribution, which implies that the acceptance probability

Recall that in Bayesian analyses of psychometric models the posterior distribution is generally only known up until a constant of proportionality (see (

A Metropolis(-Hastings)-within-Gibbs sampler, also termed single-component-Metropolis(-Hastings), combines the component decomposition approach of the Gibbs sampler with the flexibility of Metropolis(-Hastings). As noted above, Gibbs sampling involves sampling from the full conditional distributions for each parameter separately. When these full conditionals are not of known form, a Metropolis(-Hastings) step may be taken where, for each parameter, a candidate value is drawn from a proposal distribution

This section reviews key developments in the growing literature on psychometric modeling using MCMC. In tracing the foundational developments and current applications, the emphasis is placed on models and modeling scenarios where the power of MCMC is leveraged to facilitate estimation that would prove difficult of not intractable for traditional procedures, highlighting the flexibility of MCMC and the resulting freedom it provides.

FA and SEM models with linear equations relating the latent and observed variables and (conditional) normality assumptions may be easily handled by traditional ML or LS estimation routines (Bollen [

The great advantage of MCMC for SEM lies in its power to estimate nonstandard models that pose considerable challenges for ML and LS estimation (Lee [

The implication is that, though traditional estimation routines that evolved with the standard FA and SEM paradigm may be suitable for simple models, extending the standard models to more complex situations may necessitate the use of more flexible MCMC procedures. Moreover, contrary to a common belief, the computation necessary to implement MCMC estimation in such complex models is generally less intense than that necessary to conduct ML estimation (Ansari and Jedidi [

In this section, we survey applications of MCMC to models in which a set of discrete, possibly ordinal observables are structured as indicators of continuous latent variables from both FA and IRT perspectives, highlighting aspects in which existing estimation traditions limit our modeling potential.

The FA tradition models discrete data using continuous latent variables by considering the observables to be discretized versions of latent, normally distributed data termed latent response variables. Traditional FA estimation methods have relied on calculating and factoring polychoric correlations, which involves the integration over the distribution of the latent response variables. This approach suffers in that the FA routines were explicitly developed for continuous rather than discrete data. Wirth and Edwards [

The dominant estimation paradigm in IRT involves marginal maximum likelihood (MML; Bock and Aitkin [

A foundation for MCMC for IRT, and psychometric modeling more generally, was given by Albert [

The turning point in the application of MCMC for psychometric modeling came with the work of Patz and Junker [

To highlight an arena where the intersection of different modeling paradigms and their associated traditional estimation routines poses unnecessary limits, consider multidimensional models for discrete observables. The equivalence between IRT and FA versions of the models has long been recognized (Takane and de Leeuw [

MCMC may be seen as a unifying framework for estimation that frees the analyst from these restrictive—and conflicting—misconceptions. Examples of the use of MCMC in the multidimensional modeling from both FA and IRT perspectives include the consideration of dichotomous data (Béguin and Glas [

Similar to the case of FA of continuous data, ML estimation can typically handle traditional, unrestricted LCA models that model discrete observables as dependent on discrete latent variables. And here again, MCMC may still be advantageous for such models in handling missingness, large data sets with outliers, and constructing credibility intervals for inference when an assumption of multivariate normality (of ML estimates or posterior distributions) is unwarranted (Hoijtink [

Turning to more complex models, MCMC has proven useful in estimating models with covariates (Chung et al. [

These models may be also be cast as Bayesian networks, which allow for the estimation of a wide variety of complex effects via MCMC. Examples include compensatory, conjunctive, disjunctive, and inhibitor relationships for dichotomous and polytomous data assuming for dichotomous or ordered latent student skills or attributes (Almond et al. [

The discussion has so far been couched in terms of traditional divisions between models, highlighting applications that pose difficulties for estimation routines typically employed. An advanced approach to model construction takes a modular approach in which the statistical model is constructed in a piecewise manner, interweaving and overlaying features from the traditional paradigms as necessary (Rupp [

The machinery of MCMC can be brought to bear in addressing recurring complications inherent in psychometric applications. MCMC naturally handles missing data (e.g., Chung et al. [

To illustrate the need for a comprehensive model estimation paradigm that is sensitive to the various data structures—and the capability of MCMC to fill that need—consider the National Assessment of Educational Progress (NAEP), which is characterized by (a) inferences targeted at the level of (sub)populations (rather than individuals) that are hierarchically organized, (b) administration of dichotomously and polytomously scored items, (c) complex sampling designs for subjects and items, and (d) covariates at each level of the analysis.

Beginning with the piecewise traditional approach, multiple-imputation approaches (Beaton [

In contrast, Johnson and Jenkins [

The Gibbs, Metropolis-Hastings, and Metropolis samplers are described in the context of psychometric models with latent variables to illustrate the flexibility and power of MCMC in estimating psychometric models under a Bayesian paradigm. It is emphasized that the partitioning of the parameter space of Gibbs samplers and the requirement that the stationary distribution need only be specified up to a constant of proportionality in Metropolis(-Hastings) aligns these MCMC routines with the key features of the characteristics and challenges posed by the desired posterior distribution in Bayesian psychometric modeling.

Many of the examples are given to highlight how MCMC can be leveraged to (a) estimate complex statistical psychometric models that cannot be practically estimated by conventional means and (b) overcome the limitations of other approaches in situations in which the traditions of modeling and estimation paradigms unnecessarily restrict the scope of the models. Other examples highlight how MCMC can be gainfully employed in settings where alternative estimation routines already exist, such as in the analysis of small samples, missing data, and possible underidentification, and where divide-and-conquer strategies systematically understate the uncertainty in estimation.

Despite these advances, it is far from clear that MCMC will become as prevalent as, let alone replace, traditional likelihood-based or least-squares estimation. For straightforward applications of paradigmatic factor analytic, structural equation, item response, and latent class models, traditional estimation is fairly routine, accurate, and accessible via widely-available software (e.g., Mislevy and Bock [

It is readily acknowledged that MCMC is difficult, both computationally in terms of necessary resources and conceptually in terms of constructing the chains, making relevant choices, and understanding the results. As to the computing challenge, the availability of software for conducting MCMC is a burgeoning area. Programs are available for conducting MCMC for IRT, FA, SEM, and diagnostic classification models (Arbuckle [

The criticism that MCMC is conceptually difficult is somewhat ironic, given that—for complex statistical models that reflect substantively rich hypotheses—it may actually be

Nevertheless, there is no debating that a certain level of technical sophistication is required to properly conduct an MCMC analysis. To this end didactic treatments of MCMC generally (Brooks [

Further assistance in this area is provided by research that focuses on specific aspects of MCMC estimation. For example, the research community is aided by dedicated treatments and research on the complex issues of convergence assessment (see Sinharay [

The computations necessary for Bayesian procedures, including posterior predictive model checking (Gelman et al. [

This is not to assert that these approaches to data-model fit are necessarily superior to traditional approaches or are firmly established without debate. For example, posterior predictive checks (Gelman et al. [

With regard to this last point, a number of historically reoccurring features, assumptions, and beliefs about psychometric models (e.g., linear relationships, independence and normality of errors, few latent variables in IRT, few discrete observables in FA) have evolved in part from limitations on estimation routines. The flexibility of MCMC frees the analyst from the bonds associated with other estimation approaches and allows the construction of models based on substantive theory. Indeed, the lasting impact of Patz and Junker’s [

The author wishes to express his gratitude toward Bob Mislevy and John Behrens for stimulating conversations on the utility of MCMC. Thanks also to Bob Mislevy, Dubravka Svetina, and an anonymous reviewer for their helpful comments regarding earlier versions of the manuscript.