A short note on the maximal point‐biserial correlation under non‐normality

The aim of this paper is to derive the maximal point‐biserial correlation under non‐normality. Several widely used non‐normal distributions are considered, namely the uniform distribution, t‐distribution, exponential distribution, and a mixture of two normal distributions. Results show that the maximal point‐biserial correlation, depending on the non‐normal continuous variable underlying the binary manifest variable, may not be a function of p (the probability that the dichotomous variable takes the value 1), can be symmetric or non‐symmetric around p = .5, and may still lie in the range from ?1.0 to 1.0. Therefore researchers should exercise caution when they interpret their sample point‐biserial correlation coefficients based on popular beliefs that the maximal point‐biserial correlation is always smaller than 1, and that the size of the correlation is always further restricted as p deviates from .5.     

Significance tests to determine the direction of effects in linear regression models

Previous studies have discussed asymmetric interpretations of the Pearson correlation coefficient and have shown that higher moments can be used to decide on the direction of dependence in the bivariate linear regression setting. The current study extends this approach by illustrating that the third moment of regression residuals may also be used to derive conclusions concerning the direction of effects. Assuming non‐normally distributed variables, it is shown that the distribution of residuals of the correctly specified regression model (e.g., Y is regressed on X) is more symmetric than the distribution of residuals of the competing model (i.e., X is regressed on Y). Based on this result, 4 one‐sample tests are discussed which can be used to decide which variable is more likely to be the response and which one is more likely to be the explanatory variable. A fifth significance test is proposed based on the differences of skewness estimates, which leads to a more direct test of a hypothesis that is compatible with direction of dependence. A Monte Carlo simulation study was performed to examine the behaviour of the procedures under various degrees of associations, sample sizes, and distributional properties of the underlying population. An empirical example is given which illustrates the application of the tests in practice.     

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.     

Blocking by fixed and variable stimuli: Effects of stimulus distribution on blocking

An experiment with rats compared the ability of fixed and variable duration cues to produce blocking. Rats in group B (Blocking) were trained that both fixed- (F) and variable- (V) duration cues would be followed by food delivery. In a subsequent training stage F and V continued to be reinforced, but F was accompanied by X, and V by Y. In the test phase responding to X and Y was examined. Control group O (Overshadowing) received identical treatment, except that F and V were nonreinforced in the first training stage. In group B there was evidence for blocking, but only of X, which had been conditioned in compound with the fixed-duration F; there was no evidence for blocking of Y, which had been conditioned in compound with the variable-duration V. It is suggested that this result may occur because fixed cues reach a higher, more stable asymptote of associative strength than do their variable equivalents.     

Correlations between rats' spatial location and intracranial stimulation administration affects rate of acquisition and asymptotic level of time allocation preference in the open field

The present experiment explores the effects of the response (1-sec occupancy of a target area in an open field)-reinforcer (intracranial stimulation) contingency on time allocation in the open field in rats. The probability of reinforcement given response (X) and the probability of reinforcement given nonresponse (Y) were varied randomly across sessions within a subject. The 21 contingency treatments explored included all possible combinations of values (0, .1, .2, .3, .4, .5) of X and Y such that X ≥ Y. The results indicate that rate of acquisition and asymptotic level of time allocation preference to the target area are negatively related to the value of Y (for any given value of X). Variations in X (for any given value of Y) were less effective. Evaluation of proposed contingency metrics revealed that the Weber fraction (X − Y)/X most closely approximates performance, and that the value of the difference detection threshold derived from the Weber fraction is a constant.     

The nontruncated marginal of a truncated bivariate normal distribution

Inference is considered for the marginal distribution ofX, when (X, Y) has a truncated bivariate normal distribution. TheY variable is truncated, but only theX values are observed. The relationship of this distribution to Azzalini's skew-normal distribution is obtained. Method of moments and maximum likelihood estimation are compared for the three-parameter Azzalini distribution. Samples that are uniformative about the skewness of this distribution may occur, even for largen. Profile likelihood methods are employed to describe the uncertainty involved in parameter estimation. A sample of 87 Otis test scores is shown to be well-described by this model.     

The rowwise correlation between two proximity matrices and the partial rowwise correlation

This paper discusses rowwise matrix correlation, based on the weighted sum of correlations between all pairs of corresponding rows of two proximity matrices, which may both be square (symmetric or asymmetric) or rectangular. Using the correlation coefficients usually associated with Pearson, Spearman, and Kendall, three different rowwise test statistics and their normalized coefficients are discussed, and subsequently compared with their nonrowwise alternatives like Mantel'sZ. It is shown that the rowwise matrix correlation coefficient between two matricesX andY is the partial correlation between the entries ofX andY controlled for the nominal variable that has the row objects as categories. Given this fact, partial rowwise correlations (as well as multiple regression extensions in the case of Pearson's approach) can be easily developed.The author wishes to thank the Editor, two referees, Jan van Hooff, and Ruud Derix for their useful comments, and E. J. Dietz for a copy of the algorithm of the Mantel permutation test.     

Intuitive prediction: ecological validity versus representativeness

Insufficiently regressive intuitive predictions have been attributed to mistaken reliance on the representativeness heuristic. In contrast, we suggest that intuitive predictions stem from a conceptualization of ‘goodness of prediction’ that differs from the accepted statistical definition in terms of error minimization, namely, ecological validity—that is, representation of the substantive characteristics of the predicted variable Y and its distribution as well as of the relationship between Y and the predictor X—rather than minimization of prediction errors. Simultaneous satisfaction of the above representation requirements is achieved by multivalued prediction: The prediction of different Y′ values for the same X value, resulting in conditional distributions Y′|X for at least some X values. Empirical results supporting this hypothesis are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Maximum likelihood estimation of the joint covariance matrix for sections of tests given to distinct samples with application to test equating

Consider an old testX consisting ofs sections and two new testsY andZ similar toX consisting ofp andq sections respectively. All subjects are given testX plus two variable sections from either testY orZ. Different pairings of variable sections are given to each subsample of subjects. We present a method of estimating the covariance matrix of the combined test (X ₁, ...,X _s ,Y ₁, ...,Y _p ,Z ₁, ...,Z _q ) and describe an application of these estimation techniques to linear, observed-score, test equating.The author is indebted to Paul W. Holland and Donald B. Rubin for their encouragement and many helpful comments and suggestions that contributed significantly to the development of this paper.This research was supported by the Program Statistics Research Project of the ETS Research Statistics Group.     

Flexible Approaches to Computing Mediated Effects in Generalized Linear Models: Generalized Estimating Equations and Bootstrapping

In behavioral research, interest is often in examining the degree to which the effect of an independent variable X on an outcome Y is mediated by an intermediary or mediator variable M. This article illustrates how generalized estimating equations (GEE) modeling can be used to estimate the indirect or mediated effect, defined as the amount by which the regression coefficient of X on Y changes after adjusting for M. Advantages of this method are: (a) it applies to the class of generalized linear models, including linear, logistic, and Poisson regression as special cases; (b) it allows multiple independent variables and mediators in the same model; and (c) asymptotically valid standard errors and confidence intervals are obtained using standard software. This methodology is compared with the bootstrap, another general methodology that can be applied to the same broad class of models, and is evaluated using simulation in both linear and logistic regression scenarios. The methods are utilized to examine the degree to which the effect of low birthweight status on internalizing symptoms at age 20 is mediated through IQ at age 8.     

Local measures of association: Estimating the derivative of the regression line

A local measure of association that allows both heteroscedasticity and a non‐linear association was developed during the 1990s. The basic goal is to measure the strength of the association between X and Y, given X, when Y=θ(X)+τ(X)ε for some unknown functions θ(X) and τ(X). Application of this method requires the estimation of the derivative of θ(X). The focus in this paper is on four alternatives to a very slight modification of the method used by Doksum et al. when estimating this derivative. The main result is that in simulations, a certain robust analogue of their method dominates in terms of mean squared error, even under normality. The bias of the method is found to be small but a little larger than the bias associated with the method used by Doksum et al. The method is based in part on bootstrap bagging followed by a lowess smooth.     

The duplication of love's reasons

If X loves Y does it follow that X has reasons to love a physiologically exact replacement for Y? Can love's reasons be duplicated? One response to the problem is to suggest that X lacks reasons for loving such a duplicate because the reason-conferring properties of Y cannot be fully duplicated. But a concern, played upon by Derek Parfit, is that this response may result from a failure to take account of the psychological pressures of an actual duplication scenario. In the face of the actual loss of a loved one and the subsequent appearance of a duplicate, how could we resist the inclination to love? Drawing upon duplication scenarios from Parfit and from Stanislaw Lem's Solaris, this paper will argue that there could be reasons for X to come to love a duplicate of Y but that these would not be identical with the reasons that X had (and may still have) to love Y. Nor (in the case of an agent with a normal causal history) could they be reasons for a love that violates the requirement that love is a response to a relationship and therefore takes time to emerge.     

The psychological side of Hempel’s paradox of confirmation

People often test hypotheses about two variables (X andY), each with two levels (e.g.,X1 andX2). When testing “IfX1, thenY1,” observing the conjunction ofX1 andY1 is overwhelmingly perceived as more supportive than observing the conjunction ofX2 andY2, although both observations support the hypothesis. Normatively, theX2&Y2 observation provides stronger support than theX1&Y1 observation if the former is rarer. Because participants in laboratory settings typically test hypotheses they are unfamiliar with, previous research has not examined whether participants are sensitive to the rarity of observations. The experiment reported here showed that participants were sensitive to rarity, even judging a rareX2&Y2 observation more supportive than a commonX1&Y1 observation under certain conditions. Furthermore, participants’ default strategy of judgingX1&Y1 observations more informative might be generally adaptive because hypotheses usually regard rare events.     

Accuracy of maximum-likelihood estimates of correlation for a biserial model

Tables are given for the calculation of variances of maximum-likelihood estimates in a multiple biserial model. For the estimates of individual correlations between the dichotomized variable and each of the graduated variables additional calculation is necessary; variances for estimates of the multiple correlation and the point of dichotomy are available directly. The required formulas and notation are those of a recent paper by Hannan and Tate.Research sponsored by the National Science Foundation under grant NSF-GP219.Aero-Space Division (Applied Mathematics), Seattle, Washington.     

Weekly Cycles in Daily Report Data: An Overlooked Issue

Daily diaries and other everyday experience methods are increasingly used to study relationships between two time‐varying variables X and Y. Although daily data potentially often have weekly cyclical patterns (e.g., stress may be higher on weekdays and lower on weekends), the majority of daily diary studies have ignored this possibility. In this study, we investigated the effect of ignoring existing weekly cycles. We reanalyzed an empirical dataset (stress and alcohol consumption) and performed Monte Carlo simulations to investigate the impact of omitting weekly cycles. In the empirical dataset, ignoring cycles led to the inference of a significant within‐person X–Y relation whereas modeling cycles suggested that this relationship did not exist. Simulation results indicated that ignoring cycles that existed in both X and Y led to bias in the estimated within‐person X–Y relationship. The amount and direction of bias depended on the magnitude of the cycles, magnitude of the true within‐person X–Y relation, and synchronization of the cycles. We encourage researchers conducting daily diary studies to address potential weekly cycles in their data. We provide guidelines for detecting and modeling cycles to remove their influence and discuss challenges of causal inference in daily experience studies.     

What mediation analysis can (not) do

The present article is concerned with a common misunderstanding in the interpretation of statistical mediation analyses. These procedures can be sensibly used to examine the degree to which a third variable (Z) accounts for the influence of an independent (X) on a dependent variable (Y) conditional on the assumption that Z actually is a mediator. However, conversely, a significant mediation analysis result does not prove that Z is a mediator. This obvious but often neglected insight is substantiated in a simulation study. Using different causal models for generating Z (genuine mediator, spurious mediator, correlate of the dependent measure, manipulation check) it is shown that significant mediation tests do not allow researchers to identify unique mediators, or to distinguish between alternative causal models. This basic insight, although well understood by experts in statistics, is persistently ignored in the empirical literature and in the reviewing process of even the most selective journals.     

What Invariance Is and How to Test for It

Causal assessment is the problem of establishing whether a relation between (variable) X and (variable) Y is causal. This problem, to be sure, is widespread across the sciences. According to accredited positions in the philosophy of causality and in social science methodology, invariance under intervention provides the most reliable test to decide whether X causes Y. This account of invariance (under intervention) has been criticised, among other reasons, because it makes manipulations on the putative causal factor fundamental for the causal methodology; consequently, the argument goes, the account is ill-suited to those contexts where manipulations are not performed, for instance, the social sciences. The article aims to extend the account of invariance (under intervention), in a way that manipulations on the putative causal factors are not methodologically fundamental, and yet invariance remains key for causal assessment both in experimental and non-experimental contexts.     

Children Use Temporal Cues to Learn Causal Directionality

The ability to learn the direction of causal relations is critical for understanding and acting in the world. We investigated how children learn causal directionality in situations in which the states of variables are temporally dependent (i.e., autocorrelated). In Experiment 1, children learned about causal direction by comparing the states of one variable before versus after an intervention on another variable. In Experiment 2, children reliably inferred causal directionality merely from observing how two variables change over time; they interpreted Y changing without a change in X as evidence that Y does not influence X. Both of these strategies make sense if one believes the variables to be temporally dependent. We discuss the implications of these results for interpreting previous findings. More broadly, given that many real‐world environments are characterized by temporal dependency, these results suggest strategies that children may use to learn the causal structure of their environments.     

Pseudocontingencies derived from categorically organized memory representations

Pseudocontingencies (PCs) allow for inferences about the contingency between two variables X and Y when the conditions for genuine contingency assessment are not met. Even when joint observations X _i and Y _i about the same reference objects i are not available or are detached in time or space, the correlation r(X _i ,Y _i ) is readily inferred from base rates. Inferred correlations are positive (negative) if X and Y base rates are skewed in the same (different) directions. Such PC inferences afford useful proxies for actually existing contingencies. While previous studies have focused on PCs due to environmental base rates, the present research highlights memory organization as a natural source of PC effects. When information about two attributes X and Y is represented in a hierarchically organized categorical memory code, as category-wise base rates p(X) and p(Y), the reconstruction of item-level information from category base rates will naturally produce PC effects. Three experiments support this contention. When the yes base rates of two respondents in four questionnaire subscales (categories) were correlated, recalled and predicted item-level responses were correlated in the same direction, even when the original responses to specific items within categories were correlated in the opposite direction.     

The effects of response-reinforcer contingency on time allocation in the open field

The present experiment explores the effects of the response (1-sec occupancy of a target area in an open field)—reinforcer (intracranial stimulation) contingency on time allocation in the open field in rats. The probability of reinforcement given a response (X) and the probability of reinforcement given the absence of a response (Y) were varied randomly across sessions within a subject. The following (X, Y) values were utilized: (.05, 0), (.15, 0), (.25, 0), (.15, .05), and (.15, .15). The results of this experiment indicate that rate of acquisition of time allocation preference is uniformly rapid during all contingency treatments wherein Y = 0 and is negatively related to the value of Y when X = .15. The relationship between the asymptote of the time allocation acquisition function and the value of X (when Y = 0) is positively sloped and negatively accelerated, while the relationship between asymptote and the value of Y (when X = .15) is negatively sloped with zero acceleration. Proposed contingency metrics are evaluated.