The impact of status hierarchy on individual behavior and team processes
Doornenbal, B.M.
2021
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Doornenbal, B. M. (2021). The impact of status hierarchy on individual behavior and team processes.
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CHAPTER 5
OPERATIONALIZING HIERARCHIES:
WHEN AND WHY AS STEEPNESS-SKEWNESS AND ACYCLICITY?
ABSTRACT
The steepness-skewness interaction and Krackhardt’s hierarchy measure (KHM) are operationalizations of two prominent hierarchy conceptualizations: disparity and acyclicity. This chapter creates clarity about when and why to measure hierarchy either as the steepness-skewness interaction or as KHM. By highlighting differences and similarities between disparity and acyclicity and their associated measures, our aim is to reduce inconsistent use of hierarchy
operationalizations and avoid mistaken research conclusions caused by misfits between operationalizations and conceptualizations. To empirically demonstrate the difference between the hierarchy operationalizations and to provide a null hypothesis against which researchers can compare their findings, the statistical relationship between the steepness-skewness interaction and KHM is tested based on survey data and simulation data. Collectively, this chapter facilitates informed decision-making about how to operationalize hierarchy.
INTRODUCTION
Team research describes hierarchy as one of the key factors affecting team processes and outcomes (Magee & Galinsky, 2008). Hierarchy is often conceived as a factor affecting teams such that (in)formally higher ranked individuals have more influence over the team and its members (Bunderson et al., 2016; Magee & Galinsky, 2008). Although meta-analytical evidence points towards an overall negative impact of hierarchy (Greer et al., 2018), studies have shown both functional and dysfunctional effects (Anderson & Willer, 2014; Bunderson et al., 2016). Scholars explained that study findings are affected by contextual
characteristics. Studies showed that the negative influence of hierarchy is stronger when the team context is such that teams are more vulnerable to conflicts, for example when there is are greater diversity of skills (Bunderson & Sutcliffe, 2002; Greer et al., 2018).
In addition to the moderating influence of contextual characteristics, scholars proposed that the impact of hierarchy depends on how the hierarchy is operationalized . In support of this proposition, a study showed that hierarchy is differently related to team outcomes depending on whether hierarchy is computed as the Coefficient of Variation or as the Gini-coefficient . A simulation study provided further support for the importance of the hierarchy operationalization by showing that hierarchy computed as the standard deviation underestimates the impact of hierarchy relatively to hierarchy computed as the Coefficient of Variation and hierarchy computed as the Gini-coefficient (Wei et al., 2016). In this simulation study, the scholars argued that, compared to other common indicators, hierarchy
computed as the standard deviation is less sensitive to the degree to which a few team members are at the top of the hierarchy (Wei et al., 2016), which is a vital aspect of the concept of hierarchy (Harrison & Klein, 2007).
Despite the demonstrated importance of the operationalization of hierarchy , team scholars still apply different hierarchy operationalizations (Bunderson et al., 2016; Greer et al., 2018). I expect that part of these inconsistencies are appropriate, because they are the result of different understandings of the concept of a hierarchy. Often, however, scholars use a hierarchy operationalization without explicitly mentioning to what extent it aligns with their conceptualization of a hierarchy (Bunderson et al., 2016; Harrison & Klein, 2007). This practice has created uncertainty about the alignment between how scholars conceptualize and operationalize hierarchy. Misalignments are problematic because they can lead to misleading study outcomes (Harrison & Klein, 2007). A possible reason not to explain the alignment and even to choose operationalizations inconsistently (Allison, 1978; Wei et al., 2016), is that choosing an operationalization based on a conceptualization is complex in general, but especially for operationalizing within-team differences (Harrison & Klein, 2007). Hierarchy operationalizations, moreover, strongly correlate when applied in larger (+30 members) groups (Wei et al., 2016), which can make them seem
interchangeable.
To reduce inconsistent hierarchy operationalizations and increase the alignment between conceptualizations and operationalizations, several scholars provided useful directions. Harrison and Klein (2007) provided guidelines for (theoretically) conceptualizing and (practically) operationalizing within-team
differences, such as hierarchy. They suggested to conceptualize hierarchy as disparity, which they denoted as “differences in concentration of valued social assets or resources such as pay and status among unit members—vertical differences that, at their extreme, privilege a few over many.” (Harrison & Klein, 2007, p. 1200) They described that disparity is greater when the distribution of valued social assets or resources is positively skewed, with one member at highest endpoint of the continuum of the valued social assets or resources and others at lowest (see Figure 1). The Coefficient of Variation and the Gini-coefficient were suggested as operationalizations or measures of disparity. Bunderson and colleagues (2016) discussed several theoretical hierarchy conceptualizations and introduced a new one: acyclicity, which they denoted as “cascading relations of dyadic influence” (Bunderson et al., 2016, p. 1266) They described that acyclicity is greater when the influence within a hierarchy is cascading down a complete rank-order, from the top to the bottom of the hierarchy (see Figure 1). Krackhardt’s hierarchy measure (KHM, 1994) was suggested as operationalization of acyclicity. Chapter 2 of this dissertation reflected on the disparity operationalizations proposed by Harrison and Klein (2007), and introduced an improved
operationalization, the steepness-skewness interaction. As showed in Chapter 2 (and in Chapter 3), the steepness-skewness interaction is an accurate measure of the concept of disparity – the degree to which valued social assets or resources are positively skewed, with one member at highest endpoint of the continuum of the valued social assets or resources and others at lowest.
Because KHM and the steepness-skewness interaction were proposed rather independently – Bunderson and colleagues (2016) did not discuss the
steepness-skewness interaction while proposing KHM and Chapter 2 does not discuss KHM while proposing the steepness-skewness interaction – it is unclear why and when to choose these hierarchy operationalizations. This unclarity could lead to even more misalignments between conceptualizations and
operationalizations and thus hinder the advancement of the hierarchy literature (Bunderson et al., 2016; Greer et al., 2018; Wei et al., 2016). Therefore, more clarity is needed about when scholars should operationalize hierarchy as KHM or as the steepness-skewness interaction. Because the theoretical perspective of the study should drive the choice of the operationalization (Harrison & Klein, 2007), there is a particular need for clarity about the conceptual differences underlying KHM and the steepness-skewness interaction.
Addressing this need, the goal of this chapter is to create more clarity about when and why to measure hierarchy either as the steepness-skewness interaction or as KHM. I create this clarity by comparing the steepness-skewness interaction and KHM (Krackhardt, 1994), and the hierarchy conceptualizations they are indicators of: respectively disparity (Harrison & Klein, 2007) and acyclicity
(Bunderson et al., 2016). To empirically substantiate this comparison, I examine the statistical association between the two hierarchy operationalizations – the
steepness-skewness interaction and KHM – across organizations. I enrich this comparison by conducting two studies in which data is simulated to test the relationship between the steepness-skewness interaction and KHM. In both of these studies, I simulate data containing the rank order between each pair of individuals. In the first simulation study, which I call acyclicity simulation, the data are partially ordered (in which persons rank as either low or high), such as in
acyclicity studies. The rank orders are simulated based on variance in acyclicity. In the second simulation study, which I call disparity simulation, the data are completely ordered (in which persons rank on a continuum from low to high), such as in disparity studies. The rank orders are simulated based on variance in disparity. The examination of the statistical association between the steepness-skewness interaction and KHM helps to clarify the extent to which these measures can be used interchangeably.
By creating more clarity about when and why to operationalize hierarchy as the steepness-skewness interaction and as KHM, this chapter primarily contribute to the hierarchy literature by facilitating informed decision-making about how to operationalize hierarchy. As Harrison and Klein argue (2007), informed decision-making reduces mistaken research conclusions caused by misfits between operationalizations and conceptualizations. Consistency between
conceptualizations and operationalizations help to further accumulate insights about the impact of hierarchy. More insights are needed, among others, to test to what extent different properties of a hierarchy elicit different group processes and outcomes (Greer et al., 2018). By testing the association between the steepness-skewness interaction and KHM across organizations, I examine the statistical relationship between operationalizations (or measures) of respectively disparity and acyclicity. The primary objective of this examination is to clarify the extent to which the steepness-skewness interaction and KHM can be used interchangeably. A weaker relationship would imply that the operationalizations of disparity and acyclicity are more distinct, hence that choosing different measures (statistics) may lead to findings that suggest a different impact on team outcomes. The simulation
studies also contribute to the hierarchy literature by providing a null hypothesis on the relationship between hierarchy operationalizations against which researchers can compare their findings (Krackhardt, 1994). Comparing real-world observations with null-hypotheses can help to assess the generalizability of these observations as it helps to examine to which degree study findings may be caused by a chance process. A null-hypothesis indicating a weaker statistical relationship would mean that the disparity and acyclicity operationalizations are more distinct, hence suggests that studies on disparity and acyclicity are more likely to lead to different insights.
CONCEPTUALIZING AND OPERATIONALIZING HIERARCHY In the team literature, scholars acknowledge that construct clarity helps in comparing and contrasting studies on within-team differences such as hierarchy (Bunderson et al., 2016; Greer et al., 2018; Harrison & Klein, 2007). Clear constructs are “simply robust categories that distill phenomena into sharp distinctions that are comprehensible to a community of researchers” (Suddably, 2010, p. 346). To facilitate construct clarity, Harrison and Klein (2007)
recommended conceptualizing hierarchy as disparity, which they denoted as the relative distribution of socially valued resources within a unit. Underlying this conceptualization, they described three assumptions: members within a unit can differ in the extent to which they hold socially valued resources (assumption 1), units can differ in the extent to which their socially valued resources are distributed equally among unit members (assumption 2), and differences among units in the extent to which the socially valued resources are distributed equally among unit
member lead to predictable consequences (assumption 3). Most hierarchy studies have adopted a conceptualization in line with disparity. Across these studies, scholars focused on different socially valued resources underlying hierarchy (Greer et al., 2018). Mostly, scholars focused on resources that led to team influence (Harrison & Klein, 2007; Magee & Galinsky, 2008).
Conceptualizing hierarchy as disparity or acyclicity
Although scholars mostly agree that a minimum hierarchy is a situation in which members do not differ in the extent to which they hold socially valued resources (see assumption 1), there are different ideas about when the distribution of socially valued resources within units is more unequal (assumption 2), and thus when hierarchy has a greater impact (assumption 3). Most scholars conceptualize hierarchy similar to the disparity conceptualization of Harrison and Klein (2007). From the disparity perspective, a low hierarchy is present when the variance among unit members in socially valued resources is minimal (e.g. when there are no clear highest and lowest ranked team members with respect to resources), whereas a maximum hierarchy is present when a privileged number of members hold all socially valued resources. The logic behind this difference between low (ladder-shaped) and maximum (pyramid-shaped ) hierarchy relates to the asymmetry in socially valued resources at the unit level (see assumption 3). In low hierarchies, only a small proportion of the unit’s members are disadvantaged compared to the majority, whereas the overwhelming majority of unit members are disadvantaged compared to a few members in a maximum hierarchy. In line with this logic, Harrison and Klein (2007) described a moderate hierarchy as a uniform
distribution in which there is little variance among unit member in socially valued resources and differences are compressed such that none of the members have a large disadvantage compared to their closest higher-ranked unit member.
Bunderson and colleagues (2016) introduced hierarchy as acyclicity. From the acyclicity perspective, a low hierarchy is present when the dyadic relationships within the unit form a cyclical chain of influence (in which influence keeps on flowing in circles), whereas a maximum hierarchy is present when the dyadic relationships form a non-cyclical chain through which members pass down influence. The logic behind the difference between low (cyclic influence, incomplete order) and maximum hierarchy (cascading influence, complete order) relates to the asymmetry in socially valued resources between all pairs of individuals in a unit (see assumption 2). In low hierarchy, the asymmetry forms a cyclical chain of influence in which no one has final say in decision-making, whereas in maximum hierarchy the asymmetry is such that clarity exist about who has more say in decisions. In line with this logic, Bunderson and colleagues (2016) described moderate hierarchy as a setting in which some unit members have more say, which is the case when part of the dyadic relationships form a structure in which the influence is passed down.
The difference between disparity and acyclicity becomes especially salient by visualizing them and by considering the differences in situations of maximum hierarchy. As illustrated in Figure 1, a maximum disparity and a maximum acyclicity are distinct in that acyclicity suggests a total order in the hierarchy in which individuals rank from a person at the top to a person at the bottom, whereas a maximum disparity suggests a partial order in the hierarchy in which one person
stands at the top of the hierarchy and the remaining at the bottom. For maximum acyclicity, disparity can be moderate and even not bottom-heavy (see moderate disparity in Figure 1). If the differences in rank are smaller at the top, maximum acyclicity can even be small in terms of disparity. For maximum disparity, acyclicity is not at its maximum because most (low-ranked) team members do not have hierarchical dyadic relationships through which influence is passed down. For near maximum disparity, however, acyclicity can be (near) maximum, for example when the large subgroup of low ranked individuals is ordered, from least low to lowest.
Figure 1: Disparity vs. Acyclicity.
Note. Taken from Harrison and Klein (2007) and Bunderson et al. (2016).
To conceptualize hierarchy appropriately, scholars should consider the theoretical focus of the study (Bunderson et al., 2016; Harrison & Klein, 2007). A focus on the degree to which a few members hold all socially valued resources is
what Harrison and Klein (2007) conceived as disparity. In contrast, a focus on the degree to which socially valued resources are distributed such that influence cascades down as a continuous flow from the top to the bottom of the hierarchy, is what Bunderson and colleagues (2016) conceived as acyclicity. Aligning with these conceptualizations helps to accumulate knowledge about hierarchy across studies.
From conceptualization to operationalization
The hierarchy operationalization introduced in this dissertation, the steepness-skewness interaction, and KHM (Krackhardt, 1994) are differently appropriate for studying hierarchy. As shown in Chapter 2, the steepness-skewness interaction is an appropriate operationalization when hierarchy is conceptualized as disparity. As illustrated in Figure 2, higher levels of disparity are present when hierarchies are steeper and more positively skewed. Steepness and skewness are measured as respectively the standard deviation and the skewness in socially valued resources. Harrison and Klein (2007) proposed operationalizing disparity as either the Coefficient of Variation or the Gini-coefficient. However, as shown in Chapter 2 and illustrated in Table 1, a higher Coefficient of Variation or Gini-coefficient does not always reflect a bottom-heavy hierarchy (pyramid-shaped
), which is a central feature of disparity (Harrison & Klein, 2007). Operationalizing hierarchy as the steepness-skewness interaction overcomes these measurement validity shortcomings. Hence, the steepness-skewness interaction is a more precise measure for the concept of disparity.
Although the steepness-skewness interaction is introduced primarily to operationalize disparity, I also expect it to be sensitive to differences in acyclicity.
More acyclicity is present when more of the dyadic relationships form a non-cyclical chain, in which the influence clearly flows down the hierarchy. The steepness-skewness interaction is computed based on member’s overall social position, for example based on the average influence of the members within each dyadic relationship. When more people disagree about whether a person is influential, the overall position of the member becomes less extreme. For example, team member C has an influence score of 2 when team members A and B ascribe that member influence, but member C has an influence score of 1 when only A or B ascribes that member influence. When acyclicity is lower, the team disagrees more about the influence of each person, which means that more individuals will have a moderate level of influence and therefore that the hierarchy is less steep. Furthermore, whenever the skewness is not negative (i.e. hierarchy shaped as inverted pyramids ), such as for moderate and high disparity (see Figure 1), the acyclicity is greater. Operationalizing individuals’ influence as the amount of their (in)direct outflowing lines, the steepness and skewness in the moderate acyclicity configuration (Figure 1) are respectively 1.29 and 0.00, whereas they are 1.50 and 0.14 in the high acyclicity configuration (0.00 and 0.00 in the low acyclicity configuration). Irrespective of the chosen operationalization of acyclicity, I thus expect the steepness-skewness interaction to be sensitive to this acyclicity measure.
Figure 2: Steepness and skewness scores for different hierarchies.
Note: Greater disparity at the top right. Figure taken from Chapter 2. The numbers in the circles denote individual’s status level.
Table 1: Status skewness across teams with similar concentration. Coefficient Team Individuals’ status
level
Steepness Skewness Coefficient of Variation Gini 1 3 – 3 – 3 – 5 1.00 + 2.00 0.29 .07 2 3 – 6 – 6 – 6 1.50 – 2.00 0.29 .07 3 1 – 7 – 7 – 7 3.00 – 2.00 0.55 .14 4 1 – 1 – 1 – 7 3.00 + 2.00 1.20 .30 Note. Indices calculated based on 4-person teams. Formula 15 from Biemann and Kearney (2010) is used to calculate the Gini coefficient. Table taken from Chapter 2.
Bunderson and colleagues (2016) proposed KHM (Everett & Krackhardt, 2012; Krackhardt, 1994) as an appropriate operationalization when hierarchy is conceptualized as acyclicity. KHM takes a social network theory approach in which hierarchy is measure as the amount of differences in the socially valued resources that each pair of unit members holds. Specifically, Krackhardt (1994) proposes measuring hierarchy as: KHM = 1 – [V/ maxV], where V is the total amount of member pairs that are symmetrically connected (that is, where A influences B and B influences A, directly or indirectly) and maxV is the total amount of member pairs that are connected (that is, where at least A influences B or B influences A, directly or indirectly). For example, when three pairs of individuals are symmetrically connected (V = 3) in a three-person team in which each member is connected to each other member (maxV = 3), KHM is zero (1 – [3/3] = 0). From the perspective that socially valued resources result in influence (Magee & Galinsky, 2008), KHM focuses on the flow of influence between each pair of individuals, which is consistent with the concept of acyclicity (Bunderson et al., 2016).
Despite its advantages in measuring acyclicity (Bunderson et al., 2016), KHM seems deficient as a measure of disparity and should not be used as a measure of disparity. More disparity is present when a privileged few hold more socially valued resources than the majority (Harrison & Klein, 2007). KHM does not indicate to how many individuals the influence flows to (Everett & Krackhardt, 2012; Krackhardt, 1994). Maximum acyclicity according to KHM could indicate hierarchies in which the influence flows down to all but one member, which Harrison and Klein would indeed describe as maximum disparity (see maximum disparity in Figure 1), but also hierarchies in which the influence flows down to one
lowest ranked member, which Harrison and Klein would describe as moderate disparity (see moderate disparity in Figure 1). Another reason why KHM is less suitable for operationalizing disparity is that disparity focuses on the extent to which a majority is disadvantaged in socially valued resources. Although acyclicity implies a non-zero distance in socially valued resources, KHM focuses on ordinals and does not reflect the size of the distance between individuals (Krackhardt, 1994). Within a pair of ordered individuals, the social distance can range from very small to very large. Thus, KHM does not indicate the extent to which a majority is disadvantaged in socially valued resources. KHM is therefore likely to be insufficiently consistent with the concept of disparity.
EMPIRICAL EXAMINATION
To substantiate the theoretical explanation of when and why to operationalize as the steepness-skewness interaction and KHM, this empirical examination investigates the statistical relationship between the two
operationalizations across organizations. The aim of this examination is to clarify the extent to which the steepness-skewness interaction and KHM can be used interchangeably. As previously described, disparity and acyclicity are theoretically distinct. I therefore expect that the steepness-skewness interaction and KHM cannot be used interchangeably. Two research questions form the structure of the examination:
Research Question 1: What is the strength of the statistical association between the steepness-skewness interaction and KHM?
Research Question 2: What KHM values can be expected across different combinations of steepness and skewness values?
Sample, measures, and procedure
Survey data are used to analyze the relationship between the steepness-skewness interaction and KHM. The data were collected by Master Students in 2013. The data come from various organizations, mainly in the Retail Trade and Services and Manufacturing and Energy sectors. In total, data were collected from 170 teams across 128 organizations. Incomplete teams were excluded from the analysis, resulting in 93 teams from 70 organizations.
In order to test the relationship between steepness-skewness interaction and KHM, I focused on the multiple correlation (R2) between the
steepness-skewness interaction and KHM. The multiple correlation was tested by performing an ordinary least square (OLS) regression analysis in which KHM scores were regressed to values in steepness, skewness, and steepness-skewness interaction. In this test, a higher R2 value suggests a greater correlation between the
hierarchical measures and thus a lower discriminant validity between the KHM and the steepness-skewness interaction.
Steepness, skewness, and KHM were measured using a round robin approach in which participants were asked to answer the following question for each of their fellow team members: “How much influence do you think this team member has within the team?” Answers were given on a 5-point scale (1 = “very little influence”, 5 = “A lot of influence”). The average response was 3.30 (SD = .60, skewness = -0.01), suggesting that people with an influence score of 3 or lower
were considered as relatively low in influence, whereas people with an influence score of 4 and 5 were considered as relatively high in influence.
Binary data are required to compute KHM values. Therefore, the 5-point influence scale was dichotomized by classifying individuals as influential when they were scored 4 or higher on the 5-point scale. Across all dyads, 383 (41.6%) individuals scored 4 or higher and were thus perceived as influential. KHM values were computed using the previously described formula of Krackhardt (1994): 1 – [V/ maxV], where V is the total amount of member pairs that are symmetrically connected and maxV is the total amount of member pairs that are connected. To compute steepness and skewness values, it is not necessary to dichotomize the influence data. Instead, the influence that individuals were ascribed by their team members was mean-aggregated for each individual. By computing the standard deviation of the aggregated influence scores and the skewness of the mean-aggregated influence scores, respectively the steepness and skewness values were computed for each team.
As reported in Table 2, the average hierarchy skewness within the sample was around zero (-0.01). Across teams with steeper hierarchies, the KHM values were greater (r = .327, p < .01). KHM values were greater as well across teams with lower influence values (r = -.327, p < .01). The average influence of teams was not statistically significant related to steepness (r = -.017, n.s.) and skewness (r = -.134, n.s.).
Table 2: Description of Variables and Correlations among them.
Variables Mean SD 1 2 3 4 5 1. Team size 3.49 0.85 2. Mean Influence 3.31 0.50 -.072 3. Steepness 0.73 0.45 -.172 -.017 4. Skewness -0.01 0.36 -.061 -.134 -.133 5. KHM 0.62 .42 -.021 -.327** .327*** .056 6. Steepness x Skewness -0.03 0.28 -.015 -.201* -.206* .852*** .061 Note. N = 98. SD = Standard Deviation, KHM = Krackhardt’s hierarchy measure. * p < .05. ** p < .01. *** p < .001
RESULTS
As reported in Table 3, team size, steepness, skewness and the steepness-skewness interaction explain around 10% (R2) of the variance in KHM. However, the
interaction effect between steepness and skewness on KHM is non-significant (B = 0.25, SE = 0.28, n.s.). Answering RQ1, the statistical association between the steepness-skewness interaction and KHM is small.
To provide insight in what KHM values can be expected across different combinations of steepness and skewness values (RQ2), Figure 3 is included. At the top right of this figure, the hierarchies have a greater disparity – thus are pyramid-shaped ( ). At the top left, hierarchies are more pyramid-shaped as inverted pyramids ( ). Figure 3 shows the estimated KHM values for the observed range of steepness and skewness scores. The KHM values were estimated based on the fit between KHM
and the steepness-skewness interaction (see Table 3). For example, the estimated KHM value for a steepness value of 1 and a skewness value of 0.5 was 0.822, as 0.822 = 0.390 (intercept) + 1 * -0.330 (coefficient steepness) + 0.5 * -0.045 (coefficient skewness) + 0.5 (steepness * skewness) * 0.248 (coefficient steepness-skewness interaction) + 0 (error). Because KHM ranges theoretically between zero and one, these values were set as the minimum and maximum values. Figure 3 depicts higher KHM values by a darker hue. To highlight the KHM values, KHM values were also plotted as contours (isolines), in steps of 0.05. Overall, Figure 3 shows that the estimated KHM values were somewhat higher for steeper hierarchies with a more positive skewness (RQ2) – thus for greater disparity. Nonetheless, as presented in Table 3, the steepness-skewness interaction and KHM were weakly related.
Table 3: KHM regressed to the steepness-skewness interaction Variables Intercept 0.390 0.077 Steepness 0.330*** 0.092 Skewness -0.045 0.213 Steepness x Skewness 0.248 0.278 Adjusted R2 .10 F 4.44*
Note. N = 98. B = unstandardized regression coefficients. SE = standard error. * p < .05. ** p < .01. *** p < .001.
Figure 3: Contour plots in which the contours are average KHM values as a function of steepness and skewness.
Collectively, the empirical examination further clarifies that the steepness-skewness interaction and KHM are largely distinct. The statistical association between the steepness-skewness interaction and KHM is small (R2 = .10, RQ1).
Although the average KHM values were somewhat higher for steeper hierarchies with a more positive skewness (RQ2), KHM values varied strongly for fixed combinations of steepness and skewness values.
The findings warrant several considerations. First, the relationship was tested based on non-binary ordinal network data. These data contained
information about both the flow of influence between each pair of individuals and the extent to which individuals differed in influence. Because both these aspects help to measure disparity, they were both used in measuring disparity. However, as soon as only binary network data are available that does not contain information
on the degree to which individual influence each other, less variance in disparity can be measured. Hence, if scholars would compute disparity based binary network data, they may find a stronger relationship between the steepness-skewness interaction and KHM. Second, the relationship was studied across different team sizes. The size of teams restricts the range of possible hierarchy scores (Lindell & Brandt, 2000). Possibly, scholars find a stronger relationship between the steepness-skewness interaction and KHM in smaller teams, when the range of possible hierarchy scores is smaller.
SIMULATION STUDIES
To account for the two limitations described in the results section of the empirical examination, the subsequent two simulations studies further investigate the statistical relationship between the steepness-skewness interaction and KHM. Different from the previous empirical examination, the simulation studies test the relationship between the steepness-skewness interaction and KHM using two different data types, continuous and binary data (see limitation 1), and for fixed team sizes (see limitation 2). The data will be simulated such that the relationship between the two hierarchy operationalizations can be tested across a wide range of hierarchy distributions. By testing the relationship across a wide range of controlled settings, the study results will provide a null hypothesis on the relationship between the hierarchy operationalizations against which researchers can compare their findings (Krackhardt, 1994).
In both simulation studies, network data are used to test the relationship between the two operationalizations. The difference between the two simulation
studies is the simulated data. In the first simulation study, called acyclicity simulation, acyclicity is simulated similar to how Krackhardt simulated acyclicity (1994), based on binary network data. These data are partially ordered, in which persons rank either low or high. The rank orders are simulated based on variance in the likelihood that persons rank as high. In the second simulation study, called disparity simulation, disparity is simulated based on continuous network data sampled from different skew-normal distributions (Azzalini, 2013). These data are completely ordered, in which persons rank on a continuum from low to high, such as in disparity studies. The rank orders are simulated based on variance in the skewness of the distribution of influence. The simulation studies will be used to answer the two research questions that were also addressed in the previous empirical examination:
Research Question 1: What is the strength of the statistical association between the steepness-skewness interaction and KHM?
Research Question 2: What KHM values can be expected across different combinations of steepness and skewness values?
Procedure and analysis
In both simulation studies, hierarchies were simulated starting with the generation of digraphs (i.e. directed graphs) containing the influence each person has on each of its peers. The influence information within the digraphs was generated by sampling values from probability distributions. Two values were sampled for each pair of individuals, the influence from A on B and from B on A.
Hence, for each team, N2 – N values were sampled, in which N is equal to the size
of the team. For example, 20 values representing influence units were sampled for 5-person teams. Tables 4 and 5 show example digraphs in which the influence flows from the individuals on the top (i.e. from the columns) down to the
individuals on the right (i.e. the rows). Table 4 illustrates, for example, that member three (fourth column) has direct influence over member two (second row) during the first simulation, but not during the second simulation, see entry 1 / 0 in cell (4,2). Member two (third column) has only direct influence over member three (third row) during the second simulation, see entry 0 / 1 in cell (3,3). During both simulation rounds, members two and three have asymmetrical influence over each other. However, during the second simulation round, member three has an indirect influence on member two, through influence on member four, see cell (4,4) for the influence of member three on member four and see cell (5,2) for the influence of member four on member two. The flow of (indirect) influence is also plotted in Figures 4 and 5.
Figure 4: Acyclicity plotted based on the example digraphs reported in Table 4. Note: The hierarchy plotted on the left is based on the first digraph simulation (KHM = 0.9), the hierarchy plotted on the right is based on the second digraph simulation (KHM = 0.0).
Table 4: Two example digraphs with the same KHM for acyclicity simulation
Member (from) 1 2 3 4 5 (to) 1 1 / 0 1 / 0 1 / 0 0 / 0 2 0 / 0 1 / 0 1 / 1 1 / 0 3 0 / 0 0 / 1 0 / 0 0 / 0 4 0 / 0 1 / 1 0 / 1 0 / 1 5 0 / 0 0 / 1 0 / 0 0 / 1 Average influence 0.00 / 0.00 0.50 / 0.75 0.50 / 0.25 0.50 / 0.50 0.25 / 0.25 Note: The two digraphs are separated by the slash sign: Before …/ = first simulation, after /… = second simulation. N = 5, P = .35: chance of influence is 35%, KHM = 0.9 / 0.0, Steepness = 0.22 / 0.19, Skewness = -0.60 / 0.28.
Next, the digraphs were used to compute: steepness, skewness, and KHM values. Different aspects were focused on to compute these values. While computing KHM values (to measure acyclicity), the focus was on the influence between pairs of individuals. In contrast, while computing steepness and skewness values (to measure disparity), the focus was on the distribution of the average influence scores. The computed scores were used to examine the multiple correlation (R2) between the steepness-skewness interaction and KHM, similar to in
the empirical examination. Again, the multiple correlation was examined by conducting ordinary least squares (OLS) regression analyses in which KHM scores were regressed to values in steepness, skewness, and the steepness-skewness interaction. In these analyses, higher R2 values suggest a stronger overlap, and thus
lower discriminant validity between the KHM and the steepness-skewness interaction.
Figure 5: Hierarchies plotted based on the example digraphs reported in Table 5. Note: The hierarchy plotted on the left is based on the first digraph simulation (KHM = 0.0), the hierarchy plotted on the right is based on the second digraph simulation (KHM = 0.00).
Table 5: Two example digraphs with the same KHM for disparity simulation
Member (to) 1 2 3 4 5 (from) 1 1.31/0.36 -0.71/1.63 1.10/-1.20 2.08/-0.38 2 1.2/-0.71 -0.86/0.93 0.86/-1.57 1.03/0.22 3 -0.70/-1.03 0.12/0.30 1.36/1.05 0.60/-0.56 4 1.39/0.38 -0.17/0.65 0.74/0.77 0.05/-1.61 5 0.84/1.33 0.44/0.08 0.11/0.46 0.87/-0.30 Average influence 0.68/-0.01 0.42/0.35 -0.18/0.95 1.05/-0.50 0.94/-0.58 Note: The two digraphs are separated by the slash sign: Before …/ = first simulation, after /… = second simulation. N = 5, alpha = .35, KHM = 0.00 / 0.00, Steepness = 0.49 / 0.63, Skewness = -0.51 / 0.29 The KHM scores were calculated after setting values higher than average influence (0.58 / 0.04) to 1 and values lower than average influence to zero.
The two simulation studies differed from each other in the sampling procedure. During the acyclicity simulation, binary values (representing influence and no influence) were sampled and randomly ascribed to pairs of individuals. While sampling values, the probability (P) that any individual had influence over each other individual was manipulated, similar to Krackhardt’s simulation approach (1994). Table 4 presents two example digraphs in which P was set to .35. This value of P means that the sampled values were on average 35% of the instances one (i.e.
team member A has influence over team member B) and 65% of the instances zero (i.e. team member A has no influence over team member B). Note in Table 4 that the total amount of influence, the asymmetry (and thus the acyclicity), and the distribution of the average influence (and thus the disparity) can differ across digraphs with a similar P. On average, larger values of P result in lower KHM scores (Krackhardt, 1994). Values of P larger than .5 result in more negatively skewed hierarchies (i.e. hierarchies shaped like inverted pyramids ), whereas values lower than .5 result in more positively skewed hierarchies (i.e. pyramid-shaped hierarchies ). For extreme values of P, the hierarchy is less ordered, such that either most individuals have no influence over each other (very low P), or most individuals influence each other (very high P). Across the simulations, the value of P varied from .01 to .99, in increments of .01.
During the disparity simulation, N vectors of N continuous values (representing the degree of influence) were sampled for each digraph. In a fixed order, the sampled values were ascribed to pairs of individuals (ith value of each
vector to ith individual). The values were sampled from skew-normal distributions
(Azzalini, 1985, 2013), for which the skewness was manipulated through a shape parameter (alpha). Before using the digraphs to compute hierarchy scores, one value in each vector was set to zero: the value that represented the influence that persons have over themselves (e.g. the values on the diagonal in Table 5). Table 5 shows two example digraphs in which the alpha was set to .35. Note that the asymmetry (and thus the acyclicity) and the distribution of the average influence (and thus the disparity) can differ across digraphs with a similar alpha as well. On average, a more negative alpha results in more negatively skewed hierarchies (i.e.
hierarchies shaped like inverted pyramids ), whereas a more positive alpha results in more positively skewed hierarchies (i.e. pyramid-shaped hierarchies ). Across the simulations, the shape parameter (alpha) of the skew-normal
distributions varied from -5 to 5, in increments of .10.
Because team size restricts the range of possible hierarchy scores (Krackhardt, 1994; Wei et al., 2016), the digraphs, and thus the hierarchies, were simulated for different team sizes (N): 3, 5, 8, 13, and 21. Teams with more than 21 members were not explored, because those team sizes are uncommon in the team hierarchy literature. For each of the 1000 simulation conditions, that is for each P and N (acyclicity simulation) and each alpha and N (disparity simulation), 500 digraphs were created. A seed value (1234 in' R ' statistical software) was used for pseudo-random sampling for each simulation condition to ensure reproducible results.
RESULTS Research Question 1
Addressing the overall strength of the statistical association between the steepness-skewness interaction and KHM (RQ1), Figure 6 and Table 6 report the 95% confidence intervals (CI) of the multiple correlation (R2) between the
steepness-skewness interaction and KHM for each team size N and simulation study. Theoretically, R2 values range from 0 (zero overlap between the
measurements) to 1 (complete overlap between the measurements). Higher values suggest a weaker discriminant validity between the steepness-skewness
overall weak relationship, and thus a high discriminant validity, between the two measurements.
Figure 6: Multiple correlation (R2) between the steepness-skewness interaction
and KHM for each N and study. Note: Each point represents the mean R2 and the
whiskers around each mean encompass the 95% confidence intervals.
Besides reporting the R2 values, Figures 7 and 8 report the Coefficient of
Variation of acyclicity (as triangles), steepness (as lines), and skewness (as crosses). As with the standard deviation, the Coefficient of Variation describes the average level of variability. Different from the standard deviation, the Coefficient of Variation is relative to the mean (it being defined as the ratio of the standard deviation of x to the mean of x), which make these easier to compare when the mean values differ. The Coefficient of Variation values are reported because they help to interpret the R2 values. When the Coefficient of Variation is close to zero,
lower R2 values can be expected because it suggests that the outcome variable has
little variance to be explained and/or the predictors have little variance to explain the outcome variable with.
Table 6: Multiple correlation (R2) between the steepness-skewness interaction and
KHM. Simulation N R2 CL L CLU Acyclicity 3 .38 .32 .44 5 .14 .09 .19 8 .08 .05 .12 13 .05 .02 .07 21 .02 .01 .04 Disparity 3 .04 .03 .04 5 .04 .03 .04 8 .03 .02 .03 13 .01 .01 .01 21 .00 -.00 .00 KHM for each N and study. Note: The table reports 95% confidence intervals. CLL =
lower confidence limit, CLU = upper confidence limit, N = team size.
The results of the acyclicity simulation study (Figure 7) suggest a low multiple correlation between KHM score and the steepness-skewness interaction (RQ1) when the probability (P) for which each team member has influence over each other team member is greater than 10% or the team size is larger than 5. The lower correlation across larger teams might be partly the result of the small
variance in KHM scores. In many simulation conditions, KHM values show no variance, in contrast to the values in steepness and skewness. As a result, the R2
values in those simulation conditions are zero. The variance in KHM scores is especially small for large values of P, when individuals have more influence over each other, which is in line with Krackhardt’s statement that P heavily constraint the possible range of acyclicity values (1994).
For teams of 3 or 5 members, a strong statistical relationship is present between the steepness-skewness interaction and KHM when P is respectively smaller than .24 or .11 respectively (see Figure 7). Assuming that KHM is an accurate measure to operationalize acyclicity, this suggests that the steepness-skewness interaction is sensitive to variance in acyclicity in smaller teams that have only few people in higher social positions.
Figure 7: Multiple correlation (R2) between the steepness-skewness interaction
and KHM as a function of P for each N in the acyclicity simulation. Note: Coefficient of Variation (CV) values greater than 1 are visualized as 1.
Figure 8: Multiple correlation (R2) between the steepness-skewness interaction
and KHM as a function of alpha for each N in the disparity simulation. Note: Coefficient of Variation (CV) values greater than 1 are visualized as 1.
The results of the disparity simulation study (Figure 8) further support the overall weak relationship between the steepness-skewness interaction and KHM
(RQ1). Across all simulation conditions, the largest R2 value was .11. In many
negative alpha simulation conditions, in which hierarchies are negatively skewed (shaped like inverted pyramids ), KHM values showed no variance.
Consequently, the R2 values for many of the negative alpha simulation conditions
were zero. This finding suggests that KHM is insensitive to variance across negatively skewed hierarchies. For teams of 3 or 5 members, the steepness-skewness interaction explained some variance in KHM scores when the value of alpha was positive, and hierarchies were thus positively skewed (pyramid-shaped
).
Research Question 2
To explore what KHM values can be expected across a broad range of different combinations of steepness and skewness values (RQ2), Figures 9 to 12 were included. These figures depict KHM values against steepness and skewness values. Two different types of visualizations are used: contour plots and scatter plots. The contour plots (Figures 9 and 11) provide insight into the average value of KHM for different combinations of steepness and skewness values. Higher average KHM values are depicted by a darker hue. To highlight the differences in KHM values, a selection of average KHM values are visualized as contours (isolines) against steepness and skewness values. The average KHM values are computed based on the regression fit between KHM and the steepness-skewness interaction, similar to in the previous empirical examination. Again, the computed KHM values were restricted to its theoretical range (from zero to one) and values were computed only for the simulated range of steepness and skewness values.
The scatter plots (Figures 10 and 12) show the hierarchy scores found (i.e. steepness, skew, and KHM values) for each simulated team. The teams are represented as dots. The coordinates of the dots indicate the values of skewness (x-axis) and steepness (y-axis) of the teams and the hue of the dots the KHM values. The scatter plots reveal the heterogeneity of the relationship between KHM and the steepness-skewness interaction. When KHM values differ more for
combinations of steepness and skewness values, the hue of the dots is more heterogeneously scattered – forming a cloud of less evenly distributed colors.
Both contour and scatter plots were included for each team size (N) and for both studies in respectively Figures 9 and 10 (acyclicity simulation) and in Figures 11 and 12 (disparity simulation). At the top right of these figures, the hierarchies have a greater disparity – thus are pyramid-shaped ( ). At the top left, hierarchies are more shaped as inverted pyramids ( ). Note that the axes of the plot differ because the range in average steepness and skewness values relates respectively negatively and positively to team size.
The contour plots of the acyclicity simulation (Figure 9) suggest that KHM values are inconsistently related to combinations of steepness and skewness values (RQ2), depending on the team size. In smaller teams, with 3 or 5 members, hierarchies with the highest KHM values (see darker areas of plots) often have a more positive skewness in combination with a larger steepness (top right of plots). In contrast, in larger teams (incl. teams of 8 members), hierarchies with the highest KHM values often have a more positive skewness in combination with a smaller steepness (bottom right of plots). Both for smaller and for larger teams, the
contour plots show higher KHM values for more positively skewed hierarchies (right of plots) – that is, for pyramid-shaped hierarchies ( ).
Figure 9: Contour plots of the acyclicity simulation in which the contours are average (interpolated) KHM values as a function of steepness and skewness. Note: N = team size.
Figure 10: Scatter plots of the distribution of steepness, skewness, and KHM simulated during the acyclicity simulation. Note: N = team size.
The scatter plots of the acyclicity simulation (see Figure 10) show darker scatters for higher skewness values, suggesting higher KHM values for more positive skewness values. The scatter plots further suggest that range restriction causes the inconsistency across the contour plots (see Figure 10). As can be seen by the shape of the distribution of the scatters, steepness restricts the possible range of skewness. In teams of 3 or 5 members, more extreme skewness values are
possible for teams with a higher steepness. In contrast, for teams of 8, 13, or 21 members, more extreme skewness values are possible for teams with a smaller steepness. Given the positive relationship between KHM and skewness, teams with higher KHM values are likely to have higher steepness values in teams of 3 or 5 members because the range of skewness for these teams is broader for higher steepness values. Analogously, teams with higher KHM values are likely to have lower steepness values in teams of 8, 13, or 21 members because the range of skewness values for these teams is broader (note the less grouped skewness scores for teams with more than 5 members in Figure 10) for lower steepness values.
The contour plots of the disparity simulation (Figure 11), in which steepness and skewness were manipulated, show very little variance in average KHM values – as visualized by the small variance in the hue of the dots. The average KHM values, visualized as contours, are especially small (< .06) in teams of 8, 13, and 21 members. In teams of 3 or 5 members, the largest observed average KHM values are respectively .9 and .6, at the top right of the figures. As larger disparity is more to the top-right of the figures, the contour plots suggest that – only in smaller teams – greater disparity (computed as SK*ST) results in higher average KHM values (RQ2).
Although the contour plots suggest that greater disparity results in higher average KHM values in smaller teams, the scatter plots of the disparity simulation (see Figure 12) show an inconsistent relationship between the steepness-skewness interaction and KHM values (RQ2). At the top-right of the plots, which suggest greater disparity, teams can have lower (<.5) KHM values. At the middle-bottom of
the plots, for low values of steepness and skewness, teams can have larger KHM values. In addition to showing an inconsistent relationship between KHM values and steepness and skewness values across smaller teams, the scatter plots illustrate the small range of average KHM values visualized in the contour plots for the team of 8, 13, and 21 members. For these teams, no high KHM values can be seen – as visualized by the light gray hue of the dots.
Figure 11: Contour plots of the disparity simulation in which the contours are average (interpolated) KHM values as a function of steepness and skewness. Note: N = team size.
Figure 12: Scatter plots of the distribution of steepness, skewness, and KHM simulated during the disparity simulation. Note: N = team size.
Collectively, the two simulation studies show similar results. Answering RQ1, the results of the simulations studies suggest a weak overall relationship between the steepness-skewness interaction and KHM. In some simulation conditions, mostly in positively skewed hierarchies, the two hierarchy measures showed some overlap. Answering RQ2, hierarchies with higher KHM values were often steeper and more positively skewed – that is, more pyramid shaped ( ). For teams of 8, 13
or 21 members, however, KHM related weakly to the steepness-skewness interaction. The relationship between KHM (values) and the steepness-skewness interaction was possibly weaker in these teams because N heavily constraints values of KHM (Krackhardt, 1994). When N increases, it becomes more likely that a member has influence over at least one fellow member, meaning a lower acyclicity.
By comparing the steepness-skewness interaction and KHM across a wide range of random team hierarchies, rather than across ordered hierarchies that organizational contexts imply, the simulation studies provide a null hypothesis against which scholars can compare observations in empirical settings (Krackhardt, 1994). In the previously empirical examination, for example, the multiple
correlation between the steepness-skewness interaction and KHM was .09. The results from the simulation studies suggest that this correlation is as expected considering the distribution of influence in the empirical examination, in which on average 41.6% of the individuals received influence within the dyads and the average skewness was around zero (-0.01).
CONCLUSION
The goal of this chapter was to create more clarity about when and why to measure hierarchy either as the steepness-skewness interaction or as Krackhardt’s hierarchy measure (KHM). I argued that scholars should choose a hierarchy operationalization depending on how they conceptualize hierarchy. Throughout this chapter, I elaborated on differences and similarities between two prominent hierarchy conceptualizations, disparity and acyclicity, and explained that respectively the steepness-skewness interaction and KHM are appropriate
operationalizations for these conceptualizations. To substantiate the explanation empirically, I demonstrated the difference between the steepness-skewness interaction and KHM based on organizational data and simulated data.
Figure 13: Summary of when and why to operationalize hierarchy as the steepness-skewness interaction and KHM.
The empirical examination and the simulation studies revealed a weak overall relationship between the steepness-skewness interaction and KHM. The strongest overlap between the two operationalizations was found for 3 person
teams and when the data contained no information about the extent to which individuals have influence over each other. Under these circumstances, teams with higher KHM values often have a steeper hierarchy with a more positive skewness, thus have pyramid-shaped hierarchies ( ). In contrast, when hierarchy skewness is more negative (hierarchies more shaped as inverted pyramids ), the relationship between the steepness-skewness interaction and KHM was very weak. The relationship was weaker as well for larger teams. In larger teams, more complex rank orders between dyads are possible. The steepness-skewness interaction is not sensitive to these complex rank orders because it focuses on the overall rank of individuals. Collectively, the empirical examination and the simulation studies clarifies that the steepness-skewness interaction and KHM should not be used interchangeably.
By answering when and why scholars should consider the steepness-skewness interaction and KHM as operationalizations of hierarchy, this chapter contributes to the hierarchy literature by facilitating informed decision-making about how to operationalize hierarchy. Informed decision-making reduces mistaken research conclusions due to misfits between operationalizations and conceptualizations (Harrison & Klein, 2007). A misfit such as operationalizing acyclicity as the steepness-skewness interaction (an operationalization that is appropriate for the measurement of disparity), will result in findings that are based on variance in disparity instead of acyclicity. The simulation studies highlighted the difference between KHM and the steepness-skewness interaction and provided a null hypothesis against which scholars can compare real-world observations. Comparing real-world observations with null-hypotheses is useful while evaluating
the degree to which study findings are caused by a chance process. This evaluation is needed to understand the generalizability of findings. A null-hypothesis suggesting a weaker statistical relationship would imply that the operationalizations of disparity and acyclicity are more distinct, hence that studies on disparity and acyclicity are more likely to produce different outcomes. To assist scholars in choosing concepts and measurements of hierarchy, Figure 13
summarizes this chapter’s recommendations to when and why hierarchy should be operationalized as the steepness-skewness interaction and KHM. Following these recommendations will help accumulating studies on the impact of hierarchy, and may ultimately benefit the understanding of hierarchy’s impact.
