ANOVA and ANCOVA of pre- and post-test,ordinal data

With random assignment to treatments and standard assumptions, either a one-way ANOVA of post-test scores or a two-way, repeated measures ANOVA of pre- and post-test scores provides a legitimate test of the equal treatment effect null hypothesis for latent variable . In an ANCOVA for pre- and post-test variablesX andY which are ordinal measures of and , respectively, random assignment and standard assumptions ensure the legitimacy of inferences about the equality of treatment effects on latent variable . Sample estimates of adjustedY treatment means are ordinal estimators of adjusted post-test means on latent variable .     

On the estimation of polychoric correlations and their asymptotic covariance matrix

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.The research was supported by the Swedish Council for Research in the Humanities and Social Sciences (HSFR) under the programMultivariate Statistical Analysis. The author thanks a reviewer for pointing out an error in the original version of the paper.     

Sensitivity analysis in factor analysis: Difference between using covariance and correlation matrices

Influence curves of some parameters under various methods of factor analysis have been given in the literature. These influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis. The differences between the influence curves based on the covariance and the correlation matrices are derived in this paper. Simple formulas for the differences of the influence curves, based on the two matrices, for the unique variance matrix, factor loadings and some other parameter are obtained under scale-invariant estimation methods, though the influence curves themselves are in complex forms.The authors are most grateful to the referees, the Associate Editor, the Editor and Raymond Lam for helpful suggestions for improving the clarity of the paper.     

The approximation of two-mode proximity matrices by sums of order-constrained matrices

A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered define Q-forms (or anti-Q-forms) for a two-mode matrix, where after suitable and separate row and column reorderings, the entries within each row and within each column are nondecreasing (or nonincreasing) to a maximum (or minimum) and thereafter nonincreasing (or nondecreasing). Several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy.     

Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value

A method is presented for constructing a covariance matrix Σ*₀ that is the sum of a matrix Σ(γ₀) that satisfies a specified model and a perturbation matrix,E, such that Σ*₀=Σ(γ0) +E. The perturbation matrix is chosen in such a manner that a class of discrepancy functionsF(Σ*₀, Σ(γ₀)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ₀ as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator. We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.     

A comparison of generalized and modified sample biserial correlation estimators

In a recent paper, Bedrick derived the asymptotic distribution of Lord's modified sample biserial correlation estimator and studied its efficiency for bivariate normal populations. We present a more detailed examination of the properties of Lord's estimator and several competitors, including Brogden's estimator. We show that Lord's estimator is more efficient for three nonnormal distributions than a generalization of Pearson's sample biserial estimator. In addition, Lord's estimator is reasonably efficient relative to the maximum likelihood estimator for these distributions. These conclusions are consistent with Bedrick's results for the bivariate normal distribution. We also study the small sample bias and variance of Lord's estimator, and the coverage properties of several confidence interval estimates.The author would like to thank the referees for several suggestions that improved the paper.     

Correspondence analysis and optimal structural representations

Many well-known measures for the comparison of distinct partitions of the same set ofn objects are based on the structure of class overlap presented in the form of a contingency table (e.g., Pearson's chi-square statistic, Rand's measure, or Goodman-Kruskal'sτ _b ), but they all can be rephrased through the use of a simple cross-product index defined between the corresponding entries from twon ×n proximity matrices that provide particular a priori (numerical) codings of the within- and between-class relationships for each of the partitions. We consider the task of optimally constructing the proximity matrices characterizing the partitions (under suitable restriction) so as to maximize the cross-product measure, or equivalently, the Pearson correlation between their entries. The major result presented states that within the broad classes of matrices that are either symmetric, skew-symmetric, or completely arbitrary, optimal representations are already derivable from what is given by a simple one-dimensional correspondence analysis solution. Besides severely limiting the type of structures that might be of interest to consider for representing the proximity matrices, this result also implies that correspondence analysis beyond one dimension must always be justified from logical bases other than the optimization of a single correlational relationship between the matrices representing the two partitions.     

Acquisition and generalization of manual signs by hearing-impaired adults with mental retardation

Behavioral procedures have been used to teach manual communication skills to individuals with mental retardation, although few studies have examined the assessment of generalization of such skills beyond the teaching environment. In this study, we investigated the effectiveness of directed rehearsal for teaching and facilitating the generalization of manual signs by six hearing-impaired adults with mental retardation. The effects of directed rehearsal, alone and combined with positive reinforcement for correct responses, were compared to a no-training control condition in an alternating treatments design. Although directed rehearsal was superior to the no-training control condition in teaching manual signs, the procedure was more effective when combined with positive reinforcement. The combined procedure was used to facilitate the generalization of learned signs across a number of variables. The results showed that there were high levels of generalization by all participants across novel phrases, settings, and trainers but variable levels of generalization to another response mode by some subjects. Further, increased levels in overall vocal communication were found for most participants.     

Equivalent models in covariance structure analysis

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified modelsM ₁ andM ₂ when fitting the more restricted modelM ₀. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters ofM ₁ andM ₂ (which characterizes modelM ₁₂), is a necessary and sufficient condition for the local equivalence ofM ₁ andM ₂. This condition is satisfied, in practice, when the analysis dealing with the fitting ofM ₀, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting ofM ₁ orM ₂, orM ₁₂ are all equal for any set of sample data, except on differences due to rounding errors.This research was supported by the Foundation of Social-Cultural Sciences which is subsidized by the Dutch Scientific Organization (N.W.O.) under project number 500-278-003. The author wishes to thank Anne Boomsma, Ivo Molenaar, Albert Satorra, and Tom Snijders for their stimulating and crucial comments during the research, and the Editor, Paul Bekker, Henk Broer, and anonymous reviewers for their helpful suggestions.     

More “tricks of the trade”: Reparameterizing LISREL models using negative variances

This paper shows how LISREL may be used to estimate simplex models which impose constraints on the variances of endogenous variables. This technique allows us to estimate both the parameters and the standard errors of the correlated measurement error model proposed by Wiley and Wiley (1974).We would like to thank Jim Wiley for his many helpful comments and suggestions on an earlier draft. We are grateful also to an anonymous reviewer for supplying the EQS program presented in Figure 4.