The standardized root mean squared residual (SRMR) is commonly reported to evaluate approximate fit of latent variable models. As traditionally defined, SRMR summarizes the discrepancy between observed covariance elements and implied covariance elements. However, current applications of latent variable models often include additional features like overidentified mean structures and covariates, to which the traditional SRMR definition is not applicable. To date, SRMR extensions for models with covariates have received limited attention. Nonetheless, mainstream software provide SRMR for models with covariates, but values differ based on model specification and differ across programs. The goal of this paper is to formalize SRMR definitions for models with covariates. We develop possible SRMR definitions corresponding to different model specifications with covariates, discussing the advantages and disadvantages of each. Importantly, some SRMR definitions are susceptible to confounding misfit and model size such that SRMR values systematically decrease and suggest better fit when covariates are present, even if covariates have null effects. The primary conclusion is that there may not be a single unifying SRMR definition for covariates, but practically, researchers reporting SRMR with covariates should be aware (a) which definition is being used and (b) which information is and is not included in the particular definition.

The deployment of statistical models—such as those used in item response theory—necessitates the use of indices that are informative about the degree to which a given model is appropriate for a specific data context. We introduce the InterModel Vigorish (IMV) as an index that can be used to quantify accuracy for models of dichotomous item responses based on the improvement across two sets of predictions (i.e., predictions from two item response models or predictions from a single such model relative to prediction based on the mean). This index has a range of desirable features: It can be used for the comparison of non-nested models and its values are highly portable and generalizable. We use this fact to compare predictive performance across a variety of simulated data contexts and also demonstrate qualitative differences in behavior between the IMV and other common indices (e.g., the AIC and RMSEA). We also illustrate the utility of the IMV in empirical applications with data from 89 dichotomous item response datasets. These empirical applications help illustrate how the IMV can be used in practice and substantiate our claims regarding various aspects of model performance. These findings indicate that the IMV may be a useful indicator in psychometrics, especially as it allows for easy comparison of predictions across a variety of contexts.

A person fit test based on the Lagrange multiplier test is presented for three item response theory models for polytomous items: the generalized partial credit model, the sequential model, and the graded response model. The test can also be used in the framework of multidimensional ability parameters. It is shown that the Lagrange multiplier statistic can take both the effects of estimation of the item parameters and the estimation of the person parameters into account. The Lagrange multiplier statistic has an asymptotic χ2-distribution. The Type I error rate and power are investigated using simulation studies. Results show that test statistics that ignore the effects of estimation of the persons’ ability parameters have decreased Type I error rates and power. Incorporating a correction to account for the effects of the estimation of the persons’ ability parameters results in acceptable Type I error rates and power characteristics; incorporating a correction for the estimation of the item parameters has very little additional effect. It is investigated to what extent the three models give comparable results, both in the simulation studies and in an example using data from the NEO Personality Inventory-Revised.

In this paper, it is shown that various violations of the 2-PL model and the nominal response model can be evaluated using the Lagrange multiplier test or the equivalent efficient score test. The tests presented here focus on violation of local stochastic independence and insufficient capture of the form of the item characteristic curves. Primarily, the tests are item-oriented diagnostic tools, but taken together, they also serve the purpose of evaluation of global model fit. A useful feature of Lagrange multiplier statistics is that they are evaluated using maximum likelihood estimates of the null-model only, that is, the parameters of alternative models need not be estimated. As numerical examples, an application to real data and some power studies are presented.

Two new tests for a model for the response times on pure speed tests by Rasch (1960) are proposed. The model is based on the assumption that the test response times are approximately gamma distributed, with known index parameters and unknown rate parameters. The rate parameters are decomposed in a subject ability parameter and a test difficulty parameter. By treating the ability as a gamma distributed random variable, maximum marginal likelihood (MML) estimators for the test difficulty parameters and the parameters of the ability distribution are easily derived. Also the model tests proposed here pertain to the framework of MML. Two tests or modification indices are proposed. The first one is focused on the assumption of local stochastic independence, the second one on the assumption of the test characteristic functions. The tests are based on Lagrange multiplier statistics, and can therefore be computed using the parameter estimates under the null model. Therefore, model violations for all items and pairs of items can be assessed as a by-product of one single estimation run. Power studies and applications to real data are included as numerical examples.

It is widely believed that a joint factor analysis of item responses and response time (RT) may yield more precise ability scores that are conventionally predicted from responses only. For this purpose, a simple-structure factor model is often preferred as it only requires specifying an additional measurement model for item-level RT while leaving the original item response theory (IRT) model for responses intact. The added speed factor indicated by item-level RT correlates with the ability factor in the IRT model, allowing RT data to carry additional information about respondents’ ability. However, parametric simple-structure factor models are often restrictive and fit poorly to empirical data, which prompts under-confidence in the suitablity of a simple factor structure. In the present paper, we analyze the 2015 Programme for International Student Assessment mathematics data using a semiparametric simple-structure model. We conclude that a simple factor structure attains a decent fit after further parametric assumptions in the measurement model are sufficiently relaxed. Furthermore, our semiparametric model implies that the association between latent ability and speed/slowness is strong in the population, but the form of association is nonlinear. It follows that scoring based on the fitted model can substantially improve the precision of ability scores.

Taking a falsificationist perspective, the present paper identifies two major shortcomings of existing approaches to comparative model evaluations in general and strategy classifications in particular. These are (1) failure to consider systematic error and (2) neglect of global model fit. Using adherence measures to evaluate competing models implicitly makes the unrealistic assumption that the error associated with the model predictions is entirely random. By means of simple schematic examples, we show that failure to discriminate between systematic and random error seriously undermines this approach to model evaluation. Second, approaches that treat random versus systematic error appropriately usually rely on relative model fit to infer which model or strategy most likely generated the data. However, the model comparatively yielding the best fit may still be invalid. We demonstrate that taking for granted the vital requirement that a model by itself should adequately describe the data can easily lead to flawed conclusions. Thus, prior to considering the relative discrepancy of competing models, it is necessary to assess their absolute fit and thus, again, attempt falsification. Finally, the scientific value of model fit is discussed from a broader perspective.

We analyze behavior in two basic classes of decision tasks: description-based and experience-based. In particular, we compare the prediction power of a number of decision learning models in both kinds of tasks. Unlike most previous studies, we focus on individual, rather than aggregate, behavioral characteristics. We carry out an experiment involving a battery of both description- and experience-based choices between two mixed binary prospects made by each of the participants, and employ a number of formal models for explaining and predicting participants’ choices: Prospect theory (PT) (Kahneman & Tversky, 1979); Expectancy-Valence model (EVL) (Busemeyer & Stout, 2002); and three combinations of these well-established models. We document that the PT and the EVL models are best for predicting people’s decisions in description- and experience-based tasks, respectively, which is not surprising as these two models are designed specially for these kinds of tasks. Furthermore, we find that models involving linear weighting of gains and losses perform better in both kinds of tasks, from the point of view of generalizability and individual parameter consistency. We therefore, conclude that, overall, when both prospects are mixed, the assumption of diminishing sensitivity does not improve models’ prediction power for individual decision-makers. Finally, for some of the models’ parameters, we document consistency at the individual level between description- and experience-based tasks.

Circular data originates in a wide range of scientific fields and can be analyzed on the basis of directional statistics and special distributions wrapped around the circumference. However, both propensity to transform non-linear to linear data and complexity of directional statistics limited the generalization of the circular paradigm in the animal breeding framework, among others. Here, we generalized a circular mixed (CM) model within the context of Bayesian inference. Three different parametrizations with different hierarchical structures were developed on basis of the von Mises distribution; moreover, both goodness of fit and predictive ability from each parametrization were compared through the analyses of 110 116 lambing distribution records collected from Ripollesa sheep herds between 1976 and 2017. The naive circular (NC) model only accounted for population mean and homogeneous circular variance, and reached the lowest goodness-of-fit and predictive ability. The CM model assumed a hierarchical structure for the population mean by accounting for systematic (ewe age and lambing interval) and permanent environmental sources of variation (flock-year-season and ewe). This improved goodness of fit by reducing both the deviance information criterion (DIC; −2520 units) and the mean square error (MSE; −12.4%) between simulated and predicted lambing data when compared against the NC model. Finally, the last parametrization expanded CM model by also assuming a hierarchical structure with systematic and permanent environmental factors for the variance parameter of the von Mises distribution (i.e. circular canalization (CC) model). This last model reached the best goodness of fit to lambing distribution data with a DIC estimate 5425 units lower than the one for NC model (MSE reduced 13.2%). The same pattern revealed when models were compared in terms of predictive ability. The superiority revealed by CC model emphasized the relevance of heteroskedasticity for the analysis of lambing distribution in the Ripollesa breed, and suggested potential applications for the sheep industry, even genetic selection for canalization. The development of CM models on the basis of the von Mises distribution has allowed to integrate flexible hierarchical structures accounting for different sources of variation and affecting both mean and dispersion terms. This must be viewed as a useful statistical tool with multiple applications in a wide range of research fields, as well as the livestock industry. The next mandatory step should be the inclusion of genetic terms in the hierarchical structure of the models in order to evaluate their potential contribution to current selection programs.

In order to ensure the construct validity of the three-factor model of the Multi-dimensional Scale of Perceived Social Support (MSPSS), and based on the assumption that it helps users differentiate between sources of social support, in this study a revised version was created and tested. The aim was to compare the level of model fit of the original version of the MSPSS against the revised version - which contains a minor change from the original. The study was conducted on 486 medical students who completed the original and revised versions of the MSPSS, as well as the Rosenberg Self-Esteem Scale (Rosenberg, 1965) and Beck Depression Inventory II (Beck, Steer, & Brown, 1996). Confirmatory factor analysis was performed to compare the results, showing that the revised version of MSPSS demonstrated a good internal consistency - with a Cronbach's alpha of .92 for the MSPSS questionnaire, and a significant correlation with the other scales, as predicted. The revised version provided better internal consistency, increasing the Cronbach's alpha for the Significant Others sub-scale from 0.86 to 0.92. Confirmatory factor analysis revealed an acceptable model fit: χ2 128.11, df 51, p < .001; TLI 0.94; CFI 0.95; GFI 0.90; PNFI 0.71; AGFI 0.85; RMSEA 0.093 (0.073-0.113) and SRMR 0.042, which is better than the original version. The tendency of the new version was to display a better level of fit with a larger sample size. The limitations of the study are discussed, as well as recommendations for further study.

The aim of the present study was to investigate the daily measured traits milk yield, water intake and dry matter intake with fixed and random regression models added with different error covariance structures. It was analysed whether these models deliver better model fitting in contrast to conventional fixed and random regression models. Furthermore, possible autocorrelation between repeated measures was investigated. The effect of model choice on statistical inference was also tested. Data recording was performed on the Futterkamp dairy research farm of the Chamber of Agriculture of Schleswig-Holstein. A dataset of about 21 000 observations from 178 Holstein cows was used. Average milk yield, water intake and dry matter intake were 34.9, 82.4 and 19.8 kg, respectively. Statistical analysis was performed using different linear mixed models. Lactation number, test day and the parameters to model the function of lactation day were included as fixed effects. Different structures were tested for the residuals; they were compared for their ability to fit the model using the likelihood ratio test, and Akaike’s and Bayesian’s information criteria. Different autocorrelation patterns were found. Adjacent repeated measures of daily milk yield were highest correlated (p1 = 0.32) in contrast to measures further apart, while for water intake and dry matter intake, the measurements with a lag of two units had the highest correlations with p2 = 0.11 and 0.12. The covariance structure of TOEPLITZ was most suitable to indicate the dependencies of the repeated measures for all traits. Generally, the most complex model, random regression with the additional covariance structure TOEPLITZ(4), provided the lowest information criteria. Furthermore, the model choice influenced the significance values of one fixed effect and therefore the general inference of the data analysis. Thus, the random regression + TOEPLITZ(4) model is recommended for use for the analysis of equally spaced datasets of milk yield, water intake and dry matter intake.