Kraha et al.Interpreting multiple regression!cc dep=YDb=AzenData.savSet=AzenInd=X1 X2 X3.Step 11: Retrieve commonality results. Commonality files are written to AzenCommonalityMatrix.sav <strong>and</strong> AzenCCByVariable.sav.APS files are written toAzenaps.sav.Step 12: Relative Weights (Step 13 to Step 16).Step 13: Before executing, download mimr_raw.sps <strong>and</strong> save to working directory from http://www.springerlink.com/content/06112u8804155th6/supplementals/Step 14: Open or activate <strong>the</strong> AzenData.sav <strong>data</strong>set file, by clicking on it.Step 15: If applicable, change <strong>the</strong> reliabilities of <strong>the</strong> predictor variables as indicated (4 in <strong>the</strong> given example).Step 16: Highlight all <strong>the</strong> syntax <strong>and</strong> run; <strong>the</strong>se steps will yield relative importance weights.<strong>Frontiers</strong> in Psychology | Quantitative Psychology <strong>and</strong> Measurement March 2012 | Volume 3 | Article 44 | 91
METHODS ARTICLEpublished: 09 July 2012doi: 10.3389/fpsyg.2012.00231A simple statistic for comparing moderation of slopes<strong>and</strong> correlationsMichael Smithson*Psychology, The Australian National University, Canberra, ACT, AustraliaEdited by:Jason W. Osborne, Old DominionUniversity, USAReviewed by:Fern<strong>and</strong>o Marmolejo-Ramos,University of Adelaide, AustraliaKristopher Jason Preacher, V<strong>and</strong>erbiltUniversity, USA*Correspondence:Michael Smithson, Psychology, TheAustralian National University,Building 39 Science Road, Canberra,ACT, Australia.e-mail: michael.smithson@anu.edu.auGiven a linear relationship between two continuous r<strong>and</strong>om variables X <strong>and</strong> Y that maybe moderated by a third, Z, <strong>the</strong> extent to which <strong>the</strong> correlation ρ is (un)moderated by Z isequivalent to <strong>the</strong> extent to which <strong>the</strong> regression coefficients β y <strong>and</strong> β x are (un)moderatedby Z iff <strong>the</strong> variance ratio σ 2 2y /σ x is constant over <strong>the</strong> range or states of Z. O<strong>the</strong>rwise,moderation of slopes <strong>and</strong> of correlations must diverge. Most of <strong>the</strong> literature on this issuefocuses on tests for heterogeneity of variance in Y, <strong>and</strong> a test for this ratio has not beeninvestigated. Given that regression coefficients are proportional to ρ via this ratio, accuratetests, <strong>and</strong> estimations of it would have several uses.This paper presents such a test for botha discrete <strong>and</strong> continuous moderator <strong>and</strong> evaluates its Type I error rate <strong>and</strong> power underunequal sample sizes <strong>and</strong> departures from normality. It also provides a unified approachto modeling moderated slopes <strong>and</strong> correlations with categorical moderators via structuralequations models.Keywords: moderator effects, interaction effects, heteroscedasticity, regression, correlationINTRODUCTIONLet X <strong>and</strong> Y have a bivariate normal distribution, X ∼ N (µ x , σx 2),<strong>and</strong> Y ∼ N (µ y , σy 2 ). Suppose also that <strong>the</strong> correlation between X<strong>and</strong> Y is a function of a moderator variable Z. Under homogeneityof variance (HoV),moderation of correlations implies moderationof regression coefficients (or means, in ANOVA), <strong>and</strong> vice versa.For example, establishing <strong>the</strong> existence of a moderator effect fromZ in a linear regression model with X <strong>and</strong> Z predicting Y byfinding a significant regression coefficient for <strong>the</strong> product termX × Z suffices to infer a corresponding moderator effect of Z on<strong>the</strong> correlation between X <strong>and</strong> Y.Heterogeneity of variance (HeV) due to Z, however, can altermoderator effects so that correlation <strong>and</strong> regression coefficients arenot equivalently moderated. We may have moderation of slopes,for instance, without moderation of correlations, moderation ofcorrelations with no moderation of slopes, moderation of slopes<strong>and</strong> correlations in opposite directions, or even moderation ofregression coefficients in opposite directions (e.g., what appearsto be a positive moderator effect when X predicts Y becomes anegative effect when Y predicts X).Although some scholars have warned about <strong>the</strong> impacts of heteroscedasticityon <strong>the</strong> analysis of variance (e.g., Grissom, 2000)<strong>and</strong> linear regression, most contemporary textbook advice <strong>and</strong>published evidence on this matter comforts researchers with <strong>the</strong>notion that ANOVA <strong>and</strong> regression are fairly robust against it.Howell (2007, p. 316), for instance, states that despite Grissom’spessimistic outlook “<strong>the</strong> homogeneity of variance assumptioncan be violated without terrible consequences” <strong>and</strong> advises thatfor symmetrically distributed populations <strong>and</strong> equal numbers ofparticipants in each cell, <strong>the</strong> validity of ANOVA is likely if <strong>the</strong>ratio of <strong>the</strong> largest to <strong>the</strong> smallest variance is no greater than4. Tabachnick <strong>and</strong> Fidell (2007, pp. 121–123) are even morerelaxed, recommending an upper limit on this ratio of 10 beforeraising an alarm. A recent investigation into <strong>the</strong> robustness ofone-way ANOVA against violations of normality (Schmider et al.,2010) also is relatively reassuring on that count. A fairly recentcomparison of several tests of homogeneity of variance (Correaet al., 2006) generally finds in favor of <strong>the</strong> Levene test butleaves <strong>the</strong> issue of <strong>the</strong> impact of HoV on moderator effectsunexamined.Never<strong>the</strong>less, this problem is well-known. Arnold (1982) drewa distinction between <strong>the</strong>“form”<strong>and</strong>“degree”of moderator effects,whereby <strong>the</strong> “form” is indexed by moderation of slopes (or means,in ANOVA) whereas <strong>the</strong> “degree” is indexed by moderation ofcorrelations. He argued from first principles <strong>and</strong> demonstratedempirically that it is possible to find a significant differencebetween correlations from two independent-samples but fail tofind a corresponding significant regression interaction term, <strong>and</strong>vice versa. A related treatment was presented independently bySharma et al. (1981), who referred to “degree” moderators as“homologizers” (a term taken from Zedeck, 1971). They pointedout that homologizers that act through <strong>the</strong> error-term in aregression instead of through <strong>the</strong> predictor itself.Stone <strong>and</strong> Hollenbeck (1984) dissented from Arnold (1982),arguing that only moderated regression is needed to assess moderatingeffects, regardless of whe<strong>the</strong>r <strong>the</strong>y are of form or degree.Their primary claims were that moderated slopes also can beinterpreted as differing strengths of relationship, <strong>and</strong> that <strong>the</strong>subgrouping method advocated by Arnold raises concerns abouthow subgroups are created if <strong>the</strong> moderator is not categorical.Arnold (1984) rebutted <strong>the</strong>ir claim regarding <strong>the</strong> slope as a measureof relationship strength, reiterating <strong>the</strong> position that slopes,<strong>and</strong> correlations convey different types of information about suchrelationships. He also declared that both moderated regressionwww.frontiersin.org July 2012 | Volume 3 | Article 231 | 92
Change language