Misleading interaction and curvilinear terms.

This article examines the relationships between interaction (product) terms and curvilinear (quadratic) terms in regression models in which the independent variables are correlated. The author uses 2 substantive examples to demonstrate the following outcomes: (a) If the appropriate quadratic terms are not added to the estimated model, then the observed interaction may indicate a synergistic (offsetting) relationship between the independent variables, whereas the true relationship is, in fact, offsetting (synergistic). (b) If the appropriate product terms are not added to the equation, then the estimated model may indicate concave (convex) relationships between the independent variables and the dependent variable, whereas the true relationship is, in fact, convex (concave). (c) If the appropriate product and quadratic terms are not examined simultaneously, then the observed interactive or curvilinear relationships may be nonsignificant when such relationships exist. The implications of these results for the examination of interaction and quadratic effects in multiple regression analysis are discussed. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Analysis of multiplicative combination rules when the causal variables are measured with error.

Evaluates the validity of the observational method used to test multiplicative combination rules with respect to 2 measurement issues: measurement level (i.e., the effects produced by allowing monotonic transformations of the measures) and measurement error (i.e., the effects produced by using unreliable measures of the causal variables). The evaluation is based on a theoretical distinction between the structural model (the set of equations relating theoretical constructs to each other) and the measurement model (the set of equations relating the theoretical constructs to the observed measures). It is concluded that hierarchical regression analysis is inadequate for determining whether the structural model is additive or multiplicative for 2 reasons: First, an additive structural model may produce multiplicative effects through a nonlinear measurement model. Second, a multiplicative structural model may produce nondetectable multiplicative effects because of multiplicative measurement error. Some alternatives to hierarchical regression analysis are described. (35 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Failure to detect moderating effects: Is multicollinearity the problem?

We show that Morris, Sherman, and Mansfield's (1986) contention that multicollinearity causes ordinary least squares–moderated multiple regression (OLS–MMR) to underestimate the importance of moderator effects is incorrect for their own data. Multicollinearity was reduced to the point that it was negligible by transforming predictor variables and moderator variables to standard scores prior to computing cross-product terms. We show the resulting cross-product terms both mathematically and empirically to have near-zero correlations with standardized predictors and moderators. Yet, as Arnold and Evans (1979) showed, the results of OLS–MMR are unchanged by this linear transformation of scale. Morris et al's (1986) finding of significant moderator effects when using principal-components regression (PCR) is probably a result of some artifact of PCR. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Moderated regression analysis and Likert scales: Too coarse for comfort.

One of the most commonly accepted models of relationships among 3 variables in applied industrial and organizational psychology is the simple moderator effect. However, many authors have expressed concern over the general lack of empirical support for interaction effects reported in the literature. It is demonstrated in the current sample that use of a continuous, dependent-response scale instead of a discrete, Likert-type scale, causes moderated regression analysis effect sizes to increase an average of 93%. Use of relatively coarse Likert scales to measure fine dependent responses causes information loss that, although varying widely across subjects, greatly reduces the probability of detecting true interaction effects. Specific recommendations for alternate research strategies are made. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A tetrad test for causal indicators.

The authors propose a confirmatory tetrad analysis test to distinguish causal from effect indicators in structural equation models. The test uses "nested" vanishing tetrads that are often implied when comparing causal and effect indicator models. The authors present typical models that researchers can use to determine the vanishing tetrads for 4 or more variables. They also provide the vanishing tetrads for mixtures of causal and effect indicators, for models with fewer than 4 indicators per latent variable, or for cases with correlated errors. The authors illustrate the test results for several simulation and empirical examples and emphasize that their technique is a theory-testing rather than a model-generating approach. They also review limitations of the procedure including the indistinguishable tetrad equivalent models, the largely unknown finite sample behavior of the test statistic, and the inability of any procedure to fully validate a model specification. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Effect Size and Power in Assessing Moderating Effects of Categorical Variables Using Multiple Regression: A 30-Year Review.

The authors conducted a 30-year review (1969-1998) of the size of moderating effects of categorical variables as assessed using multiple regression. The median observed effect size (f2) is only .002, but 72% of the moderator tests reviewed had power of .80 or greater to detect a targeted effect conventionally defined as small. Results suggest the need to minimize the influence of artifacts that produce a downward bias in the observed effect size and put into question the use of conventional definitions of moderating effect sizes. As long as an effect has a meaningful impact, the authors advise researchers to conduct a power analysis and plan future research designs on the basis of smaller and more realistic targeted effect sizes. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The Role of Nonlinear Factor-to-Indicator Relationships in Tests of Measurement Equivalence.

Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups or time. This article considers the potential problems that can arise for tests of measurement invariance when the true factor-to-indicator relationship is nonlinear (quadratic) and invariant but the linear factor model is nevertheless applied. The factor loadings and indicator intercepts of the linear model will diverge across groups as the factor mean difference increases. Power analyses show that even apparently small quadratic effects can result in rejection of measurement invariance at moderate sample sizes when the factor mean difference is medium to large. Recommendations include the identification of nonlinear relationships using diagnostic plots and consideration of newly developed methods for fitting nonlinear factor models. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Ridge regression: Application to nonexperimental data.

The value of regression analysis as a statistical tool may be significantly diminished when the set of independent or predictor variables are approximately collinear. The presence of approximate linear relationships in a set of predictors is not unusual for data that arise in social science research. This situation, often referred to as multicollinearity, may be difficult to detect and is even more difficult to remedy. A method called ridge regression has recently been developed and may be useful for solving some of the problems associated with multicollinearity. The ridge estimators have smaller variances (in comparison with those of ordinary regression), but they are biased. When the reduction in variance exceeds the square of the bias, ridge estimates are preferred. In this article, the ridge method is described. Its application is demonstrated by analyzing data from an employee satisfaction survey, (30 departmental units, each with 35 employees and a supervisor), and the ridge solution is compared with the regular regression solution. (27 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Psychological androgyny and the Masculinity?×?Femininity interaction.

Clarifies 3 proposed models of psychological androgyny, since recent discussions have been flawed by misunderstandings of the nature of statistical interaction. The additive model predicts main effects of masculinity (M) and femininity (F). The balance model predicts only M?×?F interaction. The emergent properties model predicts by effects and M?×?F interaction. The interaction effect assessed in a 2?×?2 ANOVA by a multiplicative M?×?F term in regression is compared with the absolute value of the M?×?F difference in regression. It is concluded that ANOVA and multiplicative interaction terms are generally satisfactory operationalizations of balance (though each has strengths and weaknesses), whereas the absolute difference term yields unpredictable results and should be avoided in most situations. (16 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Inference for odds ratio regression models with sparse dependent data

Suppose the number of 2 x 2 tables is large relative to the average table size, and the observations within a given table are dependent, as occurs in longitudinal or family-based case-control studies. We consider fitting regression models to the odds ratios using table-level covariates. The focus is on methods to obtain valid inferences for the regression parameters beta when the dependence structure is unknown. In this setting, Liang (1985, Biometrika 72, 678-682) has shown that inference based on the noncentral hypergeometric likelihood is sensitive to misspecification of the dependence structure. In contrast, estimating functions based on the Mantel-Haenszel method yield consistent estimators of beta. We show here that, under the estimating function approach, Wald's confidence interval for beta performs well in multiplicative regression models but unfortunately has poor coverage probabilities when an additive regression model is adopted. As an alternative to Wald inference, we present a Mantel-Haenszel quasi-likelihood function based on integrating the Mantel-Haenszel estimating function. A simulation study demonstrates that, in medium-sized samples, the Mantel-Haenszel quasi-likelihood approach yields better inferences than other methods under an additive regression model and inferences comparable to Wald's method under a multiplicative model. We illustrate the use of this quasi-likelihood method in a study of the familial risk of schizophrenia.     

Comparing meta-analytic moderator estimation techniques under realistic conditions.

One of the most problematic issues in contemporary meta-analysis is the estimation and interpretation of moderating effects. Monte Carlo analyses are developed in this article that compare bivariate correlations, ordinary least squares and weighted least squares (WLS) multiple regression, and hierarchical subgroup (HS) analysis for assessing the influence of continuous moderators under conditions of multicollinearity and skewed distribution of study sample sizes (heteroscedasticity). The results show that only WLS is largely unaffected by multicollinearity and heteroscedasticity, whereas the other techniques are substantially weakened. Of note, HS, one of the most popular methods, typically provides the most inaccurate results, whereas WLS, one of the least popular methods, typically provides the most accurate results. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Genetic counseling in autism and pervasive developmental disorders

Although alcohol intake and hepatitis B and C virus (HBV and HCV) infections are the major determinants of liver cirrhosis (LC) in western countries, the joint effect of these factors on LC risk has not yet been adequately studied. Data from three case-control studies performed in Italy were used. Cases were 462 cirrhotic patients admitted to Hospitals for liver decompensation. Controls were 651 inpatients admitted for acute diseases unrelated to alcohol. Alcohol consumption was expressed as lifetime daily alcohol intake (LDAI). Three approaches were used to explore the interaction structure. The Breslow and Storer parametric family of relative risk functions showed that an intermediate structure of interaction from additive to multiplicative was the most adequate one. The Rothman synergism index showed that the interaction structure between LDAI and viral status differed significantly from the additive model in particular for high levels of alcohol intake. When multiple regression additive and multiplicative models were compared after adjustment for the known confounding variables. a trend of the interaction structure towards the multiplicative model was observed at increasing levels of consumption. Better methods are needed for assessing mixed interaction structures in conditions characterized by multifactorial etiologies like cirrhosis of the liver.     

Estimating, testing, and comparing specific effects in structural equation models: The phantom model approach.

The phantom model approach for estimating, testing, and comparing specific effects within structural equation models (SEMs) is presented. The rationale underlying this novel method consists in representing the specific effect to be assessed as a total effect within a separate latent variable model, the phantom model that is added to the main model. The following favorable features characterize the method: (a) It enables the estimation, testing, and comparison of arbitrary specific effects for recursive and nonrecursive models with latent and manifest variables; (b) it enables the bootstrapping of confidence intervals; and (c) it can be applied with all standard SEM programs permitting latent variables, the specification of equality constraints, and the bootstrapping of total effects. These features along with the fact that no manipulation of matrices and formulas is required make the approach particularly suitable for applied researchers. The method is illustrated by means of 3 examples with real data sets. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Comparative efficacy of individual and group psychotherapy: A meta-analytic perspective.

Recent reviews of the group psychotherapy literature indicate that group is a beneficial and cost-effective treatment format. However, collective findings on the differential efficacy of group when compared with individual therapy remain problematic, incomplete, or controversial. To remedy this problem, the authors conducted a meta-analysis of 23 outcome studies that directly compared the effectiveness of the individual and group therapy formats when they were used within the same study. Results were consistent with previous reports that indicated no difference in outcome between the group and individual formats. This finding generally held true when client, therapist, methodology, treatment, and group variables were examined for possible relationship with effect sizes comparing group and individual therapy. Results bolster past findings that group therapy can be used as an efficacious cost-effective alternative to individual therapy under many different conditions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Modeling distributions of immediate memory effects: No strategies needed?

Many models of immediate memory predict the presence or absence of various effects, but none have been tested to see whether they predict an appropriate distribution of effect sizes. The authors show that the feature model (J. S. Nairne, 1990) produces appropriate distributions of effect sizes for both the phonological confusion effect and the word-length effect. The model produces the appropriate number of reversals, when participants are more accurate with similar items or long items, and also correctly predicts that participants performing less well overall demonstrate smaller and less reliable phonological similarity and word-length effects and are more likely to show reversals. These patterns appear within the model without the need to assume a change in encoding or rehearsal strategy or the deployment of a different storage buffer. The implications of these results and the wider applicability of the distributionmodeling approach are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Regression models for the determination of the absorbed dose rate with an extrapolation chamber for flat ophthalmic applicators

The average surface absorbed dose rate, given by flat ophthalmic applicators (90Sr/90Y, 925 MBq) is determined in equivalent soft tissue using an extrapolation chamber with two flat parallel electrodes of variable separation; the input electrode is fixed in relation to the collector electrode of constant area. When estimating the extrapolation curve slope using a linear regression model, it has been observed that average surface dose rate values were underestimated by up to 19%, as compared to estimations of these values by means of a second degree polynomial regression model, while an improvement of up to 37% is observed in the standard error of the slope in the quadratic model, as compared to that of the linear model. With the aim of validating the results of these models, goodness of fit tests to a Normal (the Shapiro-Wilk test) as well as homogeneity tests on treatment variance (the Bartlett test) were applied. The analysis of variance (ANOVA) tables of fit and residual error breakdown are given: table 3a and 3b for linear fit; 7a and 7b for quadratic fit, and table 10 to error breakdown. Also presented is the global uncertainty of the average dose rate, taking into account the reproducibility of the experimental set-up. It may be inferred that by using this type of measurement for the extrapolation curve slope, quadratic regression models allow for a greater degree of accuracy and precision in determining surface dose rate values. The effective area of the collector electrode and the effective electrode separation in the chamber are also determined by measuring the chamber's electric capacity. Finally, there is an attempt to relate the use of the regression models to the experimental conditions during the measurement of ionization currents (diameter of collector electrode, electrical field gradient, radiation field uniformity, radiation field intensity, etc.). In this particular case, deviations in the distance inverse square law and the "screening" effects during the collection of negative charges (both for primary radiation and the ionization generated by it), are presented as necessary, but insufficient, conditions to explain thoroughly the quadratic behavior of ionizing currents.     

基于偏最小二乘回归的高炉焦比影响因素分析

影响高炉焦比的基本因素之间通常存在显著的相关性,常规最小二乘回归法估计回归系数存在较大的偏差甚至出现反常现象.偏最小二乘法能够克服自变量集合多重相关性对回归建模的不良影响,提高模型可靠性.基于偏最小二乘回归法建立了10成分模型,通过比较标准回归系数分析各因素对焦比的影响程度和自变量与因变量的正、负相关性, 调整主要因素, 使焦比接近或达到最优值.     

Assessment of ecologic regression in the study of lung cancer and indoor radon

Ecologic regression studies conducted to assess the cancer risk of indoor radon to the general population are subject to methodological limitations, and they have given seemingly contradictory results. The authors use simulations to examine the effects of two major methodological problems that affect these studies: measurement error and misspecification of the risk model. In a simulation study of the effect of measurement error caused by the sampling process used to estimate radon exposure for a geographic unit, both the effect of radon and the standard error of the effect estimate were underestimated, with greater bias for smaller sample sizes. In another simulation study, which addressed the consequences of uncontrolled confounding by cigarette smoking, even small negative correlations between county geometric mean annual radon exposure and the proportion of smokers resulted in negative average estimates of the radon effect. A third study considered consequences of using simple linear ecologic models when the true underlying model relation between lung cancer and radon exposure is nonlinear. These examples quantify potential biases and demonstrate the limitations of estimating risks from ecologic studies of lung cancer and indoor radon.