Latent class analysis with ordered latent classeCroon, Marcel
doi: 10.1111/j.2044-8317.1990.tb00934.xpmid: N/A
In this paper a latent class model is described in which the latent classes are ordered imposing inequality constraints on item response and cumulative response probabilities from subsequent latent classes. These inequality constraints are derived from the basic assumption that, when the latent classes may be ordered from low to high along the latent continuum, the probability of a ‘positive’ response should increase monotonically as one moves along this continuum. An algorithm to obtain the maximum likelihood estimates of the model parameters is proposed and is applied to a real data set.
An approximately standardized person test for assessing consistency with a latent trait modelKlauer, Karl Christoph; Rettig, Klaus
doi: 10.1111/j.2044-8317.1990.tb00935.xpmid: N/A
Latent trait models predict that an examinee's ability level is invariant over subtests of the total test, that is, that all subtests measure the same latent trait. Three statistics are proposed and evaluated as significance tests of this invariance hypothesis. They are computed on the basis of a single response vector and are asymptotically standardized, that is, their conditional distributions given a particular level of ability do not depend upon the absolute value of the examinee's ability parameter. In a Monte Carlo study, one of the statistics turned out to perform satisfactorily for test lengths of 80 items. The use of the test statistic is illustrated in two applications.
Selection effects and unidimensionality in item response theoryGrayson, D. A.
doi: 10.1111/j.2044-8317.1990.tb00936.xpmid: N/A
In the factor‐analytic perspective on unidimensional item response theory (IRT), the latent trait is construed as a single common factor to the items, each item having an independent residual which determines the item characteristic curve (ICC). It is possible that such residual terms have a component which is ‘stable’ and reflects some attribute of the subjects being measured other than the trait in common to all the items (the common factor). That is, heuristically, the ‘residual’ may consist of ‘specific’ and ‘error’ components. As such, comparison of subpopulations on such a test may confound selection effects on the common factor with selection effects on such stable components of the residuals, vitiating inference from manifest test response differences to common factor differences. A technique for exploratory use in assessing the presence of such (undesirable) stable residuals using repeated measures data is discussed, together with applications. The technique is of broader potential application, allowing one to use repeated measures data to search for ‘user‐defined’ violations of unidimensionality. It assumes the existence of a sufficient score for the common factor, and is thus an exact technique when a Rasch model is assumed.
Non‐linear redundancy analysis †Burg, Eeke; Leeuw, Jan
doi: 10.1111/j.2044-8317.1990.tb00937.xpmid: N/A
A non‐linear version of redundancy analysis is introduced. The technique is called REDUNDALS. It is implemented within the computer program for canonical correlation analysis called CANALS (Van der Burg & De Lccuw, 1983). The REDUNDALS algorithm is of an alternating least squares (ALS) type. The technique is defined as minimization of a squared distance between criterion variables and weighted predictor variables. The matrix of weights can be restricted to a specified rank. With the help of optimal scaling the variables are transformed non‐linearly (cf. Young, 1981). An application of redundancy analysis is provided.
A multimode direct product model for covariance structure analysisVerhees, J.; Wansbeek, T. J.
doi: 10.1111/j.2044-8317.1990.tb00938.xpmid: N/A
In the psychometric literature, there is evidence that the modes (facets, dimensions, etc.) in multimode data interact multiplicatively. A basic expression of this idea is that a covariance matrix may then be written as repeated Kronecker product of k, say, parameter matrices, where k is the number of modes. For such a ‘factorial covariance structure’ we give an integrated treatment of ML, WLS and ULS estimators. A modification of the ULS estimator appears to be non‐iteratively computable.
The rank difference test: A new and meaningful alternative to the Wilcoxon signed ranks test for ordinal dataKornbrot, Diana Eugenie
doi: 10.1111/j.2044-8317.1990.tb00939.xpmid: N/A
A new distribution free inferential procedure for comparing paired observations, called the ‘rank difference test’, is presented. The new procedure is based on ranks, and is applicable as a robust alternative to the related f test in all situations where the Wilcoxon signed ranks test is applicable. In additions it may be applied to ordinal data or operational measures which do not meet the assumptions underlying the Wilcoxon. Since the rank difference statistic, D, can take on half‐integer, as well as integer, values, it has continuity advantages over the Wilcoxon for small samples. The exact null distribution of the rank difference test statistic, D, is derived and tabulated for samples sizes from 2 to 7 pairs. The null distribution of the rank difference test statistic is shown to be asymptotically the same as that of the Wilcoxon test statistic for large sample sizes. Permutation methods are used to derive the null distribution for sample sizes from 8 to 20 based on 400000 simulations for each sample size. The practical application of the rank difference test is evaluated by comparing its power and performance with that of the Wilcoxon, the sign test and the related t test, for several classic sets of data in the literature, and for several sets of simulated data.
Analysing unbalanced repeated measures designsKeselman, Joanne C.; Keselman, H. J.
doi: 10.1111/j.2044-8317.1990.tb00940.xpmid: N/A
For repeated measures designs containing at least one between‐subjects factor (split‐plot designs), exact univariate F tests of within‐subjects effects depend on the assumption of multisample sphericity. Monte Carlo methods were used to examine univariate and multivariate data‐analytic strategies for unbalanced split‐plot repeated measures designs in which the data do not conform to the condition of multisample sphericity. For varying degrees of assumption violation, empirical Type I error and power rates were determined for varying numbers of levels of the within‐subjects factor, total sample size, and degree of sample size inequality. Only the approximate univariate procedures provided robust tests of within‐subjects main effects in unbalanced designs. Robust univariate and multivariate tests of within‐subjects interaction effects were only achieved when the design was balanced and the number of repeated measurements was moderate.
Relative improvement over chance (RIOC) for 2×2 tablesCopas, John B.; Loeber, Rolf
doi: 10.1111/j.2044-8317.1990.tb00942.xpmid: N/A
A novel index of association, called Relative Improvement Over Chance (RIOC), is described for 2×2 tables that corrects for chance and for limitations in maximum predictability due to unequal distributions of the marginals in such tables. Criteria for comparing the utility of this and existing indexes of association are reviewed. The theoretical distribution of RIOC for large and small sample sizes is presented, and tested against the results from computer generated tables. Methods are presented for calculating confidence intervals for RIOC, for testing the significance of individual values of RIOC, and for testing the difference between several values of RIOC from different studies.
Variants of chi‐square for 2 × 2 contingency tablesRichardson, John T. E.
doi: 10.1111/j.2044-8317.1990.tb00943.xpmid: N/A
Four variants of the chi‐square statistic are evaluated in terms of their adequacy as tests of association in 2 × 2 contingency tables. When both sets of marginal totals are fixed in advance, none of these variants is wholly satisfactory, even with moderately large samples. When one set of marginal totals is fixed in advance while the other is free to vary, or when neither set of marginal totals is fixed in advance, a variant of chi‐square proposed by Upton (1982) is to be preferred both theoretically and in practice. The variant proposed by Yates (1934) which incorporates a correction for continuity and which is often recommended in statistics textbooks is theoretically unsound, shows an extremely conservative bias, and lacks sufficient power.