JournalofManagement
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,
w o r k groups are nested in departments, departments are nested in organizations,
and organizations are nested in environments. Given this characteristic, a natural
concern is h o w these level issues influence organizational research (e.g., Rousseau, 1985). Even though recent theoretical discussions and empirical investiga-
Direct all correspondenceto: DavidA. Hofmann,Departmentof Management, Collegeof Business Administration, Texas A & M University, CollegeStation, Texas 77843;e-maih <dhofmann@tamu.edu>.
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 crosslevel 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 m o d e 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 crosslevel 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)
~llaj + U l j
(5)
~ l j = T10 +
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
626
D.A. HOFMANN AND M.B. GAVIN
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~ectof 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 Raudenbush (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"
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
CENTERING DECISIONS
627
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., XijX.. 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 XU (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., Longford, 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
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
628
D.A. HOFMANN AND M.B. GAVIN
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 determined 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(Aj) + .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(DijDj) + random error].
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
CENTERING DECISIONS
629
Table 1. Simulated Data Results for Group Mean and Grand Mean Centering
Variables included in:
Level-I Model
Level-2 Model
Parameter estimate
(Gj )
G r a n d m e a n centering
Null
Aij
Aij, Bij
aij, Bij, Cij
Aij, Bi) Cij,Dij
Gj
Gj
Gj
Gj
Gj
.445"*
.209**
.064
.007
-.028
G r o u p mean centering
Null
aij-a j
Aij-Aj, Bij-Bj
Aij-A~ Bij-Bj,%-C j
Aij-Aj, Bo-Bj, Cij-Cj,Dij-Dj
Gj
Gj
Gj
Gj
Gj
.445**
.445**
.445**
.445**
.445**
G r o u p mean centering with A,B,C,D means in level-2 model
Null
aij-Aj,
aij-Aj, Bij-Bj,
Ao-Aj, Bij-Bj, Cij-Cj
ao-a j, Bij-Bj, Cq-Cj,Dij-Dj
Gj
Gj,Aj
Gj,Aj, Bj
Gj,A)Bj, Cj
Gj,A~B~ Cj,Dj
.445**
.300**
.132
.119
.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
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
630
D.A. HOFMANN AND M.B. GAVIN
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 Gi 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. Crosslevel 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 =
Level-2:
~10j = Y00 + U0j
(7)
~lj=710
(8)
~0j + ~ l j ( X i j
- X . . ) + rij
(6)
where 130jand ~ll~are the level-1 regression parameters estimated for each group,
Y00 is the grand ~aean of 130j.., and Y10 is the grand mean of the 131:
j parameter (i •e ,"
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.
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
CENTERING DECISIONS
631
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.J into. the model as a predictor of the intercept term. Group mean
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
Level-2:
[~0j = ~t00+ 'YolGj + u
(10)
131j = ~'1o + ~'11Gi + u
(11)
(9)
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):
Level-l:
Yij =
Level-2:
130j = ~'oo + ~'01Xj + "I02Gj + ~'o3 (XjGj) + u
[~oj +
~ l j = ~tl0 +
~lj(Xij-Xj) + rij
~[IIGj + u
(12)
(13)
(14)
In this model, the cross-level interaction is represented by the 711 parameter, and
the between group interaction is represented by the ~(03parameter.
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
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
632
D.A. HOFMANN AND M.B. GAVIN
i
'=
v
e~
i ' ' 1 ' ~
i"
"
L,
i
-d
~J
"I2tl
+
e4
r~
~.-~+
+~
+
!
+
+
÷
+
÷
÷
II
II
If
II
i
i
i
÷
÷
÷
+
II
II
a~
JOURNAL OF MANAGEMENT, VOL. 24, NO. 5, 1998
°
i'
i"
~
CENTERING DECISIONS
633
(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 crosslevel 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. Alternatively, 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 crosslevel and between group interactions.
Implications of Centering Decisions for Alternative Cross-Level Paradigms
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 preference 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 descriptions 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
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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 individuallevel 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
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635
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 = Ill0 + 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 package by specifying a multiparameter contrast effect (see Bryk et al., 1994).
Mediational
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
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D.A. HOFMANN AND M.B. GAVIN
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 bearing 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
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637
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 Raudenbush, 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 withinschool 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-forperformance 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
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D.A. HOFMANN AND M.B. GAVIN
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 proffered. 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 Cronbach, 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
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639
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 throughout 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 (<hofmann@tamu.edu>; <mgavin@okway.okstate.edu>).
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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