Effects of predictor intercorrelations and reliabilities on moderated multiple regression

ORGANIZATIONAL

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41, 248-258 (1988)

Effects of Predictor Intercorrelations and Reliabilities Moderated Multiple Regression

on

WILLIAM l? DUNLAP AND EDWARD R. KEMERY Tulane fJniversily

Results of Monte Carlo simulation suggest that detection of moderator effects in moderated multiple regression is hampered by poor reliability in either the independent variable, X, or the moderator variable, W. This tinding was anticipated from the fact that reliability of a product term, xw, is determined in part by the product of the reliabilities of its constituents. An interesting finding was that the probability of detection of a product (interaction) term increases as the correlation between x and w increases; it is known that the reliability of a product term increases similarly. An unexpected finding was the inflated probabilities of Type I errors that occurred for direct effects of x or w when they were not directly related to the criterion in the underlying model. Detection of spurious direct effects was exacerbated by increased correlation between x and w. It is clear that moderated multiple regression is adversely affected by measurement error, but that the impact is complex. It is apparent that researchers in this area must strive to improve the reliabilities of predictor variables if they are to have a reasonable chance of discovering moderator effects.

0 1988 Academic

Press, Inc.

Several psychological theories postulate that certain variables influence people in complex ways. For example, person-environment interactionists (e.g., Endler & Magnussen, 1976) assert that the relationship between personal characteristics (x) and outcomes (J) is not constant for all individuals. Rather, it is thought that person-outcome relationships vary as a function of the environment (w). If this is the case, w is said to moderate the relationship between x and y, and, therefore, w is a moderator variable. Historically, moderator variables have been given a variety of labels (Zedeck, 1971), including population control variables (Gaylord & Carroll, 1948), subgrouping variables (Frederiksen & Melville, 1954), referent variables (Toops, 1959), predictability variables (Ghiselli, 1956), and modifier variables (Grooms & Endler, 1960). Interest in moderator hypotheses has led to a number of attempts to define techniques for detecting moderator effects. One such procedure, moderated multiple regression analysis (MMRA), described by Saunders (1956), involves the use of two prediction equations to test for moderator Requests for reprints should be addressed to the first author at the Department of Psychology, Tulane University, New Orleans, LA 70118. 248 0749-5978188$3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

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effects. Based on the general linear model, the procedure involves comparing the variance explained by a model containing a predictor variable, the moderator variable, and their cross-product, with a model containing only the predictor and the moderator. Specifically, the coefficient of determination is obtained from the following model: y = b,x + b,w + bjXW + a,

(1)

where y is the criterion variable, x is the predictor variable, and w is the moderator variable. The variable xw represents the cross-product of .Y and w, which carries the moderator information (Cohen, 1978). The coefficient of determination for this model is compared with the coefficient from the following model: y = b*Ix + h**W + a*,

(2)

F = (R,* - R,*)(N - pc- l)/(l - R,*),

(3)

according to the F ratio where R,* refers to the coefficient of determination from the complete model (Eq. (l)), R,* refers to the coefficient of determination from the reduced model (Eq. (2)), N is the number of subjects, and pc is the number of terms in the complete model. The degrees of freedom for the F ratio are 1, and N - 4 for MMRA. It has been argued that MMRA provides an unambiguous test of moderator effects (Arnold & Evans, 1979); however, this assertion has recently been questioned (Busemeyer & Jones, 1983; Evans, 1985; Morris, Sherman, & Mansfield, 1986). Morris et al. (1986) argued that MMRA may produce an inflated Type II error rate because the cross-product term is highly related to both x and w. They conclude that multicollinearity introduced by the cross-product term serves to mask moderator effects. To overcome this apparent problem, they suggest the use of principal components regression analysis, a biased regression technique where factors with extremely small associated eigenvalues are excluded from the analysis. Dunlap and Kemery (1987), however, pointed out the multicollinearity “problem” is more apparent than real. Using a framework provided by Bohrnstedt and Marwell (1977), they demonstrated that because multicollinearity is a function of means and standard deviations of the raw scores of x and w, it can be substantially reduced by standardizing the variables. As pointed out by Cohen (1978), standardizing will result in a change in the p weights of the regression equation; however, the proportion of variance explained and the associated significance tests are invariant with respect to this transformation. Thus, the failure to find expected moderator effects does not seem to be attributable to multicollinearity.

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Busemeyer and Jones (1983, p. 559), based on a mathematical analysis of MMRA, concluded that the presence of measurement error in the predictor variables would drastically reduce the power to detect a significant contribution from the product term. They further concluded that MMRA may produce spurious findings, such that, even with a strictly multiplicative underlying model, the regression model could contain both additive and multiplicative effects (Busemeyer & Jones, 1983, p. 552). Evans (1985) investigated the possibility that common method variance may produce spurious moderator effects in MMRA. He concluded, however, based on computer-generated large-sample studies, that artifactual moderator effects cannot be created by either correlated or independent errors. Evans (1985) did find evidence that when method variance is introduced, moderator effects can be attenuated as a result of the use of fallible measures. In light of recent research interest in moderator effects, it is important to known under which conditions investigators are likely to be misled by MMRA. Busemeyer and Jones (1983) highlighted the severe unreliability of the cross-product term in MMRA and warned researchers to recognize that the statistical power to detect multiplicative effects may be extremely low when variables are measured with error. Evans (1985) also demonstrated an increase in Type II errors when more unreliability was introduced into the analysis. The influence of unreliability is not surprising based on the work of Bohrnstedt and Mat-well (1977), who demonstrated mathematically that the reliability of a cross-product variable may be expressed as r xwxw = (r,,r,,

+ r2,J4 1 + r’,,),

where r,,, is the reliability of the cross-product term, r,, is the reliability of the predictor, r,, is the reliability of the moderator, and rxw is the correlation between the predictor and moderator. When the independent variables are orthogonal (r,,+ = 0), the reliability of the cross-product term is simply the product of the independent variable reliabilities, which means that the reliability of this new variable cannot be higher than the reliability of either independent variable. When intercorrelation exists among the independent variables, the cross-product term’s reliability is somewhat higher than the product of their reliabilities. The expected effects of independent variable unreliability have yet to be empirically studied in a comprehensive manner. The work of Evans (1985) primarily concerns the case in which correlated errors are present, but the predictors themselves are assumed to be orthogonal. In many applied investigations, however, the latter assumption is untenable. Therefore, the effect of unreliability on power of MMRA when true

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scores of variables making up the cross-product term are correlated is a more general question. The following investigation was conducted to delineate the magnitude of the effects predicted mathematically by Busemeyer and Jones (1983), that power to detect multiplicative effects would be diminished by measurement error and that spurious additive effects not present in the underlying model might be detected. Specifically, we used Monte Carlo simulation to assess the power and Type I error rate associated with MMRA analysis under various degrees of predictor unreliability and predictor intercorrelation. METHOD In the description that follows, “true” scores will be indicated by capital letters, and “observed” scores, scores which have error added, by lowercase letters. The three interaction models studied in detail were (a) the pure product condition, Y = XW, where only interaction is present and there are no direct, or main, effects of either X or W; (b) the uncorrelated moderator condition, Y = X + XW, where both the independent variable and the product determine Yin equal proportions, but W is unrelated to Y; and (c) the main effects plus interaction condition, Y = X + W + XW, where Y is determined equally by the direct effects of both X and W and by their product. Unit weighting of the various model components was used, so that any difference in sensitivity of multiple regression analysis to terms of different types would be apparent. Independent random standard normal deviates were generated via the IBM random normal score generator, RNOR. For each data set, tirst independent variables, X and W, were generated, then intercorrelated by adding a common independent variable, C, via the following equations: X =

X(1 - yxw)1’2 + Cr,,‘“,

W = W( 1 - rxw)1’2 + Crxw”2,

(5) (6)

where C is an independently generated normal score common to both X and W, and rxw is the desired correlation between true scores. Next, the true score product, XW, was computed from X and W, and Y was derived using one of the three models described above. Error was then added to Y, X, and W, such that y had reliability equal to 0.7, and the reliability of x, r,,, and of w, r,,,,,,, were equal to either 0.2, 0.5, 0.8 or 1.0, as determined by the various conditions studied. After the introduction of error to determine the reliabilities, the actual correlation between x and w is automatically reduced via r xw = (r,r,,)1’2rxw,

(7)

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so that the actual intercorrelations between x and w are those presented in Table 1. Finally xw was computed from x and w. For each problem generated, the sample size was set equal to 30. Although this sample size is considerably smaller than likely in most correlational research, it was felt that if problems or difficulties were to be seen with moderated multiple regression, they would be most apparent with small 12.For each data set, moderated multiple regression was accomplished by computing R2, (R2y * x, w,xw) and three reduced R2r’s (R2y * w,xw, R2y * x,xw, and R2y . x,w). The F ratios for complete vs reduced model tests on 1 and 26 degrees of freedom were computed by Eq. (3).

For each set of problem parameters (indicated by the first two columns in Tables 2, 3, and 4) 10,000 simulations were performed, and the proportion of F ratios exceeding the critical value at the 0.05 significance level was tallied, which contitutes the data presented in the three tables. RESULTS

The proportion of F ratios significant at the 0.05 level for the effects of x, w, and xw for the pure product model are shown in Table 2. Because the model in this case contains no direct (or main) effects of either X or W, all findings under columns headed by x or w, represent Type I errors; whereas, the proportions of significant F ratios under xw represent power to detect the interaction term, because the XW interaction is in fact present for this model. The last row of Table 2, where x and w reliabilities equal one, confirms that in this case moderated multiple regression analysis works properly; that is, the power to detect the product term is high TABLE

1

ADJUSTEDCROSS-CORRELATIONS, r,,, BETWEEN~NDEPENDENT~ARIABLE,~, AND MODERATORVARIABLE,W,ASFUNCTIONSOFTRUESCORECORRELATION rxw AND RELIABILITIES r, AND r,,

r,

r,,

(r,r,,Y

0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.8 0.8 1.0

0.2 0.5 0.8 1.0 0.5 0.8 1.0 0.8 1.0 1.0

0.200 0.316 0.400 0.447 0.500 0.633 0.707 0.800 0.894 1.000

0.2

0.5

0.8

,040 ,063 .080 .089 ,100 .127 ,141 .160 .179 ,200

,100 .158 .200 .224 .250 ,316 ,354 .400 ,447 .500

.160 .253 .320 .358 .400 ,506 ,566 ,640 ‘716 ,800

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AND RELIABILITY

TABLE

2

PROP~RTIONSOFSIONIFICANTF RATIOSFORMODEFXATEDMULTIPLEREGRESSIONTESTS OF~NDEPENDENT~ARIABLE,~. MODERATORVARIABLE, MI,ANDTHEPRODUCT,XW. AS FUNCTIONSOFCROSS-CORRELATION,r,,. ANDRELIABILITIES, r,, AND r,,. FORTHE PURE PRODUCT MODEL Y = XW WITH u EQUAL 0.05 AND N EQUAL 30

rxw 0.0 r xx

r,,

0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.8 0.8 I.0

0.2 0.5 0.8 I.0 0.5 0.8 1.0 0.8 I.0 1.0

0.5

0.2

0.8

-

x

M’

XM’

X

M’

XM’

L

I,’

.VU’

,068 ,063 ,054 ,042 ,086 ,062 ,049 ,084 ,047 ,052

,061 ,104 ,140 ,165 ,084 ,121 ,140 ,085 ,105 ,052

,147 ,286 ,402 ,490 ,542 ,741 ,838 ,933 ,983 ,998

,071 ,061 .052 ,048 ,082 ,070 .047 .082 ,047 ,048

,068 ,104 .142 .I76 ,083 ,115 ,143 ,082 .I05 ,049

.I62 ,293 ,427 ,497 ,561 ,764 ,845 ,938 .982 ,998

,071 .058 ,051 ,050 ,090 .065 .056 ,079 ,052 ,049

,070 ,114 ,165 .I98 ,088 ,120 ,156 .077 .I01 ,048

,174 ,335 ,484 .560 .613 ,802 ,876 ,950 ,985 ,999

I

.075 .060 ,058 .073 .087 ,066 ,017 ,071 .056 ,051

M

,075 .I25 ,194 ,240 ,084 ,132 .I89 ,080 ,102 ,047

IM‘

,228 .418 .563 ,635 ,706 ,867 .926 .976 ,993 ,999

and the Type 1 error rates for x and MIare approximately 0.05, the (Ylevel used.

In general it can be seen that as predicted, low reliability in x and/or H’ leads to decreased power for detecting significant interaction. The intriguing finding that was not predicted is that Type I error rates for .y and w are often inflated when the reliabilities are less than one, and this inflation can be substantial even though the reliabilities are quite high. When the reliability of one variable is low and that of the other variable is high, it is the variable with high reliability that has the dramatically inflated Type I error rate, which is additionally counterintuitive. The latter problem only becomes worse with increasing correlation between x and w. Throughout, however, it is clear that the product or interaction term is the most likely to be found significant (although its power is reduced), so at worse, under these conditions, one might spuriously detect main effects of the predictor variable with highest reliability. Proportions of significant F ratios for moderated multiple regression applied to the uncorrelated moderator model, Y = X + XW, are shown in Table 3. Because W is uncorrelated with Y, the proportions under columns headed by w represent Type I errors, whereas, proportions under x or xw represent power. The last row in Table 3 confirms that the analysis behaves properly when the reliabilities are perfect; that is, x and xw have high probabilities of detection and the Type I error rate for MJis about 0.05. Also, correlation between X and W has little effect on the analysis.

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TABLE 3 PROWRTION OF SIGNIFICANT F RATIOS FOR MODERATED MULTIPLE REGRESSION TESTS OF INDEPENDENT VARIABLE, x, MODERATOR VARIABLE, w, AND THE PRODUCT, xw, AS FUNCTIONS OF CROSS-CORRELATION, rxw, AND RELIABILITIES, r, AND r,,,,,,, FOR THE UNCORRELATED MODERATOR MODEL Y = X + XW

rxw 0.0 r,

r,,

0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0

0.2 0.5 0.8 1.0 0.2 0.5 0.8 1.0 0.2 0.5 0.8 1.0 0.2 0.5 0.8 1.0

x

W

.270 ,285 .295 .310 ,610 .641 .678 .702 .835 ,884 .925 .953 ,914 .952 .983 .994

.058 ,077 .092 .I05 ,056 .065 .088 .097 .052 .057 .069 ,074 .049 .045 .050 .050

0.2 xw

,108 .182 .250 .308 ,193 .375 .540 .631 ,306 .587 .799 .891 .397 .726 .921 .979

0.5

X

W

xw

X

W

,270 ,273 .278 .287 .607 .636 .657 ,678 ,833 ,876 .910 .945 .909 .950 .983 ,993

.066 .091 .118 .137 .058 .071 .094 .115 .053 .060 .070 ,083 .046 .050 .049 ,048

.113 .I88 .260 ,325 .202 .388 ,550 II45 ,322 .596 .805 ..890 .407 ,740 ,922 ,978

.241 .237 ,222 ,215 .564 .555 .555 ,544 .783 .826 .857 .883 .872 ,932 ,968 ,988

.092 ,170 ,238 .306 .075 .I13 ,165 .202 .056 .070 .085 .104 .047 ,045 ,050 .048

0.8 xw

,132 .233 .326 ,384 ,248 .455 .622 ,715 .371 .678 .849 ,924 ,462 .791 .939 .982

X

W

,204 ,163 .I23 .I14 .484 ,419 ,322 .251 .720 .711 ,634 ,551 .829 .855 .882 ,864

.137 .302 .476 .580 .094 ,192 .337 .459 .067 ,085 ,133 ,195 .063 .069 .055 ,051

xw

.168 .317 .433 .519 ,318 .575 .749 .824 .466 ,770 ,907 ,954 .562 .866 .972 .988

High reliability in w and low reliability in x lead to an increased Type I error rate for the direct effect of w, when w in this model should have no direct effect. Correlation between X and W exacerbates this problem to the point where w may be more likely to be found significant than x (see last three columns where rxw equals 0.8). When rxw is zero, power to detect the product term is always lower than power to detect the main effect of x, as would be predicted from Eq. (4) for the reliability of a product term. Interestingly, the power difference is even present when the reliabilities of x and w are one; therefore, the lower power for a product term relative to a main effect term is not entirely a function of reliability. Presumably the latter effect is attributable to some other factor, such as the shape of the sampling distribution of product terms. When reliability of x is high and reliability of w is low, as predicted, the product term will be considerably less powerful than the main effect. Results for the final model, Y = X + W + XW, are presented in Table 4. Because all effects are present in the model, all of these data are estimates of power. When reliabilities of both x and w are one and the correlation between x and w is zero, all three terms are tested with sub-

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AND RELIABILITY

TABLE 4 POWER FOR MODERATED MULTIPLE REGRESSION TESTS OF THE SIGNIFICANCE OF INDEPENDENT VARIABLE, x, MODERATOR VARIABLE, ~1, AND THE PRODUCT, xw, AS FUNCTIONS OF CROSS-CORRELATION, rxw, AND RELIABILITIES, r, AND rww, FOR THE MAIN EFFECTS PLUS INTERACTION CONDITION Y = X + W + XW

0.0

0.5

0.2

0.8

‘rx

rww

x

M’

xw

x

w

XM’

x

M

XM’

X

4‘

.YU’

0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.8 0.8 1.0

0.2 0.5 0.8 1.0 0.5 0.8 1.0 0.8 1.0 1.0

,209 ,230 .238 .257 .500 ,568 .614 .829 .888 ,973

,199 ,453 .663 ,771 ,506 ,738 ,844 ,830 .929 ,915

.089 ,146 .215 ,213 ,290 ,443 ,553 .693 .811 ,932

.237 .244 ,245 ,254 ,556 ,574 ,596 ,844 ,875 ,972

,247 ,529 ,171 ,862 ,548 ,798 ,898 ,840 ,939 ,973

,095 .148 .223 .280 .290 .459 ,578 .692 ,820 ,933

,273 ,250 ,212 ,191 ,574 ,517 ,464 ,818 ,792 ,935

,282 .641 ,872 ,950 ,581 ,853 ,950 .819 ,940 ,940

.098 ,173 ,274 ,353 ,338 .523 ,655 ,743 ,864 .949

,301 ,221 ,147 ,097 .570 ,364 ,200 ,709 ,436 ,722

,312 ,712 ,938 ,983 ,572 ,880 ,974 ,705 .918 .727

,128 ,231 .359 ,463 ,440 ,634 ,771 ,825 .918 ,967

stantial power; however, as noted before, the power of xw is a bit less than that of x or w. Interestingly, this condition is reversed when yXWis large. Power for the product term is seldom greater than for the least powerful direct effect term, either x or w, except when rXw is high, in which case the power of the product term may substantially exceed that of the least powerful direct effect. In short, all terms suffer power reduction when reliabilities are low, but, the product term in general appears most vulnerable to loss of power as a result of low reliability of its constituent terms. DISCUSSION

In general the results of our Monte Carlo simulation do not support the contention that MMRA provides unambiguous tests of moderator effects. Under certain conditions, the likelihood of detecting moderator effects will decrease, while under others, the likelihood of detecting spurious main effects will increase, which bears out the mathematical predictions of Busemeyer and Jones (1983). These problems appear to be directly attributable to the reliabilities of the variables included in the analysis. The impact of poor reliability on the detection of moderator effects is largely consistent with predictions drawn from the Bohrnstedt and Marwell (1977) equation (see Eq. (4) above), which shows that the reliability of a product is equal to the product of the reliabilities of its constituent terms when those constituents are independent, but increases as a func-

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tion of the correlation between constituents. Power to detect significant product (moderator) effects is directly proportional to the product reliability. Reliability has always been acknowledged as a problem in prediction. Our results indicate, however, that measurement error has a somewhat greater impact on more complex models. If a construct cannot be gauged consistently, then researchers should be extremely hesitant to embed that construct in a complex model, particularly in conjunction with other fallible measures. If, however, theory necessitates the use of such a measure, then researchers should take into account the power to detect moderator effects, and adjust sample size accordingly. The finding of inflated Type I errors in tests of the direct effects of independent and moderator variables for underlying models where no direct effects are present was unexpected. Although Evans (1985) had previously demonstrated that the addition of error (reduced reliability) did not create spurious product effects with an underlying additive model, the converse does not appear to be the case. Poor reliability does create spurious direct effects when the data generation equation is purely multiplicative (cf. Busemeyer & Jones, 1983). Furthermore, the greatest increase in spurious direct effects is for the variable with highest reliability, and is partly a function of using that variable in conjunction with another variable having lower reliability. This finding is counterintuitive in the sense that the likelihood of model misidentification is greater for the variable with the greater precision. The practical problem imposed by increased Type I errors for direct effects when the underlying model is strictly multiplicative is unclear, because it is difficult to conceive of an applied situation where precise independence exists between the criterion and either predictor variable. These findings imply, however, that less-than-perfect reliability will lead to overemphasis on direct relationships, at the likely expense of multiplicative ones. The fact that the present study used x and w in the form of standard scores might at first seem to limit the generalizability of these findings; actually, however, it does not. Arnold and Evans (1979) have shown quite clearly that the squared multiple correlation coefficients in MMRA are invariant under linear transformation of the variables involved; therefore expressing the predictors in standard score form merely serves to limit correlations between x and w and the product term, xw, and has an effect on neither coefficients of determination nor their tests of significance (Cohen, 1978). Although not of great magnitude, it was clear that another mechanism was at work to lower power for the detection of product terms even when the reliabilities of x and w were perfect. (See the bottom rows of Tables 3 and 4.) In several instances, the power to detect the product effect was

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lower than the power to detect main effects. This smaller effect is most likely a result of the nonnormal distribution of product scores. Interestingly, when both x and w are measured perfectly and are correlated at 0.8, the power of MMRA to detect direct effects is reduced beyond that when the variables are less intercorrelated (see the last row in Table 4). This Iinding probably results from the product term having a greater effective weight than either predictor when the predictors are correlated. The formula for the variance of a product variable, given by Bohrnstedt and Goldberger (1969), is V(xw) = l?(x)V(w) + Ez(w)V(x) + 2E(x)E(w)C(x,w) + V(x)V(w) + cyx,w),

(8)

where V refers to the respective variances, E refers to the respective expected values, and C refers to the covariance. Because standard scores were used in the present study, Eq. (8) reduces to V(xw) = 1 + &+.

(9)

Therefore, even though unit weighting was specified in the model, the effective weighting of the product variable was greater than one, except for the condition in which rxw was zero. The effect of this greater weighting was to increase power for the product term, at the expense of power to detect the direct effects. A final problem was that the above simulations were all conducted with small samples of 30, so that subtle differences in power would be apparent. A legitimate question remains as to whether the phenomena revealed still persist when sample sizes are large. To address such concerns, the Monte Carlo study was repeated for selected points from Tables 2 through 4 with sample sizes of 500. With large samples, the effects on power became more patent. That is, if the effect was present in the underlying data generation model, it was highly likely to be detected. even when reliabilities were low. Therefore, large sample sizes appear to go a long way toward correcting the loss of power for detection of product terms. On the other hand, the probabilities of Type 1 errors in finding significant direct effects of variables not in the underlying model were vastly increased. Busemeyer and Jones (1983) had predicted on mathematical grounds that certain artifacts of MMRA would not be helped by large sample sizes. The fact that model misidentification was only made worse by large N provides the warning that large samples are by no means a universal panacea. Researchers using MMRA should be aware of this tendency of power for the detection of moderator effects to be lost, at the expense of overemphasis on direct effects of predictor variables.

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