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.     

Abacs for determination of a correlation coefficient corrected for restriction of range

Abacs approximating the product-moment correlation for both explicit and implicit selection are presented. These abacs give accuracy to within .01 of the corresponding analytic estimate.     

Estimation of the correlation coefficient in the case of a bivariate normal population when one of the variables is dichotomized

It is shown that the problem of estimation of the correlation coefficient of a bivariate normal population when one of the variables is dichotomized may be attacked with “probit analysis” methods. This represents an extension of the work of Gillman and Goode (3), as it was possible to find by this approach an approximation to the large-sample variance of the resulting estimateG ofρ. An empirical investigation was undertaken with the object of obtaining some information about the distribution ofG for large sample size. Methods for determining the “pass-fail” cut-off are considered.     

Approximate interval estimation for a certain intraclass correlation coefficient

When the raters participating in a reliability study are a random sample from a larger population of raters, inferences about the intraclass correlation coefficient must be based on the three mean squares from the analysis of variance table summarizing the results: between subjects, between raters, and error. An approximate confidence interval for the parameter is presented as a function of these three mean squares.Dr. Fleiss is also with the Biometrics Research Unit of the New York State Psychiatric Institute. This work was supported in part by grant DE 04068 from the National Institute of Dental Research.     

Nomogram for the tetrachoric correlation coefficient

This article offers a new nomogram for the tetrachoric correlation coefficient, together with a correcting table. The development of the nomogram is described and directions for its use are included.     

Estimation of a correlation coefficient from an uncertainty measure

In this paper graphs are given for estimating a correlation _XY from a double-entry table. 2H _r , twice the statisticH _r , relative uncertainty (the average of the ratios of the conditional uncertainty of each variable to its uncertainty) was computed. Mean 2H _r 's were computed from 100 samples for each of several sample sizes and numbers of categories. Graphs relating mean 2H _r to , the estimated correlation, for 3–17 categories to each variable and 3 sample sizes—50, 100, and 500—were constructed. 2.5 per cent and 97.5 per cent fiducial limits curves were constructed for establishing confidence limits on _XY and .The research reported in this paper was conducted by personnel of the Aerospace Medical Research Laboratories, Aerospace Medical Division, Air Force Systems Command, Wright-Patterson AFB, Ohio. Further reproduction is authorized to satisfy the needs of the U. S. Government. This paper has been identified by the Aerospace Medical Research Laboratories as AMRL-TR-66-54.     

The reliability of a point biserial coefficient of correlation

An attempt is made to elaborate upon Lev's results concerning the reliability of the point biserial coefficient of correlation in a manner that will be helpful to the psychological statistician. Procedures required in the use of the non-centralt tables prepared by Johnson and Welch are described as they relate to the determination of the fiducial limits for a point biserial coefficient. A normal approximation technique for the estimation of fiducial limits is also suggested. Numerical evidence is presented which shows that relative to a given level of significance the width of the fiducial interval estimated from a point biserial coefficient of any size is smaller than that of the fiducial interval corresponding to an ordinary Pearsonian coefficient of the same magnitude.     

Bootstrap standard error and confidence intervals for the correlation corrected for range restriction: a simulation study

The standard Pearson correlation coefficient is a biased estimator of the true population correlation, rho, when the predictor and the criterion are range restricted. To correct the bias, the correlation corrected for range restriction, rc, has been recommended, and a standard formula based on asymptotic results for estimating its standard error is also available. In the present study, the bootstrap standard-error estimate is proposed as an alternative. Monte Carlo simulation studies involving both normal and nonnormal data were conducted to examine the empirical performance of the proposed procedure under different levels of rho, selection ratio, sample size, and truncation types. Results indicated that, with normal data, the bootstrap standard-error estimate is more accurate than the traditional estimate, particularly with small sample size. With nonnormal data, performance of both estimates depends critically on the distribution type. Furthermore, the bootstrap bias-corrected and accelerated interval consistently provided the most accurate coverage probability for rho.     

Testing compensatory health beliefs in a UK population

Compensatory health beliefs, beliefs that healthy behaviours can compensate or neutralise unhealthy behaviours, have been proposed as one way of understanding why people engage in health-risk behaviours (Kn?uper, B., Rabiau, M., Cohen, O., & Patriciu, N. (2004). Compensatory health beliefs scale development and psychometric properties. Psychology and Health, 19, 607-624). However, measuring compensatory health beliefs has proved a challenge, with several recent studies being unable to replicate the psychometric properties of Kn?uper et al.'s (2004) scales. The aims of this study were to: (1) test the factor structure of the compensatory health beliefs scale in the UK, (2) examine the predictive validity of the scale by testing the relationships between compensatory health beliefs and health behaviours over a six-month time interval and (3) assess the 6-month test-retest reliability of the scale. A total of 393 participants completed measures of compensatory health beliefs and health behaviours at two time points separated by six months. The findings were potentially problematic for research into compensatory health beliefs: the factor structure was not confirmed, there was little evidence of predictive validity, and test-retest reliability was poor. Further research is required to understand the operation of compensatory health beliefs and to develop the measurement of compensatory health beliefs.     

Testing compensatory health beliefs in a UK population

Compensatory health beliefs, beliefs that healthy behaviours can compensate or neutralise unhealthy behaviours, have been proposed as one way of understanding why people engage in health-risk behaviours (Knäuper, B., Rabiau, M., Cohen, O., & Patriciu, N. (2004). Compensatory health beliefs scale development and psychometric properties. Psychology and Health, 19, 607–624). However, measuring compensatory health beliefs has proved a challenge, with several recent studies being unable to replicate the psychometric properties of Knäuper et al.'s (2004) scales. The aims of this study were to: (1) test the factor structure of the compensatory health beliefs scale in the UK, (2) examine the predictive validity of the scale by testing the relationships between compensatory health beliefs and health behaviours over a six-month time interval and (3) assess the 6-month test–retest reliability of the scale. A total of 393 participants completed measures of compensatory health beliefs and health behaviours at two time points separated by six months. The findings were potentially problematic for research into compensatory health beliefs: the factor structure was not confirmed, there was little evidence of predictive validity, and test–retest reliability was poor. Further research is required to understand the operation of compensatory health beliefs and to develop the measurement of compensatory health beliefs.     

The impact of visual exploration of judgments of whether a gap is crossable

When observers look down into a gap in the ground plane, their judgments of the widest gap they can step across (gapmax) decrease as gap depth increases (Y. Jiang & L. S. Mark, 1994). This study investigated the possibility that Jiang and Mark's viewing conditions did not afford observers a sufficient opportunity to perform exploratory movements needed to detect information about gap width. Experiment 1 showed that the gap depth by gaze interaction disappeared only when restrictions were not imposed on observers' exploratory activities (eye, head, and body movements). Experiment 2 showed that observers tended to see the vertical surface as slanted away from them, which made the bottom of the surface appear farther away from them than the top. Only when observers were able to view the gap binocularly under conditions that did not restrict exploratory activity did their slant perception improve and their gapmax judgments no longer covary with gap depth. The data indicate that the exploratory movements of prospective actors are essential for the pickup of information about their action capabilities.