Predicting social influence with faction sizes and relative status

Social Science Research 42 (2013) 1346–1356

Contents lists available at SciVerse ScienceDirect

Social Science Research journal homepage: www.elsevier.com/locate/ssresearch

Predicting social influence with faction sizes and relative status David Melamed a,⇑, Scott V. Savage b a b

Department of Sociology, University of South Carolina, Columbia, SC 29208, United States Department of Sociology, University of California, Riverside, CA 92521, United States

a r t i c l e

i n f o

Article history: Received 8 December 2012 Revised 3 June 2013 Accepted 6 June 2013 Available online 18 June 2013 Keywords: Factions Group processes Mathematical sociology Social influence Status

a b s t r a c t Building on a recent theoretical development in the field of sociological social psychology, we develop a formal mathematical model of social influence processes. The extant theoretical literature implies that factions and status should have non-linear effects on social influence, and yet these theories have been evaluated using standard linear statistical models. Our formal model of influence includes these non-linearities, as specified by the theories. We evaluate the fit of the formal model using experimental data. Our results indicate that a one-parameter mathematical model fits the experimental data. We conclude with the implications of our research and a discussion of how it may be used as an impetus for further work on social influence processes. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction The science of social influence has identified several factors that promote influence. Sociologists have shown that both status characteristics (e.g., Berger et al., 1977; Berger and Webster, 2006) and the structure of relationships among individuals (Friedkin, 1998; Friedkin and Johnsen, 2011) shape influence processes. Psychologists have shown that the distribution of opinions in a group or faction sizes shapes social influence (e.g., Latane, 1981; Tanford and Penrod, 1984). Until recently, little research investigated the combined effects of these different factors. Two separate lines of research, however, have changed this. First, Kalkhoff et al. (2010) integrated two sociological accounts of social influence by formally linking status and network processes. They showed that weighting interpersonal associations within a network by status generates more precise predictions of social influence, thereby combining interpersonal influence with status-based influence in one formal framework. Second, Melamed and Savage (2013) united psychological and sociological research on social influence by showing how faction sizes and status processes combine to affect social influence. One key difference between these two approaches is that the latter postulates a non-linear relationship between status and social influence. It is this second line of inquiry and the non-linear relationship between status and social influence that concerns us. Melamed and Savage’s (2013) theory of social influence links faction sizes and status processes to social influence through the intervening mechanism of uncertainty reduction. It argues that both status information and faction sizes combine to affect perceptions of uncertainty about an outcome, and as a result, social influence. Here, social influence refers to a change in an individual’s thought or behaviors as a result of interacting with others (Rashotte, 2007). One of the implications of their theory is that people rely on status processes most when faction size fails to reduce uncertainty (i.e., when the group is evenly divided). Using standard statistical procedures, they found that faction sizes and relative status interact in their effect on social influence. ⇑ Corresponding author. Address: Department of Sociology, University of South Carolina, Columbia, SC 29208, United States. E-mail address: [email protected] (D. Melamed). 0049-089X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ssresearch.2013.06.002

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

1347

In the present paper, we extend this line of inquiry by developing a formal mathematical model consistent with Melamed and Savage’s theory of social influence. The model estimates a single scaling parameter from the data. We then fit the model to experimental data. Results indicate that the single parameter formal model fits the experimental data. Below we review Melamed and Savage’s theory of social influence. We then formalize it, describe the data, and evaluate the model. We conclude with a general discussion and directions for future research.

2. Theories of social influence The theoretical model developed in Melamed and Savage (2013) integrates research on faction sizes with Status Characteristics Theory. Consequently, we briefly review these literatures before describing the logical argument. Building on the seminal works of Asch (1951, 1956) and Moscovici (1980), several psychological models have been developed to predict how much influence an individual experiences when confronted with different faction sizes or numerical distributions of opinions. Here, a faction refers to a subset of individuals within a group that share a common opinion. Although there may be more than two opinions on many matters, presently we focus on a binary task implying that each actor may belong to one of two possible factions. Social Impact Theory (Latane, 1981), the Social Influence Model (Tanford and Penrod, 1984), and Self-Attention Theory (Mullen, 1983) are all attempts to mathematically model how the number of others in disagreement impacts or affects individuals. These models have in common the fact that the focus is on the distribution of opinions, and that it has a large effect on predictions of social influence. More recent research in this area has examined how faction sizes interact with other factors, such as the size of the minority faction (Gordijn et al., 2002), argument quality (Tormala and DeSensi, 2009), and prior attitudes of group members (Erb et al., 2002). Very little research, however, investigates how social status and factions combine to shape social influence. This is problematic, because as repeated tests of Status Characteristics Theory (SCT) have demonstrated, status characteristics also affect social influence (see Berger and Webster, 2006, for a review). An attribute is a status characteristic if it involves at least two states that are differentially evaluated, and each state has a distinct specific expectation state. This description characterizes specific characteristics, such as mathematical or gardening abilities. States of diffuse status characteristics, such as education or sex, are also associated with similarly evaluated general expectation states. The theory applies to collectively-oriented and task-focused groups. Given those conditions, the theory argues that any status-valued attribute, such as race, sex, or specific abilities, that differentiates the group members will become salient and affect the formation of expectation states for each person in the group, with individuals higher on status characteristics having more positive expectation states. Subsequently, the theory argues that social influence (and other behavioral indicators of inequality) will be a direct function of expectation states.1 Given the same scope conditions as SCT (e.g., task and collectively-oriented groups working on a binary veridical task), Melamed and Savage (2013) argue that people process both the distribution of opinions and the distribution of status when determining the correct solution. Based on principles of cognitive efficiency, they argue that people process the distribution of opinions first, which is consistent with the extant social cognition literature (e.g., Fiske and Taylor, 1991). If the distribution of opinions does not result in a state of consensus, they argue that people subsequently seek out other pertinent information, such as status, which, by definition, implies competence. They further argue that when the distribution of opinions reduces the least amount of uncertainty – that is, in situations with evenly divided factions – other factors should have the strongest effect. Thus Melamed and Savage’s theoretical model implies that the effect of status on social influence should be strongest when the group is evenly divided in terms of their initial opinions, and the effect should be weakest when there is consensus (or near consensus). It also implies that these processes apply to settings where individuals of different statuses report their initial opinions independently and concurrently and know the distribution of these initial opinions before making a final decision. Each iteration of a jury deliberation is an example of such a setting.2 Melamed and Savage (2013) evaluated their theoretical argument with experimental data. Their results supported the theory’s logic, with status, faction sizes, and their interaction all predicting social influence. Their results came from estimating a series of generalized linear mixed models predicting whether a participant was influenced on each of twenty ‘‘critical’’ trials. The predictors were (1) the participant’s faction size, which varied at the trial level, (2) the relative status of the participant’s faction, which is an a priori quantification based on the mathematics of SCT, and (3) the interaction of (1) and (2) (plus a constant term). Other theories of social influence are more mathematically-oriented. SCT (Berger et al., 1977), Social Impact Theory (Latane, 1981) and Social Influence Network Theory (Friedkin, 1998) estimate very little from the empirical data in order to make predictions. Social Influence Network Theory, for example, only estimates the participant’s perception of how influential each person was (c.f., Kalkhoff et al., 2010). Thus our aim is to ‘‘civilize’’ (Skvoretz, 1983) Melamed and Savage’s theory of influence by quantifying it. Below, we describe how we do so. 1

For a formal presentation of the theory, see: Berger et al. (1977: 107–130), Kalkhoff and Thye (2006: 221–222), or Webster et al. (2004: 742–743). The independent and concurrent reporting of initial opinions does not always happen. In many situations, high status actors express their opinions first, effectively influencing and silencing lower status others before they can express their first impressions. Our research does not speak to these situations. We leave it for future work to examine the evolution of social influence (see the discussion section). 2

1348

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

3. Formalizing the model Melamed and Savage’s (2013) theoretical model in its current form makes general assertions about the effects of status and faction size. Such hypotheses can be tested generally using standard statistical analyses that incorporate appropriate variables (e.g., faction sizes and some indicator for relative status). We try to improve upon this by creating a formal mathematical model that can generate precise point predictions of influence for situations where more than two individuals work on a binary task.3 We leave it for future research to generalize this model to other types of outcomes (e.g., multinomial or continuous). Our mathematical model draws upon the formalisms of SCT to account for the joint effects of status processes and faction sizes. Like Melamed and Savage’s theoretical model, the formal mathematical model begins with the distribution of opinions. As the number of influential others increases, the effect of the number of influential others should increase rapidly (Latane, 1981), until it reaches a certain critical mass where we expect the marginal effect to begin to decrease. This implies that the shape of the effect of the number of influential others should resemble an S-shaped, third-order production function; that is, the effect is non-linear and as the number of influential others increases the effect starts small, increases rapidly, and then decreases quickly. Since the present aim is to generate such an S-shaped function, the model begins with a function that is similar to the logistic cumulative distribution function (see e.g., Forbes et al., 2011: 127). This function is as follows:

 1 ðQðN=2ÞÞ PðsÞ ¼ 1 þ eð a Þ

ð1Þ

In Eq. (1), P(s) refers to the probability of a stay response (rejecting influence from others and staying with an initial opinion), e refers to the mathematical constant, Q refers to the size of the focal individual’s faction, N refers to the size of the entire group, and a is scaling parameter that needs to be estimated from the data (preferably via maximum likelihood estimation). Modeling a focal faction size in this manner implies that as the faction increases in size, so does the probability of resisting influence from others, and the rate of acceleration and deceleration of the effect of the focal faction size is determined by the parameter a. Modeling the effect of the distribution of opinions like the logistic cumulative distribution function allows the effect to be non-linear, and therefore allows the distribution of status to be included in a non-linear and non-additive fashion. Following Melamed and Savage (2013), we sum the expectation standing for each actor in the focal actor’s faction to quantify the relative status of the focal actor’s faction (please see Appendix A on how to compute expectation standings). In short, the expectation standing is an a priori estimation of the amount of status possessed by each individual in a setting. Estimated values are a normed version of each individual’s aggregate expectation state values, which are obtained by modeling the graphic representation of a status situation (in addition to Appendix A please see, Berger et al., 1977; Fisßek et al., 1991; Kalkhoff et al., 2010; Whitmeyer, 2003). Eq. (2) extends Eq. (1) to include the effects of status via the total expectation standing of the focal individual’s faction (Eq).

PðsÞ ¼ 1

.

1 þ eð

ðQ 1ÞðN=2Þ

a

 h .  . i ðQþ1ÞðN=2Þ ðQ1ÞðN=2Þ Þ þ E 1 þ eð a Þ  Eq 1 þ eð a Þ q

ð2Þ

Eq. (2) simultaneously models the distribution of opinions and the distribution of status. Because of the general shape of the function in Eq. (1), the effect of status in this model is largest when the group is evenly split (i.e., as Q approaches N/2), and smallest near either state of consensus (i.e., as Q approaches N and as Q approaches 1). Functionally status operates as a weight, conditioning the effect of the focal individual’s faction size. One additional noteworthy point out about Eq. (2) is that it still only requires one term to be estimated from the data (a). To illustrate, Fig. 1 presents hypothetical predicted P(s) values for a group of six individuals.4 The box plots in the figure illustrate how the predictions change as the summed expectation standing for the focal faction changes for all possible faction sizes. When no one agrees with a focal individual (i.e., a faction of 1) variation in the expectation standing of the faction leads to a .2 change in the P(s). Likewise, when all but one agree with a focal individual (i.e., a faction of 5) variation in the expectation standing of the faction leads to a .2 change in the P(s). When the faction size is 3, variation in the expectation standing of the faction leads to a .42 change in the predicted P(s). Thus, when the factions are evenly divided, the distribution of opinions cannot be relied upon to reduce uncertainty and the predicted effect of status is strongest. We evaluate the fit of the formal model to determine the extent to which the theory generates predictions that are consistent with the available empirical evidence. It is not our aim to create a model solely to capture the variation that is observed in an artificial laboratory setting. Rather, we set out to construct a model that reflects the data generating mechanisms outlined by the theory and operationalized in a laboratory setting (Balkwell, 1991a). If the model cannot fit those data, then future applications of the model are groundless. Thus to evaluate our formal mathematical model, we analyze Melamed and Savage’s (2013) experimental data. A full description of the experimental protocols may be found in that 3 In this respect, our research is in line with much theoretical work in the expectation states tradition which produces a priori numerical predictions that are then tested against experimental data (e.g., Berger et al. 1985; Fisßek et al., 1991; Berger and Fisßek 2006). 4 The predicted values in Fig. 1 are estimated assuming an alpha of 1. In general, as alpha increases from 1, the effect of status in the model decreases, and as alpha decreases from 1, the effect of status in the model increases.

1349

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

1.0 .20

0.8 .34

.11

.19

0.6 .42

.24

0.4 .34

0.2

.20

.19

.11

0.0 Faction of 1

Faction of 2

Faction of 3

Faction of 4

Faction of 5

Fig. 1. Box plots of predicted P(s) values for a hypothetical group of six individuals. Lower and upper limits reflect an expectation standing of .05 and .95, lower and upper limits of the box reflect an expectation standing of .25 and .75, and the dashed line reflects an expectation standing of .5. A faction of 6 is not reported as that would constitute a state of consensus. An alpha (a) of 1 is assumed.

Table 1 Summary of experimental design. The between-subjects manipulation constitutes the rows and a sub-set of the within-subjects manipulation constitutes the columns. Status structure of the group

H,H,H,M H,H,M,L H,M,L,L M,L,L,L

Status and number of Os in agreement by trial 1

2

3

4

5

6

7

H, H H, H H, L L, L

H, H H, L H, L L, L

H, H, H H, H, L H, L, L L, L, L

H L L L

Ø Ø Ø Ø

H, H, H H, H, L H, L, L L, L, L

H H H L

Note: L = Low Status; M = Medium Status; H = High Status. The participant’s status is in bold.

publication. We review key elements of the protocol and procedures below. We then describe how the model was used to estimate predicted values and evaluate the predictions. 4. Method The experiment used a variant of the standard experimental setting (SES; Berger et al., 1977: 43–48; Berger, 2007) for tests of Status Characteristics Theory.5 The SES was adapted to account for four participant groups rather than the typical dyad. Melamed and Savage (2013) only analyzed groups of four. Subsequent data will be required to evaluate the formal model in groups of varying sizes. 4.1. Design and subjects The experiment was a 4  4 mixed design crossing the status distribution of the group with the number of others agreeing with the participant’s initial choice on each of 25 trials. The manipulations in the experiment are between subjects (the status distribution) and within subjects (number in agreement). The participant’s (referred to as P) location in the status distribution was varied by assigning scores from a fictitious ability test to each of the four individuals in the group. Table 1 summarizes the experimental design by presenting the number and relative status of Os in agreement for the first seven trials of the experiment.6 In all conditions the participant was given an average score. The partners (referred to as Os, for others) were assigned either (1) all above average scores, (2) two above average scores and one below average score, (3) one above average score and two below average scores, or (4) all below average scores (see the rows of Table 1). On each of 25 trials, zero, one, two or three of the partners agreed with the participant’s initial choice. The sequence of which was randomly generated and consistent for every participant in the experiment. In trial number two, for example, two others agreed with the participant. In condition 1, both others were high status. Thus the participant’s faction disagreed with a high status O. In conditions 2 and 3, one other was high status and the other was low status. So in condition 2, the participant’s faction disagreed with a high status O, while in condition 3 the participant’s faction disagreed with a low status O. In condition 4, both others in agreement were low status, and the disagreeing O was also low status. Over the course of the 25 trials, all combinations of number in agreement and 5 6

Portions of the ‘‘Method’’ section have been adapted from Melamed and Savage (2013). We only report information for 7 of the 25 for the sake of brevity.

1350

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

relative status of those in agreement were varied. Participants were undergraduate females from a large public university. Eighty participants were randomly assigned to one of four status distributions.7 4.2. Procedures Upon arriving at the experimental laboratory, participants were directed to isolated rooms and informed that they would be interacting with several partners over a computer network in a two phase experiment. In reality, partners were simulated actors that behaved in predetermined ways, which enabled control of their behavior. After the brief introduction to the study, the remainder of the experiment was computer mediated. The instructions addressed three important aspects of the experiment. First, participants were informed that they would be paid for their performance in the experiment. Instructions stated that the minimum the individual could earn was $6.00 but that she could earn an additional $.125 for each trial she got correct in phase two and $.125 for each trial that each of her three partners got correct in phase two.8 The earnings were used to motivate participants to be both task and collectively oriented: making earnings contingent on performance should increase task orientation, while making earnings dependent on partners should increase collective orientation. This operationalization of valued outcomes meets the SCT assumption that actors are working on a valued task in a manner consistent with other experimental work in sociology which operationalizes valued outcomes in monetary terms (e.g., Markovsky et al., 1988; Molm, Collett and Schaefer, 2007; Ridgeway and Correll, 2006). Second, the instructions introduced the experimental task: contrast sensitivity. Contrast sensitivity was described as a relatively new ability that is not associated with other abilities such as mathematical or verbal skills. Participants were shown a rectangular image, composed of black and white areas and were asked to select which color constituted the majority of the image. They were told that there is a correct answer to each trial and that the test in phase one, which was designed for student populations, would measure the presence of contrast sensitivity ability. In reality, contrast sensitivity is not a real ability and there is no correct answer (Moore, 1968). Third, the instructions were used to create the initial conditions of the theoretical model. The instructions described the team portion of the experiment as a ‘‘critical choice’’ situation in which taking others’ opinions into consideration leads to an increased likelihood of making a correct decision. This portion of the instructions was intended to increase participants’ collective orientation. Phase one of the experiment consisted of a practice trial with on-screen instructions followed by twenty trials of contrast sensitivity problems. Participants were given ten seconds to decide whether an image consisted of more white or black area, after which they were prompted to make a final choice. After phase one, the participant’s and the three partners’ scores were reported to the participant. This constituted the manipulation of the distribution of status. To ensure participants received the manipulation and that this information was salient to them, they were asked to write down these phase one scores on a table that was provided to them. Phase two of the experiment consisted of twenty-five trials of contrast sensitivity problems. The phase two task differed slightly from the phase one task. In phase two, participants were first given ten seconds to initially indicate which of two images contained the most white area. Then they were shown all three of their partners’ initial opinions, given ten more seconds to study the images, and were asked for their final opinions. After phase two, participants completed a brief questionnaire and were debriefed. 4.3. Manipulations The between participants manipulation was the distribution of relative status. Feedback from the phase one contrast sensitivity test was used test to manipulate status between the participant and her three partners. Phase one consisted of twenty trials so the scores ostensibly ranged from zero to twenty. A conventional interpretation for scores was used: zero–ten correct indicates a poor performance and lack of ability, eleven–fifteen correct indicates an average ability level, and sixteen–twenty correct indicates a superior ability level (e.g., Berger and Fisßek, 1970: 295). In all four conditions the participant was told that she scored a fourteen, one more than the ‘‘national average.’’ High status others were assigned a score of eighteen or nineteen, and low status others were assigned a score of seven or eight. In the first condition the participant interacted with three high status others. In the second condition the participant interacted with two high status others and one low status other. In the third condition the participant interacted with one high status other and two low status others. In the fourth condition the participant interacted with three low status others. Thus in condition one, the participant was low status and accounted for a small amount of the relative status in the group; in condition four, the participant was high status and accounted for a large amount of the relative status in the group. The within-participants manipulation was the number of partners that agreed with the participant’s initial opinion in phase two of the experiment. No matter what the participant’s initial choice was, the program systematically informed participants whether zero, one, two, or all three of the others agreed with the participant’s initial opinion on each of the twentyfive trials. The number of individuals who agreed with the participant, as well as which of the others would agree with the 7 8

Six of the original cases were eliminated and re-ran. They were eliminated for violating scope conditions or failing manipulation checks. In reality, all participants were paid $15.00 for participating in the study. They did not know this until the debriefing phase of the experiment.

1351

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356 Table 2 Number of trials, observed proportion of stay responses, and expected proportion of stay response by experimental conditions. Condition Number of trials

Observed P(s)

Expected P(s)

|Observed  Expected|

1. H, H, H None H H, H

Status of Os in agreement with P

140 120 140

0.05 0.37 0.99

0.02 0.36 0.84

0.03 0.01 0.15

2. H, H, L None H L H, H H, L

140 80 40 40 100

0.09 0.79 0.18 1.00 0.96

0.03 0.50 0.13 0.97 0.78

0.06 0.29 0.05 0.03 0.18

3. H, L, L None H L L, L H, L

140 40 80 40 100

0.06 0.98 0.29 0.70 1.00

0.05 0.80 0.20 0.65 0.95

0.01 0.18 0.09 0.05 0.05

4. L, L, L None L L, L

140 120 140

0.26 0.83 0.99

0.35 0.80 0.95

0.09 0.03 0.04

Note: L = Low Status; H = High Status.

participant for trials involving one or two others agreeing, was randomly assigned. In all conditions the participants experienced the same sequence of number of others agreeing with them. On trials when others agreed with the participant, the same others agreed with all of the participants. On trial number one, for example, two others agreed with the participant and they were always the first and third others in the group. Per the between participants manipulation, however, the status of the others that agreed or disagreed with them varied systematically. 5. Analytic strategy Our unit of analysis is not the experimental participant, but rather configurations of faction size and relative status. Our outcome is the proportion of stay responses in phase two within each of these configurations. This is analogous to SCT researchers modeling the proportion of stay responses in their experiments (e.g., Webster and Driskell, 1978), except that we have to also account for a within-subjects manipulation. We exclude from our analyses trials in which all others agree with the participants’ initial opinions, as consensus presents no opportunity for social influence.9 Eliminating consensus configurations left sixteen configurations of faction sizes and relative status. For example, in condition one, where the participant is interacting with three high status others, there are three configurations: (1) none of the others might agree with the participant’s initial opinion, (2) one of them might, or (3) two of the others might. Since all three others have the same status, it does not matter which of them agree or disagree on any given trial. In general, there are three configurations in conditions one and four (because all of the O’s have equal status) and five configurations in conditions two and three (because when one or two agrees with P they may be either high or low status). Twenty participants in each condition with sixteen configurations of faction sizes and relative status provides a sample space of 320 units of analysis. 6. Results Table 2 presents the number of trials and the proportion of stay responses (P(s)) by the between and within factors and provides clear descriptive support for Melamed and Savage’s (2013) theoretical model. Within conditions, as the number in agreement with the participant increases, so does the proportion of stay responses. Likewise, as the relative status of the others in agreement increase, so does the proportion of stay responses. Previous analyses of these data also show that the effect of status on influence is at its maximum when only one other agrees with the participant (Melamed and Savage, 2013). With this descriptive evidence, we now turn to the formal evaluation of our mathematical model. First, we obtained the maximum likelihood solution to a, the dispersion term, by maximizing the kernel of the likelihood function for a.10 This produced an estimate of .083 (e.g., Eliason, 1993: 10, Eq. (1.9)). Using this estimated value of a, we then relied on the formal model 9 This is analogous to SCT experiments that only analyze the ‘‘critical trials’’ (Berger et al., 1977: 48; Kalkhoff and Thye, 2006: 229–230), in which the two actors disagree. 10 All of the expectation state mathematics computed in this paper use the Fisßek et al. (1992) path function equations.

1352

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

to generate point predictions for the P(s) values for all of the combinations of faction sizes and relative status (see Appendix B for two examples of using the model to generate point predictions). Table 2 presents these estimates. Table 2 also presents the absolute difference between the observed and expected P(s) values. Comparing the observed values to the predicted values in Table 2 illustrates that the predictions from the model appear to be quite good. There is, however, some local-lack-of-fit, although in only four cells of the sixteen is the difference between predicted and observed P(s) values above .10. To formally evaluate the fit of the predicted values we used the predicted P(s) value to generate a model of expected cell counts of stay responses and trials influenced11 for each of the sixteen units of analysis and computed a v2 goodness-of-fit test between the predicted values and the observed. The results of this test indicate that the predicted cell counts are significantly different from the observed cell counts (v2ð15Þ ¼ 94:59, p < .001). The model does not fit the data when comparing the observed Chi-squared test statistic to the theoretical distribution on 15 degrees of freedom. As recently pointed out by Fisßek and Barlas (2013), however, when data is aggregated to compute goodness-of-fit, the Chi-squared value becomes inflated (see e.g., Tavaré and Altham, 1983).12 In this case, aggregating repeated trials on participants may be inflating the Chi-squared test statistic. To account for this inflation, Fisßek and Barlas recommend comparing the observed Chi-squared statistic to a Monte Carlo distribution of possible Chi-squared values that result from permuting the original data. To adjust for the inflation in the Chi-squared test statistic, we adapted Fisßek and Barlas’ (2013) procedure to our setting. The permutation procedure we adopted proceeded as follows: (i) we permuted the observed P(s) values, (ii) maximized the kernel of the likelihood function to obtain a, (iii) estimated the fitted values, (iv) computed the Chi-squared test statistic, and (v) repeated steps i through iv 10,000 times.13 The result is a distribution of Chi-squared statistics that are based on permutations of the data. Comparing the observed Chi-squared value to the permuted distribution allows us to compute a p-value that adjusts for the aggregation of participant data. The estimated p-value is .9996, indicating that the model fits the data quite well. Thus after accounting for the inflation of the Chi-squared test statistic, the formal mathematical model of social influence fits the experimental data very well. Despite the model fitting the data, scrutinizing the differences between the observed and the expected P(s) values in Table 2 reveals a weakness in our model. In those spots where the difference between the observed and the expected P(s) score is at least .1, the model is under-estimating the P(s). In fact, out of 16 configurations of status and numbers, the formal model under-estimates the P(s) in all but one of the configurations. Although we can only speculate as to why this is the case, one possibility is that our model does not contain a self-bias parameter. Analyses of the standard experimental setting for tests of Status Characteristics Theory find time-and-time-again that the baseline propensity to reject influence is above chance (.63, see Kalkhoff and Thye, 2006: 237), indicating a non-negligible self-bias. Perhaps incorporating such a self-bias parameter would enhance the performance of our mathematical model. We do not attempt such a model here because the model as it stands fits the experimental data. 7. Discussion and conclusion Melamed and Savage’s (2013) theoretical model of social influence addresses a shortcoming in the social psychological literature by offering a theoretical explanation for how numbers modify the effect of status on influence. Specifically, it argues that in a collective task group consisting of more than two people working together to solve a problem without a clear solution, the need to reduce uncertainty will lead people to rely on information about the distribution of opinions and the status of individuals who hold particular opinions to decide whether or not to submit to the pressures of social influence. It further contends that status becomes increasingly important as the number of people who hold different opinions becomes evenly split. Previous empirical research establishes these points with a four parameter statistical model, thereby highlighting the fact that the effects of status on social action are highest in highly uncertain situations (Melamed and Savage, 2013). Despite this support, the generalized linear mixed model previously used to test the theory only roughly approximates the theory’s explanation for how numbers and status combine to affect social influence. The logic of the theory points to the non-linear effect of faction size, with the effect of an increase in the number of others who agree initially being small, increasing rapidly, and then decreasing quickly. It also indicates that the effects of the distribution of status should become more pronounced as the opinions of group members about the correctness of a decision become evenly divided. Both of these observations suggest that we can better model the effects of the distribution of opinions and status by relying on an S-shaped production function that conditions the effect of the focal individual’s faction size with status. Our formal mathematical model does this and as a result, more accurately reflects the logic of Melamed and Savage’s (2013) theoretical model. Only estimating one parameter from the data is a marked improvement over the standard model, which estimates four parameters, and occurs because of the functional form of our formal mathematical model. Our model consists of a highly non-linear cumulative distribution function modified to account for status organizing processes, while a standard regression model assumes that the effects of status and factions are linear and additive and that their interaction is multiplicative. Put 11

On any given trial, a participant could have stayed with their initial opinion, or deferred to those attempting to influence her. We thank an ‘anonymous’ reviewer for pointing out that the Chi-squared statistic is inflated due to data aggregation. 13 The R code for either the algorithm for the ML solution to alpha or the permutation test is available from the first author upon request. The latter requires the former. 12

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

1353

differently, assuming that the right hand side of the statistical model is multivariate normal is not necessarily a valid assumption given what we know about social influence processes, and correcting for this may account for why the formal model fits the data better than the more traditional regression model. If this is the case, it suggests that many applications of ‘‘canned’’ statistical models may be sensitive to the actual form of the relationship between the factors giving rise to the data, and assuming that points in the sample space are drawn from a multivariate normal distribution may lead to biased conclusions. Relying on the language of mathematics to accurately model the logic of existing sociological theories, then, may help us better understand the social world and make predictions about it. Various existing lines of research, such as that of network exchange theory (for a review, see Willer (1999)), reveal the merits of this approach. The importance of the non-linearities in our model brings us back to the difference between our approach to social influence and that of Kalkhoff et al. (2010). While we focus on how the effect of status is moderated by the distribution of opinions, they focus on how status may operate as a weight to interpersonal associations. One clear benefit to their approach is that the focus on a network of interpersonal associations allows them to model social influence in any size group whether or not individuals are directly connected. We assume that all relevant parties are members of a faction and that their opinions are known. One shortcoming relevant to both our approach and Kalkhoff et al.’s (2010) is that initial opinions and relative status are assumed to be known while each group member is making his or her final decision.14 In many situations higher status individuals indicate their choices first, which pressures lower status individuals to defer to them. Put differently, we focus on a limited range of influence situations – situations in which each individual knows the initial opinions of others and then makes a final decision. Research in this area should also focus on the evolution of social influence, or how relative status and the order of professed opinions affect influence. In addition, future research should test the utility of our mathematical model for groups of different sizes. Although we find evidence in support of our mathematical model, we recognize that with a four-person group it is difficult to establish empirically between the fit of different functions.15 Research on collective action tells us that it is important to understand differences among production functions if we are to understand the choices of interdependent actors (Oliver et al., 1985). Future research, then, should examine whether the mathematical function underlying our model is appropriate for groups larger than four. In sum, we presented and experimentally evaluated a theoretically derived formal model of how numbers and status combine to affect social influence in collectively oriented, task-focused groups. The formal model does not fit the data, but it does fit the data better than a statistical model, which estimates three additional parameters from the data. Thus we view the formal model as a contribution to our understanding of social influence. The issue of local-lack-of-fit, however, prompts future theoretical and mathematical revisions, which we hope will further our understanding of social influence processes even more. Acknowledgments An earlier version of this paper won the Graduate Student Paper Award from the Mathematical Sociology Section of the American Sociological Association (2012). This research was supported by the mathematical sociology section of the American Sociological Association (Dissertation-in-Progress Award to David Melamed, 2010) and the National Science Foundation (Dissertation Improvement grant SES-1029068) to Linda Molm and David Melamed. The first author thanks Ronald Breiger for his support on a Defense Threat Reduction Agency grant (HDTRA1-10-1-0017). We also thank Will Kalkhoff and three anonymous reviewers for helpful feedback that improved the manuscript. Appendix A. Computing expectation standings The graphic representation of SCT visually represents the elements and relations found in the theory’s logical core. Fig. A1 represents the status situation found in Condition 2 of the experiment, although for simplicity, only one of the high status Os is illustrated in the graph. Condition 2 (and 3) has three salient states of contrast sensitivity ability, as an average focal actor (P) interacts with two high status others (Os) and one low status other (O). Because the mathematics of the expectation state theories yields positive values for high status individuals and symmetrically negative values for low status individuals, we treat P as having a neutral value (i.e., zero) and the high and low status Os as having positive and negative values, respectively. While we could have given P the same value as a low status O, that interpretation of the scores is inconsistent with recent research finding that magnitudes of difference on status characteristics matter for small group inequalities (Melamed, 2011, 2013). Holding P’s expectation state value as zero allowed us to compute the expectation state values for each of the Os. As Fig. A1 indicates, the high status O (O+) and her low status partner (O) are linked to differentially evaluated states of the instrumental task characteristic (C) through the possession relation.16 The different states of the instrumental task 14

See Friedkin and Johnsen (2003) for an exception. We thank an anonymous reviewer for making this point. 16 The instrumental task characteristic is a specific status characteristic that is the ability which relays competence at the groups’ task. If the group is working on a math problem, for example, then mathematics ability is the instrumental task characteristic. 15

1354

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

Fig. A1. Graphic representation of the experimental situation. —— = Possession Relation, — — = Dimensionality Relation, - - - - - = Relevance Relation.

characteristic are linked in Fig. A1 by the dimensionality relation. Each of the states of the instrumental task characteristic, in turn, is linked to a like-signed state of the task outcome (T) through the relevance relation, thereby indicating that possessing the positively evaluated state of the instrumental task characteristic should help one contribute to task success, while possessing the negatively evaluated state of the instrumental task characteristic should undermine one’s ability to contribute to task success. Importantly, the paths in the graphic representation have valences. Whether the path contributes positively or negatively to the expectation state value depends on whether the overall valence of the path is positive or negative. All relations are positive, except for the dimensionality relation, which is negative. The overall valence of a path is the product of the outcome state (either positive (T(+)) or negative (T())) and the sign of the path (either positive or negative through the dimensionality relation). Thus O+ has a positive two-path (O+–C(+)–T(+)) and a positive three-path through the dimensionality bond (O+–C(+)–() () C –T()). Graphic representations are necessarily symmetrical, implying that O has a negative two-path and a negative three-path (for more on the construction of graphic representations, please see: Berger et al., 1977: 100–113). Experimental evidence suggests that behavior in collective task groups unfolds as if individuals generate a positive subset of status information, a negative subset of status information, and then combine the two subsets into an aggregate expectation state value (Berger et al., 1992). The equations for these two subsets are given in Eqs. (A1.1) and (A1.2). In (A1.1) and (A1.2), x refers to a single individual in the graphic representation, f(i) refers to a function of a path of length i and f(n) refers to the nth or final path function. Eq. (A1.3) represents the aggregate expectation state value for actor x. Presently there are three ways to estimate the path functions (Balkwell, 1991b; Berger et al., 1977; Fisßek et al., 1992). We rely on Fisßek et al.’s (1992) path function equations, but all three can be used to generate expectation standings. Eq. (A1.4) presents the Fisßek et al. path function equation in its general form.

eþx ¼ ½1  ð1  f ðiÞÞ . . . ð1  f ðnÞÞ

ðA1:1Þ

ex ¼ ½1  ð1  f ðiÞÞ . . . ð1  f ðnÞÞ

ðA1:2Þ

ex ¼ eþx þ ex

ðA1:3Þ 2i

f ðiÞ ¼ 1  e2:618

ðA1:4Þ

Using Eq. (A1.4), we can generate the path functions from the graphic representation presented in Fig. A1. In Fig. A1, O+ has a positive two-path and a positive three-path. Substituting 2 for i in Eq. (A1.4) yields .632 and substituting 3 for i yields .317. Thus O+’s positive subset is [1  (1  .632)(1  .317)] = .749, and because O+ has no negative paths, O+’s negative subset is 0. Therefore O+’s aggregate expectation state value is .749. Through symmetry, the exact opposite is true for O, who has an aggregate expectation state value of .749. Thus O+’s aggregate expectation state value is .749, O’s aggregate expectation state value is .749, and P’s aggregate expectation state value is 0. The three states, therefore, are equally spaced through the symmetry of the graphic representation. The expectation standing is a simple normed version of the expectation state values. Fisßek et al. (1991) define the expectation standing as follows:

1 þ ex sx ¼ Pn j¼1 ð1 þ ej Þ

ðA1:5Þ

where sx refers to a focal individual’s expectation standing, ex refers to a focal individual’s aggregate expectation state value, and the summation is over all of the people in the situation. In condition two, therefore, P’s expectation standing is .211 (i.e., 1/(1 + .251 + 1.749 + 1.749) = .221). Each of the high status Os have an expectation standing of .368 (i.e., 1.749/ (1 + .251 + 1.749 + 1.749) = .368), and the low status O has an expectation standing of .053 (i.e., .251/ (1+.251 + 1.749 + 1.749) = .053). The formal model presented in Eq. (2) uses the total expectation standing for a focal faction. If, for example, one high status O agreed with the participant’s initial opinion, and they collectively disagreed with a high and a low status O, the total expectation standing of the faction would be .589 (i.e., .221 + .368 = .589).

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

1355

Appendix B. Generating point predictions from the formal model In this appendix, we derive the predicted values found in Table 2. Specifically we calculate the values found in the first row and third column and in the last row and third column of the table. To derive the predicted value (.02) for situations in which three high status Os disagree with the participant’s initial opinion (first row, third column), we substitute into Eq. (A2.1) the appropriate numbers for Q and N as well as the expectation standing, Eq. In this case, Q is one (the participant is the only member of the focal faction), N is four, and Eq is .16 (see Appendix A on computing expectation standings). The value of a was estimated via maximum likelihood estimation to be .083. Solving for Eq. (A2.2) yields .02.

 1  1  1  ðQ1ÞðN=2Þ ðQþ1ÞðN=2Þ ðQ1ÞðN=2Þ PðSÞ ¼ 1 þ eð a Þ þ Eq þ eð a Þ  Eq þ eð a Þ

ðA2:1Þ

 1  1  1  ð11Þð4=2Þ ð1þ1Þð4=2Þ ð11Þð4=2Þ :02 ¼ 1 þ eð :083 Þ þ :16 þ eð :083 Þ  :16 þ eð :083 Þ

ðA2:2Þ

When two low status Os agree with the participant’s initial opinion and they collectively disagree with one low status O (the last row and third column of Table 2), the predicted probability of a stay response is .95. In this case, the total expectation standing of the faction is .857 (i.e., P’s expectation standing is .57, plus two expectation standings for the low status Os of.143 = .857). Substituting this value into Eq. (A2.1) along with the appropriate values for Q (3), and N (4) and a (.083) yields Eq. (A2.3), which solves to .95.

 1  1  1  ð31Þð4=2Þ ð3þ1Þð4=2Þ ð31Þð4=2Þ :95 ¼ 1 þ eð :083 Þ þ :857 þ eð :083 Þ  :857 þ eð :083 Þ

ðA2:3Þ

References Asch, Solomon, 1951. Effects of group pressure upon the modification and dostortion of judgements. In: Guetzkow, H. (Ed.), Groups, Leadership, and Men. Carnegie Press, Pittsburgh, PA. Asch, Solomon, 1956. Studies of independence and conformity: a minority of one against a unanimous majority. Psychological Monographs 70 (9), 416. Balkwell, James, 1991a. From expectations to behavior: an improved postulate for expectation states theory. American Sociological Review 56, 355–369. Balkwell, James W., 1991b. Status characteristics and social interaction. Advances in Group Processes 8, 135–176. Berger, Joseph, 2007. The standardized experimental situation in expectation states research: notes on history, uses, and special features. In: Webster, M., Sell, J. (Eds.), Laboratory Experiments in the Social Sciences. Elsevier, New York, pp. 353–378. Berger, Joseph, Fisßek, M. Hamit, 1970. Consistent and inconsistent status characteristics and the determinations of power and prestige orders. Sociometry 33 (3), 287–304. Berger, Joseph, Fisßek, M. Hamit, 2006. Diffuse status characteristics and the spread of status value: a formal theory. American Journal of Sociology 111 (4), 1038–1079. Berger, Joseph, Webster, Murray, 2006. Expectations, status, and behavior. In: Burke, P. (Ed.), Contemporary Social Psychological Theories. Stanford University Press, Palo Alto, CA. Berger, Joseph, Fisßek, M. Hamit, Norman, Robert, Zelditch Jr., Morris, 1977. Status Characteristics and Social Interaction: An Expectations States Approach. Elsevier, New York. Berger, Joseph, Fisßek, M. Hamit, Norman, Robert, Wagner, David G., 1985. Formation of reward expectations in status situations. In: Berger, J., Zelditch, M., Jr. (Eds.), Status, Rewards, and Influence. Jossey-Bass, San Francisco, CA. Berger, Joseph, Balkwell, James, Norman, Robert, Smith, Roy, 1992. Status inconsistency in task situations: a test of four status processes principles. American Sociological Review 57 (6), 843–855. Eliason, Scott R., 1993. Maximum Likelihood Estimation: Logic and Practice. Sage, Thousand Oaks, CA. Erb, Hans-Peter, Bohner, Gerd, Rank, Susanne, Einwiller, Sabine, 2002. Processing minority and majority communications: the role of conflict with prior attitudes. Personality and Social Psychology Bulletin 28 (9), 1172–1182. Fisßek, M. Hamit, Barlas, Zeynap, 2013. Permutation tests for goodness-of-fit testing of mathematical models to experimental data. Social Science Research 42 (2), 482–495. Fisßek, M. Hamit, Berger, Joseph, Norman, Robert, 1991. Participation in heterogeneous and homogeneous groups: a theoretical integration. American Journal of Sociology 97 (1), 114–142. Fisßek, M. Hamit, Norman, Robert, Nelson-Kilger, Max, 1992. Status characteristics and expectations states theory: a priori model parameters and test. Journal of Mathematical Sociology 16, 285–303. Fiske, Susan T., Taylor, Shelley, 1991. Social Cognition. McGraw-Hill, New York. Forbes, Catherine, Evans, Merran, Hastings, Nicholas, Peacock, Brian, 2011. Statistical Distributions, fourth ed. Wiley, Hoboken, NJ. Friedkin, Noah, 1998. A Structural Theory of Social Influence. Cambridge University Press, New York. Friedkin, Noah, Johnsen, Eugene C., 2003. Attitude change, affect control, and expectation states in the formation of influence networks. In: Thye, S.R., Skvoretz, J. (Eds.), Advances in Group Processes. Elsevier, New York, pp. 1–29. Friedkin, Noah, Johnsen, Eugene C., 2011. Social Influence Network Theory: A Sociological Examination of Small Group Dynamics. Cambridge University Press, New York. Gordijn, Ernestine H., de Vries, Nanne K., Dreu, Carsten K.W., 2002. Minority influence on focal and related attitudes: changes in size, attributions, and information processes. Personality and Social Psychology Bulletin 28 (10), 1315–1326. Kalkhoff, William, Thye, Shane, 2006. Expectation states theory and research. Sociological Methods and Research 35 (2), 219–249. Kalkhoff, William, Friedkin, Noah, Johnsen, Eugene, 2010. Status, networks and opinions: a modular integration of two theories. In: Thye, S. (Ed.), Advances in Group Processes. JAI Press, New York, pp. 1–38. Latane, Bibb, 1981. The psychology of social impact. The American Psychologist 36 (4), 343–356. Markovsky, Barry, Willer, David, Patton, Travis, 1988. Power relations in exchange networks. American Sociological Review 53 (2), 220–236. Melamed, David, 2011. Graded status characteristics and expectation states. In: Thye, S., Lawler, E. (Eds.), Advances in Group Processes. Emerald Group, Cambridge, MA, pp. 1–31. Melamed, David, 2013. Do magnitudes of difference on status characteristics matter for small group inequalities? Social Science Research 42 (1), 217–229. Melamed, David, Savage, Scott V., 2013. Status, numbers and influence. Social Forces 91 (3), 1085–1104.

1356

D. Melamed, S.V. Savage / Social Science Research 42 (2013) 1346–1356

Molm, Linda, Collett, Jessica L., Schaefer, David R., 2007. Building solidarity through generalized exchange: a theory of reciprocity. American Journal of Sociology 113 (1), 205–242. Moore, James, 1968. Status and influence in small group interactions. Sociometry 31, 47–63. Moscovici, Serge, 1980. Towards a theory of conversion behavior. In: Berkowitz, Leonard (Ed.), Advances in Experimental Social Psychology, vol. 13. Academic Press, Inc., New York, pp. 209–237. Mullen, Brian, 1983. Operationalizing the effect of group on the individual: a self-attention perspective. Journal of Experimental Social Psychology 19 (4), 296–322. Oliver, Pamela, Marwell, Gerald, Teixeira, Ruy, 1985. A theory of the critical mass. I. interdependence, group heterogeneity, and the production of collective action. American Journal of Sociology 91 (3), 522–556. Rashotte, Lisa, 2007. Social influence. In: Ritzer, G. (Ed.), The Blackwell Encyclopedia of Sociology. Blackwell, Hoboken, NJ, pp. 4426–4429. Ridgeway, Cecilia, Correll, Shelley J., 2006. Consensus and the creation of status beliefs. Social Forces 85 (1), 431–453. Skvoretz, John, 1983. Salience, heterogeneity and consolidation of parameters: civilizing Blau’s primitive theory. American Sociological Review 48, 360–375. Tanford, Sarah, Penrod, Steven, 1984. Social influence model: a formal integration of research in majority and minority influences processes. Psychological Bulletin 95 (2), 189–225. Tavaré, Simon, Altham, Patricia E., 1983. Serial dependence of observations leading to contingency tables, and corrections to chi-squared statistics. Biometrika 70 (1), 139–144. Tormala, Zakary L., DeSensi, Victoria L., 2009. The effect of minority/majority source status on attitude certainty: a matching perspective. Personality and Social Psychology Bulletin 35 (1), 114–125. Webster Jr., Murray, Driskell Jr., James, 1978. Status generalization: a review and some new data. American Sociological Review 43, 220–236. Webster Jr., Murray, Whitmeyer, Joseph, Rashotte, Lisa Slattery, 2004. Status claims, performance expectations, and inequality in groups. Social Science Research 33, 724–745. Whitmeyer, Joseph, 2003. The mathematics of expectations states theory. Social Psychology Quarterly 66 (3), 238–253. Willer, David, 1999. Network Exchange Theory. Praeger, Westport, CT.