Centering Decisions in Hierarchical Linear Models: Implications for Research in Organizations

Journal of Management 1998, Vol. 24, No. 5, 623-641

Centering Decisions in Hierarchical Linear Models: Implications for Research in

Organizations David A. Hofmann

Texas A & M University

Mark B. Gavin Oklahoma State University

Organizational researchers are increasingly interested in modeling the multilevel nature of organizational data. Although most organizational researchers have chosen to investigate these models using traditional Ordinary Least Squares approaches, hierarchical linear models (i.e., random coefficient models) recently have been receiving increased attention. One of the key questions in using hierarchical linear models is how a researcher chooses to scale the Level-1 independent variables (e.g., raw metric, grand mean centering, group mean centering), because it directly influences the interpretation of both the level-1 and level-2 parameters. Several scaling options are reviewed and discussed in light of four paradigms of multilevel~cross-level research in organizational science: incremental (Le., group variables add incremental prediction to individual level outcomes over and above individual level predictors), mediational (i.e., the influence of group level variables on individual outcomes are mediated by individual perceptions), moderational (i.e., the relationship between two individual level variables is moderated by a group level variable), and separate (i.e., separate within group and between group models). The paper concludes with modeling recommendations for each of these paradigms and discusses the importance of matching the paradigm under which one is operating to the appropriate modeling strategy.

Organizations are inherently hierarchical. Individuals are nested in work groups, work groups are nested in departments, departments are nested in organizations, and organizations are nested in environments. Given this characteristic, a natural concern is how these level issues influence organizational research (e.g., Rous- seau, 1985). Even though recent theoretical discussions and empirical investiga-

Direct all correspondence to: David A. Hofmann, Department of Management, College of Business Administra- tion, Texas A & M University, College Station, Texas 77843; e-maih <[email protected]>.

Copyright © 1998 by JAI Press Inc. 0149-2063

623

624 D.A. HOFMANN AND M.B. GAVIN

tions have increased the attention given to level of analysis concerns (e.g., see Baratta & McManus, 1992; I-Iofmann & Stetzer, in press; House, Rousseau, & Thomas-Hunt, 1995; Klein, Dansereau, & Hall, 1994; Martocchio, 1994; Mathieu, 1991; Mathieu & Kohler, 1990; Mellor, Mathieu, & Swim, 1994; Thomas, Shankster, & Mathieu, 1994; Vancouver, Millsap, & Peters, 1994), there still remains confusion and controversy (e.g., George & James, 1993, 1994; Klein et al., 1994; Yammarino & Markham, 1992). Although much of the controversy surrounding level concerns has been focused on general problems and issues (e.g., aggregation decisions), our focus is on one particular issue; that is, how to appropriately model relationships between variables reflecting different levels of analysis using hierarchical linear models. Within this focus, we will be primarily concerned with decisions researchers make in scaling (i.e., centering) the level-1 independent variables and the implications of these decisions for parameter estimation and interpretation.

When theoretical questions involve variables at different levels of analysis, one is confronted with a cross-level model. Rousseau (1985: 14) defined cross- level models as those that specify "the effects phenomena at one level have on those of another." In organizational research, these models most often, though by no means exclusively, involve an outcome variable at the individual level and predictors at both an individual and higher level of analysis [see Klein et al.'s (1994) discussion of "mixed determinant" models]. 1

The purpose of the current paper is to review and discuss: (a) recent methodological developments in assessing cross-level models (i.e., hierarchical linear models or random coefficient models; Bryk & Raudenbush, 1992; Goldstein, 1995; Longford, 1993), (b) how decisions regarding the scaling of independent variables can influence the interpretation of these cross-level models, and (c) the implications of the various scaling options as they relate to four multilevel paradigms in organizational research.

Cross- level mode l s

Before considering the most recent methodological approaches to investigating cross-level models, it is important to review briefly the way in which these models have been historically investigated within the organizational sciences. The vast majority of organizational researchers have adopted Ordinary Least Squares regression approaches to predicting an outcome variable with both individual and group level variables (Mossholder & Bedeian, 1983). The regression equation would take the following form:

Y6 = b0 + blXij + b2Gj + eij (1)

where Yij is the individual level outcome variable, X i. is the individual score on a • :/

given individual level variable, and Gj is a group level variable with the value for the group assigned down to each group member. In equation 1, b 0 is the intercept, b I and b 2 are the regression coefficients, and eij is the residual. In this model, b 2 represents the influence of the group level variable on the individual outcome.

JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998

CENTERING DECISIONS 625

One can easily extend this equation to include an interaction term (e.g., Bedeian, Kemery, & Mossholder, 1989) which indicates whether the slope of Yij regressed onto Xij varies significantly across groups as a function of Gj (see Boyd & Iversen, 1979; Iversen, 1991). This regression equation would take the following form:

Yij = bo + blXl7 + b2Gj + b3XljGj + eij (2)

where b 3 represents the interaction effect. This interaction assesses the degree to which the relationship between Xij and Yij is moderated by Gj.

It is important to recognize, however, that in cross-level investigations such as those described above, individuals within the same group are all exposed to similar group stimuli. Therefore, they are likely to be more similar to one another than individuals in other groups. OLS regression techniques, however, assume that the random errors are independent, normally distributed, and have constant variance. As Bryk and Raudenbush (1992) noted, this assumption will likely be violated because the random error component within nested data will, in addition to an individual component, include group level random error which renders observations within groups dependent (i.e., since a portion of the random error is group random error which is constant across individuals within a given group). In addition, this group level random error is also likely to vary across groups, thereby violating the constant variance assumption (see Bryk & Raudenbush, 1992). In addition to violating this assumption, the assignment of group level variables down to the individual level results in statistical tests that are based on the number of individuals instead of the number of groups. Thus, the standard errors associated with the tests of the group-level variables may be underestimated (Bryk & Raudenbush, 1989; Tate & Wongbundhit, 1983).

Recent methodological advances have produced more efficient and more statistically appropriate approaches to cross-level models than the OLS techniques discussed above (i.e., random coefficient models; Bryk & Raudenbush, 1992; De Leeuw & Kreft, 1986; Goldstein, 1995; Longford, 1993; Mason, Wong, & Entwisle, 1983). Essentially, these models adopt a two-level approach to cross- level investigations. At level-I, a within group model is estimated separately for each group. In this analysis, an individual level outcome is regressed onto the individual level predictor(s). The parameter estimates from the first level (i.e., intercepts and slopes) are then used as outcome variables in the level-2 analysis in which they are modeled as a function of group level variables. For example, equation 1 and 2 above could be written from an intercepts- and slopes-as-outcomes perspective as follows (in keeping the nomenclature in Bryk & Raudenbush, 1992, [~ will be used instead of b when referring to hierarchical linear models):

Level-l : YU = P0j + Pl Cij + rij (3)

Level-2: [~0j = Y00 + Y01Gj + U0j (4)

~l j = T10 + ~llaj + U l j (5)



where YO' Xij' Gj, are defined as above; ~0j, and ~lj are level-1 intercepts and slopes, respectively, estimated separately for each group (as noted by the subscript j); ~/00 and 710 are the level-2 intercept terms, and 701 and ~'11 are the level-2 slopes relating Gj to the intercept and slope terms from the Level-1 equation, respectively, r i, , U 0. , and Ulj are the level-1 and level-2 residuals. Equation 4 represents the ma~n ef~t~ect of Gj on Y6 (i.e., the analog to b e in equation 1) and equation 5 represents the interaction of Gj and Xij (i.e., the analog to b 3 in equation 2).

Although the difference in estimation strategy between the OLS and hierarchical linear model approaches appears rather straightforward (i.e., OLS adopts a single level approach, whereas hierarchical linear models adopt a two-level approach), the differences run much deeper. As described by Bryk and Rauden- bush (1992), hierarchical linear models are random coefficient models in the sense that the level-1 parameters are allowed to vary across groups. Furthermore, the variance and covariance of the level-2 residuals, or the variance components, are also estimated (see Bryk & Raudenbush, 1992). This approach differs from the traditional OLS approach in which all of the regression parameters are fixed and level-2 variance components are not separable from the individual level residual. In addition, it should be mentioned that the HLM software (i.e., Bryk, Raudenbush, & Congdon, 1994) uses a maximum likelihood estimation of the variance components (i.e., the level-1 and level-2 residuals), generalized least squares estimates of the level-2 regression parameters, and can yield empirical Bayes estimates of the level-1 regression parameters (Bryk & Raudenbush, 1992; see also Raudenbush, 1988). Although these estimation differences are important to understand, the remaining discussion will rely more on a conceptual, as opposed to a statistical, understanding of hierarchical linear models.

Choosing a metric for level-1 predictors

Given that hierarchical linear models use the level-1 parameters as outcome variables in the level-2 analysis, the meaning and interpretation of these parameters becomes critical. Technically, as is consistent with the regression focus, the slope value represents the expected change in Yij given a unit increase in Xiy and the intercept term can be interpreted as the expected value of Yi' when Xij is zero (Cohen & Cohen, 1983). In the organizational sciences, however, having an intercept value equal to the expected value of Y6 when Xij is zero may not be particularly meaningful, because a zero value for Xij itself may not be meaningful. For example, what does it mean for an individual to have zero commitment, satisfaction, mood, or efficacy and, likewise, what does it mean for an organization to have zero structure, technology, formalization, or centralization? This begs the question of whether one can rescale the level-1 independent variables to render the intercept term more interpretable or meaningful.

Alternative Scaling Decisions Researchers within the multilevel modeling literature have addressed the

scaling of the level-1 predictors within the context of different "centering"



options. 2 For the purposes of this discussion, three different scaling options will be considered: (a) raw metric scaling where no centering occurs, and the level-1 predictors are used in their original metric, (b) grand mean centering where the grand mean of the level-1 predictor is subtracted from each level-1 case (i.e., Xij- X.. where X.. is the grand mean of Xij !, or !c) group mean centering where the relevant group mean of the level-1 predictor is subtracted from each case 0.e., Xij - Xj where Xj represents the mean for group j). Under these various scaling optaons, the intercept term takes on a different meaning. Specifically, raw metric scaling yields an intercept equal to the expected value of Y(/when Xuis zero. Grand mean centering yields an intercept equal to the expected value of Yi for an individual with an "average" level of X U (i.e., the expected value for ~ for a person with a score on X equal to the mean across all individuals in the sample). Group mean centering yields an intercept equal to the expected value of Y/j for an individual whose value on Xij is equal to their group's mean.

Although Bryk and Raudenbush (1992) mentioned that choices regarding the centering of Xij do have serious implications for: (a) the interpretation of the intercept term, (b) the variance in the intercept term across groups, and (c) the covariance of the intercept term with other parameters, the specific implications for research in the organizational sciences have not been explicitly addressed. Given the increasing interest in multilevel theories/models of organizational phenomena (e.g., Bryk & Raudenbush, 1992; Deadrick, Bennett, & Russell, 1997; Griffin, 1997; Hofmann, 1997; Kidwell, Mossholder, & Bennett, 1997; Vancouver, 1997; Vancouver et al., 1994) as well as the subtleties surrounding these three scaling options (Kreft, 1995), it is imperative that the implications of each scaling approach be addressed early in the application of hierarchical linear models in order to avoid misspecifications, misinterpretations, and misleading conclusions.

Finally, as we elaborate below, these scaling decisions have far greater implications for organizational researchers than merely the interpretation, variance, or covariance of the intercept term. Specifically, these scaling decisions can determine whether there is a match or a mismatch between the paradigm from which the researcher is operating and the paradigm implicitly represented in their data analysis.

Scaling Options: Statistical and Conceptual Differences From a statistical perspective, there have been recent debates (e.g., Long-

ford, 1989; Plewis, 1989; Raudenbush, 1989a, 1989b) as well as statistical demonstrations regarding the effects scaling decisions have on the estimation of random coefficient models (Kreft, De Leeuw, & Aiken, 1995). In the most recent treatment of centering issues, Kreft et al. (1995; see also Kreft, 1995) investigated the equivalence of different centering options (i.e., raw metric, grand mean, and group mean) for several different random coefficient models where equivalence was defined as two models producing the same expectations and dispersions for the outcome variable.

In summary, Kreft et al, (1995) concluded that grand mean centering and raw metric approaches produced equivalent models. Kreft et al. (1995: 10), however, noted that even though these two models were equivalent, grand mean centered



models did provide a "computational advantage" (see also Raudenbush, 1989a, 1989b) because, in most cases, grand mean centering reduced the correlation between the intercept and slope estimates across groups. This reduction of the covariation between the random intercepts and slopes can help to alleviate potential level-2 estimation problems due to multicollinearity (see Cronbach, 1987 for a related discussion for single level moderated regression). Group mean centering will, however, in most all cases not be equivalent to either raw metric or grand mean centering. In other words, even though all of the comparisons between grand mean and raw metric approaches produced equivalent models, in all but a few special circumstances, group mean centering did not produce an equivalent model to either the grand mean or raw metric approach. Despite these differences, Kreft et al. (1995: 17) concluded that "there is no statistically correct choice" among the three models, but rather, that the choice between grand mean (which is preferred over raw metric approaches) and group mean centering "must be deter- mined by theory."

Although the grand mean/raw metric and group mean centering options have been shown to be statistically non-equivalent, it should also be pointed out that these models answer inherently different conceptual and theoretical questions. As Bryk and Raudenbush (1992) discussed, when group mean centering is adopted, the level-1 intercept variance is equal to the between group variance in the outcome measure. As a result, the level-2 regression coefficients, under group mean centering, simply represent the group level relationship between the level-2 predictor and the outcome variable of interest (i.e., the relationship between the level-2 predictor, Xj , and I~). Alternatively, when grand mean centering is adopted, the variance in the intercept term represents the between group variance in the outcome measure adjusted for the level-1 predictor(s). Therefore, with this approach, the level-2 regression coefficients represent the group level relationship between the level-2 predictor and the outcome variable less the influence of the level-1 predictor(s).

Scaling Options: An illustration 3 In order to further demonstrate this conceptual distinction, an illustrative

data set was created with four, level-1 predictors (i.e., Aij, Bij, Cij, and Dij) and one group level predictor (i.e., Gj). The data set was created by generating 10 individual level members for 15 groups (i.e., 15 groups of size 10). The group level variable, Gj, was created as an equally weighted function of the group means, Aj and Bj [e.g., a j = .9(A j) + .9(Bj) + random error]. Furthermore, the between groups variance m the outcome variable, I~, was an equally weighted function of Aj and Bj [e.g., Yj = .6(Aj) + .6(Bj) + random error]. Thus, at the group level, we created a data set where Gj was related to Yj only through its association with Aj and Bj. In other words, the association of Gj and I~ was fully mediated by Aj and Bj, such that if Aj and Bj were included in the model, Gj would no longer be associated with I~. The within group variance in Y, was associated with the within group variance A, B, C, and D [e.g., Yij-j = .8(Aij - Aj) + .4(Bij - Bj) -.2(C0. - Cj) -.2(Dij- D j) + random error].



Table 1. Simulated Data Results for Group Mean and Grand Mean Centering

Variables included in:

Level-I Model Level-2 Model Parameter estimate (Gj ) Grand mean centering

Null Gj .445"*

Aij Gj .209**

Aij, Bij Gj .064 aij, Bij, Cij Gj .007 Aij, Bi) Cij, Dij Gj -.028

Group mean centering

Null Gj .445**

aij-a j Gj .445**

Aij-Aj, Bij-B j Gj .445**

Aij-A ~ Bij-Bj,%-C j Gj .445**

Aij-A j, Bo-B j, Cij-Cj, Dij-D j Gj .445**

Group mean centering with A,B,C,D means in level-2 model

Null Gj .445**

aij-A j, Gj, Aj .300**

aij-A j, Bij-B j, Gj, Aj, Bj .132

Ao-A j, Bij-B j, Cij-C j Gj, A)Bj, Cj .119

ao-a j, Bij-B j, Cq-Cj, Dij-D j Gj, A~B~ Cj, Dj .099

Notes: Data set consisted of 15 groups of 10. The between groups slope relating Gj to Yj was equal to .445. **p < .01

Grand mean centering. As noted above, under grand mean centering, the variance in the intercept term ([~0j) represents the between group variance in the outcome variable adjusted for the level-1 variables (i.e., after partialling out or controlling for the level-1 variables). The results presented in the top of Table 1 clearly demonstrate that after grand mean centered A and B are entered in the level-1 model, the effect of Gj is no longer significant. This is consistent with the interpretation of the intercept term, under grand mean centering, as the between group variance in the outcome variable from which the level-1 variables have been partiaUed. It is also consistent with the nature of the demonstration data, in the sense that Gj was related to Y only through its association with Aj and Bj (i.e., after controlling for Aj and Bj, the effect of Gj should no longer be significant). Therefore, this illustration highlights the fact that the between group variance in A and B is partialled out of the outcome variable when grand mean centered A and B are included as level-1 predictors.

Group mean centering. In group mean centered models, the intercept variance simply represents the between group variance in the outcome measure. The middle section of Table 1 clearly illustrates that the relationship between Gj and



the intercept term remains unaltered even as A and B are included in the level-1 model. Specifically, the parameter estimate for Gj remains constant at .445 which represents the between group relationship between Yj and Gj (i.e., the between group slope). It can be seen that the addition of group mean centered A, and group mean centered B does not alter the magnitude or significance of the relationship between Gj and the intercept term (where the magnitude of this relationship is equal to the weighted between groups regression between G i and Yj). Therefore, these results illustrate that under group mean centering, no between group variance is partialled out of I1//and, therefore, all of the level-2 intercept relationships simply represent the between group relationship between Yj.. and the group level variable (i.e., the intercept variance is the between group variance in Y/j).

Group mean centering with means in level-2. Recall that the demonstration data set was generated such that if the between group variance in A and B was included in the model, the association between Gj and Y would no longer be significant. Given that Gj was significant in all o f the models estimated using group mean centering, it is clear that the between group variance in A and B was excluded from the model. One way to remedy this situation, when using group mean centering, is to reintroduce the between group variance of A and B back into the level-2 intercept model; that is, include group means A and B (Aj and B.) as

• J level-2 predictors of the intercept term. The bottom part of Table 1 dlustrates that when the between group variance in A and B is reintroduced back into the group mean centered model, the effect of Gj is no longer significant.

Cross-level interactions. To this point, only the interpretation of the intercept term has been discussed and the demonstration assumed a "main effects" model, in the sense that only the level-2 intercept model was of interest. Cross- level interaction models, however, specify level-2 predictors of the variance in level-1 slopes (~lj)" In this case, the researcher is interested in determining whether the within group relationship (i.e., the within group slope) between the individual level predictor and outcome varies as a function of the between group predictors (i.e., cross-level moderation; equation 5). It is critical, therefore, that one obtain an unbiased estimate of the within group slope.

As pointed out by Bryk and Raudenbush (1992), under grand mean centering one does not always obtain a clear estimate of the within group slope• Take the following hierarchical linear model:

Level-l: Y/j = ~0j + ~ l j ( X i j - X . . ) + rij ( 6 )

Level-2: ~10j = Y00 + U0j (7)

~ l j = 7 1 0 ( 8 )

where 130j and ~ll~ are the level-1 regression parameters estimated for each group, Y00 is the grand ~aean of 13 .., and Y10 is the grand mean of the 131: parameter (i e , 0j j • " since there are no level-2 predictors, the intercept terms of the level-2 equations are equal to the grand mean of the outcome variables). Xij - X.. signifies that X/j has been centered around its grand mean.



As noted by Bryk and Raudenbush, in this model )tl0 does not provide an unbiased estimate of the pooled within group slope, but rather a mix of the within group slope and between group slope. If, however, one includes Xj in the level-2 intercept model (i.e., equation 7 becomes [~0i = Y00 + 'YolXj + U0j), then ~tl0 can be interpreted as the pooled within group sloiae. This occurs because when Xj is entered in the level-2 equation it partials out the between group variance in Xij. Thus, this renders the ~/10 parameter an unbiased estimate of the pooled within group slope (see Bryk & Raudenbush, 1992: Table 5.9, 5.10). Thus, under grand mean centering, one can obtain an unbiased estimate of the within group slope by introducing X. into the model as a predictor of the intercept term. Group mean J . centering, alternatively, always produces an unbiased estimate of the within group slope (Raudenbush, 1989b).

When moving to models that include cross-level interactions, the differences between grand mean and group mean centering become even more critical. Take, for example, the following model:

Level-l: Y/j = 130j + ~lj(Xij - X..) + eij (9)

Level-2: [~0j = ~t00 + 'YolGj + u (10)

131j = ~'1o + ~'11Gi + u (11)

where each of the level-2 models specify a group level predictor of ~0j and [31j, and where the level-1 model includes Xij centered around its grand mean. Th-e relevant part of this model is the cross-level interaction (i.e., equation 11). Raudenbush (1989b) noted that equation 11 confounds the cross-level interaction [i.e., (Xq - Xj)Gj; i.e., the moderation of the within group slope by the group level variable] and the between group interaction (i.e., X,Gj; i.e., the group level interaction between Xj and Gj). This occurs because, as ~oted above, the [~lj parameter is a mix of the within group and between group slope. In order to separate out the cross-level and between group interaction, the following model could be estimated using group mean centering (Raudenbush, 1989b):

L e v e l - l : Yij = [~oj + ~l j (X i j -X j ) + rij ( 1 2 )

Level-2: 130j = ~'oo + ~'01Xj + "I02Gj + ~'o3 (XjGj) + u (13)

~ l j = ~tl0 + ~[IIGj + u (14)

In this model, the cross-level interaction is represented by the 711 parameter, and the between group interaction is represented by the ~(03 parameter.

In order to illustrate this distinction, two additional data sets were generated, one containing a cross-level interaction and one containing a between group interaction. Specifically, each of these two data sets consisted of 50 groups of 20 observations (i.e., individuals) within each group and three variables of interest: the outcome variable (Yij), the level-1 predictor (Xij), and the group level predictor



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i ' ' 1 ' ~

i

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(Gj). For the cross-level interaction data set, the outcome variable, Y(/, was a function of the interaction between the slope of (Xij - X.) and the group level variable, • :/ Gj (1 e , the slope between X r - X. and Yr was moderated by G-) For the between- . : . ~ J .~ . j " . . group interaction data, the outcome variable, Yij, was a function of the interaction between Xj and Gj (i.e., the slope between Xj and I~ was moderated by Gj).

Table 2 presents the results comparing the estimation of equation 9, 10, and 11 with the results from estimating equations 12, 13, and 14. Of particular importance, the results show that when there was either a cross-level interaction or a between group interaction, equation 11 yielded a significant finding for a cross- level interaction (i.e., ~/11 is significant). This occurred despite the fact that one of the two data sets did not contain a cross-level interaction, but a between group interaction. As can be seen in Table 2, when equations 12-14 are estimated, the cross-level interaction and between group interaction are separated out and appropriately identified. More specifically, the data with a cross level interaction produced a significant ~/11 parameter and a non-significant ~03 parameter. Alterna- tively, the data with a between group interaction produced a significant "/03 parameter and a non-significant ~11 parameter. Although the exact circumstances presented in Table 2 may only rarely occur, the point of this demonstration is that only group mean centering enables the researcher to differentiate between cross- level and between group interactions.

I m p l i c a t i o n s o f C e n t e r i n g D e c i s i o n s fo r A l t e r n a t i v e C r o s s - L e v e l P a r a d i g m s

Thus far, we have discussed the differences between various centering options both statistically and conceptually. Given the discussion above, the general conclusion is that group mean centering approaches to multilevel data are different from grand mean or raw metric approaches--both statistically and conceptually. As Kreft et al. (1995) have pointed out, there is no statistical prefer- ence between these various centering options, but rather, the choice needs to be made based on theoretical and conceptual considerations. From our viewpoint, there has emerged over the last several years four dominant paradigms from which researchers have investigated relationships between variables that span multiple levels of analysis (i.e., cross-level paradigms). For the purposes of this discussion, we will refer to these different paradigms as incremental, mediational, moderational, and separate models.

It should be noted here that these four paradigms are merely used as descrip- tions of the way in which cross-level research has been conducted within the organizational sciences. It is not our purpose to specify the adequacy of the theory underlying the examples cited for each of the following paradigms. Instead, our focus is merely on the different types of cross-level models that have been investigated within the organizational sciences and how the centering decisions discussed above relate to these different models the theoretical justification for these models notwithstanding. Although given recent discussions regarding the general lack of theoretical acuity within the organizational sciences (see e.g., DiMaggio, 1995; Pfeffer, 1993; Sutton & Staw, 1995; Weick, 1995), it is certainly our position that prior to engaging sophisticated analytical techniques, such as



hierarchical linear models, it is critical to have a well developed underlying theoretical model that warrants the investigation of the complex relationships investigated by these methods.

Incremental The incremental paradigm simply states that group level variables act as

main effects in the prediction of individual-level outcomes. Typically, cross-level researchers adopting this paradigm investigate the influence of group level variables on individual-level outcomes after controlling for various individual-level predictors. In essence, the researcher is interested in whether the group level variable provides incremental prediction of an individual-level outcome over and above individual-level predictors.

There are a number of examples in the research literature that have adopted this approach. For example, several investigations of employee absence/lateness behavior have adopted this paradigm (e.g., Blau, 1995; Martocchio, 1994; Mathieu & Kohler, 1990). In each of these studies, the researchers were interested in the degree to which group level variables (i.e., absence climate) added incremental prediction over and above individual-level predictors of absence. Along similar lines, Ostroff (1992) investigated the joint prediction of organizational climate and personal orientations on work related attitudes and behaviors, and Hofmann and Stetzer (1996) investigated the extent to which safety climate and group process incrementally predicted unsafe behavior after controlling for role overload.

Subsumed under this paradigm would be traditional contextual models (see Firebaugh, 1980) where researchers are interested in investigating whether the aggregate of the individual level predictor incrementally predicts the individual level outcome (i.e., does Xj incrementally predict Yij after controlling for X~/; e.g., Baratta & McManus, 1992). When using fixed effects techniques (e.g., OLS techniques), Firebaugh (1980) noted that after Xj is entered into the regression equation, b I is equal to the pooled within group slope and b 2 is equal to the between group slope minus the within group slope (bb/n - bw/n). A significant contextual effect occurs if the between group slope significantly differs from the within group slope (i.e., if b 2 is significant). The underlying rationale for defining a significant contextual effect only when bb/n and bw/n differ significantly is related to Alwin's (1976) proof which revealed that when there is no group level effect (e.g., under random grouping situations), bb/n is equal to bw/n. Thus, when there is no group effect, one would expect bb/n to be equal to bw/n. Therefore, when these two coefficients do differ significantly, one concludes that there is a meaningful "contextual effect," thereby indicating that both the individual and group level variables are necessary to fully describe the relationship of interest.

The incremental perspective suggests that group level variables will add incremental prediction to an individual outcome after controlling for individual- level predictors (no variance in the within group slopes is expected). Thus, in order for this model to be correctly tested, one needs to adequately control for the influence of individual-level variables. As illustrated in Table 1, grand mean centering would appropriately control for the level-1 variables. Group mean



centering would not adequately control for these level-1 effects unless the means are added back in the level-2 intercept model.

In the special case of contextual models (where the group level predictor is the aggregate of the individual-level predictor), either grand mean or group mean centering options could be used. It should be recalled, however, that a contextual effect occurs when the between group slope is significantly different from the within group slope. Under grand mean centering, the statistical test associated with the aggregate variable is a test of the difference between the between and within group slope (i.e., the significance test for the level-2 slope parameter associated Xj predicting the intercept; see Bryk & Raudenbush, 1992). If group mean centering is used, the test for a contextual effect takes a slightly different form. More specifically, Raudenbush (1989a) suggested an alternative contextual model where a within group centered model is estimated such as the following:

Level-l: Yij = ~oj + ~llj (Xij" Xj) + rij (15)

Level-2: ~10j = Y00 + Y01Xj + u (16)

~ l j = I l l0 + U (17)

Given the Level-1 model has been group mean centered (i.e., Xij - Xj), the variance of the Level-2 outcome measure ~0j is equal to the between group variance in I16. Therefore, the Level-2 parameter Y01 is equal to the between group regression of Yi onto X i (i.e., [~b/n), and the Level-2 parameter Y10 represents the within group regressioffof Yii onto X 6 - Xj pooled across groups (i.e., pooled [~w/n)- A significant contextual effect would occur if Y01 differs significantly from qfi0. One can test the difference between these two parameters in the HLM software pack- age by specifying a multiparameter contrast effect (see Bryk et al., 1994).

Mediat ional The mediational paradigm proposes that group-level variables influence

individual behaviors and attitudes only indirectly through other mediating mechanisms. Probably the most prevalent example of this paradigm is research and theory surrounding psychological climate (see James, James, & Ashe, 1990). This theory proposes that objective contextual factors influence individual outcome variables through the meanings that individuals attach to these contextual variables. For example, the level of centralization in a given work unit might influence an individual's job satisfaction through the mediating variable of perceived autonomy. Similarly, the technology of a work unit might influence individual behavior via its influence on perceived job complexity or perceptions of role requirements (e.g., role overload, role ambiguity). Other non-psychological climate examples include investigations of job characteristics mediating the relationship between structural relationships and individual level outcomes (e.g., Brass, 1981; James & Jones, 1976; Oldham & Hackman, 1981; Rousseau, 1978), role strain as a mediator of the unit cohesion - satisfaction relationship (Mathieu, 1991), self-efficacy and affective reactions as mediators of the relationship



between aggregate situational constraints and performance (Mathieu, Martineau, & Tannenbaum, 1993), and job affect as a mediator of the relationship between unit technology and both job perceptions and intentions to quit (Hulin & Roznowski, 1985).

Within the psychological climate literature, James and colleagues (e.g., James et al., 1990) have noted that these individual psychological perceptions (i.e., the mediating variables) can consist of "shared" perceptions. In other words, given the influence of, for example, social information processing mechanisms (Salancik & Pfeffer, 1978), individuals within the same group may develop similar perceptions of and attach similar meanings to the group-level variable. In situations where these perceptions and/or meanings are sufficiently shared, James and colleagues (e.g., James et al., 1990) suggested that one can use aggregated individual perceptions to describe the context in psychologically meaningful terms. Whether one uses individual versus aggregated mediators, however, has no bear- ing on the underlying paradigm (or model) under investigation. Specifically, the paradigm is still one in which individual perceptions (i.e., either individual or "shared") are proposed to mediate the relationship between objective contextual variables and individual reactions.

The mediational paradigm also specifies controlling for individual level variables and a main effects model (i.e., within group slopes are assumed to not vary meaningfully across groups). Therefore, either grand mean centering or group mean centering with the means reintroduced into the level-2 intercept model would provide an appropriate test of this model. From a psychological climate perspective, it may also be the case that these individual perceptions are "shared" and, therefore, can be aggregated to the group level of analysis. This does not present a problem in hierarchical linear models. In other words, the researcher is still interested in assessing whether individual perceptions mediate the relationship between objective organizational characteristics and individual outcomes. Whether the individual perceptions are entered in the level-1 model, the level-2 model, or both makes no difference as long as they are accounted for prior to investigating the influence of the group level variable.

Moderational

The moderational paradigm proposes that group level variables moderate the relationship between two individual-level variables (see Bedeian et al., 1989). Research from this paradigm could include investigations of how the organizational context can influence the magnitude of the relationship between two individual-level variables. For example, the influence of mood on helping behavior (e.g., George, 1991) could be moderated by the proximity of coworkers. Although somewhat intuitive, it is easy to envision that the relationship between mood and helping behavior is likely to be non-existent in situations where coworkers are in distal locations. Alternatively, if coworkers are in close proximity, this relationship is likely to be much stronger.

Along similar lines, the situational constraints literature (e.g., Peters & O'Connor, 1980) would reside within this paradigm. Peters and O'Connor (1980) suggested that a number of situational constraints influenced the relationship



among individual differences and work outcomes. In other words, the relationship between individual differences and work outcomes (i.e., a relationship among two individual-level variables) is moderated by a situational (i.e., contextual) variable. In another example, Hofmann and Griffin (1992) hypothesized that the average salary of a school district would moderate the relationship between job satisfaction and turnover intentions. The results demonstrated significant cross-level moderation such that the relationship between satisfaction and turnover intentions was stronger in districts with, on average, lower salaries. In other words, satisfaction was a much better predictor of turnover intentions when coupled with low average salaries. In wealthier districts, the relationship between satisfaction and turnover was attenuated. Along another line of inquiry, Mellor, Mathieu, and Swim (1994) recently investigated the potential moderation of union structure on the relationship between gender and union commitment. In all of these cases, the research questions involved an individual-level relationship that was moderated by a group level variable; that is, the magnitude of the within group relationship depended upon the level of the group variable.

The moderational paradigm specifically models variance in the slope parameters across groups. As noted above (Table 2), grand mean centering investigations can sometimes confound a cross-level interaction with a between group interaction. If one is interested in separating out these two different types of interactions, then group mean centering would be the appropriate choice (see Rauden- bush, 1989b).

Separate Although this paradigm has not been discussed much in organizational

science, Cronbach (1976) gave it an extended treatment. In this paradigm, separate structural models are proposed for both the within-group and between-group components of the outcome variable. Cronbach (1976) discussed the influence of student and school effects on student achievement. For example, the within- school structural model might include student characteristics (e.g., motivation, ability) while the between-school structural model might include school characteristics (e.g., socio-economic-status of the surrounding community, average funding per student). With regard to the outcome measure, the within group model would be investigating within school differences in student achievement, whereas the between group model would investigate between school differences in student achievement.

Other literature relevant here would be models that specify a within group comparison process where there may also be a main effect of the context. Within the organizational sciences, these paradigms have often been referred to as "frog pond" (e.g., Firebaugh, 1980), heterogeneity (Klein et al., 1994), or parts effects (Dansereau, Alutto & Yammarino, 1984). ~ Although he did not find support for a frog pond effect, Markham (1988) did discuss the pay-for- performance relationship from this viewpoint. In this case, it would not be so much individuals' actual performance level, but rather, their performance relative to other group members that determines their raise. Related to this notion, Hulin (1966) and Blood and Hulin (1967) discussed the influence of the



surrounding community on individuals' satisfaction. Specifically, Hulin (1966) found that pay satisfaction was negatively related to the prosperity of the community. In other words, in a poor community a middle level worker is better off than if he/she was in a wealthy community. The separate paradigm would simply take this approach and extend it to include a main effect for the community (i.e., on average, individuals in wealthy communities are more satisfied). In both of these cases, the group context is important in the sense that it defines one's relative position via a comparison to everyone else in the group. Within the frog pond approach, it is this relative difference within groups that is important (i.e., how do individuals compare to their colleagues).

Although not involving comparison processes per se, there may be other situations within which one would expect different relationships to emerge within as opposed to between units. For example, Lincoln and Zeitz (1980) noted that within and between organizational relationships, once separated, can produce somewhat different conclusions. They found, for example, that within organizations, administrative status was positively related to the involvement in decision making. At the organizational level, however, organizations with large administrative functions were more centralized in their decision making. Thus, the organizational effect was negative.

Under the separate paradigm, the researcher is interested in estimating separate within and between group structural models. Thus, the variance in the outcome measure needs to be partitioned into its within and between components. Only group mean centering will provide the appropriate within and between group partitioning of the outcome variance and allow separate structural models for each component of the variance.

Conclusion

From the discussion and illustrations above, several conclusions can be prof- fered. First, for random regression models, contextual models, and random slope models, raw metric and grand mean centering options provide equivalent models (see Kreft et al., 1995). This notwithstanding, Kreft et al. (1995) recommended the use of grand mean centering instead of raw metric approaches, because it usually results in a reduction of the covariance between the intercepts and slopes, thereby reducing potential problems associated with multicollinearity (see Cron- bach, 1987 for a related discussion). Second, in most all cases, group mean centering will produce models that are not equivalent to either raw metric or grand mean centering approaches. Third, all three centering options are statistically appropriate despite the fact that these various centering options are not equivalent (Kreft et al., 1995). Finally, it appears that the choice of centering options must be a function of the conceptual paradigm and research question under investigation.

The overriding theme of the current paper is that researchers should be aware, both conceptually and theoretically, of the paradigm from which they are operating, because these paradigms have implications for the choice of centering. In addition, it is important to point out that certain centering options can produce a mismatch between the paradigm from which the researcher is operating and the



implicit paradigm operationalized analytically. Thus, centering options should be chosen carefully and thoughtfully, with a view less towards the statistical differences and more towards the conceptual questions under investigation.

Notes

1. Although the discussion within the current context will primarily refer to "individual" and "group" levels of analysis, hierarchical linear models (i.e., random coefficient models) and the implications of the discussion herein are equally applicable to a large variety of situations where hierarchical units are under investigation (see Hofmann, 1997, for additional organizationally related topics where these models would be of interest). For the sake of clarity and consistency with other treatments of levels issues (e.g., Klein et al), however, we will refer to "individual" and "group" levels of analysis.

2. There has been much discussion regarding the influence of various scaling, or centering, options and the interpretation of single level regression and moderated regression (e.g., Busemeyer & Jones, 1983; Cronbach, 1987). Although the centering options described herein will not alter the overall fit of the model within a given level-1 unit (i.e., R2's will be identical), they do influence the interpretation or the meaning of the parameter estimates. It is these changed meanings, particularly in the intercept term, which is the primary focus through- out our demonstrations.

3. The data used for all demonstrations (i.e., Table 1, and 2) are available from either author via regular mall or e-mail (<[email protected]>; <[email protected]>).

4. Bryk and Raudenbush (1989) discussed frog pond effects as cross-level moderators. We, however, feel that the organizational literature has largely discussed frog pond effects investigating the pooled within group relationship where this relationship does not vary meaningfully across groups. One could easily extend the frog pond model to hypothesize that it is one's relative position that counts and the relationship between one's relative position and the outcome variable differs across groups. In this case, the frog pond model can be extended to include a cross-level moderator in keeping with the approach described above (see Bryk & Raudenbush, 1989). In either case, our centering recommendations would be the same.

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Centering Decisions in Hierarchical Linear Models: Implications for Research in Organizations