Testing equality of correlated proportions with incomplete data on both responses

Two test statistics are proposed for testing the equality of two correlated proportions when some observations are missing on both responses. The performance of these tests in terms of size and power is compared with other tests by means of Monte Carlo simulations. The proposed tests are easily computed and compare favorably with other tests.     

On testing the equality of proportions in the paired case with incomplete data

The problem of testing two correlated proportions with incomplete data is considered by means of Monte Carlo simulations studies. A test proposed in this paper, which can be regarded as a generalization of McNemar's test, is recommended in all cases with incomplete data and not too small samples.     

Trend in correlated proportions

A random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable. Because the observed proportions may be correlated under these conditions, estimating their trend with respect to the independent variable is no longer a standard problem for probit, logit or loglinear analysis. Using a qualitative analogue of a random regressions model, we employ instead marginal maximum likelihood to estimate the average latent trend line. Likelihood ratio tests of the hypothesis of no trend in the average line, and the hypothesis of no differences in average trend lines between experimental treatments, are proposed. We illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms.Supported by a grant from the MacArthur Foundation and National Science Foundation Grant BNS85-11774.The authors are indebted to James Heckman for calling our attention to the Clark algorithm.     

On using multiple imputation for exploratory factor analysis of incomplete data

A simple multiple imputation-based method is proposed to deal with missing data in exploratory factor analysis. Confidence intervals are obtained for the proportion of explained variance. Simulations and real data analysis are used to investigate and illustrate the use and performance of our proposal.     

Note on the correction for continuity in testing the significance of the difference between correlated proportions

Methods of correcting for continuity in tests of significance of the difference between correlated proportions are presented. These corrections should increase the range of usefulness of the formulas developed by McNemar (1).     

Note on the sampling error of the difference between correlated proportions or percentages

Two formulas are presented for judging the significance of the difference between correlated proportions. The chi square equivalent of one of the developed formulas is pointed out.     

Dual imputation model for incomplete longitudinal data

Missing values are a practical issue in the analysis of longitudinal data. Multiple imputation (MI) is a well‐known likelihood‐based method that has optimal properties in terms of efficiency and consistency if the imputation model is correctly specified. Doubly robust (DR) weighing‐based methods protect against misspecification bias if one of the models, but not necessarily both, for the data or the mechanism leading to missing data is correct. We propose a new imputation method that captures the simplicity of MI and protection from the DR method. This method integrates MI and DR to protect against misspecification of the imputation model under a missing at random assumption. Our method avoids analytical complications of missing data particularly in multivariate settings, and is easy to implement in standard statistical packages. Moreover, the proposed method works very well with an intermittent pattern of missingness when other DR methods can not be used. Simulation experiments show that the proposed approach achieves improved performance when one of the models is correct. The method is applied to data from the fireworks disaster study, a randomized clinical trial comparing therapies in disaster‐exposed children. We conclude that the new method increases the robustness of imputations.     

A critique of the correlated trait-correlated method and correlated uniqueness models for multitrait-multimethod data

The correlated trait-correlated method (CT-CM) and correlated uniqueness (CU) confirmatory factor analysis models for multitrait-multimethod data are critiqued. Although the CU model often returns convergent and admissible factor solutions when the CT-CM model does not, the CU model is shown to have theoretical and substantive shortcomings. On the basis of this critique, the authors recommend that the CT-CM model be regarded as the generally preferred model and that the CU model be invoked only when the CT-CM model fails.     

Recovery of structure in incomplete data by alscal

A Monte Carlo study was carried out in order to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures when portions of the data are missing. All sets of simulated proximity measures were based on 30 stimuli and three dimensions, and selection of missing elements was done randomly. Properties of the simulated data varied according to (a) the number of individuals, (b) the level of random error, (c) the proportion of missing data, and (d) whether the same entries or different entries were deleted for each individual. Results showed that very accurate recovery of true distances, stimulus coordinates, and weight vectors could be achieved with as much as 60% missing data as long as sample size was sufficiently large and the level of random error was low.     

One-sided nonparametric tests for ordinal data

Baumgartner, Weiss, and Schindler (1998) introduced a novel non-parametric test for the two-sample comparison that is superior to commonly used tests such as the Wilcoxon rank-sum test. A modification of the novel test statistic can be used for one-sided comparisons based on ordinal data. Such comparisons frequently occur in psychological research, and the Wilcoxon test is often recommended for their analysis. Here, the two tests were compared in a simulation study. According to this study the tests have a similar type I error rate, but the modified Baumgartner-Weiss-Schindler test is more powerful than the Wilcoxon test.     

A least squares solution for paired comparisons with incomplete data

A precise and rapid procedure has been devised for dealing with a matrix of incomplete data in paired comparisons. This method should increase the general applicability of paired comparisons since experiments involving large numbers of stimuli may now be shortened to feasible experimental proportions. Also, we may now use sets of stimuli which cover a wide range, resulting in a considerable number of 100 per cent vs. 0 per cent judgments, and still give a precise solution depending equally on each of the observations.This study was supported in part by the Office of Naval Research Contract N6onr-270-20 with Princeton University and by the National Science Foundation Grant G-642. The opinions expressed are, of course, those of the author and do not represent attitudes or policies of the Office of Naval Research or of the National Science Foundation.The author wishes to acknowledge helpful suggestions and comments received from Frederic M. Lord, Warren S. Torgerson, and Ledyard R Tucker. Thanks are also due to Mrs. Gertrude Diederich for some of the tabulating and computing for the illustrative problem.     

An incomplete data approach to the analysis of covariance structures

In this paper, linear structural equation models with latent variables are considered. It is shown how many common models arise from incomplete observation of a relatively simple system. Subclasses of models with conditional independence interpretations are also discussed. Using an incomplete data point of view, the relationships between the incomplete and complete data likelihoods, assuming normality, are highlighted. For computing maximum likelihood estimates, the EM algorithm and alternatives are surveyed. For the alternative algorithms, simplified expressions for computing function values and derivatives are given. Likelihood ratio tests based on complete and incomplete data are related, and an example on using their relationship to improve the fit of a model is given.This research forms part of the author's doctoral thesis and was supported by a Commonwealth Postgraduate Research Award. The author also wishes to acknowledge the support of CSIRO during the preparation of this paper and the referees' comments which led to substantial improvements.     

Tests of homogeneity of means and covariance matrices for multivariate incomplete data

Existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns. The likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects. A generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples. Three factors were varied for a simulation: number of variables, percent missing completely at random, and sample size. One thousand data sets were simulated for each condition. The generalized least squares test of homogeneity of means performed close to an ideal Type I error rate for most of the conditions. The generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also.Preliminary results on this research were presented at the 1999 Western Psychological Association convention, Irvine, CA, and in the UCLA Statistics Preprint No. 265 (http://www.stat.ucla.edu). The assistance of Ke-Hai Yuan and several anonymous reviewers is gratefully acknowledged.     

Clustering binary data in the presence of masking variables

A number of important applications require the clustering of binary data sets. Traditional nonhierarchical cluster analysis techniques, such as the popular K-means algorithm, can often be successfully applied to these data sets. However, the presence of masking variables in a data set can impede the ability of the K-means algorithm to recover the true cluster structure. The author presents a heuristic procedure that selects an appropriate subset from among the set of all candidate clustering variables. Specifically, this procedure attempts to select only those variables that contribute to the definition of true cluster structure while eliminating variables that can hide (or mask) that true structure. Experimental testing of the proposed variable-selection procedure reveals that it is extremely successful at accomplishing this goal.     

On the Wilson tests

A general critical analysis of the median tests proposed by Wilson for certain analysis of variance hypotheses is presented. Specifically, discrepancies between the purported and actual approximate distributions of some of the test statistics are noted. Validity and power of the resulting tests are discussed.This work was sponsored in part by the Office of Naval Research while the author was at Stanford University. Reproduction in whole or in part is permitted for any purpose of the United States Government. The author wishes to thank Professors Fred C. Andrews, Lincoln E. Moses, and David L. Wallace for their helpful criticisms and suggestions in the writing of this paper.     

Evaluating order-constrained hypotheses for circular data using permutation tests

Psychological researchers in different fields sometimes encounter circular or directional data. Circular data are data measured in the form of angles or two-dimensional orientations. As an example, experiments investigating the development of spatial memory and the influence of visual experience on haptic orientation perception are presented. Three permutation tests are proposed for the evaluation of ordered hypotheses. The quality of the permutation tests is investigated by means of several simulation studies. The results of these studies show the expected increase in power when the permutation tests for ordered hypotheses are compared to a common non-directional test for circular data. The differences in power between the three tests for ordered alternatives are small.     

Testing for the presence of 1/f noise in continuation tapping data.

A number of recent papers have suggested that the series of time intervals produced in continuation tapping may have fractal properties. This proposition, nevertheless, was only based on the visual appraisal of graphical results, and was not statistically supported. In the present study, we applied the ARMA/ARFIMA modeling procedures proposed by Wagenmakers, Farrell, and Ratcliff (2005) to test for the presence of long-range dependencies in continuation tapping data. Our results demonstrate the presence of long-range dependencies in most series and offer strong support for the hypothesis that fluctuations in tapping series are fractal in nature.     

The estimation of variance-covariance and correlation matrices from incomplete data

Employing simulated data, several methods for estimating correlation and variance-covariance matrices are studied for observations missing at random from data matrices. The effect of sample size, number of variables, percent of missing data and average intercorrelations of variables are examined for several proposed methods.The author is indebted to Professors Leonard A. Marascuilo, Gus W. Haggstrom, especially Henry F. Kaiser for their invaluable suggestions throughout this work. Appreciation is also extended to the Computer Center facility of the University of California at Berkeley for the use of computer time to complete the necessary computations.     

Relating magnitudes: the brain's code for proportions

Whereas much is known about how we categorize and reason based on absolute quantity, data exploring ratios of quantities, as in proportions and fractions, are comparatively sparse. Until recently, it remained elusive whether these two representations of number are connected, how proportions are implemented by neurons and how language shapes this code. New data derived with complementary methods and from different model systems now shed light on the mechanisms of magnitude ratio representations. A coding scheme for proportions has emerged that is remarkably reminiscent of the representation of absolute number. These novel findings suggest a sense for ratios that grants the brain automatic access to proportions independently of language and the format of presentation.