Summed Score Likelihood–Based Indices for Testing Latent Variable Distribution Fit in Item Response Theory

Abstract

In standard item response theory (IRT) applications, the latent variable is typically assumed to be normally distributed. If the normality assumption is violated, the item parameter estimates can become biased. Summed score likelihood–based statistics may be useful for testing latent variable distribution fit. We develop Satorra–Bentler type moment adjustments to approximate the test statistics’ tail-area probability. A simulation study was conducted to examine the calibration and power of the unadjusted and adjusted statistics in various simulation conditions. Results show that the proposed indices have tail-area probabilities that can be closely approximated by central chi-squared random variables under the null hypothesis. Furthermore, the test statistics are focused. They are powerful for detecting latent variable distributional assumption violations, and not sensitive (correctly) to other forms of model misspecification such as multidimensionality. As a comparison, the goodness-of-fit statistic M₂ has considerably lower power against latent variable nonnormality than the proposed indices. Empirical data from a patient-reported health outcomes study are used as illustration.

Introduction

Item response theory (IRT) provides powerful methods supporting educational and psychological measurement (Thissen & Steinberg, 2009). The latent variable in IRT models is usually assumed to follow a normal distribution for the purpose of item parameter estimation (Bock & Aitkin, 1981; Bock & Lieberman, 1970). However, this assumption might be violated in some situations (Woods, 2006; Woods & Lin, 2009). Woods (2006) described several potential situations where $\theta$ may be nonnormal. For example, as severe symptoms of psychological disorders rarely exist in the general population and most people have low levels of psychopathological symptoms, latent variables reflecting these symptoms may be positively skewed. Another possible cause arises in the situation when the population is heterogeneous. For instance, when two or more subpopulations with different means and variances are grouped together, potentially multimodal population distributions may be the result. Calibrating the items with respect to the combined population renders the normality assumption suspect. When the assumption of normal latent variable distribution is violated, the item parameter estimates might be biased, leading to bias in subsequent inferences based on these item parameter estimates. Take computer adaptive testing as an example, the item parameter estimates are used for both item selection and test scoring. Thus, bias in the estimation of item parameters might result in significant bias in the reported test scores.

Although alternative approaches exist for estimating the latent variable distribution in standard IRT models (Bock & Aitkin, 1981; Woods & Lin, 2009; Woods & Thissen, 2006), these approaches are computationally more demanding and specialized software is necessary. For example, in our experience, the empirical histogram representation of the latent prior distribution is often less stable numerically than the standard normal prior. Thus, it is worthwhile to test the assumption of latent variable normality before more “expensive” approaches are applied. Summed score likelihood–based statistics may be useful for testing latent variable distribution fit. One problem is that the statistics do not asymptotically follow a chi-squared distribution. We propose a Satorra–Bentler type moment adjustment method (Satorra & Bentler, 1994) in this article. The statistics’ tail-area probability can be approximated by making use of the item parameter error covariance matrix and a Jacobian. The properties of the adjusted and unadjusted statistics are examined by simulation and empirical studies. Additionally, a modified Lord–Wingersky algorithm for computing the Jacobian matrix is presented in the appendix.

Item Response Theory Models

In standard IRT models, the conditional item response probabilities (also referred to as item tracelines or item characteristic curves) are represented as a function of latent variable $\theta$ and item parameters. For example, the three-parameter logistic (3PL) model can be written as

$$T_i(1|\theta )=g_i+\frac{1-g_i}{1+\exp [-(c_i+a_i\theta )]},$$

where $T_i(1|\theta )$ represents item i’s traceline for the 1 category (indicating correct/endorsement response in most contexts) as a function of $\theta$. The item parameters include: $g_i$, which is the pseudoguessing probability for the item (the lower asymptote parameter); $a_i$, which is the slope (the discrimination parameter), and $c_i$, which is the item intercept parameter. The classical difficulty (threshold) parameter is obtained as$-c_i/a_i$. If $g_i$ is 0, the model reduces to a two-parameter logistic (2PL) model, and if all the item slopes are constrained to be equal to a common slope ($a_i\equiv a$), the one-parameter logistic (1PL) model is the result. The incorrect/nonendorsement response probability is equal to$T_i(0|\theta )=1-T_i(1|\theta )$.

For an item with $K_i$ ordered polytomous responses, the graded response model is often used. Let the response categories be coded as $k=0,\dots K_i-1$. The cumulative response probability for item i in categories k and above is

$$T_i^{+}(k|\theta )=\frac{1}{1+\exp [-(c_{\mathrm{ik}}+a_i\theta )]},$$

for $k=1,\dots K_i-1$. Having defined the boundary cases $T_i^{+}(0|\theta )=1$ and $T_i^{+}(K_i|\theta )=0$, the category response probabilities can be written as

$$T_i(k|\theta )=T_i^{+}(k|\theta )-T_i^{+}(k+1|\theta ),$$

for $k=0,\dots K_i-1$. Let $U_i$ be a random variable whose realization $u_i$ is a response to item i. Regardless of the number of categories, the probability mass function of $U_i$, conditional on $\theta$, is that of a multinomial with trial size 1:

$$P(U_i=u_i|\theta )=\underset{k=0}{\overset{K_i-1}{\Pi }}{\left[T_i(u_i|\theta )\right]}^{1_k\left(u_i\right)},$$

where $1_k\left(u_i\right)$ is an indicator function such that

$$1_k\left(u_i\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill \mathrm{if}k=u_i\hfill \\ \multicolumn{1}{c}{0,}& \hfill \mathrm{otherwise}\hfill \end{array}.$$

The Latent Variable Distribution in IRT

Estimating the latent variable distribution along with item parameters using the empirical histogram (Bock & Aitkin, 1981; Mislevy, 1984; Zimowski, Muraki, Mislevy, & Bock, 1996) is an established strategy for detecting and correcting latent variable nonnormality in IRT. Newer semiparametric density estimation procedures offer more efficient alternatives. These include the Ramsay Curve IRT (Woods & Thissen, 2006), and Davidian Curve IRT (Monroe & Cai, 2014; Woods & Lin, 2009), as well as its multidimensional extension (Monroe, 2014). In practice, however, estimating latent variable densities often requires specialized software. More complex latent variable distributions also involve more parameters to be estimated from the data, increasing the need for larger calibration sample sizes to achieve stable estimation. Finally, even as nonnormal latent densities may be modeled, for example, using a Ramsay curve IRT model (Woods & Thissen, 2006), and the relative model fit may be evaluated against a baseline using likelihood ratio tests, it does not circumvent the need for absolute goodness-of-fit indices to establish the adequacy of the least restrictive model in the class of models being compared (see Maydeu-Olivares & Cai, 2006, for further explanation). It would be highly desirable to establish a set of statistics that can be used to diagnose the extent to which a normal (or nonnormal) latent variable distribution may in fact be a reasonable characterization before more “expensive” methods and software programs for semiparametric density estimation are used.

In developing such a group of test statistics for latent variable distribution fit, several desiderata should be taken into account. First, the statistics should be easily computable, preferably using only standard by-products of the item calibration process. Second, the statistics should have well-grounded heuristic motivation and theoretical justification. Third, the frequency calibration of the statistics under the null hypothesis should be sufficiently accurate. Finally, the statistics should have adequate power that is focused on latent variable distribution assumption violation and sufficient diagnostic specificity, rather than becoming a surrogate of overall model fit tests.

The guiding insight has been provided elsewhere in the literature. For unidimensional IRT modeling, the observed and model-implied summed score distribution can be a basis for inferring the adequacy of the latent variable distribution specification in the IRT model (Thissen & Wainer, 2001). After model fitting, residual summed score probabilities may be used to construct chi-square test statistics. While the idea itself is not new (see, Ferrando & Lorenzo-seva, 2001; Hambleton & Traub, 1973; Lord, 1953; Ross, 1966; Sinharay, Johnson, & Stern, 2006, among others), we use the recently developed theory of limited-information goodness-of-fit testing to formally demonstrate that the summed score likelihood–based fit index proposed here belongs to the general family of multinomial limited-information tests.

The Multinomial Sampling Model and Maximum Likelihood Estimation

Let there be I items in a test. Under the conditional independence assumption, the IRT model specifies the conditional response pattern probability as the following product:

$$P({\cap }_{i=1}^I{U}_i=u_i|\theta )=\underset{i=1}{\overset{I}{\Pi }}P(U_i=u_i|\theta ).$$

Assuming that $g\left(\theta \right)$ is the distribution of the latent variable (also known as the prior distribution), the marginal response pattern probability is the following integral:

$$P({\cap }_{i=1}^I{U}_i=u_i)=\int \underset{i=1}{\overset{I}{\Pi }}P(U_i=u_i|\theta )g\left(\theta \right)d\theta ={\mathbf{\pi }}_{\mathbf{u}}\left(\mathbf{\gamma }\right),$$

where $\mathit{u}=(u_1,\dots ,u_I)$ is the response pattern, and $\mathbf{\gamma }$ is a $d\times 1$ vector that collects together the free item parameters from all I items. The parenthetical notation ${\mathbf{\pi }}_{\mathbf{u}}\left(\mathbf{\gamma }\right)$ in Equation (7) is used to emphasize the fact that it is the model. The marginal response probability depends on the item parameters, the item-level response models, and the assumed latent variable distribution.

Recall that $K_i$ is the number of categories for item i. For I items, the IRT model generates a total of $C=\underset{i=1}{\overset{I}{\Pi }}{K}_i$ cross-classifications or possible item response patterns in the form of a contingency table. Based on a sample of N respondents, let the observed proportion associated with pattern $\mathit{u}$ be denoted as $p_u$. The sampling model for this contingency table is a multinomial distribution with C cells and N trials. The multinomial log-likelihood for the item parameters $\mathbf{\gamma }$ is proportional to

$$\log L\left(\mathbf{\gamma }\right)\propto N\sum _{\mathit{u}}{p}_{\mathit{u}}\log {\mathbf{\pi }}_{\mathit{u}}\left(\mathbf{\gamma }\right),$$

where the summation is over all C response patterns. Maximization of the log-likelihood (e.g., with the expectation–maximization algorithm; Bock & Aitkin, 1981) leads to the maximum marginal likelihood estimator $\hat{\gamma }$.

On finding $\hat{\gamma }$, the IRT model generates model-implied probabilities for each response pattern ${\pi }_{\mathit{u}}\left(\hat{\gamma }\right)={\hat{\pi }}_u$. Suppose the model-implied response pattern probabilities ${\hat{\pi }}_u$ are collected into a $C\times 1$ vector $\hat{\pi }$ of all model-implied response pattern probabilities. By analogy, let a $C\times 1$ vector $\mathbf{\pi }$ contain the true (population) response pattern probabilities. Similarly, the observed proportions $p_{\mathit{u}}$ can be collected into a $C\times 1$ vector p. For example, for three dichotomously scored items there are $2^3=8$ item response patterns, and the response pattern probabilities and observed proportions are

$$\mathbf{\pi }\mathbf{=}\mathbf{\left(}\begin{array}{c}\hfill {\mathbf{\pi }}_{000}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{001}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{010}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{011}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{100}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{101}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{110}}\\ \multicolumn{1}{c}{{\mathbf{\pi }}_{111}}\end{array}}\end{array}}\end{array}}\end{array}\mathbf{\right)}\mathbf{,}\mathbf{\hspace{1em}}\hat{\mathbf{\pi }}\mathbf{=}\mathbf{\left(}\begin{array}{c}\hfill {\hat{\mathbf{\pi }}}_{000}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\hat{\mathbf{\pi }}}_{001}\hfill \\ \multicolumn{1}{c}{{\hat{\mathbf{\pi }}}_{010}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\hat{\mathbf{\pi }}}_{011}\hfill \\ \multicolumn{1}{c}{{\hat{\mathbf{\pi }}}_{100}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\hat{\mathbf{\pi }}}_{101}\hfill \\ \multicolumn{1}{c}{{\hat{\mathbf{\pi }}}_{110}}\\ \multicolumn{1}{c}{{\hat{\mathbf{\pi }}}_{111}}\end{array}}\end{array}}\end{array}}\end{array}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\begin{array}{c}\hfill {\mathbf{\pi }}_{000}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{001}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{010}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{011}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{100}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{101}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{110}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}}\\ \multicolumn{1}{c}{{\mathbf{\pi }}_{111}\mathbf{\left(}\hat{\mathbf{\gamma }}\mathbf{\right)}}\end{array}}\end{array}}\end{array}}\end{array}\mathbf{\right)}\mathbf{,}\mathbf{\hspace{1em}}\mathit{p}\mathbf{=}\mathbf{\left(}\begin{array}{c}\hfill {\mathbf{p}}_{000}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{p}}_{001}\hfill \\ \multicolumn{1}{c}{{\mathbf{p}}_{010}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{p}}_{011}\hfill \\ \multicolumn{1}{c}{{\mathbf{p}}_{100}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{p}}_{101}\hfill \\ \multicolumn{1}{c}{{\mathbf{p}}_{110}}\\ \multicolumn{1}{c}{{\mathbf{p}}_{111}}\end{array}}\end{array}}\end{array}}\end{array}\mathbf{\right)}\mathbf{.}$$

From results in discrete multivariate analysis (e.g., Bishop, Fienberg, & Holland, 1975), $\hat{\gamma }$ is consistent, asymptotically normal, and asymptotically efficient, which can be summarized as follows:

$$\sqrt{N}(\hat{\gamma }-\mathbf{\gamma }){\to }^D{\mathcal{N}}_d(0,{\mathcal{F}}^{-1}),$$

where $\mathcal{F}=\mathbf{\Delta }\prime {\left[\mathit{diag}\left(\mathbf{\pi }\right)\right]}^{-1}\mathbf{\Delta }$ is the $d\times d$ Fisher information matrix, with the Jacobian matrix $\mathbf{\Delta }$ defined as the $C\times d$ matrix of all first-order partial derivatives of the response patterns probabilities with respect to the item parameters:

$$\mathbf{\Delta }=\frac{\partial \mathbf{\pi }\mathbf{\left(}\mathbf{\gamma }\mathbf{\right)}}{\partial \mathbf{\gamma }\prime }.$$

Distribution of Residuals Under Maximum Likelihood Estimation

Based on Equation (10), it can be shown that the asymptotic distribution of the difference $(\mathit{p}-\mathbf{\pi })$ is C-variate normal:

$$\sqrt{N}(\mathit{p}-\mathbf{\pi }){\to }^D{\mathcal{N}}_C(0,\mathbf{\Xi }),$$

where $\mathbf{\Xi }=\mathrm{diag}\left(\mathbf{\pi }\right)-\mathbf{\pi }\mathbf{\pi }\mathbf{\text{'}}$ is the covariance matrix associated with the multinomial. The residual vector $(\mathit{p}-\hat{\pi })$ is asymptotically C-variate normal under maximum likelihood estimation:

$$\sqrt{N}(\mathit{p}-\hat{\pi }){\to }^D{\mathcal{N}}_C(0,\mathbf{\Gamma }),$$

where $\mathbf{\Gamma }=\mathbf{\Xi }-\mathbf{\Delta }{\mathcal{F}}^{-1}\mathbf{\Delta }\prime$, and the second term reflects variability due to estimation of item parameters.

Lower Order Marginal Probabilities

The IRT model implies marginal probabilities. Consider the three-item example from above. There are three first-order marginal probabilities ${\stackrel{·}{\pi }}_i(i=1,\dots ,3)$, one per item. There are also three second-order marginal probabilities ${\stackrel{··}{\pi }}_{\mathrm{ij}}$ for the unique item pairs $(1\le j<i\le 3)$. In general, these probabilities correspond to the I univariate and $I(I-1)/2$ bivariate margins that can be obtained from the full C-dimensional contingency table using a reduction operator matrix (see e.g., Maydeu-Olivares & Joe, 2005). An example is given below:

$${\hat{\pi }}_2=\left(\begin{array}{c}\hfill {\stackrel{·}{\pi }}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\pi }}_2}\\ \multicolumn{1}{c}{{\stackrel{·}{\pi }}_3}\\ \multicolumn{1}{c}{{\stackrel{··}{\pi }}_{21}}\\ \multicolumn{1}{c}{{\stackrel{··}{\pi }}_{31}}\\ \multicolumn{1}{c}{{\stackrel{··}{\pi }}_{32}}\end{array}\right)=\mathbf{L}\hat{\pi }=\left(\begin{array}{cccccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\hat{\pi }}_{000}\hfill \\ \multicolumn{1}{c}{{\hat{\pi }}_{001}}\\ \multicolumn{1}{c}{{\hat{\pi }}_{010}}\\ \multicolumn{1}{c}{{\hat{\pi }}_{011}}\\ \multicolumn{1}{c}{{\hat{\pi }}_{100}}\\ \multicolumn{1}{c}{{\hat{\pi }}_{101}}\\ \multicolumn{1}{c}{{\hat{\pi }}_{110}}\\ \multicolumn{1}{c}{{\hat{\pi }}_{111}}\end{array}\right)$$

where L is a fixed operator matrix of 0s and 1s that reduces the response pattern probabilities and proportions into marginal probabilities and proportions up to order 2. ${\hat{\pi }}_2$ is the vector of first- and second-order marginal probabilities. Correspondingly ${\mathit{p}}_2=\mathbf{L}\mathit{p}$ is the vector of first- and second-order observed marginal proportions.

More general versions of the reduction operator matrices for multiple categorical IRT models can be derived using similar logic (see, e.g., Cai & Hansen, 2013; Maydeu-Olivares & Joe, 2006). Note that L has full row rank. It implies that the marginal residual vector $({\mathit{p}}_2-{\hat{\pi }}_2)=\mathbf{L}(\mathit{p}-\hat{\pi })$ is a full-rank linear transformation of the multinomial residual vector $(\mathit{p}-\hat{\pi })$. Therefore, the marginal residual vector $({\mathit{p}}_2-{\hat{\pi }}_2)$ is asymptotically normal:

$$\sqrt{N}({\mathit{p}}_2-{\hat{\pi }}_2)=\sqrt{N}\mathbf{L}(\mathit{p}-\hat{\pi }){\to }^D{\mathcal{N}}_Q(0,{\mathbf{\Gamma }}_2),$$

and ${\mathbf{\Gamma }}_2=\mathbf{L}\mathbf{\Gamma }\mathbf{L}\mathbf{\text{'}}=\mathbf{L}\mathbf{\Xi }\mathbf{L}\mathbf{\text{'}}-\mathbf{L}\mathbf{\Delta }{\mathcal{F}}^{-1}\mathbf{\Delta }\mathbf{\text{'}}\mathbf{L}\mathbf{\text{'}}={\mathbf{\Xi }}_2-{\mathbf{\Delta }}_2{\mathcal{F}}^{-1}{\mathbf{\Delta }}_2^{\prime }$, where ${\mathbf{\Xi }}_2=\mathbf{L}\mathbf{\Xi }\mathbf{L}\mathbf{\text{'}}$, and ${\mathbf{\Delta }}_2=\mathbf{L}\mathbf{\Delta }$ is the Jacobian for the marginal probabilities. The dimensionality Q of the normal random variable is equal to the number of first- and second-order marginal residuals. For example, in the case of dichotomous items, the number is $Q=I+I(I-1)/2=I(I+1)/2$.

Summed Score Probabilities

In addition to the response pattern and marginal probabilities, the IRT model also generates model-implied summed score probabilities. For a test with I items and $k=0,\dots ,K_i-1$ coded categories for item i, there are a total of $S=1+\sum _{i=1}^I(K_i-1)$ summed scores ranging from 0 to $S-1$. Suppose the observed summed probabilities based on a sample of size N are equal to ${\overline{p}}_s$ for $s=0,\dots ,S-1$. Under maximum likelihood estimation of item parameters, the corresponding IRT model-implied summed score probabilities are formally defined as

$${\overline{\pi }}_s=\sum _{\mathit{u}}{1}_s\left(\mathit{u}\right){\hat{\pi }}_{\mathit{u}},$$

where $\mathit{u}=\sum _{i=1}^I{u}_i$ is a notational shorthand for the summed score associated with response pattern u, and the indicator function takes a value of 1 if and only if $s=\mathit{u}$:

$$1_s\left(\mathit{u}\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill \mathrm{if}s=\mathit{u}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \mathrm{otherwise}\hfill \end{array}.$$

Equation (16) shows that the IRT model-implied probability for summed score s is a sum over all such response pattern probabilities leading to summed scores, in other words, it may also be obtained by a reduction operator matrix.

Let S be a matrix of fixed 0s and 1s such that the premultiplication of π by S yields the summed score probabilities. Each row of S can be understood as a set of binary logical relations. An element in row j of S is equal to 1 if and only if the corresponding response pattern in π leads to summed score $j-1$. In general, for I items, there are S rows and C columns in S. In particular, S has full row rank and the rows of S are mutually orthogonal.

Returning to the three-item example, there are four summed scores in this case: 0, 1, 2, and 3. The $4\times 8$ matrix S (below) relates the summed score probabilities to the original multinomial probabilities:

$$\overline{\pi }=\left(\begin{array}{c}\hfill {\overline{\pi }}_0\hfill \\ \multicolumn{1}{c}{{\overline{\pi }}_1}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\overline{\pi }}_2\hfill \\ \multicolumn{1}{c}{{\overline{\pi }}_3}\end{array}}\end{array}\right)=\mathbf{S}\mathbf{\pi }\mathbf{=}\mathbf{\left(}\begin{array}{cc}\hfill \begin{array}{cc}\hfill \begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\hfill & \hfill \begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}}& \hfill \begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}\hfill \end{array}\hfill & \hfill \begin{array}{cc}\hfill \begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill \end{array}\hfill & \hfill \begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}}& \hfill \begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\hfill \end{array}\hfill \end{array}\mathbf{\right)}\mathbf{\left(}\begin{array}{c}\hfill {\mathbf{\pi }}_{000}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{001}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{010}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{011}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{100}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\mathbf{\pi }}_{101}\hfill \\ \multicolumn{1}{c}{{\mathbf{\pi }}_{110}}\\ \multicolumn{1}{c}{{\mathbf{\pi }}_{111}}\end{array}}\end{array}}\end{array}}\end{array}\mathbf{\right)}\mathbf{,}\mathbf{\hspace{1em}}\widehat{\overline{\mathbf{\pi }}}\mathbf{=}\mathbf{\left(}\begin{array}{c}\hfill {\widehat{\overline{\mathbf{\pi }}}}_0\hfill \\ \multicolumn{1}{c}{{\widehat{\overline{\mathbf{\pi }}}}_1}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\widehat{\overline{\mathbf{\pi }}}}_2\hfill \\ \multicolumn{1}{c}{{\widehat{\overline{\mathbf{\pi }}}}_3}\end{array}}\end{array}\mathbf{\right)}\mathbf{=}\mathbf{S}\hat{\mathbf{\pi }}\mathbf{.}$$

The observed summed score proportions can be obtained in a similar way:

$$\overline{\mathit{p}}=\left(\begin{array}{c}\hfill {\overline{\mathit{p}}}_0\hfill \\ \multicolumn{1}{c}{{\overline{\mathit{p}}}_1}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\overline{\mathit{p}}}_2\hfill \\ \multicolumn{1}{c}{{\overline{\mathit{p}}}_3}\end{array}}\end{array}\right)=\mathbf{S}\mathit{p}=\left(\begin{array}{cc}\hfill \begin{array}{cc}\hfill \begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\hfill & \hfill \begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}}& \hfill \begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}\hfill \end{array}\hfill & \hfill \begin{array}{cc}\hfill \begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill \end{array}\hfill & \hfill \begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill \end{array}}& \hfill \begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill p_{000}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill p_{001}\hfill \\ \multicolumn{1}{c}{p_{010}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill p_{011}\hfill \\ \multicolumn{1}{c}{p_{100}}\\ \multicolumn{1}{c}{\begin{array}{c}\hfill p_{101}\hfill \\ \multicolumn{1}{c}{p_{110}}\\ \multicolumn{1}{c}{p_{111}}\end{array}}\end{array}}\end{array}}\end{array}\right).$$

From Equation (13), under maximum likelihood estimation, the summed score residual vector $\overline{\mathit{p}}-\hat{\overline{\pi }}$ is asymptotically S-variate normally distributed:

$$\sqrt{N}(\overline{\mathit{p}}-\hat{\overline{\pi }})=\sqrt{N}(\mathit{Sp}-\mathit{S}\hat{\pi })=\sqrt{N}\mathbf{S}(\mathit{p}-\hat{\pi }){\to }^D{\mathcal{N}}_S(0,\overline{\mathbf{\Gamma }}),$$

and $\overline{\mathbf{\Gamma }}=\mathbf{S}\mathbf{\Gamma }\mathbf{S}\mathbf{\text{'}}=\mathbf{S}\mathrm{diag}\left(\mathbf{\pi }\right)\mathbf{S}\mathbf{\text{'}}-\mathbf{S}\mathbf{\pi }\mathbf{\pi }\mathbf{\text{'}}\mathbf{S}\mathbf{\text{'}}\mathbf{-}\mathbf{S}\mathbf{\Delta }{\mathcal{F}}^{\mathbf{-}1}\mathbf{\Delta }\mathbf{\text{'}}\mathbf{S}\mathbf{\text{'}}\mathbf{=}\mathbf{diag}\mathbf{\left(}\overline{\mathbf{\pi }}\mathbf{\right)}\mathbf{-}\overline{\mathbf{\pi }}{\overline{\mathbf{\pi }}}^{\mathbf{\text{'}}}\mathbf{-}\overline{\mathbf{\Delta }}{\mathcal{F}}^{\mathbf{-}1}{\overline{\mathbf{\Delta }}}^{\mathbf{\text{'}}}$, with $\overline{\mathbf{\Delta }}=\mathbf{S}\mathbf{\Delta }$.

The reason for introducing the reduction operator matrix S is primarily a theoretical one. It facilitates the subsequent derivations of summed score likelihood–based indices for testing latent variable distribution fit. Pragmatically, the Lord–Wingersky (1984) algorithm should be used to compute the model-implied summed score probabilities. If summed score to scale score conversion tables are computed (see, Thissen & Wainer, 2001), the probabilities become automatic by-products.

Goodness-of-Fit Statistics for IRT models

Existing overall goodness-of-fit indices may be used for testing latent variable distribution fit in IRT. The full-information test statistics such as likelihood ratio G² and Pearson’s X² use residuals based on the full response pattern cross-classifications to test the IRT model against the general multinomial alternative. The comparison between ${\hat{\pi }}_{\mathit{u}}$ and $p_{\mathit{u}}$ (on logarithmic or linear scales) leads to well-known goodness-of-fit statistics such as the likelihood ratio G² and Pearson’s X²:

$$G^2=2N\sum _u{p}_{\mathit{u}}\log \frac{p_{\mathit{u}}}{{\hat{\pi }}_{\mathit{u}}},X^2=N\sum _{\mathit{u}}\frac{{(p_{\mathit{u}}-{\hat{\pi }}_u)}^2}{{\hat{\pi }}_u}.$$

Under the null hypothesis that the IRT model fits exactly, these two statistics have the same asymptotic reference distribution, which is a central chi-square with degrees of freedom (df) equal to $C-1-d$ (Bishop et al., 1975). For subsequent development, it is instructive to rewrite Pearson’s statistic as a quadratic form in multinomial residuals: $X^2=N{(\mathit{p}-\hat{\pi })}^{\prime }{\left[\mathrm{diag}\left(\hat{\pi }\right)\right]}^{-1}(\mathit{p}-\hat{\pi }).$

Unfortunately, as the number of items increases, the number of response patterns increases exponentially. For more than a dozen or so dichotomous items (or perhaps a handful of polytomous items), the contingency table on which the multinomial is defined becomes sparse for any realistic N. Consequently, the asymptotic chi-square approximations for the full-information test statistics break down (see, e.g., Bartholomew & Tzamourani, 1999) and the utility of the full-information overall goodness-of-fit indices for routine IRT applications becomes questionable.

Recently, limited-information overall fit statistics such as Maydeu-Olivares and Joe’s (2005)M₂ have been developed. Limited-information fit statistics use residuals based on lower order (e.g., first and second order) margins of the contingency table. These lower order margins are far better filled when compared with the sparse full contingency table. There is growing awareness that limited-information tests can maintain correct size and can be more powerful than the full-information tests (Cai, Maydeu-Olivares, Coffman, & Thissen, 2006; Joe & Maydeu-Olivares, 2010).

Under the assumption that the number of first- and second-order margins is larger than the number of free parameters $(Q>d)$ and that ${\mathbf{\Delta }}_2$ has full column rank (local identification), $M_2$ can be written as

$$M_2=N{({\mathit{p}}_2-{\hat{\pi }}_2)}^{\prime }{\tilde{\mathbf{\Delta }}}_2{\left[{\tilde{\mathbf{\Delta }}\prime }_2{\mathbf{\Xi }}_2{\tilde{\mathbf{\Delta }}}_2\right]}^{-1}{\tilde{\mathbf{\Delta }}\prime }_2({\mathit{p}}_2-{\hat{\pi }}_2),$$

where ${\tilde{\mathbf{\Delta }}}_2$ is a $Q\times (Q-d)$ orthogonal complement of ${\mathbf{\Delta }}_2$ such that ${\tilde{\mathbf{\Delta }}\prime }_2{\mathbf{\Delta }}_2=0$. From Equation (15), $({\mathit{p}}_2-{\hat{\pi }}_2)$ is asymptotically normal with zero means and covariance matrix ${\mathbf{\Xi }}_2-{\mathbf{\Delta }}_2{\mathcal{F}}^{-1}{\mathbf{\Delta }\prime }_2$, which implies that the covariance matrix of ${\tilde{\mathbf{\Delta }}}_2^{\prime }({\mathit{p}}_2-{\hat{\pi }}_2)$ is ${\tilde{\mathbf{\Delta }}\prime }_2{\mathbf{\Xi }}_2{\tilde{\mathbf{\Delta }}}_2$. Thus, $M_2$ is asymptotically chi-square distributed with $Q-d$ degrees of freedom. In the current simulation study, $M_2$ will be used as a benchmark because of its numerous desirable properties identified in the literature (see, e.g., Cai & Hansen, 2013). Performance of the proposed latent variable distribution fit indices will be evaluated against $M_2$.

While an overall test may be used to detect specification errors of latent variable distributions, the fact that they are also sensitive to other forms of model error (e.g., unmodeled multidimensionality) makes it difficult to pinpoint the source of misspecification. To that end, more specific diagnostic indices have been created for IRT. For example, Chen and Thissen’s (1997) local dependence indices are particularly sensitive to violations of the local independence assumption. Orlando and Thissen’s (2000) item fit diagnostics is another example where the extent to which the IRT model fits the empirical operating characteristics for an item (e.g., whether monotonicity holds) can be examined. The next section develops a set of indices that specifically target latent variable distribution fit for IRT models.

The Summed Score Likelihood–Based Indices and Statistical Adjustments

There are two important lines of reasoning for the derivation of these model fit indices. The first is a recognition based on heuristics: IRT model–implied summed score probabilities may provide useful diagnostic information about the latent variable distributional assumption (Thissen & Wainer, 2001). The second recognition is that the summed score likelihood–based indices are formally limited-information test statistics.

A Heuristic Motivation

When the latent variable distribution assumed in the IRT model does not represent the population distribution of the respondents adequately, the model-implied summed score probabilities ${\overline{\pi }}_s$ will depart from the observed summed score probabilities ${\overline{p}}_s$. Hence all that is needed is to find appropriate test statistics that can summarize the degree to which the model-implied and observed summed score probabilities diverge. It is also preferable if the indices are approximately chi-square distributed test statistics. Pearson’s X² introduced in the previous section meets this requirement.

Recall that the total number of summed scores is $S=1+\sum _{i=1}^I(K_i-1)$. The Pearson-type ${\overline{X}}^2$ below yields a direct comparison between the model-implied summed score probabilities ${\hat{\overline{\pi }}}_s$ and the observed summed score probabilities ${\overline{p}}_s$:

$${\overline{X}}^2=N\sum _{s=0}^{S-1}\frac{{({\overline{p}}_s-{\hat{\overline{\pi }}}_s)}^2}{{\hat{\overline{\pi }}}_s},$$

where ${\overline{p}}_s$ and ${\overline{\pi }}_s$ represent the observed and model-implied summed score probability for score s, respectively. This test statistic is different from the full-information test statistic shown in Equation (21) because it is based on summed score probabilities as opposed to response pattern probabilities.

In preliminary studies (Li & Cai, 2012) we had conjectured that under a wide variety of conditions ${\overline{X}}^2$ may have similar asymptotic distributions whose tail-area probabilities can be approximated by a central chi-squared random variable with $S-1-2$ degrees of freedom under the null hypothesis that the latent variable distribution $g\left(\theta \right)$ is correctly specified in the IRT model. This conjecture will be tested in the sequel with simulations.

The rationale behind the specific degrees of freedom is as follows: The S summed scores’ probabilities must sum to 1. The first minus 1 is to reflect that constraint. Had the item parameters been known, the degrees of freedom would have been exactly $S-1$. When the item parameters are estimated (assuming with maximum marginal likelihood), an additional penalty must be introduced to reflect the effect of parameter estimation. While the location and scale of the latent variable $\theta$ are typically fixed for model identification, the model-implied summed score distribution does not have an inherent location and scale. The location and scale is determined as a result of estimating the item parameters. Hence, the estimation of item parameters amounts to adding at least two more constraints for the model-implied summed score probability distribution. The details are of course more complex, and will be explained next.

A More Formal Derivation

While the proposed test statistics are not associated with particular marginal probabilities in the same manner as Maydeu-Olivares and Joe’s (2005)M₂, they are nevertheless related to the response pattern probabilities via the reduction operator matrix S defined earlier (see Equations [18]). It is the choice of this particular reduction operator that leads to more focused tests targeting latent variable distribution fit (see Joe & Maydeu-Olivares, 2010). For IRT models with constrained equal item discrimination parameters (e.g., the 1PL model), it is widely recognized that the summed scores are sufficient statistics for the latent variables in the model. Though the summed score sufficiency property does not hold for other IRT models such as the 2PL or the graded model, researchers have nevertheless found that summed score is an important source of information regarding the ordering of individuals along the latent variable continuum (e.g., van der Ark, 2005). One could even base parameter estimation on summed score groups (Chen & Thissen, 1999).

Using the reduction operator S, the derivations above imply that the Pearson-type statistic ${\overline{X}}^2$ can be rewritten as

$${\overline{X}}^2=N\sum _{s=0}^{S-1}\frac{{({\overline{p}}_s-{\hat{\overline{\pi }}}_s)}^2}{{\hat{\overline{\pi }}}_s}=N(\overline{\mathit{p}}-\hat{\overline{\pi }})\prime {\left[\mathrm{diag}\left(\hat{\overline{\pi }}\right)\right]}^{-1}(\overline{\mathit{p}}-\hat{\overline{\pi }}),$$

where $(\overline{\mathit{p}}-\hat{\overline{\pi }})=\mathbf{S}(\mathit{p}-\hat{\pi })$ is the summed score residual vector (see Equation [20]). Under the null hypothesis that the IRT model is correctly specified, one can obtain the probability limit of the weight matrix as $\mathrm{plim}\left({\left[\mathrm{diag}\left(\hat{\overline{\pi }}\right)\right]}^{-1}\right)={\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}$ by the consistency of the maximum likelihood estimator (see Equation 9), the continuity of the mapping from $\mathbf{\gamma }$ to the summed score probabilities, and the continuity of the matrix inverse. Following results on quadratic forms of random vectors (e.g., Mathai & Provost, 1992), the asymptotic expected value of ${\overline{X}}^2$ is equal to

$$\mathrm{tr}\left\{\overline{\mathbf{\Gamma }}{\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}\right\}=\mathrm{tr}\left\{[\mathrm{diag}\left(\overline{\pi }\right)-\overline{\pi }\overline{\pi }\prime ]{\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}\right\}-\mathrm{tr}(\overline{\mathbf{\Delta }}{\mathcal{F}}^{-1}\overline{\mathbf{\Delta }}\prime {\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1})=S-1-\mathrm{tr}\{{\mathcal{F}}^{-1}\overline{\mathbf{\Delta }}\prime {\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}\overline{\mathbf{\Delta }}\}={\mu }_1.$$

From Equations (24) and (25) we can see that the statistic ${\overline{X}}^2$cannot be asymptotically chi-square distributed. Even though it is a quadratic form in asymptotically normally distributed random vectors, a key condition for its chi-squaredness is not met. That is, the product of the probability limit of the weight matrix ${\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}$ and the covariance matrix of the normal random vector $\overline{\mathbf{\Gamma }}$ is not idempotent in general, that is, $\overline{\mathbf{\Gamma }}{\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}\overline{\mathbf{\Gamma }}{\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}\ne \overline{\mathbf{\Gamma }}{\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}$. On the other hand, Equation (25) shows that the asymptotic expected value of ${\overline{X}}^2$ is equal to $S-1$ minus a constant that depends on the trace of ${\mathcal{F}}^{-1}\overline{\mathbf{\Delta }}\prime {\left[\mathrm{diag}\left(\overline{\pi }\right)\right]}^{-1}\overline{\mathbf{\Delta }}$, which reflects additional uncertainty due to estimation of item parameters. With the first-order moment of ${\overline{X}}^2$, the Satorra–Bentler type moment adjustment approaches can be applied to adjust the statistic, so that the tail area of its distribution can be better approximated by a chi-square distribution (Cai et al., 2006; Satorra & Bentler, 1994).

Adjustment of Statistics

According to Satorra and Bentler’s (1994) article, test statistics that do not asymptotically follow a chi-squared distribution can be corrected, by matching the mean (or the mean & variance) to fixed degrees of freedom. Let $\mathrm{df}$ indicate the degrees of freedom of interest, and ${\mu }_1$ indicate the asymptotic expected value of ${\overline{X}}^2$. The moment-adjusted statistic is

$${\overline{X}}_C^2={\overline{X}}^2{\left(\frac{{\mu }_1}{\mathrm{df}}\right)}^{-1}.$$

Theoretically, the constant df can take on an arbitrary value. For the purpose of comparison, df will take the value of $S-1-2$ in this article.

One challenge to obtain the adjusted statistics is calculating the first-order moment in Equation (25). Some commercial software for IRT (e.g., flexMIRT®; Cai, 2013) provides the Fisher information matrix $\mathcal{F}$ and the model-implied summed score probabilities $\overline{\pi }$ in the output file, but currently none of them produces the Jacobian matrix $\overline{\mathbf{\Delta }}$. Numerical calculation of the Jacobian matrix can be computationally demanding, especially when the number of items (n) is large. Take the 2PL IRT model as an example. It requires the computation of $2\times 2^n$ first-order derivatives to obtain $\overline{\mathbf{\Delta }}$. For a test of 12 items, 8,192 first-order derivatives need to be computed. When n increases to 24, that requires 33,554,432 first-order derivatives to be computed. To solve this problem, a modification of Lord–Wingersky algorithm (Lord & Wingersky, 1984) for calculating the Jacobian matrix is developed (see the appendix). Once the Jacobian matrix is computed, the first-order moments of ${\overline{X}}^2$ can be computed.

Simulations

Simulations were undertaken to evaluate the summed score likelihood–based indices ${\overline{X}}^2$ and ${\overline{X}}_C^2$, by comparing them with Maydeu-Olivares and Joe’s $M_2$. There were 108 conditions (2*2*3*3*3), with 1,000 replications in each condition (Table 1). Manipulated factors were the IRT model type (2PL or graded), the number of items (12 or 24), the sample size (500, 1,000, or 1,500), dispersion of item parameters (equal, random, or dispersed), and the distribution of latent variable (unidimensional normal, unidimensional nonnormal, or multidimensional multivariate normal).

Table 1.

Manipulated Factors and Conditions for Simulation Study.

Factor (levels)	Conditions
Types of IRT model (2)	2PL, graded
Number of items (2)	12, 24
Sample size (3)	500, 1,000, 1,500
Values of item parameters (3)	Equal slopes and equal intercepts
	Random slopes and random intercepts
	Dispersed slopes and dispersed intercepts
Latent variable distribution (3)	Normally distributed unidimensional
	Nonnormally distributed unidimensional
	Correlated bivariate normally distributed

Note. The factors are fully crossed in a $2\times 2\times 3\times 3\times 3$ design with 1,000 attempted replications per cell.

In the null condition, response pattern data were simulated with a latent variable having unidimensional normal distribution. In the alternative conditions, response pattern data were simulated either with a nonnormally distributed latent variable or with a bivariate normally distributed latent variable. The nonnormal $\theta$s were generated from a distribution obtained from a 1:4 mixture of two normally distributed densities ( $M_1=1,S{D}_1=0.4$; $M_2=0,S{D}_2=1$). The multidimensional $\theta$ distribution is standard bivariate normal with correlation equal to 0.9, representing substantial overlap between the two dimensions. Half of the items loaded on each dimension in a pure between-item multidimensional model. In other words, each item is only directly influenced by a single dimension, but the dimensions are correlated.

There were three conditions for item parameters. For the “Equal Slopes and Equal Intercepts” condition, all the slope parameters are fixed to 1, and all the intercept parameters are fixed to 0. For the “Random Slopes and Random Intercepts” condition, parameters for 24 items were randomly generated with properties mimicking standard educational and psychological assessments. Discrimination (a) parameters were drawn from a log-normal distribution $(M=0.5,\mathrm{SD}=0.2)$, the threshold values (b) were drawn from a normal distribution $(M=0,\mathrm{SD}=0.75)$, the intercepts (c) were calculated as (−ab). Parameters for the first 12 items were used for shorter tests. For the “Dispersed Slopes and Dispersed Intercepts” condition, item slope parameters were designed to spread from 1 to 3 in equal increments, while item thresholds spread from −2 to 2 across the 12 or 24 items.

The fitted models were standard unidimensional IRT models. In the null conditions, the data-generating models and the fitted models were the same. In the alternative conditions, the fitted models were misspecified for ignoring either latent variable nonnormality or multidimensionality. Bock and Aitkin’s (1981) expectation–maximization algorithm was used to obtain maximum likelihood estimates, and the Lord–Wingersky (1984) algorithm was used to compute the model-implied summed score probabilities.

To compare the performance of the fit statistics, empirical Type I Error rates were computed in the null conditions, and empirically observed power were computed in the alternative conditions at three alpha levels: .01, .05, and .10. In addition, another model fit index, Maydeu-Olivares and Joe’s $M_2$ was used as a benchmark.

Results

Type I Error Rates

Tables 2 and 3 present the simulation study results for the unidimensional normal case under the null hypothesis. The extent to which the tail areas of the proposed statistics’ distribution are well approximated is examined by comparing the observed Type I error rates against the nominal alpha levels. The results indicate that, when the slope and threshold parameters are equal across items, the adjusted and unadjusted summed score likelihood–based indices both work well. Empirical rejection rates and their corresponding alpha levels are close to each other. However, when the item parameters become dispersed, the adjusted statistic ${\overline{X}}_C^2$ performs better than the unadjusted statistic ${\overline{X}}^2$. These results hold across different numbers of items and different sample sizes.

Table 2.

Selected Simulation Results Under the Null Hypothesis: Normally Distributed Unidimensional Latent Variable in 2PL Models.

n	N	Index	df	Equal slopes and intercepts					Random slopes and intercepts					Dispersed slopes and intercepts
				M	Var	$\mathrm{ERR}$ a		KS	M	Var	$\mathrm{ERR}$ a		KS	M	Var	$\mathrm{ERR}$ a		KS
						.01	.05	p			.01	.05	p			.01	.05	p
12	500	${\overline{X}}^2$	10	10.0	19.7	.01	.04	.94	9.2	18.1	.01	.04	.00	9.1	18.0	.01	.03	.00
		${\overline{X}}_C^2$	10	10.0	19.9	.01	.05	.98	10.1	21.5	.01	.06	.51	10.4	23.4	.02	.06	.08
		$M_2$	54	54.3	119	.02	.06	.33	54.5	112.8	.01	.06	.15	54.2	116.4	.01	.06	.25
12	1,500	${\overline{X}}^2$	10	10.0	18.8	.01	.05	.64	9.3	17.2	.01	.03	.00	8.9	15.9	.01	.03	.00
		${\overline{X}}_C^2$	10	10.0	18.9	.01	.05	.49	10.1	20.1	.01	.05	.96	10.0	20.2	.01	.04	1.00
		$M_2$	54	53.8	112.9	.01	.06	.83	53.8	101.2	.01	.04	.79	53.8	107.0	.01	.05	.36
24	1,500	${\overline{X}}^2$	22	21.8	45.3	.01	.06	.15	21.5	40.2	.01	.04	.03	21.4	44.8	.01	.05	.03
		${\overline{X}}_C^2$	22	21.9	45.5	.01	.06	.26	22.1	42.3	.01	.05	.75	22.2	48.0	.02	.06	.70
		$M_2$	252	252.9	483.3	.02	.05	.20	251.9	502.2	.01	.05	.81	251.6	526.7	.01	.04	.55

^aEmpirical rejection rates (ERRs) at α levels .01 and .05.

Table 3.

Selected Simulation Results Under the Null Hypothesis: Normally Distributed Unidimensional Latent Variable in Graded Models.

n	N	Index	d	Equal slopes and intercepts					Random slopes and intercepts					Dispersed slopes and intercepts
				M	Var	$\mathrm{ERR}$ a		KS	M	Var	$\mathrm{ERR}$ a		KS	M	Var	$\mathrm{ERR}$ a		KS
						.01	.05	p			.01	.05	p			.01	.05	p
12	500	${\overline{X}}^2$	34	34.5	69.9	.01	.06	.07	34.1	65.7	.01	.05	.67	33.8	65.6	.01	.04	.97
		${\overline{X}}_C^2$	34	34.6	70.3	.01	.06	.02	34.4	66.8	.01	.05	.12	34.5	68.4	.01	.06	.04
		$M_2$	30	29.5	60.4	.01	.05	.06	29.8	59.0	.01	.05	.63	29.7	60.0	.01	.05	.17
12	1,500	${\overline{X}}^2$	34	33.8	69.0	.01	.04	.64	33.4	59.6	.01	.03	.01	33.5	61.8	.01	.03	.21
		${\overline{X}}_C^2$	34	33.9	69.3	.01	.04	.75	33.8	60.9	.01	.04	.10	34.4	65.0	.01	.05	.26
		$M_2$	30	31.8	80.6	.02	.08	.00	29.8	56.6	.01	.04	.77	30.1	58.6	.01	.04	.24
24	1,500	${\overline{X}}^2$	70	70.9	147.1	.01	.06	.03	70.3	136.0	.01	.05	.46	69.9	138.6	.01	.05	.36
		${\overline{X}}_C^2$	70	70.9	147.3	.01	.06	.03	70.5	137.1	.01	.06	.24	70.4	140.7	.01	.05	.15
		$M_2$	204	204.3	435.9	.01	.06	.58	202.0	369.9	.01	.03	.00	204.6	414.6	.01	.05	.29

^aEmpirical rejection rates (ERRs) at α levels .01 and .05.

Furthermore, as suggested earlier, the observed means of these indexes should be close to the expected values of the approximating chi-squared distributions (the degrees of freedom). The results in Tables 2 and 3 confirm that when item parameters are equal, the means are close to the degrees of freedom, and the variance is approximately twice the degrees of freedom. Notice that when the number of items or the sample size increases, the results improve. For the 2PL model, when the item parameters are dispersed, the moment-adjusted statistic ${\overline{X}}_C^2$ improves on ${\overline{X}}^2$ with a heuristic degrees of freedom. However, for the graded model, both ${\overline{X}}^2$ and the adjusted statistic ${\overline{X}}_C^2$ perform well in the null condition. In addition, Maydeu-Olivares and Joe’s M₂ appears to be well calibrated for the conditions we tested.

Power

From Tables 4 and 5, it is clear that the summed score likelihood–based indices have substantially higher power than $M_2$ when the latent variable distribution is nonnormal. The performance of the proposed statistics is heavily influenced by the number of items and dispersion of item parameters. For both 2PL and graded models, the power of the proposed indexes grow as the sample size and number of items increase. This is to be expected as more data bring more information about the latent variable distribution. When the item slope and threshold parameters are equal across items, the unadjusted and adjusted statistics perform equally well. However, when the item parameters are dispersed, the adjusted statistic ${\overline{X}}_C^2$ has higher power than the unadjusted statistic ${\overline{X}}^2$. Finally, Tables 6 and 7 provide some evidence that the summed score likelihood–based indices are not sensitive to model misspecification related to multidimensionality, in contrast to $M_2$. This is a desirable feature of the proposed indices, which ought to be more targeted against specific forms of model misspecification. $M_2$ on the other hand, is a more general index for global model fit assessment.

Table 4.

Selected Simulation Results Under the Alternative Hypothesis: Nonnormally Distributed Unidimensional Latent Variable in Two-Parameter Logistic Models.

N	N	Index	df	Equal slopes and intercepts				Random slopes and intercepts				Dispersed slopes and intercepts
				M	$\mathrm{Power}$			M	$\mathrm{Power}$			M	$\mathrm{Power}$
					.01	.05	.10		.01	.05	.10		.01	.05	.10
12	500	${\overline{X}}^2$	10	13.1	.07	.17	.28	11.8	.02	.10	.20	10.7	.02	.06	.13
		${\overline{X}}_C^2$	10	13.2	.07	.19	.28	13.0	.04	.17	.27	12.4	.04	.14	.24
		$M_2$	54	53.8	.01	.04	.10	55.2	.02	.07	.14	55.6	.02	.08	.13
12	1,500	${\overline{X}}^2$	10	19.3	.28	.50	.62	17.1	.16	.39	.54	14.3	.06	.21	.34
		${\overline{X}}_C^2$	10	19.4	.28	.51	.62	18.7	.23	.47	.63	16.4	.13	.34	.48
		$M_2$	54	53.8	.01	.05	.11	55.4	.01	.06	.12	55.2	.02	.06	.12
24	1,500	${\overline{X}}^2$	22	40.9	.49	.73	.83	38.1	.39	.64	.76	37.2	.36	.62	.74
		${\overline{X}}_C^2$	22	41.2	.50	.73	.84	39.4	.45	.69	.80	38.8	.42	.68	.79
		$M_2$	252	250.5	.01	.04	.10	255.9	.02	.08	.14	258.7	.02	.10	.17

Table 5.

Selected Simulation Results Under the Alternative Hypothesis: Nonnormally Distributed Unidimensional Latent Variable in Graded Models.

N	N	Index	df	Equal slopes and intercepts				Random slopes and intercepts				Dispersed slopes and intercepts
				M	$\mathrm{Power}$			M	$\mathrm{Power}$			M	$\mathrm{Power}$
					.01	.05	.10		.01	.05	.10		.01	.05	.10
12	500	${\overline{X}}^2$	34	38.0	.04	.12	.21	40.9	.06	.19	.32	38.9	.02	.13	.22
		${\overline{X}}_C^2$	34	38.2	.04	.13	.22	41.4	.07	.21	.33	39.9	.03	.15	.27
		$M_2$	30	29.4	.01	.03	.08	29.6	.01	.05	.10	30.0	.01	.06	.10
12	1,500	${\overline{X}}^2$	34	45.5	.15	.37	.49	50.9	.29	.55	.70	45.8	.14	.37	.51
		${\overline{X}}_C^2$	34	45.6	.16	.37	.49	51.6	.31	.58	.71	47.2	.18	.42	.56
		$M_2$	30	31.6	.03	.09	.15	30.3	.01	.06	.11	30.3	.01	.05	.10
24	1,500	${\overline{X}}^2$	70	93.8	.32	.57	.70	96.8	.38	.66	.79	93.6	.29	.57	.71
		${\overline{X}}_C^2$	70	94.0	.33	.57	.70	97.4	.40	.67	.80	94.5	.32	.60	.73
		$M_2$	204	202.5	.01	.04	.09	204.5	.01	.04	.09	206.8	.02	.07	.12

Table 6.

Selected Simulation Results Under the Alternative Hypothesis: Multidimensional Distributed Unidimensional Latent Variable in Two-Parameter Logistic Models.

N	N	Index	df	Equal slopes and intercepts				Random slopes and intercepts				Dispersed slopes and intercepts
				M	$\mathrm{Power}$			M	$\mathrm{Power}$			M	$\mathrm{Power}$
					.01	.05	.10		.01	.05	.10		.01	.05	.10
12	500	${\overline{X}}^2$	10	10.0	.02	.05	.09	9.8	.01	.04	.10	9.2	.00	.04	.07
		${\overline{X}}_C^2$	10	10.1	.02	.05	.09	10.5	.02	.07	.13	10.3	.02	.07	.11
		$M_2$	54	83.9	.54	.74	.81	161.2	1.00	1.00	1.00	142.4	1.00	1.00	1.00
12	1,500	${\overline{X}}^2$	10	10.2	.01	.05	.11	10.1	.02	.06	.11	9.4	.00	.04	.07
		${\overline{X}}_C^2$	10	10.2	.01	.05	.11	10.8	.03	.09	.14	10.5	.01	.07	.13
		$M_2$	54	140.8	1.00	1.00	1.00	377.3	1.00	1.00	1.00	320.5	1.00	1.00	1.00
24	1,500	${\overline{X}}^2$	22	21.8	.01	.06	.11	22.4	.01	.07	.11	22.0	.01	.05	.09
		${\overline{X}}_C^2$	22	21.8	.01	.06	.11	22.9	.02	.08	.13	22.7	.02	.07	.12
		$M_2$	252	617.8	1.00	1.00	1.00	1,281	1.00	1.00	1.00	1,544	1.00	1.00	1.00

Table 7.

Selected Simulation Results Under the Alternative Hypothesis: Multidimensional Distributed Unidimensional Latent Variable in Graded Models.

N	N	Index	df	Equal slopes and intercepts				Random slopes and intercepts				Dispersed slopes and intercepts
				M	$\mathrm{Power}$			M	$\mathrm{Power}$			M	$\mathrm{Power}$
					.01	.05	.10		.01	.05	.10		.01	.05	.10
12	500	${\overline{X}}^2$	34	34.7	.01	.06	.12	34.3	.01	.06	.10	34.1	.01	.05	.10
		${\overline{X}}_C^2$	34	34.7	.01	.06	.12	34.5	.01	.06	.11	34.7	.01	.06	.13
		$M_2$	30	50.3	.42	.60	.68	183.4	1.00	1.00	1.00	205.8	1.00	1.00	1.00
12	1,500	${\overline{X}}^2$	34	34.6	.01	.07	.12	34.2	.01	.04	.10	33.1	.01	.03	.08
		${\overline{X}}_C^2$	34	34.6	.01	.07	.12	34.4	.01	.05	.11	34.0	.01	.04	.10
		$M_2$	30	88.2	.84	.90	.92	429.7	1.00	1.00	1.00	455.2	1.00	1.00	1.00
24	1,500	${\overline{X}}^2$	70	71.3	.02	.07	.13	70.4	.01	.04	.10	70.2	.01	.06	.11
		${\overline{X}}_C^2$	70	71.3	.02	.07	.13	70.6	.01	.05	.10	70.7	.01	.07	.12
		$M_2$	204	554.5	1.00	1.00	1.00	1,961	1.00	1.00	1.00	2,265	1.00	1.00	1.00

An Application to Empirical Data

We illustrate the test statistics with empirical data. Twelve items related to positive consequences of nicotine (Tucker et al., 2014), as part of a questionnaire dealing with various attitudes, beliefs, and behaviors related to smoking (Shadel, Edelen, & Tucker, 2011), were administered to a sample of 2,717 daily cigarette smokers. Each item was rated on a 5-point ordinal scale. This study was part of the development of the National Institute of Health’s Patient Reported Outcomes Measurement Information System (PROMIS) and extensive item and dimensional analysis was conducted prior to calibration of the items as unidimensional. The density plot (Figure 1) of the latent variable distribution for this subscale shows its deviation from a standard normal distribution that there are two maximum points in the middle instead of a “bell curve” shape. Table 8 presents the contents of the 12 items from the PROMIS smoking assessment.

Figure 1.

Latent variable distribution for empirical data. Estimated (using empirical histogram) probability density of the latent variable is plotted (dotted line), when superimposed on a standard normal density (solid line).

Table 8.

Items From PROMIS Smoking Initiative.

	Item wordings
Item 1	Smoking helps me concentrate.
Item 2	Smoking helps me think more clearly.
Item 3	Smoking helps me stay focused.
Item 4	Smoking makes me feel better in social situations.
Item 5	Smoking makes me feel more self-confident with others.
Item 6	Smoking helps me feel more relaxed when I’m with other people.
Item 7	Smoking helps me deal with anxiety.
Item 8	Smoking calms me down.
Item 9	If I’m feeling irritable, a cigarette will help me relax.
Item 10	Smoking a cigarette energizes me.
Item 11	Smoking makes me feel less tired.
Item 12	Smoking perks me up.

Note. PROMIS = Patient Reported Outcomes Measurement Information System.

Results show that when we use the normal unidimensional IRT model, ${\overline{X}}^2$ equals to 208.5, and ${\overline{X}}_C^2$ equals to 179.3, indicating significant lack of latent variable normality (df = 46, p < .0001). But when the empirical histogram latent density estimation is used instead for item parameter estimation, ${\overline{X}}^2$ is equal to 51.2 and ${\overline{X}}_C^2$ is equal to 49.4 (df = 46, p > .1). In sum, we came to the conclusion that the latent variable distribution of this set of items was probably nonnormal and our proposed indices were able to detect the violation of latent variable distribution assumption.

Discussion

Normality of latent variable distribution is a critical assumption in standard maximum marginal likelihood estimation for IRT models. However, in real-world applications, the distribution of latent variables can be nonnormal. The detection of latent variable nonnormality is important for item analysis and test scoring. In this study, we propose using summed score likelihood–based indices for testing departures from normality. We also develop a Satorra–Bentler type moment adjustment approach to approximate the tail area probabilities of the indices.

In the simulation study, the performance of unadjusted and adjusted summed score likelihood–based statistics was compared with that of $M_2$. Results show that the moment-adjusted index performs well for both dichotomous data and polytomous data and maintains correct test size across number of items, sample size, and type of IRT model considered. The unadjusted statistic does not work as well, especially when item parameters are dispersed. Furthermore, the indices were particularly sensitive to latent variable nonnormality, and not sensitive to other kinds of model misfit such as multidimensionality.

An interesting finding is that the general goodness-of-fit statistic $M_2$ (Maydeu-Olivares & Joe, 2005) has almost no power against the nonnormal alternative, and hence, cannot be recommended for testing latent variable distribution fit for IRT models (see also Hansen, Cai, Monroe, & Li, 2016). This could be explained by the observation that $M_2$ is based only on first- and second-order margins of the underlying contingency table, but to detect latent variable distributional misfit, information from higher order margins may be necessary.

This study is not without its limitations. First, the distributions of the proposed indices are not exactly chi-squared. In our study, their tail-area probabilities were approximated to first order by a chi-squared variable with the availability of the item parameter error covariance matrix and a Jacobian. We focus on the first-order correction due to its simplicity and the fact that we observed empirically that the results of the second-order correction did not differ substantively from that of the first order. In the future, higher order moments could be considered to improve the performance of the adjusted statistics for situations that we have not examined. Second, only a limited number of null conditions and only two alternative population distributions were tested in the simulations. More extensive simulations are needed to fully understand the performance of the test statistics. Third, we only studied the properties of the statistics and the corrections under maximum likelihood estimation. In principle, one could derive similar statistics under limited-information estimation (e.g., with weighted least squares). Finally, this study only considered the conditions when item response data are assumed to be unidimensional. Multidimensional IRT models (MIRT, Reckase, 2009) should be considered in subsequent work. One particularly popular model in educational and psychological research is the full-information item bifactor model (Cai, Yang, & Hansen, 2011; Gibbons & Hedeker, 1992; Reise, 2012). In this model, all items load on a general dimension, and an item is permitted to load on at most one specific dimension that influences nonoverlapping subsets of items. This feature of bifactor models implies that there exists valuable relation between an observed summed score and the distribution of the latent general dimension (Cai, 2015). This relation implies an opportunity to test the underlying assumption about the distribution of general latent dimension with summed score likelihood–based statistics.

Appendix

A Modified Lord–Wingersky Algorithm for Jacobian Computations

Consider a test with n dichotomous items, calibrated by a two-parameter logistic item response theory model. Recall that $T_i(1|\theta )$ is item i’s traceline for Category 1 (Equation 1), with $T_i(0|\theta )$ = 1 −$T_i(1|\theta )$ for Category 0. Theoretically, there should be 2ⁿ response patterns. The response pattern is indicated by $\mathit{u}=(u_1,\dots ,u_n)$. Under the assumption of items’ conditional independence, the likelihood for a response pattern u can be expressed as $L(\mathit{u}|\theta )=\underset{i=1}{\overset{n}{\Pi }}{T}_i(u_i|\theta )$. For n dichotomous items, the summed score s ranges from 0 to n. $S=n+1$ is the number of all possible summed scores. Recall that $\mathit{u}=\sum _{i=1}^n{u}_i$ is a notational shorthand for the summed score associated with response $\mathit{u}$ (see Equation 16). The likelihood for summed score $s=0,\dots ,n$ is defined as

$$L(s|\theta )=\sum _{\mathit{u}=s}L(\mathit{u}|\theta )=\sum _{s=\mathit{u}}\underset{i=1}{\overset{n}{\Pi }}{T}_i(u_i|\theta ),$$

Clearly, the likelihood of a summed score s is the sum of all response pattern likelihoods for $u=s$. In Lord–Wingersky algorithm, the summed score likelihoods are built up recursively, one at a time (Lord & Wingersky, 1984). Let $L_i(s|\theta )$ indicate the likelihood for summed score s after item i has been added into the computation. In the first step, two summed score likelihoods are computed based on the tracelines of Item 1: $L_1(0|\theta )=T_1(0|\theta )$ and $L_1(1|\theta )=T_1(1|\theta )$.

In the second step, we have three summed score likelihoods based on the likelihoods from Step 1 and tracelines of Item 2:

$$\begin{array}{c}{L}_2(0|\theta )=L_1(0|\theta )T_2(0|\theta ),\hfill \\ L_2(1|\theta )=L_1(1|\theta )T_2(0|\theta )+L_1(0|\theta )T_2(1|\theta ),\hfill \\ L_2(2|\theta )=L_1(1|\theta )T_2(1|\theta ).\hfill \end{array}$$

Suppose n items have been added. The likelihoods for summed scores $(0,\dots ,n)$ are

$$\begin{array}{c}{L}_n(0|\theta )=L_{n-1}(0|\theta )T_n(0|\theta ),\hfill \\ L_n(s|\theta )=L_{n-1}(s|\theta )T_n(0|\theta )+L_{n-1}(s-1|\theta )T_n(1|\theta ),\hfill \\ L_n(n|\theta )=L_{n-1}(n-1|\theta )T_n(1|\theta ).\hfill \end{array}$$

To obtain the Jacobian matrix of summed score likelihoods with respective to item parameters, the Lord–Wingersky algorithm is adapted slightly. As previously mentioned, in the first step, there are only two summed score likelihoods based on Item 1: $L_1(0|\theta )$ and $L_1(1|\theta )$. The first-order derivatives of summed score likelihoods with respect to a generic item parameter ${\gamma }_1$ for Item 1 are

$$\begin{array}{c}\frac{\partial L_1(0|\theta )}{\partial {\gamma }_1}=\frac{\partial T_1(0|\theta )}{\partial {\gamma }_1},\hfill \\ \frac{\partial L_1(1|\theta )}{\partial {\gamma }_1}=\frac{\partial T_1(1|\theta )}{\partial {\gamma }_1}.\hfill \end{array}$$

In the second step, Item 2 is added with a generic item parameter ${\gamma }_2$. The first-order derivatives of summed score likelihoods with respect to ${\gamma }_1$ and ${\gamma }_2$ follow from the chain rule:

$$\begin{array}{c}\frac{\partial L_2(0|\theta )}{\partial {\gamma }_1}=\frac{\partial L_1(0|\theta )}{\partial {\gamma }_1}{T}_2(0|\theta ),\hfill \\ \frac{\partial L_2(1|\theta )}{\partial {\gamma }_1}=\frac{\partial L_1(1|\theta )}{\partial {\gamma }_1}{T}_2(0|\theta )+\frac{\partial L_1(0|\theta )}{\partial {\gamma }_1}{T}_2(1|\theta ),\hfill \\ \frac{\partial L_2(2|\theta )}{\partial {\gamma }_1}=\frac{\partial L_1(1|\theta )}{\partial {\gamma }_1}{T}_2(1|\theta ),\hfill \\ \frac{\partial L_2(0|\theta )}{\partial {\gamma }_2}=L_1(0|\theta )\frac{\partial T_2(0|\theta )}{\partial {\gamma }_2},\hfill \\ \frac{\partial L_2(1|\theta )}{\partial {\gamma }_2}=L_1(1|\theta )\frac{\partial T_2(0|\theta )}{\partial {\gamma }_2}+L_1(0|\theta )\frac{\partial T_2(1|\theta )}{\partial {\gamma }_2},\hfill \\ \frac{\partial L_2(1|\theta )}{\partial {\gamma }_2}=L_1(1|\theta )\frac{\partial T_2(1|\theta )}{\partial {\gamma }_2}.\hfill \end{array}$$

Generalizing to n items, the first-order derivatives of summed score likelihood functions with respect to the n item’s parameters $({\gamma }_1,\dots ,{\gamma }_n)$ are

$$\begin{array}{c}\frac{\partial L_n(0|\theta )}{\partial {\gamma }_1}=\frac{\partial L_{n-1}(0|\theta )}{\partial {\gamma }_1}{T}_n(0|\theta ),\hfill \\ \frac{\partial L_n(s|\theta )}{\partial {\gamma }_1}=\frac{\partial L_{n-1}(s|\theta )}{\partial {\gamma }_1}{T}_n(0|\theta )+\frac{\partial L_1(s-1|\theta )}{\partial {\gamma }_1}{T}_n(1|\theta ),\hfill \\ \frac{\partial L_n(n|\theta )}{\partial {\gamma }_1}=\frac{\partial L_{n-1}(n-1|\theta )}{\partial {\gamma }_1}{T}_n(1|\theta ),\hfill \\ \frac{\partial L_n(0|\theta )}{\partial {\gamma }_n}=L_{n-1}(0|\theta )\frac{\partial T_n(0|\theta )}{\partial {\gamma }_n},\hfill \\ \frac{\partial L_n(s|\theta )}{\partial {\gamma }_n}=L_{n-1}(s|\theta )\frac{\partial T_n(0|\theta )}{\partial {\gamma }_n}+L_{n-1}(s-1|\theta )\frac{\partial T_n(1|\theta )}{\partial {\gamma }_n},\hfill \end{array}$$

$$\frac{\partial L_n(n|\theta )}{\partial {\gamma }_n}=L_{n-1}(n-1|\theta )\frac{\partial T_n(1|\theta )}{\partial {\gamma }_n}.$$

The process of modified Lord–Wingersky algorithm for calculating the Jacobian matrix is illustrated with an example. Consider a simple test with three dichotomous items. The values of slope parameters are $\mathit{a}=(1.0,0.8,1.2)$, and the values of intercept parameters are $\mathit{c}=(-0.2,0.6,-1.0)$. Recall that the marginal probability for summed scores with known $g\left(\theta \right)$ is

$$p\left(s\right)=\int L(s|\theta )g\left(\theta \right)d\theta ,$$

The integrals in Equation (33) must be approximated by quadrature. We demonstrate the algorithm by showing the calculations over a set of quadrature points (Cai, 2015). We approximate the marginal probability using $Q$ quadrature points:

$$p\left(s\right)=\int L(s|\theta )g\left(\theta \right)d\theta =\sum _{q=1}^{Q}L(s|X_q)W\left(X_q\right),$$

where $X_q$ is a quadrature node and $W\left(X_q\right)$ is the corresponding quadrature weight. To obtain $W\left(X_q\right)$, a set of normalized ordinates of the prior density are applied (Cai, 2015), that is, $W\left(X_q\right)=g\left(X_q\right)/\sum _{q=1}^{Q}g\left(X_q\right)$.

Table A1 shows the recursive computations for the parameters from Item 3. It shows the values of summed score likelihoods, first-order derivatives of tracelines, and the first-order derivatives of summed score likelihoods at five equally spaced quadrature points ( $Q=5$): −2, −1, 0, 1, and 2. More quadrature points should be used for better precision (Cai, 2015). The first block presents the summed score likelihoods after the first and second items are added in. The second block presents the first-order derivatives of Item 3’s tracelines with respect to its slope parameter. The third block presents the first-order derivatives of summed score likelihoods with respect to Item 3’s slope parameter.

Table A1.

First-Order Derivatives of Summed Score Likelihoods With Regard to Item 3’s Slope Parameter at Five Rectangular Quadrature Points.

Quadrature points	−2	−1	0	1	2
Summed score likelihoods after Items 1 and 2
${\mathit{L}}_2(0|\theta )$	.658	.423	.195	.061	.014
${\mathit{L}}_2(1|\theta )$	.315	.473	.515	.385	.213
${\mathit{L}}_2(2|\theta )$	.027	.104	.291	.553	.773
Derivatives of tracelines with respect to Item 3’s slope parameter
$\frac{\partial {\mathit{T}}_3(1|\theta )}{\partial a_3}$	−.063	−.090	.000	.248	.317
$\frac{\partial {\mathit{T}}_3(0|\theta )}{\partial a_3}$	.063	.090	.000	−.248	−.317
First-order derivatives of summed score likelihoods
$\frac{\partial {\mathit{L}}_3(0|\theta )}{\partial a_3}={\mathit{L}}_2(0|\theta )\frac{\partial {\mathit{T}}_3(0|\theta )}{\partial a_3}$	.041	.038	.000	−.015	−.004
$\frac{\partial {\mathit{L}}_3(1|\theta )}{\partial a_3}={\mathit{L}}_2(1|\theta )\frac{\partial {\mathit{T}}_3(0|\theta )}{\partial a_3}+{\mathit{L}}_2(0|\theta )\frac{\partial {\mathit{T}}_3(1|\theta )}{\partial a_3}$	−.021	.005	.000	−.080	−.063
$\frac{\partial {\mathit{L}}_3(2|\theta )}{\partial a_3}={\mathit{L}}_2(2|\theta )\frac{\partial {\mathit{T}}_3(0|\theta )}{\partial a_3}+{\mathit{L}}_2(1|\theta )\frac{\partial {\mathit{T}}_3(1|\theta )}{\partial a_3}$	−.018	−.033	.000	−.042	−.177
$\frac{\partial {\mathit{L}}_3(3|\theta )}{\partial a_3}={\mathit{L}}_2(2|\theta )\frac{\partial {\mathit{T}}_3(1|\theta )}{\partial a_3}$	−.002	−.009	.000	.137	.245

Table A2 presents the first-order derivatives of summed score probabilities with respect to Item 3 (the desired Jacobian elements). $W\left(\theta \right)$ indicates quadrature weights at each $\theta$ level. “Weighted derivatives” are found by multiplying (point to point) the first-order derivatives of summed score likelihoods with $W\left(\theta \right)$. The last column “Jacobian” indicates the first-order derivatives of summed score probabilities with respect to Item 3’s slope parameter. It is the summation of the weighted derivatives over all quadrature points for each summed score likelihood.

Table A2.

First-Order Derivatives of Summed Score Probabilities With Regard to Item 3’s Slope Parameter at Five Rectangular Quadrature Points.

$\mathit{W}\left(\mathbf{\theta }\right)$	Quadrature points
	−2.054	−1.244	0.403	1.244	2.054
	First-order derivatives of summed score likelihoods
$\frac{\partial {\mathit{L}}_3(0|\mathbf{\theta })}{\partial {\mathit{a}}_3}$	.041	.038	.000	−.015	−.004
$\frac{\partial {\mathit{L}}_3(1|\mathbf{\theta })}{\partial {\mathit{a}}_3}$	−.021	.005	.000	−.080	−.063
$\frac{\partial {\mathit{L}}_3(2|\mathbf{\theta })}{\partial {\mathit{a}}_3}$	−.018	−.033	.000	−.042	−.177
$\frac{\partial {\mathit{L}}_3(3|\mathbf{\theta })}{\partial {\mathit{a}}_3}$	−.002	−.009	.000	.137	.245
	Weighted derivatives					Jacobian
$\frac{\partial {\mathit{L}}_3(0|\mathbf{\theta })}{\partial {\mathit{a}}_3}*W\left(\theta \right)$	.002	.009	.000	−.004	.000	.008
$\frac{\partial {\mathit{L}}_3(1|\mathbf{\theta })}{\partial {\mathit{a}}_3}*W\left(\theta \right)$	−.001	.001	.000	−.020	−.003	−.023
$\frac{\partial {\mathit{L}}_3(2|\mathbf{\theta })}{\partial {\mathit{a}}_3}*W\left(\theta \right)$	−.001	−.008	.000	−.010	−.010	−.029
$\frac{\partial {\mathit{L}}_3(3|\mathbf{\theta })}{\partial {\mathit{a}}_3}*W\left(\theta \right)$	.000	−.002	.000	.033	.013	.044