The goodness of fit of categorical models of psychological processes is often assessed with the log-likelihood ratio statistic (G2), but its underlying asymptotic theory is known to have limited empirical validity. We use examples from the scenario of fitting psychometric functions to psychophysical discrimination data to show that two factors are responsible for occasional discrepancies between actual and asymptotic distributions of G2. One of them is the eventuality of very small expected counts, by which the number of degrees of freedom should be computed as (J−1) × I−P−K0.06, where J is the number of response categories in the task, I is the number of comparison levels, P is the number of free parameters in the fitted model, and K0.06 is the number of cells in the implied I × J table in which expected counts do not exceed 0.06. The second factor is the administration of small numbers ni of trials at each comparison level xi (1 ≤ i ≤ I). These numbers should not be ridiculously small (i.e., lower than 10) but they need not be identical across comparison levels. In practice, when ni varies across levels, it suffices that the overall number N of trials exceeds 40 × I if J = 2 or 50 × I if J = 3, with no ni lower than 10. Correcting the degrees of freedom and using large ni are easy to implement in practice. These precautions ensure the validity of goodness-of-fit tests based on G2.

In item response theory (IRT), it is often necessary to perform restricted recalibration (RR) of the model: A set of (focal) parameters is estimated holding a set of (nuisance) parameters fixed. Typical applications of RR include expanding an existing item bank, linking multiple test forms, and associating constructs measured by separately calibrated tests. In the current work, we provide full statistical theory for RR of IRT models under the framework of pseudo-maximum likelihood estimation. We describe the standard error calculation for the focal parameters, the assessment of overall goodness-of-fit (GOF) of the model, and the identification of misfitting items. We report a simulation study to evaluate the performance of these methods in the scenario of adding a new item to an existing test. Parameter recovery for the focal parameters as well as Type I error and power of the proposed tests are examined. An empirical example is also included, in which we validate the pediatric fatigue short-form scale in the Patient-Reported Outcome Measurement Information System (PROMIS), compute global and local GOF statistics, and update parameters for the misfitting items.

Bentler and Bonett's nonnormed fit index is a widely used measure of goodness of fit for the analysis of covariance structures. This note shows that contrary to what has been claimed the nonnormed fit index is dependent on sample size. Specifically for a constant value of a fitting function, the nonnormed index is inversely related to sample size. A simple alternative fit measure is proposed that removes this dependency. In addition, it is shown that this new measure as well as the old nonnormed fit index can be applied to any fitting function that measures the deviation of the observed covariance matrix from the covariance matrix implied by the parameter estimates for a model.

We relate Thurstonian models for paired comparisons data to Thurstonian models for ranking data, which assign zero probabilities to all intransitive patterns. We also propose an intermediate model for paired comparisons data that assigns nonzero probabilities to all transitive patterns and to some but not all intransitive patterns.

There is a close correspondence between the multidimensional normal ogive model employed in educational testing and Thurstone's model for paired comparisons data under multiple judgment sampling with minimal identification restrictions. Alike the normal ogive model, Thurstonian models have two formulations, a factor analytic and an IRT formulation. We use the factor analytic formulation to estimate this model from the first and second order marginals of the contingency table using estimators proposed by Muthén. We also propose a statistic to assess the fit of these models to the first and second order marginals of the contingency table. This is important, as a model may reproduce well the estimated thresholds and tetrachoric correlations, yet fail to reproduce the marginals of the contingency table if the assumption of multivariate normality is incorrect.

A simulation study is performed to investigate the performance of three alternative limited information estimators which differ in the procedure used in their final stage: unweighted least squares (ULS), diagonally weighted least squares (DWLS), and full weighted least squares (WLS). Both the ULS and DWLS show a good performance with medium size problems and small samples, with a slight better performance of the ULS estimator.

Fit indices are highly frequently used for assessing the goodness of fit of latent variable models. Most prominent fit indices, such as the root-mean-square error of approximation (RMSEA) or the comparative fit index (CFI), are based on a noncentrality parameter estimate derived from the model fit statistic. While a noncentrality parameter estimate is well suited for quantifying the amount of systematic error, the complex weighting function involved in its calculation makes indices derived from it challenging to interpret. Moreover, noncentrality-parameter-based fit indices yield systematically different values, depending on the indicators’ level of measurement. For instance, RMSEA and CFI yield more favorable fit indices for models with categorical as compared to metric variables under otherwise identical conditions. In the present article, approaches for obtaining an approximation discrepancy estimate that is independent from any specific weighting function are considered. From these unweighted approximation error estimates, fit indices analogous to RMSEA and CFI are calculated and their finite sample properties are investigated using simulation studies. The results illustrate that the new fit indices consistently estimate their true value which, in contrast to other fit indices, is the same value for metric and categorical variables. Advantages with respect to interpretability are discussed and cutoff criteria for the new indices are considered.

Mean corrected higher order sample moments are asymptotically normally distributed. It is shown that both in the literature and popular software the estimates of their asymptotic covariance matrices are incorrect. An introduction to the infinitesimal jackknife is given and it is shown how to use it to correctly estimate the asymptotic covariance matrices of higher order sample moments. Another advantage in using the infinitesimal jackknife is the ease with which it may be used when stacking or sub-setting estimators. The estimates given are used to test the goodness of fit of a non-linear factor analysis model. A computationally accelerated form for infinitesimal jackknife estimates is given.

The problem of dependence of the nonnormed fit index on sample size in covariance structure analysis is discussed. Contrary to Bollen (1986) we show that the mean of the nonnormed fit index is independent of sample size for true and almost true models whereas Bollen's alternative index does depend on sample size.

A model of Bradley-Terry type for paired comparisons is considered. In addition to the usual parameters, the model allows for individual parameters and corresponding parameters for the choice-alternatives. The model is applied to a set of data from a Danish investigation of the attitude of blue collar workers towards alternative social gains. The proposed model is finally compared to a model recently suggested by Schönemann and Wang.

In this paper, the problem of goodness of fit test for the inverse Rayleigh distribution based on progressively Type-II censored samples is studied. We develop two test statistics via entropy and propose one new non-entropy test statistic via a pivotal method. We also study the properties of these test statistics. Critical values are obtained by simulations. Then, we do power analysis of these test statistics against various alternatives under different censoring schemes. We conclude that the tests we proposed perform well against various alternatives, especially for non-monotone hazard alternatives. Finally, one real data set is analyzed.

Relational event models are becoming increasingly popular in modeling temporal dynamics of social networks. Due to their nature of combining survival analysis with network model terms, standard methods of assessing model fit are not suitable to determine if the models are specified sufficiently to prevent biased estimates. This paper tackles this problem by presenting a simple procedure for model-based simulations of relational events. Predictions are made based on survival probabilities and can be used to simulate new event sequences. Comparing these simulated event sequences to the original event sequence allows for in depth model comparisons (including parameter as well as model specifications) and testing of whether the model can replicate network characteristics sufficiently to allow for unbiased estimates.

In the classical collective model over a fixed time period of two insurance portfolios, we are interested, in this contribution, in the models that relate to the joint distribution F of the largest claim amounts observed in both insurance portfolios. Specifically, we consider the tractable model where the claim counting random variable N follows a discrete-stable distribution with parameters (α,λ). We investigate the dependence property of F with respect to both parameters α and λ. Furthermore, we present several applications of the new model to concrete insurance data sets and assess the fit of our new model with respect to other models already considered in some recent contributions. We can see that our model performs well with respect to most data sets.

Insurance and annuity products covering several lives require the modelling of the joint distribution of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people are independent. However, extensive research over the past decades suggests otherwise. In this paper, a copula approach is used to model the dependence between lifetimes within a married couple using data from a large Canadian insurance company. As a novelty, the age difference and the gender of the elder partner are introduced as an argument of the dependence parameter. Maximum likelihood techniques are thus implemented for the parameter estimation. Not only do the results make clear that the correlation decreases with age difference, but also the dependence between the lifetimes is higher when husband is older than wife. A goodness-of-fit procedure is applied in order to assess the validity of the model. Finally, considering several annuity products available on the life insurance market, the paper concludes with practical illustrations.

The distributional behavior of futures price spreads is examined for four commodities: corn, live cattle, gold and T-bonds. Remarkably different results are found over commodities, time period, and sample size. Actual spread changes for the smaller sample size of gold and T-bonds and for corn produce more normal distributions for weekly than for daily differencing intervals, while all live cattle spreads for actual changes are normally distributed. However, the larger sample size of both gold and T-bonds and the relative spread changes for corn and live cattle do not become more normally distributed under temporal aggregation of the data.

State-of-the-art item response theory (IRT) models use logistic functions exclusively as their item response functions (IRFs). Logistic functions meet the requirements that their range is the unit interval and that they are monotonically increasing, but they impose a parameter space whose dimensions can only be assigned a metaphorical interpretation in the context of testing. Applications of IRT models require obtaining the set of values for logistic function parameters that best fit an empirical data set. However, success in obtaining such set of values does not guarantee that the constructs they represent actually exist, for the adequacy of a model is not sustained by the possibility of estimating parameters. This article illustrates how mechanical adoption of off-the-shelf logistic functions as IRFs for IRT models can result in off-the-shelf parameter estimates and fits to data. The results of a simulation study are presented, which show that logistic IRT models can fit a set of data generated by IRFs other than logistic functions just as well as they fit logistic data, even though the response processes and parameter spaces involved in each case are substantially different. An explanation of why logistic functions work as they do is offered, the theoretical and practical consequences of their behavior are discussed, and a testable alternative to logistic IRFs is commented upon.