Item and Test Information

The two ways of writing the same model

A one-factor model for binary items with the logit link is the two-parameter logistic model. The factor model writes the log odds of the higher category as ${\lambda }_{j}f-{\tau }_j$ and the response model writes it as $a_j(\theta -b_j)$, so that the factor score is the trait, and the discrimination and the difficulty follow from the loading, the threshold and the scaling of the factor (Grimm, 2026, Equations 10.5 and 10.6; Wang & Wang, 2020, Equations 2.12 and 2.13). Asparouhov and Muthén (2020) write the passage in a form that holds whatever the factor mean $\alpha$ and variance $\psi$ are, and that is the form used here. With maximum likelihood, under either link (their Equations 21 and 22):

The expression for the difficulty is the same in every case that follows. With weighted least squares and the theta parameterization, in which the residual variance ${\theta }_j$ of the underlying response variable is a parameter (their Equation 25):

and with the delta parameterization, in which the scale factor ${\Delta }_j$ takes that place (their Equation 27):

With the factor mean at zero, its variance at one and the scale factor at one, the last expression is the one Wang and Wang (2020) print as Equations 2.20 and 2.27, and the first is their Equations 2.12 and 2.24. Where a parameter is not in the file the value it is fixed at is used: one for the residual variance of the underlying response variable under the theta parameterization, one for the scale factor under the delta parameterization, and zero for the factor mean. With more than two ordered categories the same relations give one difficulty per threshold, which is the graded response model (Wang & Wang, 2020, Equations 2.24, 2.25, 2.27 and 2.28). When the run was fitted by maximum likelihood, Mplus prints the response metric in the output and the page takes the values from there, and it reports how far the same quantities computed from the loadings and the thresholds fall from them, which is the rounding of the printed values.

The information of an item and of the test

For a categorical item, information is computed from all of its response categories. If $P_r\left(\theta \right)$ is the probability of category $r$, Samejima's general expression is

(Samejima, 1974; Asparouhov & Muthén, 2020, Equation 14). For a binary logistic item this reduces to $a_j^2{P}_j\left(\theta \right)[1-P_j(\theta \left)\right]$. For ordered items the category probabilities are those of the graded response model, so every category contributes to the information rather than being discarded.

The page keeps the link fitted in Mplus. With a logit link it uses the logistic cumulative distribution; with a probit link it uses the standard normal cumulative distribution and its density. The probit discrimination therefore remains on the normal-ogive metric printed or implied by Mplus; it is not multiplied by 1.7. This is the convention stated explicitly by Asparouhov and Muthén (2020, Equation 23).

The information of the test is the sum of the item information functions. From that measurement information follow the conditional standard error and, for the standardized IRT trait with variance one, the local reliability:

This is measurement information only. Mplus's factor-score information additionally contains the normal prior term $1/\psi$ (Asparouhov & Muthén, 2020, Equation 19); adding that prior would answer a different question about posterior factor-score precision, so it is not added here.

The categories of one item

For an item with more than two ordered categories, the probability of a particular category is the difference between two adjacent cumulative response probabilities. The page evaluates those probabilities with the fitted link: logistic for logit models and normal-ogive for probit models (Asparouhov & Muthén, 2020, Equations 1–12). The figure therefore remains in the model's own response metric.

Unsupported inputs

A model with more than one factor, since an item that loads on two traits has no single difficulty. A model whose items have no thresholds, which is a model whose items were not declared categorical. A run whose messages show that estimation did not produce interpretable parameter estimates. Information is computed for binary and ordered polytomous items; an item is omitted only when the quantities needed for its response function cannot be recovered from the output.

Self-test

Append ?selftest=1 to check parameterization, factor scaling, logistic and probit response functions, item and test information, standard errors, reliability, and unsupported inputs.

• Asparouhov, T., & Muthén, B. (2020). IRT in Mplus (Version 4, technical note). Muthén & Muthén. https://www.statmodel.com/techappen.shtml
• de Ayala, R. J. (2009). The theory and practice of item response theory. Guilford Press.
• Grimm, K. J. (2026). Categorical data analysis with structural equation models. Guilford Press.
• Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 34(4, Pt. 2), 1–97.
• Thissen, D. (2000). Reliability and measurement precision. In H. Wainer (Ed.), Computerized adaptive testing: A primer (2nd ed., pp. 159–183). Lawrence Erlbaum.
• Wang, J., & Wang, X. (2020). Structural equation modeling: Applications using Mplus (2nd ed.). Wiley. https://doi.org/10.1002/9781119422730