she may think the others in the group are not leaders. This correlation, which has been called generalized reciprocity, is defined only if there is both perceiver and target variance. The second source involves a correlation between two relationship effects from the same dyad. If Jack especially thinks that Jill is a leader (i.e., more than Jack thinks others are leaders and more than others think Jill is a leader), then does Jill especially think that Jack is a leader (i.e., more than Jill thinks others are leaders and more than others think Jack is a leader)? This correlation, which has been called the dyadic reciprocity correlation, could be negative – for instance, if the group is very hierarchical. Later in this chapter, we discuss the correlations between two variables measured in a round-robin design.
The estimation of the SRM is complicated because it contains five random variables and two covariances. We consider the rather complicated statistical details in the estimation of the SRM variances and covariances in the last section of this chapter.
SOURCES OF VARIANCE AND COVARIANCE
Dabbs and Ruback (1984) studied 20 five-person, same-gender groups who were asked to get acquainted.
Dabbs, Ruback, and Evans (1987) studied 20 five-person, mixed-gender groups who were asked to discuss what they would bring from a list of items to a desert island.
Lord et al. (1980) studied 24 four-person, mixed-gender groups who worked on four problem-solving tasks.
Malloy and Janowski (1992)studied 10 six- to eight-person, mixed-gender groups.
Montgomery (1986) studied 45 four- and five-person, mixed-gender groups. Groups met for three sessions, and the last two ratings of leadership are used as two measures of leadership.
Ruback, Dabbs, and Hopper (1984)studied 20 five-person, mixed-gender groups who were asked how they could increase tourism for a community.
Across the seven studies, there were 820 participants who were placed in 180 groups.
Variance Components
The seven studies used multiple measures of leadership, so we were able to examine the extent to which the variances remained stable across these multiple measures. This allowed us not only to separate error from relationship variance, but also to differentiate four sources of ‘‘error.’’ This process yielded eight components that gave rise to the eight SRM sources of variance:
Stable group: differences among groups in how much leadership is perceived that is consistent across measures.
Unstable group: differences among groups in how much leadership is perceived that is inconsistent across measures.
Stable perceiver: differences among individuals in how much leadership they perceive in others in general that is consistent across measures.
Unstable perceiver: differences among individuals in how much leadership they perceive in others in general that is inconsistent across measures.
Stable target: differences among individuals in how much each individual is perceived as a leader that is consistent across measures.
Unstable target: differences among individuals in how much each individual is perceived as a leader that is inconsistent across measures.
Stable relationship: disagreement about the extent to which someone is a leader that is consistent across measures.
Unstable relationship: disagreement about the extent to which someone is a leader that is inconsistent across measures.
For instance, stable target variance implies that some persons are seen as leaders and others as followers across all the measures, whereas unstable target variance implies that someone who is seen as leader on one measure is not seen as a leader on the other measures.
In past SRM studies, the group variance, whether stable or unstable, has been ignored, and all remaining unstable variances have been added together and treated collectively as error variance. However, unstable relationship variance comes closest to what most investigators would treat as pure error variance. Thus, the ability to separate stable and unstable variance across measures of leadership provides a more detailed picture of the variance partitioning. If all indicators measured the same construct and were equally reliable, then ideally no unstable perceiver and target variance would exist.
Under normal circumstances, then, we would hope to have small amount of unstable perceiver and target variance.
Results
To help us better understand the meaningfulness of each component, we have included its reliability (Bonito & Kenny, 2007) assuming a six-person group with three measures of leadership – fairly typical values for the research that we reviewed. Reliability represents how much ‘‘true’’ variance is present in a measurement of a given component.
Variances
To determine whether the group has an effect on judgments of leadership, we estimated the random variance due to the group. When we examined the data from several studies, we did not find any evidence for group variance.
In fact, we often obtained estimates of negative variance – a result consistent with the view that leadership is a zero-sum game. On the basis of this evidence, we conclude that groups do not vary in the amount of leadership perceived. For this reason, we do not include the group factor, both stable and unstable, in our table of results.
Table 1 presents the results from the variance partitioning. On average, the amount of target variance is 0.427, which indicates that group members
agree substantially on the degree of leadership displayed by each partici-pant. A small variance would have indicated that targets are perceived differentially by perceivers. The reliability of the target is 0.861, the highest reliability of any component. Notably, the variance partitioning from these seven studies is very similar to the variance partitioning from studies in which there is a single measure of leadership and when leadership is considered a rank-order measure (Livi et al., 2008).
The literature demonstrates that substantial consensus has been reached regarding variance and leadership. First, agreement concerning leadership is greater than agreement found for personality traits, which averages to 0.275 (Kenny, 1994). Second, roughly equal amounts of relationship and target variance for physical attractiveness ratings have been identified (Ho¨nekopp, 2006). Thus, perceivers agree much more about leadership than they agree about physical attractiveness. Third, in an interesting study, Vargas (1986) found that college athletes nearly agreed about as much who was a leader on their team (proportion of target variance ¼ 0.404) as they agreed about athletic skill (0.413). Fourth, there is more target variance than error variance in leadership studies, another unusual result that indicates people agree about who is the leader of the group.
Relatively small but not trivial amounts of perceiver variance have been found, with the average being about 10% of the variance. The reliability of the stable perceiver effect is somewhat disappointing, being only 0.558.
Table 1. Proportions of SRM Variance from Seven Studies.
Study Stablea Unstableb
Perceiver Target Relative Perceiver Target Relative Campbell et al. (2003) 0.126 0.355 0.179 0.034 0.039 0.267 Dabbs and Ruback (1984) 0.135 0.457 0.140 0.030 0.070 0.168
Dabbs et al. (1987) 0.124 0.465 0.200 0.021 0.039 0.151
Lord et al. (1980) 0.122 0.363 0.132 0.093 0.022 0.267
Malloy and Janowski (1992) 0.061 0.449 0.241 0.033 0.069 0.148
Montgomery (1986) 0.000 0.304 0.315 0.082 0.006 0.294
Ruback et al. (1984) 0.040 0.598 0.136 0.014 0.059 0.153
Average 0.087 0.427 0.192 0.044 0.043 0.207
Reliabilityc 0.558 0.861 0.736 0.714 0.707 –d
aStable variance is variance that correlates across different measures of leadership.
bUnstable variance is variance that is specific to a given measure of leadership.
cPresuming that there are six people in the group and three measures.
dUndefined.
There is more evidence of stable relationship variance, which averages around 20% of the total variance and is much more reliable (0.736). Because perceiver variance and relationship variance both index disagreement about who is a leader in the group, it is interesting to note that target variance or agreement (0.427) is larger on average than disagreement (0.279 ¼ 0.087 þ 0.192). In summary, about 10% of the variance is due to perceiver variance, 20% to relationship variance, 40% to target variance, and 30% to error variance.
Correlations
As we outlined in the previous section, the SRM encompasses two types of reciprocity correlations. The first, called generalized reciprocity, correlates the perceiver effect with the target effect: If someone is seen as a leader by others, does that person see others as leaders? The second, called dyadic reciprocity, correlates relationship effects: If Alice thinks Bob is more of a leader than do others, does Bob see Alice as leader more than do others?
Reciprocity in the perception of leadership, both negative and positive, could arise for several reasons. First, reciprocity would emerge when leaders have legitimacy. That is, if a leader is legitimated, he or she would also be perceived as an authority. Explicit asymmetries might lead to negative reciprocity correlations (i.e., if Bob is seen as a leader, he sees others as followers). Additionally, groups under temporal pressure are more likely to be centralized, forming a more clear structure of asymmetries among members (Pierro et al., 2003). This could lead to a stronger target effect, and possibly negative reciprocities. Positive or no correlations could be revealed in groups where leadership is more distributed, as in groups characterized by decentralized or laissez-faire leadership style (Bass, 1990).
Second, certain factors could lead to positive reciprocity. Because leadership is ordinarily positively valued, then if two people like each other, they would both be biased to see each other as leaders, given that liking shows strong positive reciprocity (Kenny, 1994). This would imply a positive correlation at the dyadic level.
Table 2presents the reciprocities for a stable perceiver with a stable target (generalized reciprocity) and between the stable relationship effects (dyadic reciprocity). Although there is some variation in the correlation shown in Table 2, on average there is little correlation at both the generalized and dyadic levels. Reciprocity appears not to be present in the perception of leadership, at least in leaderless groups.
Nonetheless, it is useful to know that these correlations are near zero, for two reasons. First, when group size is 3, as in the study conducted by
Zaccaro et al. (1991), the full SRM cannot be estimated (i.e., the model is not identified). If we can assume that the dyadic reciprocity correlation equals zero, the model is identified with as few as three persons per group.
Thus, the dyadic correlation is beneficial in that, if we assume it to be zero, we are making a valid assumption. Second, the SRM may be estimated using a SEM approach, as described later in this chapter. That model is very complicated, so adopting the assumption that some correlations are zero would greatly simplify an otherwise overly complicated model.