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.     

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.     

Intolerance of Uncertainty Mediates the Relation Between Generalized Anxiety Disorder Symptoms and Anger

Previous research has shown that individuals with generalized anxiety disorder (GAD) report elevated anger compared with nonanxious individuals; however, the pathways linking GAD and anger are currently unknown. We hypothesized that negative beliefs about uncertainty, negative beliefs about worry and perfectionism dimensions mediate the relationship between GAD symptoms and anger variables. We employed multiple mediation with bootstrapping on cross-sectional data from a student sample (N = 233) to test four models assessing potential mediators of the association of GAD symptoms to inward anger expression, outward anger expression, trait anger and hostility, respectively. The belief that uncertainty has negative personal and behavioural implications uniquely mediated the association of GAD symptoms to inward anger expression (confidence interval [CI] = .0034, .1845, P_M = .5444), and the belief that uncertainty is unfair and spoils everything uniquely mediated the association of GAD symptoms to outward anger expression (CI = .0052, .1936, P_M = .4861) and hostility (CI = .0269, .2427, P_M = .3487). Neither negative beliefs about worry nor perfectionism dimensions uniquely mediated the relation of GAD symptoms to anger constructs. We conclude that intolerance of uncertainty may help to explain the positive connection between GAD symptoms and anger, and these findings give impetus to future longitudinal investigations of the role of anger in GAD.     

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.     

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.     

On the correlation of a naturally and an artificially dichotomized variable

A method is suggested for estimating the correlation of a naturally (X) and an artificially (Y) dichotomized variable. It is assumed that a normal random variable (L) underlies the artificially dichotomized variable. The proposed correlation coefficient recovers the product moment correlation coefficient between X and L from a fourfold table of X and Y. The suggested correlation coefficient ν is contrasted with the phi correlation and the biserial η. The biserial η was proposed by Karl Pearson and is conceptually related to the new correlation coefficient. However, in addition, Pearson's biserial η invokes the assumption that the marginal distribution of L is normal, which contradicts its basic assumptions and thus does not recover the true correlation of L and X. Finally, an approximation is provided to simplify the calculation of ν and its standard error.     

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.     

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.     

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.     

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 generalized Logit-Linear Item Response Model for Binary-Designed Items

This paper introduces the generalized logit-linear item response model (GLLIRM), which represents the item-solving process as a series of dichotomous operations or steps. The GLLIRM assumes that the probability function of the item response is a logistic function of a linear composite of basic parameters which describe the operations, and the coefficients depend on three design matrices X, Y and Z. The GLLIRM provides a tool for testing hypotheses on the item-solving process and generalizes existing models. An empirical application is included, in which the model is applied to evaluate sources of difficulty and pairwise item interactions in a logical analysis test. This research was supported by the Comunidad de Madrid grant CCG06-UAM/ESP-0043.     

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 .     

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.     

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.     

Indirect representation and the self-representational theory of consciousness

According to Uriah Kriegel’s self-representational theory of consciousness, mental state M is conscious just in case it is a complex with suitably integrated proper parts, M ₁ and M ₂, such that M ₁ is a higher-order representation of lower-order representation M ₂. Kriegel claims that M thereby “indirectly” represents itself, and he attempts to motivate this claim by appealing to what he regards as intuitive cases of indirect perceptual and pictorial representation. For example, Kriegel claims that it’s natural to say that in directly perceiving the front surface of an apple one thereby perceives the apple itself. Cases such as this are supposed to provide intuitive support for the principle that if X represents Y, and Y is highly integrated into complex object Z, then X indirectly represents Z. In this paper I provide counterexamples to Kriegel’s principle of indirect representation, before going on to argue that we can explain what is going on in those cases in which the subject seems to represent a complex whole by representing one its parts without positing indirect representations anyway. I then argue that my alternative approach is superior to Kriegel’s in a number of ways, thereby rendering his theory of consciousness implausible.     

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.     

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.     

Explanation of Two Anomalous Results in Statistical Mediation Analysis

Previous studies of different methods of testing mediation models have consistently found two anomalous results. The first result is elevated Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap tests not found in nonresampling tests or in resampling tests that did not include a bias correction. This is of special concern as the bias-corrected bootstrap is often recommended and used due to its higher statistical power compared with other tests. The second result is statistical power reaching an asymptote far below 1.0 and in some conditions even declining slightly as the size of the relationship between X and M, a, increased. Two computer simulations were conducted to examine these findings in greater detail. Results from the first simulation found that the increased Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap are a function of an interaction between the size of the individual paths making up the mediated effect and the sample size, such that elevated Type I error rates occur when the sample size is small and the effect size of the nonzero path is medium or larger. Results from the second simulation found that stagnation and decreases in statistical power as a function of the effect size of the a path occurred primarily when the path between M and Y, b, was small. Two empirical mediation examples are provided using data from a steroid prevention and health promotion program aimed at high school football players (Athletes Training and Learning to Avoid Steroids; Goldberg et al., 1996), one to illustrate a possible Type I error for the bias-corrected bootstrap test and a second to illustrate a loss in power related to the size of a. Implications of these findings are discussed.     

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.