Multivariate and moderated within- and between-entity analysis (WABA) using hierarchical linear multiple regression

MULTIVARIATE AND MODERATED WITHINAND BETWEEN-ENTITY ANALYSIS (WABA) USING HIERARCHICAL LINEAR MULTIPLE REGRESSION

Chester A. Schriesheim* University

of Miami

Leadership scholars have increasingly been interested in exploring multi- and cross-level relationships in their research. However, multivariate and moderator variable approaches for examining these relationships are apparently not well-developed and well-understood. Building upon the within- and between-entities analysis (“WABA”) framework of Dansereau, Alutto, and Yammarmo (1984), this article first develops and then illustrates how multivariate and moderator analyses may be conducted using the WABA approach. A further extension of WABA, suggested by George and James (1993) and incorporating an analysis for between-entities moderation of within-entities effects, is also proposed and illustrated-allowing researchers to better assess whether obtained within-entity (“parts”) parameter estimates hold across all entities (i.e., indicative of only nonrelative relationships) or not. Finally, three recently raised concerns about WABA, and directions for future research, are considered briefly.

INTRODUCTION Organizations may be conceptualized from the perspective of multiple levels of analysis (e.g., individuals, dyads, work groups, organizations; Roberts, Hulin, & Rousseau, 1978), and it is thus not surprising that leadership scholars have been increasingly interested in formulating and testing hypotheses which involve two or more levels of analysis (e.g., House, 1991). However, while cross-level and multilevel theorizing is a potentially significant development in organizational research (Rousseau, 1985), the apparent accessibility and/ or use of appropriate data-analytic techniques has not kept *

Direct all correspondence Administration, University

to: C.A. Schriesheim, of Miami, 414 Jenkins

Leadership Quarterly, 6(l), 1-18. Copyright @ 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISSN: 1048-9843

Department of Management, School Building, Coral Gables, FL 33124-9145.

of Business

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pace with the recent increase in interest about multiple levels-of-analysis. One result is that important issues which could (or should) involve multiple levels-of-analysis may have been underinvestigated or not investigated at all. For example, while there are undoubtedly numerous leadership areas where a multiple levels-of-analysis perspective might be beneficial, one where such an approach seems mandatory is the leader-member exchange (“LMX’) theory of Graen and his associates (see Graen and Scandura, 1987, for a review). This approach explicitly postulates that leadership is a within-dyads and within-groups (and not between-groups) phenomenon. However, investigations of the LMX model have not tested whether the leader-member exchange process was occurring at the dyadic level, the work group or unit level, at both levels, or at some other level-of-analysis-the analyses have typically employed traditional correlational or analysis-of-variance procedures and have ignored the levels-of-analysis issue altogether (for some exceptions to this generalization, see Nachman, Dansereau, & Naughton, 1983; Schriesheim, Neider, & Scandura, 1994; Yammarino & Dubinsky, 1992). Similarly, the same can be said for what Dansereau, Graen, and Haga (1975) have labeled the “Average Leadership Style” (“ALS’) approach. These researchers employ a group-level conceptualization of basic leadership processes; however, they typically have not tested whether their data support a within-groups treatment, a between-groups treatment, both, or neither. Instead, the applicability of between-groups analyses has usually been assumed and the data analyzed accordingly (e.g., Greene & Schriesheim, 1980). While it is not possible to be sure why appropriate data-analytic techniques have not been employed by leadership scholars (even in situations where they are clearly needed-such as in LMX or ALS research), two reasons appear likely. First, these procedures may not be widely enough known and/ or sufficiently well described to allow their utilization. Second, most contemporary leadership research is concerned with multivariate relationships, frequently involving “moderator”variables (Cohen & Cohen, 1983; Zedeck, 1971).’ However, those applications of multilevel analysis that have appeared in the literature have not employed multivariate analyses; they have also not explained how multivariate examinations might be undertaken.’ Thus, the purpose of this article is not to chastise the field for data-analytic shortcomings but to (1) present the basics involved in one approach that allows multilevel analysis, and (2) offer a multivariate extension to these procedures (which may be particularly useful for leadership researchers-especially those interested in conducting moderator variable analyses). As an additional (and, hopefully, helpful) extension, an approach is proposed for assessing whether obtained within-entities effects are relational or not (i.e., whether finding significant within-entities or “parts” effects should be interpreted as holding or not holding across entities). To accomplish these purposes, the statistical procedures and inference-drawing processes of within- and between-entities analysis (WABA; Dansereau, Alutto, & Yammarino, 1984) are first presented. Next, extension of the general WABA approach to multivariate analysis and to examining within-entities moderator effects is described. Then, a brief numeric illustration is given employing field-collected data and the WABA procedures outlined here. Finally, three concerns about WABA are discussed, along with directions for future research.

3

Multivariate WABA

GENERAL WITHIN-

AND BETWEEN-ENTITIES

ANALYSIS (WABA)

An Overview of WABA

Although most measures used in leadership research have either individuals or dyads as their referents, assuming that strictly dyadic- or individual-based analysis is appropriate can be problematic (cf. Yammarino & Dubinsky, 1992; Yammarino, Dubinsky, & Hartley, 1987; Yammarino & Markham, 1992). Additionally, traditional data-analytic techniques (such as bivariate correlation) use only one score (the raw score) for each respondent and do not test a phenomenon’s “locus” (to determine where the phenomenon is more likely to be occurring). Thus, Dansereau, Alutto, and Yammarino (1984) developed the basic within- and between-entities analytic (WABA) procedures to help determine whether empirical findings should more properly be seen as occurring within entities, between entities, both (mixed), or neither (null or non-operative). Although this theme shall not be developed further, Dansereau et al. (1984) and virtually all users of WABA to date (e.g., Markham & McKee, 1991; Yammarino & Dubinsky, 1992; Yammarino & Markham, 1992) emphasize that within and between analyses should not be conducted as a routine exploratory data-analytic method but only when strong theoretical arguments can be made for conceptualizing a particular phenomenon as possibly occurring at more than one level of analysis. Under these circumstances, formal WABA analyses may be helpful in determining whether a particular set of data are better seen as aligned with one theoretical position, another, both, or neither. WABA is an extension of work by Robinson (1950) on the analysis of data when multiple levels are present (such as individuals or dyads within work groups); it is perhaps best viewed as an integration of different correlational, analysis-of-variance (ANOVA), and analysis-of-covariance procedures to assess variation both within and between entities (cf. Finn, 1974; Haggard, 1958; Hays, 1973; Pedhazur, 1982). Basically, WABA as a data-analytic technique involves three steps. First, in WABA I, the variables of a study are examined to assessed their relative amounts of variation between entities (e.g., between work groups or units) (indicative of between-entity heterogeneity) and within entities (indicative of within-entity heterogeneity) (cf. Yammarino & Markham, 1992). Second, the relationships among the variables are examined in WABA II to assess the amounts of between- and within-entities covariance. Finally, to consider the variance (WABA I) and covariance (WABA II) results jointly, raw score correlations are decomposed into within- and between-entities components and the findings from the first two steps examined for consistency and combined with the decomposed correlation components to draw general or overall conclusions concerning a phenomenon (Dansereau et al., 1984, pp. 183-185, discuss how WABA I and II results may be integrated; additionally, it might be noted that the decomposed correlation components should probably be given substantial weight in drawing final conclusions; cf. Dansereau et al., 1984, p. 185; Yammarino & Markham, 1992, pp. 171-172; Yammarino, personal communication, December 29, 1993). The WABA Equation

In WABA, within- and between-entity indicators are computed and compared to each other, using tests of statistical and practical significance (discussed below).

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Analogous to the logic of a one-way ANOVA, data are partitioned into within-cells (deviation from cell average) and between-cells (cell average) components, with the cells representing analytic entities such as dyads, work groups, organizational units, organizations, industries, societies, and so forth. The relationships which result from these calculations may be summarized in the fundamental WABA equation as follows:

where ‘B, and ‘By are the between-entity ems for variables x and y (respectively), ’ W, and ‘WY are the corresponding within-entity etas, ‘Bxy and r W, are the corresponding between-entity and within-entity correlations, and IT, is the total (raw score) correlation (note that 174 •t r] i = 1). ‘W, and ‘WY are calculated by correlating the raw scores (lx,] or i_v,J)with the appropriate within-entity deviation scores (Ix, - Xk] or [J+,- $1) for n parts (e.g., the 1 to N respondents) within k entities (e.g., the 1 to K work units); “B, and ‘By are calculated by correlating the raw scores of the n parts (x, or y,,) with their betweenentity scores (i.e., the appropriate l&l or [&I for the entity which encompasses each part or within which each part is situated). The within-entity correlation is computed by correlating the within-entity deviation scores (i.e., [xk - &I and bn - $1 for the n parts, while the between-entity correlation is computed by assigning each part its appropriate between-entity scores (C&l and [$I) and then correlating these across the parts (i.e., between-entity correlations are simply correlations between the entity means, weighted by the number of parts within each entity). As can be seen from the WABA equation (Equation l), any raw score correlation may be conceptualized as the sum of two separate components-a between-entity (cell) component (‘Bx9ByrBxy) and a withinentity component (“W,” W,‘W,); both are the products of multiplying their appropriate etas and component correlations. Thus, raw score correlations (‘rTXY)cannot be unambiguously interpreted without examining and considering all six terms in the WABA equation (Pedhazur, 1982; Robinson, 1950). WABA I WABA I tests the variance of each variable by partitioning the original (raw) scores (e.g., lx,]) into within- and between-entity component (deviation or average) scores (e.g., lx, - %I and [%I); these component scores are then correlated with the original raw scores to yield within-entity (VW)and between-entity (71~)etas. Finally, the etas are tested relative to each other with F-tests of statistical significance and newly developed Etests of practical significance. The traditional F tests of statistical significance have K - 1 and N - K degrees of freedom for the between- and within-entity etas, respectively, where K is the number of entities and N is the total number of parts within entities. When a between-entities eta is larger than its corresponding within-entities eta, a traditional F-test is used. However, when the within-entities eta is larger, a corrected F-test is employed (Dansereau et al., 1984; Haggard, 1958). This test, simply the inverse of the traditional F-test, is computed as follows:

Multivariate

5

WABA

Corrected

F=

[(r]$) / (N-

IQ1/ [(vi?)/ (K- l)l=

l/F

(2)

The E (eta ratio) tests index the magnitude of within- versus between- effects relative to each other; they are geometrically based, not dependent upon degrees of freedom, and computed as: E=vB/

VW

(3)

WABA I I WABA II examines relationships among variables by first computing within- and between-entities correlations (using all within- or all between-entity scores for the n parts). The magnitude of these correlations is then tested for statistical (using traditional t-tests) and for practical significance (using newly developed R-tests). The t-tests have K - 2 and N - K - 1 degrees of freedom for the between- and within-entity correlations, respectively. The geometrically based R (correlation) tests of practical significance are not dependent upon degrees of freedom and are calculated as: Rg = r~ / (1 - d)l’*

(4)

and Rw=

2 l/2

rw / (1 - rw)

(5)

Finally, differences between the paired (involving the same variables) within- and between-entity correlations are tested using Fisher 2 transformation tests of statistical significance (with K - 3 and N - K - 2 degrees of freedom for the between- and withinentity correlations, respectively), and newly developed A (angular) tests of practical significance (for both the Z and A tests, sign or directionality is ignored and only differences in absolute magnitudes are tested). The A tests are geometrically based and not dependent upon degrees of freedom, and are computed as: A=@w-@~

(6)

where OW and Oe are the angles associated with the within- and between-entity correlations (respectively) (since correlations are merely the cosines of angles, OW and OS are easily obtainable; cf. Dansereau et al., 1984). Drawing Inferences Inferences are drawn from the above tests using the .05 and .Ol levels of statistical significance (for the F, Z, and t tests) and the 15” and more conservative 30” levels of practical significance (for the E, R, and A tests). The 15 o and 30” angle criteria derive from the fact that a 90” angle represents orthogonal or unrelated variables, while a 0” angle represents perfect correspondence; smaller angles thus represent stronger relationships (in the R-tests), while angular ratios different from 1.0 or larger differences between angles (in the E and A tests, respectively) indicate more meaningful differences

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in the magnitudes of relationships (these tests use only the absolute values of the etas or correlations-directionality is ignored; see Dansereau et al., 1984, for additional details). Decomposed raw-score correlation components are also computed by multiplying the product of the independent and dependent variables’ WABA I (withinor between-entities) etas with their WABA II (within- or between-entities) correlations; these results may then be examined by the A-test to determine whether the within- or between-entities component is meaningfully greater than the other (again, both the 15” and more conservative 30” levels of practical significance are employed) (Yammarino & Dubinsky, 1992; Yammarino & Markham, 1992; F.J. Yammarino, personal communication, January 16, 1994). Finally, the WABA I and II results and the decomposed correlations are compared and an overall conclusion drawn concerning the phenomenon under investigation. (Dansereau et al., 1984, pp. 183-185, present guidelines for integrating WABA I and II results; for excellent yet brief descriptions of WABA I and II procedures, see Markham & McKee, 1991; Yammarino & Dubinsky, 1992; Yammarino & Markham, 1992; for a full and in-depth treatment of WABA, see Dansereau et al., 1984.)

EXTENSIONS TO THE BASIC WABA MODEL WABA Using Hierarchical linear Multiple Regression As mentioned earlier, leadership researchers are very often interested in conducting multivariate and moderator analyses; therefore, as suggested by Dansereau et al. (1984, pp. 288-295) and Markham and McKee (1991, pp. 963-965) the basic WABA procedures outlined above need to and can be extended to the multivariate case. For linear multiple regression, the extension of WABA is relatively easy and straightforward because: (1) one way of conceptualizing linear multiple regression is that it is simply bivariate correlation/ regression in which the independent variable is a weighted linear composite of several other variables (Baggaley, 1964; Cohen & Cohen, 1983; McNemar, 1969); and (2) “the coefficient of multiple determination R2 can be viewed as a coefficient of simple determination r2 between the responses K and the fitted values ??’ (Neter, Wasserman, & Kutner, 1985, p. 241; cf. Cohen & Cohen, 1983, ch. 3). This, naturally, leads to the following multivariate extension of the basic WABA equation in the case where two independent variables are employed (XLand x2) with one dependent variable (_Y):

The terms in Equation (7) have the same meanings as do those in Equation (1); the only notable differences in Equations (1) and (7) are that composite (“multivariate’) between (‘Bxlx2) and within (” W,,,Z) independent variable etas replace the usual (bivariate) independent variable etas (“B, and ’ W,) of Equation (l), and the Equation (1) bivariate between and within correlations (‘B,,, ’ W,, and ‘Txy) are replaced by their multivariate analogs (‘BACKS,rWx+, and rTXiX2y)in Equation (7). Computationally, the terms of Equation (7) may be calculated as follows (in multivariate WABA, the “By and DWY are computed in the same manner as indicated earlier):

Multivariate

a.

b.

c.

d.

WABA

7

An ordinary least squares linear multiple regression analysis is conducted, regressing xi and x2 on y; the multiple R from this equation is the rT*iX~yterm of Equation (7) above. The unstandardized partial regression weights (bi and b2) which were obtained in step (a) are used to create anew composite independent variable; this composite variable may then be analyzed using the standard bivariate WABA approach outlined above. If within- and between-entity scores already exist, the unstandardized partial regression weights (bi and b2) which were obtained in step (a) may be used to transform the within-entity deviation scores of the independent variables (1x1, - Xnl and [xzn - X2kl) into a weighted composite within-entities variable (lbi{xi,, scores are multiplied by their - Xlkj + bZIX2n - XP~}]) (i.e., the within-entity appropriate partial regression weights and then summed); the between-entity scores ([Xnl and l&J) may likewise be transformed into a weighted composite between-entities variable ([bi{XikJ -t b2{&}]), and the original raw scores ([x1,1 and [x2,$ may be similarly transformed ([bi{xi,) + I!J~{x~~)]).The computations outlined earlier for bivariate WABA then proceed, with the new multivariate composite variables substituted into the calculations of the within and between independent variable etas and the within and between correlations. The degrees of freedom for the t, F, and Z statistics are adjusted to reflect the additional estimates which are computed, as shown in the numeric example which follows (see the footnotes to Tables 3 and 4).3

The ease of applying the above multivariate extension should be apparent, as well as the fact that it can be extrapolated further (i.e., to the case where more than two independent variables are employed). Thus, for example, one can conduct multivariate moderator WABA analyses by treating the interaction term (the cross-product of the independent and moderator variables) as a third independent variable. In conducting this type of analysis, however, it should be emphasized that caution needs to be used in the interpretation of moderated results (as is true with all data-analytic designs) (e.g., main effects should not be interpreted in the presence of significant interactions, the effects of reciprocal causation should be considered, etc.). Additionally, when computational method (c) above is used, it should be noted that: (1) between-entity interaction terms must be computed by first calculating and then averaging (within the entities) the appropriate cross-product terms (i.e., not by multiplying two between-entity average scores together); (2) within-entity interaction terms should be computed by multiplying the two raw scores together and then subtracting from this the appropriate between-entities average; and (3) only partial regression weights derived from raw score regressions should be used in multivariate WABAs-conducting separate within-entities and between-entities regressions produces different partial regression weights and violates the equality set out in Equation (7). Finally, it warrants highlighting that since the partial regression weights of variables change as other variables are added to a regression (Cohen & Cohen, 1983), separate analyses must be conducted for each step of a hierarchical or moderated analysis.

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Testing Within-Entity Effects for Moderation

As recently noted by George and James (1993), ‘WV is a common or “pooled” regression parameter which is computed across all parts; ’ W,, therefore, summarizes any within-entity differences in regressions (i.e., any interactions between parts and entities). Thus, following Cohen and Cohen (1983, ch. 8), the presence of these moderating effects can be assessed by testing whether significant incremental explained variance (A&) is produced by a regression which includes appropriate parts-by-entity interaction terms over one which does not. These two equations may be formulated as follows: y=a-l-bIdI+...

b~-l&r + ~KXi- e, and

(8)

where x and y are the independent and dependent variables (raw scores), respectively, dr to dk-1 represent effects-coded dummy variables for the K entities, and e is error. Finding a significant increase in explained variance for Equation (9) over Equation (8) should lead one to be cautious in interpreting ‘WV and to also recognize that any obtained within~ntity effects for the independent and dependent variables is relational (i.e., effects for the parts must be interpreted as occurring within the entities). This procedure thus allows researchers to assess whether within-entity effects hold across all entities (i.e., indicative of nonrelative parts relationships) or not. A NUMERIC EXAMPLE Recognizing that illustrations can be quite helpful in communicating complex subject matter, a brief example is presented below. This example explores whether employee perceptions of the quality of exchange with their supervisors moderates the relationship between perceived supervisory delegation and rated subordinate performance. The data for this illustration are drawn from Schriesheim et al. (1994), where a more complete theoretical treatment of this topic may be found, along with additional detail on the sample, procedures, and measures (along with additional variables and findings). Method Samp/eand Procedures

Survey questionnaires were administered during normal working hours to 106 employees of a flower imposing firm; full verbal instructions, in addition to written directions, were given; and university sponsorship of the survey was stressed (as was the confidentiality of all responses). After the employees had completed their surveys, their immediate supervisors completed a performance rating form for each employee. The mean age of the sample was 37.6 years, most employees (86%) had completed high school, and 59.4% were male. The mean org~ization~ tenure of employees was 10 years and the mean job tenure was 3.5 years. The respondents had 13 different supervisors, and the number of employees who reported to the same supervisor ranged from 7 to 14 (mean unit size was 9.1).

Multivariate WABA

9

Measures

The employee surveys contained demographic measures and measures of perceived delegation and perceived leader-member exchange (among other measures). The perceived Delegation measure was a &tern scale developed from the work of Schriesheim and Neider (1988) (sample item: “My supervisor does not require that I get his/ her input or approval before making decisions”; response categories: 151Always, 141Very often, 131Fairly many times, [21Occasionally, and [II None of the time), while radar-Ferber exchange (LMX) was assessed using Schriesheim, Neider, Scandura, and Tepper’s (1992) &item scale (this instrument has shown very positive psychometric properties; see also Schriesheim, Scandura, Eisenbach, and Neider, 1992) (sample item: “The way supervisor sees me, he/she would probably say that my ability to do my job well is”; response categories: [51 Exceptional, 141Good to very good, 131Average, [21 Below average, and 111 Poor). Finally, Perfixrnance was measured by asking the supervisors of the respondents to complete a 7-item performance rating measure which was based on a scale originally employed by Mott (1972). Previous research using this measure (e.g., Fulk & Wendler, 1982) suggests that it has good reliability and significant correlations with other performance indicators (such as units produced as a percentage of standard) (sample item: “Productivity-Quantity: Thinking of the various things which this person does for his/ her job, how much is he/she producing (e.g., units produced, customers served, forms completed, pallets loaded, etc.)? Check one”; response categories: I1I His/ her production is very low, [21It is fairly low, 131It is neither high nor low, [41It is fairly high, and [5] It is very high). Raw Score Analyses

Traditional raw score correlation and moderated regression analyses were first conducted, using the respondents’ raw (untransformed) data and computing variable means, standard deviations, coefficient alpha internal consistency reliabilities, and Pearson product-moment intercorrelations in the usual manner. Then, moderated linear multiple regression analysis (Cohen & Cohen, 1983; Zedeck, 1971) was employed to assess the presence of significant interaction effects between Delegation and LMX as a correlate of rated Pe~orman~. In the moderated regression procedure, Delegation was treated as the independent variable and entered into the equation first. Then, the LMX moderator was entered second. Last, a cross-product term was added to each regression, to assess the unique variance contributed by the interaction of Delegation and LMX. WARA

Although the measures employed in the current study have either individuals (Performance) or dyads (Delegation and LMX) as referents, as mentioned above, prior research indicates that higher-level (e.g., group or work unit) effects may be present in any data set (cf. Yammarino & Dubinsky, 1992; Yammarino et al., 1987; Yammarino & Markham, 1992). Additionally, as also noted earlier, traditional raw-score analyses do not allow examination of a phenomenon’s “locus” (to assess the level[sl of analysis which are supported in a particular data set; cf. Dansereau et al., 1984). Thus, WABA was employed to assess whether obtained raw-score bivariate, multivariate, and moderator effects could best be viewed as dyadic-level, group-level, both (mixed), or

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null (non-operative) phenomena. These analyses proceeded step-by-step, as outlined above. (Perhaps it should be reiterated that, due to the fact that partial regression weights change as variables are entered into an analysis, separate multivariate WABA computations were undertaken at each step of the regression-i.e., as each variable was added.) The statistical significance of the within- and between-entity multiple correlations, and the statistical significance of increases in these correlations (as variables were added to the regressions), were tested by the hierarchical multivariate F-test procedures outlined in Cohen and Cohen (1983, ch. 3). Additionally, treating the multiple correlations as bivariate composite r’s (which are based upon linear combinations of variables), the Fisher r-to-Z transformation was employed to test for significant differences between the multiple correlations obtained for the within- and betweenentity regressions. Also, following the same logic, the standard WABA A and R tests of practical significance were used, treating the multiple correlations as bivariate correlations for analytical purposes. Finally, since significant within-entities effects were uncovered, analyses were undertaken to test for the presence of within-entities moderation (i.e., interactions between parts and entities). These analyses were conducted for each interpretable within-entities relationship and used the design outlined above (Equations 8 and 9) (again, the statistical significance of increases in explained variance were tested by the hierarchical multivariate F-test procedures outlined in Cohen and Cohen, 1983, ch. 3). Results Raw Score Results Table 1 presents the raw score univariate and bivariate results, showing that the three measures had acceptable levels of internal consistency reliability in the current sample (Nunnally, 1978). Table 2 presents the raw score moderated regression results, along with the unstandardized partial regression coefficients (bi) which were obtained at each regression step. As can be seen in Table 2 (and contrary to expectations), no significant moderator effects were uncovered. However, LMX did yield statistically significant incremental explained variance at Step 2, when it was added to Delegation as a correlate

Table 1 Raw-score Variable Means, Standard Deviations, Coeffkient Alpha Reliabilities, and Intercorrelations Intercorrelations M

Variable 1. 2. 3. 4. Now

Delegation LMX Delegation X LMX Performance The correlations

17.54 23.48 419.53 26.22

shown are all significant

SD 4.97 4.04 159.23 3.62 at p < .Ol

CY .84 .82 .83

I.

2.

3.

.39 .93 .38

.69 .39

.46

4.

11

Multivariate WABA

Raw-score moderato Step and Independent Variable Added 1. Delegation 2. LMX 3. Delegation X LMX

Table 2 Regression Results for the Dependent Variable of Performance Unstandardized Partial Regression Coefficient Step I

Step 2

Step 3

Al? When Added

.28* -

.20* .25* -

-.51 -.I8 .03

.15* .06* .02

Total Rz .15* .21* .23*

of subordinate Performance (additionally, LMX and Delegation were both statistically signi~cant in this recession). Interpreting these raw score results is clearly problematic. Although we can say that both perceived Delegation and perceived LMX were positively associated with rated subordinate Performance, we do not know the level(s) of analysis at which these relationships held. Thus, for example, it might be that the raw score relationships were based upon differences among individuals or dyads in work groups (within~ntities), upon differences in work groups or supervisors (between-entities), or, possibly, both (seen as indicative of “individual differences” by Yammarino and Markham, 1992). Thus, multivariate WABA may provide considerable assistance in clarifying the meaning of raw-score-moderator findings, and this is illustrated by the WABA analyses presented below. WABA I

Table 3 presents the obtained WABA I results, with the multivariate extension developed earlier being employed for the last two variables shown (the composite Table 3 Within- and Between-entities Analysis I Variable (Performance) (Delegation) (LMX) (Delegation X LMX) (bl[Delegation] + bJLMX])b (bl[Delegation] + BJLMX] i- bJDelegation X LMX])C Nom:

Within Eta

Between Eta

E

F”

.97 .98 .9s .99 .99 .99

.26 .19 .07 .I9 .20 .19

.26+ .19+ ,071 .19+ .20+ .19+

1.80 3.43* 25.29** 3.50** 2.90* 2.92*

n Ail F-tests with one independent variable have &‘= 93,12; those with composites involving two have F$‘= 93~ 1, while those with three have df= 93,lO; all are corrected for testing the significance of within-entities effects. ’ Composite variable computed using the Step 2 partial regression coefficients of Table 2. ’ Composite variabie computed using the Step 3 partial regression coefficients of Table 2. + Significant by the 30” test. * p 5 .os. **p5.01.

X LMXl)e

_

.40*+ .42*+ .48*+ .49*+* .51*+

WirhinCorrelation”

Analysis II

’ For k = I independent variable, dflt) = 92; fork = 2, Q(F) = 2,90; for k = 3, dffl = 3,89. b For k = I independent variable, dflt) = 11; fork = 2, dflF) = 2,lO; fork = 3, df(fl = 3,9. ’ Fork = I independent variable, df(Z) = 10,91; fork = 2, dflz) = 9,90; fork = 3, df(z) = 8,89. d Composite variable computed using the Step 2 partial regression coefficients of Table 2. ’ Composite variable computed using the Step 3 partial regression coefficients of Table 2. + Significant by the 15” test. * Significant @ < .Ol) increase in R2 over previous regression (without the last variable added). *ps.o1.

(Delegation) (LMX) (Delegation X LMX) (bl[Delegationl + b~[LMxl)~ (bdDelegation1 + bz[LMXl + bJDelegation

(Performance) (Performance) (Performance) (Performance) (Performance)

Notes:

Independent Variable

Dependenf Variable

Table 4 Within- and Between-entities

-.08 -.23 -.18 -.22 -.22

BetweenCorrelarionb

-.33+ -.21 -.32+ -.29+ -.31+

A

-1.02 -0.67 -1.02 -0.93 -0.92

z’

N

Multivariate

13

WABA

variables). As shown in Table 3, the F-test results yield support for the existence of significantly more within-entities variance than between-entities variance for all but the Performance variable. Additionally, the E-tests support concluding that there is significantly more within-entities than between-entities variance for all of the variables. Taken as a set, then, these findings clearly support the conclusion that the locus of variance in the variables of this study is primarily within-entities. WABA II

Table 4 presents the obtained WABA II results for examining the locus of covariance in this study’s variables (again, the last two lines in the table give the multivariate composite findings). As shown in Table 4, all of the within-entities correlations are both statistically (by t or F-tests) and practically (by R-tests) significant, while none of the between-entities correlations is either statistically or practically significant. Furthermore, while none of the differences between these correlations are statistically significant (by Fisher Z-tests), four are practically significant (by A-tests); using a conservative interpretation criterion of requiring both statistically and practically significant differences leads to classifying all the relationships of Table 4 as “mixed,” or supportive of both within- and between-entities covariance (the results for the relationship between Performance and LMX are not considered “null” or indicative of no operative relationship, due to the statistically and practically significant withinentities correlation; cf. Dansereau et al., 1984). Finally, it should be noted that the within-entities results mirror the raw score findings in that: (1) moderator effects are not supported (adding the interaction term to the composite independent variable did not yield a significant increase in explained Performance variance), and (2) adding LMX to Delegation did yield a significant increased in explained Performance variance (see the “double dagger” [*I footnote to the fourth line entry in Table 4). Within and Between Correlation Components and Inferences

Table 5 summarizes the WABA I and II conclusions presented above, and also presents the results obtained from partitioning the raw score correlations into withinentities and between-entities components. The last column in Table 5 shows the overall or summary inference which seems most warranted based upon the evidence as a whole. As shown in Table 5, all of the between-entities correlation components are .OO or -.Ol, while the lowest within-entities component is .38. Additionally, by the A-test, all of the within- are significantly greater than their corresponding between-entities components. These results thus support concluding that the phenomena under investigation are best viewed as occurring within-entities (within work groups or supervisors). Again, these results show Performance to be associated with Delegation, with LMX, and with both together at the within-unit level of analysis. Within-entities

Moderation

Effects

Testing the four interpretable within-entities results for interactions between parts and entities obtained significant findings for Delegation (Ap = .05; F = 5.08; p < .05), LMX (Ati = .ll; F = 13.03; p < .Ol), the interaction of Delegation and LMX (AR* = .04; F= 4.57;~ < .05), and the composite Delegation plus LMX variable (AR’ = .06; F = 7.55; p < .Ol) as correlates of Performance (all df = 1,SO). Thus, these

Notes:

a ’ + *

(LMX) (Delegation X LMX) (bl[Delegationl + b2[LMXI)” (6llDeiegationl + b$LMXl + hJDelcgation

(Delegation)

Independent Variable

X LMXl)b Dyads

Dyads Dyads Dyads Dyads

WABA I Results

covariance)

Both

Both Both Both Both

WABA II Results

-.Ol

.oo .oo -.Ol -.Ol

Overall

Dyads

Dyads Dyads Dyads Dyads*

Inference

over the previous regression (without the last variable

.49+

.38+ .39+ .46+ .47+

Bet ween Component

and Overall Inferences

Within Component

Correlation Components,

Composite variable computed using the Step 2 partial regression coefficients of Table 2. Composite variable computed using the Step 3 partial regression coefficients of Table 2. Significantly greater than the between-entities component by the 15” A test. Showed a statistically significant (p 5 .Ol) increase in within-group covariance (but not in between-group added; see Table 4).

(Performance) (Performance) (Performance) (Performance) (Performance)

Dependent Variable

Table 5 Summary of WABA I and II Results, Within- and Between-entity

P

Multivariate

15

WABA

analyses help the researcher more properly interpret the WABA results reported above-by highlighting that the obtained effects are best interpreted as due to parts within entities (work units or supervisors) and that they are not due to parts alone. In summary, then, while all of the above findings do not support LMX as a moderator of Delegation-Performance relationships, they do support the importance of good leader-member exchange (LMX) and the fact that LMX is apparently a dyadic-level and within-work unit (supervisor) variable-as it should be to be consonant with LMX theory (Graen & Scandura, 1987).

CONCLUSION

AND DIRECTIONS

FOR FUTURE

RESEARCH

It is hoped that the example provided above will help clarify for interested individuals how hierarchical linear multiple regression and within-entity moderation analyses may be used to extend WABA for both multivariate analysis and interpretative purposes. However, before concluding, it seems useful to briefly discuss three additional concerns which may be raised about WABA, all of which have been discussed at some length by George and James (1993), and all of which warrant further investigation and discussion within the research community. George and James (1993) argue that the sine qua non of the existence of a group (and, hence, of being able to correctly aggregate data to the group level of analysis) is within-group agreement (and not that more variance exists between-groups than within-groups-i.e., that significant WABA I between-entity or within-entity results are not relevant for assessing the empirical appropriateness of aggregation). Furthermore, George and James (1993) correctly point out that range restriction across group scores may lead to erroneous WABA I conclusions (since this leads to the conclusion that groups are not present) and that this may frequently occur since groups are often nested within larger entities (e.g., divisions or organizations), thereby naturally tending to restrict variance in between-group scores. I agree with the George and James (1993) position about within-group agreement as being theoretically correct. Groups have been conceptualized in many ways, but virtually all small-group theorists emphasize that, to reasonably be called a “group,” any collection of people must share certain attributes, one of which is common perception of key elements in their environment (cf. Hare, 1976; Shaw, 1981). Thus, treating within-group agreement as supporting group-level analyses is, in fact, theoretically quite reasonable. On the other hand, it should be noted that if between-groups variance is small (WABA I), between-groups covariance is also going to be small (WABA II), and between-groups relationships cannot be empirically manifested. Thus, WABA as an overall data-analytic system may be seen to speak directly to the likely practical relevance of potential group-level attributes (i.e., their empirical usefulness as “predictors”), something which the assessment of within-group agreement alone cannot do (and something which cannot occur if between-groups variance is restricted). Another concern which George and James (1993) discuss with respect to current WABA practices is that individual- and group-level effects are often treated as being mutually exclusive in a given dataset, as well as the fact that it is not an unusual practice to label “mixed” effects as being indicative of “individual differences” (cf. Yammarino & Markham, 1992). Again, this author believes that George and James (1993) are correct

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but that this is not an inherent “flaw” of WABA as a data-analytic system (it is a shortcoming of how WABA results have often been interpreted). There is certainly theory and research which support phenomena as exerting effects at several levels of analysis, and reciprocal causation may also be present (cf. Kenny & LaVoie, 1985). Thus, exercising caution in interpreting WABA results is clearly essential. However, shortcomings in interpretation are not the same as shortcomings in the data-analytic system itself, and WABA analysts clearly need to consider and treat this issue with greater care in future theorization and research. A final objection which has been raised about WABA is that, if significant partsby-entity moderation effects are found (tested by Equations 8 and 9, presented above), r W, “is not interpretable as a meaningful reflection of the data” (George & James, 1993, p. 802). However, as noted above, ‘W, is a “pooled” or “overall” effect and it may therefore be interpreted as a summary statistic for overall effects. Thus, George and James (1993) are correct about taking or interpreting ‘W, too seriously, but performing the tests which were recommended earlier (Equations 8 and 9) should help the researcher better place any obtained within-entities effects in context. In conclusion, further research on and discussion about the above-mentioned concerns would be highly useful for future investigations applying WABA and for advancing our understanding of how multilevel effects might be investigated. Certainly, this is an important area for the field, and I hope that this article has added, at least somewhat, to our ability to use and understand within- and between-entities analysis. Acknowledgments: The author would like to gratefully acknowledge the constructive advice, comments, and suggestions of Fred Dansereau, Larry R. James, Linda L. Neider, and Francis J. Yammarino on earlier drafts of this article. Financial support from the School of Business Administration, University of Miami, is also gratefully acknowledged.

NOTES 1. A moderator is a variable which conditions relationships between one or more independent variables and one or more dependent variables. Mmoderated linear multiple regression analysis is typically employed to assess the presence of significant moderator effects between an independent variable and a moderator variable in “predicting” a dependent variable (Cohen & Cohen, 1983; Zedeck, 1971). In moderated regression, hierarchical procedures are employed and the independent variable is entered into the equation first. Then, the moderator variable is entered second. Last, a cross-product term is added to the regression to assess the unique variance contributed by the interaction of the independent and moderator variables. When significant interactions are found, the distributions of the “predictor” variables are frequently dichotomized into low and high groups (based upon median splits) and cell means calculated for the resulting 2 X 2 matrix to show the form of the interaction effect (Arnold, 1982, 1984). 2. It should be acknowledged here that Markham and McKee (1991) employed WABA to first determine the appropriateness of different levels of analysis in their study. Then, multiple regression was used with variables constructed so as to reflect the appropriate analytic orientation for their analyses. Thus, Markham and McKee (1991) appear to represent the first multivariate application to use WABA. However, multivariate WABA (per se) was not developed or employed.

Multivariate

17

WABA

3. Using conventional personal computer programs (e.g., Non&, 1993) for conducting WABA analyses, the author has found it computationally easier to first compute within-entity, between-entity, and raw scores and to then transform these scores into composites using approach (c) above. However, users of the DETECT program (Dansereau et al., 1986) will probably want to use approach (b) above (i.e., compute new composite independent variables and have the program execute its normal operating procedures). It should be noted, however, that the degrees of freedom for DETECT analyses have to be adjusted to take into account the additional independent variables used in forming the composite variables.

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