In psychophysiology, an interesting question is how to estimate the reliability of event-related potentials collected by means of the Eriksen Flanker Task or similar tests. A special problem presents itself if the data represent neurological reactions that are associated with some responses (in case of the Flanker Task, responding incorrectly on a trial) but not others (like when providing a correct response), inherently resulting in unequal numbers of observations per subject. The general trend in reliability research here is to use generalizability theory and Bayesian estimation. We show that a new approach based on classical test theory and frequentist estimation can do the job as well and in a simpler way, and even provides additional insight to matters that were unsolved in the generalizability method approach. One of our contributions is the definition of a single, overall reliability coefficient for an entire group of subjects with unequal numbers of observations. Both methods have slightly different objectives. We argue in favor of the classical approach but without rejecting the generalizability approach.

Dichotomous IRT models can be viewed as families of stochastically ordered distributions of responses to test items. This paper explores several properties of such distributions. In particular, it is examined under what conditions stochastic order in families of conditional distributions is transferred to their inverse distributions, from two families of related distributions to a third family, or from multivariate conditional distributions to a marginal distribution. The main results are formulated as a series of theorems and corollaries which apply to dichotomous IRT models. One part of the results holds for unidimensional models with fixed item parameters. The other part holds for models with random item parameters as used, for example, in adaptive testing or for tests with multidimensional abilities.

An asymptotic expression for the reliability of a linearly equated test is developed using normal theory. The reliability is expressed as the product of two terms, the reliability of the test before equating, and an adjustment term. This adjustment term is a function of the sample sizes used to estimate the linear equating transformation. The results of a simulation study indicate close agreement between the theoretical and simulated reliability values for samples greater than 200. Findings demonstrate that samples as small as 300 can be used in linear equating without an appreciable decrease in reliability.

Change scores obtained in pretest–posttest designs are important for evaluating treatment effectiveness and for assessing change of individual test scores in psychological research. However, over the years the use of change scores has raised much controversy. In this article, from a multilevel perspective, we provide a structured treatise on several persistent negative beliefs about change scores and show that these beliefs originated from the confounding of the effects of within-person change on change-score reliability and between-person change differences. We argue that psychometric properties of change scores, such as reliability and measurement precision, should be treated at suitable levels within a multilevel framework. We show that, if examined at the suitable levels with such a framework, the negative beliefs about change scores can be renounced convincingly. Finally, we summarize the conclusions about change scores to dispel the myths and to promote the potential and practical usefulness of change scores.

Item response theory (IRT) models are non-linear latent variable models for discrete measures, whereas factor analysis (FA) is a latent variable model for continuous measures. In FA, the standard error (SE) of individuals’ scores is common for all individuals. In IRT, the SE depends on the individual’s score, and the SE function is to be provided. The empirical standard deviation of the scores across discrete ranges should also be computed to inform the extent to which IRT SEs overestimate or underestimate the variability of the scores. Within the target range of scores the test was designed to measure, one should expect IRT SEs to be smaller and more precise than FA SEs, and therefore preferable to assess clinical change. Outside the target range, IRT SEs may be too large and more imprecise than FA SEs, and FA more precise to assess change. As a result, whether FA or IRT characterize reliable change more accurately in a sample will depend on the proportion of individuals within or outside the IRT target score range. An application is provided to illustrate these concepts.

This paper reflects on some practical implications of the excellent treatment of sum scoring and classical test theory (CTT) by Sijtsma et al. (Psychometrika 89(1):84–117, 2024). I have no major disagreements about the content they present and found it to be an informative clarification of the properties and possible extensions of CTT. In this paper, I focus on whether sum scores—despite their mathematical justification—are positioned to improve psychometric practice in empirical studies in psychology, education, and adjacent areas. First, I summarize recent reviews of psychometric practice in empirical studies, subsequent calls for greater psychometric transparency and validity, and how sum scores may or may not be positioned to adhere to such calls. Second, I consider limitations of sum scores for prediction, especially in the presence of common features like ordinal or Likert response scales, multidimensional constructs, and moderated or heterogeneous associations. Third, I review previous research outlining potential limitations of using sum scores as outcomes in subsequent analyses where rank ordering is not always sufficient to successfully characterize group differences or change over time. Fourth, I cover potential challenges for providing validity evidence for whether sum scores represent a single construct, particularly if one wishes to maintain minimal CTT assumptions. I conclude with thoughts about whether sum scores—even if mathematically justified—are positioned to improve psychometric practice in empirical studies.

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.

Prior to discussing and challenging two criticisms on coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}, the well-known lower bound to test-score reliability, we discuss classical test theory and the theory of coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}. The first criticism expressed in the psychometrics literature is that coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} is only useful when the model of essential $\tau$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau $$\end{document}-equivalence is consistent with the item-score data. Because this model is highly restrictive, coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} is smaller than test-score reliability and one should not use it. We argue that lower bounds are useful when they assess product quality features, such as a test-score’s reliability. The second criticism expressed is that coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} incorrectly ignores correlated errors. If correlated errors would enter the computation of coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}, theoretical values of coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} could be greater than the test-score reliability. Because quality measures that are systematically too high are undesirable, critics dismiss coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}. We argue that introducing correlated errors is inconsistent with the derivation of the lower bound theorem and that the properties of coefficient $\alpha$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} remain intact when data contain correlated errors.

This paper analyzes the theoretical, pragmatic, and substantive factors that have hampered the integration between psychology and psychometrics. Theoretical factors include the operationalist mode of thinking which is common throughout psychology, the dominance of classical test theory, and the use of “construct validity” as a catch-all category for a range of challenging psychometric problems. Pragmatic factors include the lack of interest in mathematically precise thinking in psychology, inadequate representation of psychometric modeling in major statistics programs, and insufficient mathematical training in the psychological curriculum. Substantive factors relate to the absence of psychological theories that are sufficiently strong to motivate the structure of psychometric models. Following the identification of these problems, a number of promising recent developments are discussed, and suggestions are made to further the integration of psychology and psychometrics.

Sijtsma and Pfadt (2021) published a thought-provoking article on coefficient alpha. I make the following arguments against their work. 1) Kuder and Richardson (1937) deserve more credit for coefficient alpha than Cronbach (1951). 2) We should distinguish between the definition of reliability and its meaning. 3) We should be wary of overfitting in the use of FA reliability. 4) Our primary concern is to obtain accurate reliability estimates rather than conservative estimates. 5) Several reliability estimators, such as ${\lambda }_2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda _{2}$$\end{document}, ${\mu }_2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu _{2}$$\end{document}, congeneric reliability and the Gilmer-Feldt coefficient are more accurate than coefficient alpha. 6) The name omega should not be used to refer to a specific reliability estimator.

A new model of confirmatory factor analysis (CFA) for multitrait-multimethod (MTMM) data sets is presented. It is shown that this model can be defined by only three assumptions in the framework of classical psychometric test theory (CTT). All other properties of the model, particularly the uncorrelatedness of the trait with the method factors are logical consequences of the definition of the model. In the model proposed there are as many trait factors as different traits considered, but the number of method factors is one fewer than the number of methods included in an MTMM study. The covariance structure implied by this model is derived, and it is shown that this model is identified even under conditions under which other CFA-MTMM models are not. The model is illustrated by two empirical applications. Furthermore, its advantages and limitations are discussed with respect to previously developed CFA models for MTMM data.

In many instances it is reasonable to assume that the population covariance matrix has positive elements. This assumption implies for the single factor analysis model that the loadings and regression weights for best linear factor prediction are positive. For the multiple factor analysis model where each variable loads on a single factor and a hierarchical factor model, it implies that the loadings and the factor correlations are positive. For the latter model it also implies that the regression weights for first- and second-order factor prediction are positive.

Under the general common factor model, the reliabilities of factor score estimators might be of more interest than the reliability of the total score (the unweighted sum of item scores). In this paper, lower bounds to the reliabilities of Thurstone’s factor score estimators, Bartlett’s factor score estimators, and McDonald’s factor score estimators are derived and conditions are given under which these lower bounds are equal. The relative performance of the derived lower bounds is studied using classic example data sets. The results show that estimates of the lower bounds to the reliabilities of Thurstone’s factor score estimators are greater than or equal to the estimates of the lower bounds to the reliabilities of Bartlett’s and McDonald’s factor score estimators.

Classical test theory reliability coefficients are said to be population specific. Reliability generalization, a meta-analysis method, is the main procedure for evaluating the stability of reliability coefficients across populations. A new approach is developed to evaluate the degree of invariance of reliability coefficients to population characteristics. Factor or common variance of a reliability measure is partitioned into parts that are, and are not, influenced by control variables, resulting in a partition of reliability into a covariate-dependent and a covariate-free part. The approach can be implemented in a single sample and can be applied to a variety of reliability coefficients.

The sum score on a psychological test is, and should continue to be, a tool central in psychometric practice. This position runs counter to several psychometricians’ belief that the sum score represents a pre-scientific conception that must be abandoned from psychometrics in favor of latent variables. First, we reiterate that the sum score stochastically orders the latent variable in a wide variety of much-used item response models. In fact, item response theory provides a mathematically based justification for the ordinal use of the sum score. Second, because discussions about the sum score often involve its reliability and estimation methods as well, we show that, based on very general assumptions, classical test theory provides a family of lower bounds several of which are close to the true reliability under reasonable conditions. Finally, we argue that eventually sum scores derive their value from the degree to which they enable predicting practically relevant events and behaviors. None of our discussion is meant to discredit modern measurement models; they have their own merits unattainable for classical test theory, but the latter model provides impressive contributions to psychometrics based on very few assumptions that seem to have become obscured in the past few decades. Their generality and practical usefulness add to the accomplishments of more recent approaches.

In this rejoinder to McNeish (2024) and Mislevy (2024), who both responded to our focus article on the merits of the simple sum score (Sijtsma et al., 2024), we address several issues. Psychometrics education and in particular psychometricians’ outreach may help researchers to use IRT models as a precursor for the responsible use of the latent variable score and the sum score. Different methods used for test and questionnaire construction often do not produce highly different results, and when they do, this may be due to an unarticulated attribute theory generating noisy data. The sum score and transformations thereof, such as normalized test scores and percentiles, may help test practitioners and their clients to better communicate results. Latent variables prove important in more advanced applications such as equating and adaptive testing where they serve as technical tools rather than communication devices. Decisions based on test results are often binary or use a rather coarse ordering of scale levels, hence, do not require a high level of granularity (but nevertheless need to be precise). A gap exists between psychology and psychometrics which is growing deeper and wider, and that needs to be bridged. Psychology and psychometrics must work together to attain this goal.

Reliable and valid assessment of sports nutrition knowledge can inform athlete nutrition education to address knowledge gaps. This study aimed to test the reliability and validity of an electronically administered sports nutrition knowledge tool – Platform to Evaluate Athlete Knowledge of Sports Nutrition Questionnaire (PEAKS-NQ). A 94-item PEAKS-NQ was piloted to 149 developmental athletes (DA) in New Zealand, with a subset invited to complete the PEAKS-NQ again to assess reliability. Reliability was evaluated using sign test, intraclass correlation and Cronbach’s α. Accredited sports dietitians (ASD; n 255) completed the PEAKS-NQ to establish construct validity via known-groups methodology and provided relevance scores to determine the scale content validity index (S-CVI). Rasch analysis was conducted to identify potentially problematic items and test reliability. Score differences between DA and ASD were analysed using independent t or non-parametric tests. DA (n 88) were 17·8 (sd 1·4) years, 61·4 % female and mostly in high school (94·3 %). ASD (n 45) were 37·8 (sd 7·6) years, 82·2 % female, with >5 years of dietetic experience (59·1 %). ASD scored higher than DA in all sections and overall (91·5 (sd 3·4) v. 67·1 (sd 10·5) %) (P < 0·001). There were no differences between retests (n 18; P = 0·14). Cronbach’s α was 0·86. S-CVI indicated good content validity (0·88). Rasch analysis resulted in a fifty-item PEAKS-NQ with high item (0·91) and person (0·92) reliability. The PEAKS-NQ is reliable and valid for assessing sports nutrition knowledge which could assist practitioners effectively tailor and evaluate nutrition education.