Fisher transformations for correlations corrected for selection and missing data

The validity of a test is often estimated in a nonrandom sample of selected individuals. To accurately estimate the relation between the predictor and the criterion we correct this correlation for range restriction. Unfortunately, this corrected correlation cannot be transformed using Fisher'sZ transformation, and asymptotic tests of hypotheses based on small or moderate samples are not accurate. We developed a Fisherr toZ transformation for the corrected correlation for each of two conditions: (a) the criterion data were missing due to selection on the predictor (the missing data were MAR); and (b) the criterion was missing at random, not due to selection (the missing data were MCAR). The twoZ transformations were evaluated in a computer simulation. The transformations were accurate, and tests of hypotheses and confidence intervals based on the transformations were superior to those that were not based on the transformations.     

Inferential procedures for correlation coefficients corrected for attenuation

A model and computational procedure based on classical test score theory are presented for determination of a correlation coefficient corrected for attenuation due to unreliability. Next, variance-covariance expressions for the sample estimates defined earlier are derived, based on application of the delta method. Results of a Monte Carlo study are presented in which the adequacy of the derived expressions was assessed for a large number of data forms and potential hypotheses encountered in the behavioral sciences. It is shown that, based on the proposed procedures, confidence intervals for single coefficients are reasonably precise. Two-sample hypothesis tests, for both independent and dependent samples, are also accurate. However, for hypothesis tests involving a larger number of coefficients than two—both independent and dependent—the proposed procedures require largens for adequate precision. Results of a preliminary power analysis reveal no serious loss in efficiency resulting from correction for attenuation. Implications for practice are discussed.Support for the research reported in this article was provided by the Natural Sciences and Engineering Research Council of Canada. The authors acknowledge with thanks the constructive comments of the editor and three anonymous reviewers.     

A proposal for handling missing data

A method for dealing with the problem of missing observations in multivariate data is developed and evaluated. The method uses a transformation of the principal components of the data to estimate missing entries. The properties of this method and four alternative methods are investigated by means of a Monte Carlo study of 42 computer-generated data matrices. The methods are compared with respect to their ability to predict correlation matrices as well as missing entries. The results indicate that whenever there exists modest intercorrelations among the variables (i.e., average off diagonal correlation above .2) the proposed method is at least as good as the best alternative (a regression method) while being considerably faster and simpler computationally. Models for determining the best alternative based upon easily calculated characteristics of the matrix are given. The generality of these models is demonstrated using the previously published results of Timm.     

Percentage of nonoverlapping corrected data

In the present study, we proposed a modification in one of the most frequently applied effect-size procedures in single-case data analysis: the percentage of nonoverlapping data. In contrast with other techniques, the calculus and interpretation of this procedure are straightforward and can be easily complemented by visual inspection of the graphed data. Although the percentage of nonoverlapping data has been found to perform reasonably well in N = 1 data, the magnitude of effect estimates that it yields can be distorted by trend and autocorrelation. Therefore, the data-correction procedure focuses on removing the baseline trend from data prior to estimating the change produced in the behavior as a result of intervention. A simulation study was carried out in order to compare the original and the modified procedures in several experimental conditions. The results suggest that the new proposal is unaffected by trend and autocorrelation and that it can be used in case of unstable baselines and sequentially related measurements.     

Estimation for structural equation models with missing data

A direct method in handling incomplete data in general covariance structural models is investigated. Asymptotic statistical properties of the generalized least squares method are developed. It is shown that this approach has very close relationships with the maximum likelihood approach. Iterative procedures for obtaining the generalized least squares estimates, the maximum likelihood estimates, as well as their standard error estimates are derived. Computer programs for the confirmatory factor analysis model are implemented. A longitudinal type data set is used as an example to illustrate the results.This research was supported in part by Research Grant DAD1070 from the U.S. Public Health Service. The author is indebted to anonymous reviewers for some very valuable suggestions. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

Best practices for missing data management in counseling psychology

This article urges counseling psychology researchers to recognize and report how missing data are handled, because consumers of research cannot accurately interpret findings without knowing the amount and pattern of missing data or the strategies that were used to handle those data. Patterns of missing data are reviewed, and some of the common strategies for dealing with them are described. The authors provide an illustration in which data were simulated and evaluate 3 methods of handling missing data: mean substitution, multiple imputation, and full information maximum likelihood. Results suggest that mean substitution is a poor method for handling missing data, whereas both multiple imputation and full information maximum likelihood are recommended alternatives to this approach. The authors suggest that researchers fully consider and report the amount and pattern of missing data and the strategy for handling those data in counseling psychology research and that editors advise researchers of this expectation.     

Missing data imputation and corrected statistics for large-scale behavioral databases

This article presents a new methodology for solving problems resulting from missing data in large-scale item performance behavioral databases. Useful statistics corrected for missing data are described, and a new method of imputation for missing data is proposed. This methodology is applied to the Dutch Lexicon Project database recently published by Keuleers, Diependaele, and Brysbaert (Frontiers in Psychology, 1, 174, 2010), which allows us to conclude that this database fulfills the conditions of use of the method recently proposed by Courrieu, Brand-D’Abrescia, Peereman, Spieler, and Rey (2011) for testing item performance models. Two application programs in MATLAB code are provided for the imputation of missing data in databases and for the computation of corrected statistics to test models.     

Imputation methods for missing data in educational diagnostic evaluation

In the diagnostic evaluation of educational systems, self-reports are commonly used to collect data, both cognitive and orectic. For various reasons, in these self-reports, some of the students' data are frequently missing. The main goal of this research is to compare the performance of different imputation methods for missing data in the context of the evaluation of educational systems. On an empirical database of 5,000 subjects, 72 conditions were simulated: three levels of missing data, three types of loss mechanisms, and eight methods of imputation. The levels of missing data were 5%, 10%, and 20%. The loss mechanisms were set at: Missing completely at random, moderately conditioned, and strongly conditioned. The eight imputation methods used were: listwise deletion, replacement by the mean of the scale, by the item mean, the subject mean, the corrected subject mean, multiple regression, and Expectation-Maximization (EM) algorithm, with and without auxiliary variables. The results indicate that the recovery of the data is more accurate when using an appropriate combination of different methods of recovering lost data. When a case is incomplete, the mean of the subject works very well, whereas for completely lost data, multiple imputation with the EM algorithm is recommended. The use of this combination is especially recommended when data loss is greater and its loss mechanism is more conditioned. Lastly, the results are discussed, and some future lines of research are analyzed.     

Fast regression estimates of missing data

A recent comparison of methods for estimating missing data concluded that when there is sufficient redundancy to justify using a more elaborate method than the mean of each variable, the principal components and regression methods are equally good and superior to the other methods investigated. Principal components was preferred because of its tremendous computational savings over the regression method. This note proposes an alternate implementation of the regression method which should be slightly faster than the principal components method.     

Estimation for the multiple factor model when data are missing

A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.This paper is based on the author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The author gratefully acknowledges the aid of Drs. Robert Bohrer, Charles Lewis, Robert Linn, Maurice Tatsuoka, and Ledyard Tucker.     

Asymptotic comparison of missing data procedures for estimating factor loadings

Large sample properties of four methods of handling multivariate missing data are compared. The criterion for comparison is how well the loadings from a single factor model can be estimated. It is shown that efficiencies of the methods depend on the pattern or arrangement of missing data, and an evaluation study is used to generate predictive efficiency equations to guide one's choice of an estimating procedure. A simple regression-type estimator is introduced which shows high efficiency relative to the maximum likelihood method over a large range of patterns and covariance matrices.     

Some simple procedures for handling missing data in multivariate analysis

For analyses with missing data, some popular procedures delete cases with missing values, perform analysis with missing value correlation or covariance matrices, or estimate missing values by sample means. There are objections to each of these procedures. Several procedures are outlined here for replacing missing values by regression values obtained in various ways, and for adjusting coefficients (such as factor score coefficients) when data are missing. None of the procedures are complex or expensive.This research was supported by NIH Special Research Resources Grant RR-3. The author expresses his gratitude to Robert I. Jennrich and the referees for their suggestions.     

Planned missing data designs in psychological research

The authors describe 2 efficiency (planned missing data) designs for measurement: the 3-form design and the 2-method measurement design. The 3-form design, a kind of matrix sampling, allows researchers to leverage limited resources to collect data for 33% more survey questions than can be answered by any 1 respondent. Power tables for estimating correlation effects illustrate the benefit of this design. The 2-method measurement design involves a relatively cheap, less valid measure of a construct and an expensive, more valid measure of the same construct. The cost effectiveness of this design stems from the fact that few cases have both measures, and many cases have just the cheap measure. With 3 brief simulations involving structural equation models, the authors show that compared with the same-cost complete cases design, a 2-method measurement design yields lower standard errors and a higher effective sample size for testing important study parameters. With a large cost differential between cheap and expensive measures and small effect sizes, the benefits of the design can be enormous. Strategies for using these 2 designs are suggested.