Network Effects in Binge Drinking by Patrick Mannion

Patrick Mannion Econ 970jl Term Paper Tutorial Leader: Jenn Larson Network Effects in Binge Drinking Sunday, May 1, 2011

I. Motivation The motivation for this paper is an annual presentation put on by the Harvard College Drug and Alcohol Peer Advisors, commonly known as DAPA. The group attempts to instruct the student body on issues of health and safety related to drug and alcohol use. A particular focus of DAPA during the annual presentation given to freshmen is to teach safe drinking habits, along with the facts and figures of alcohol use and abuse. A number and phrase that is continually reinforced is the concept of “binge drinking,” which constitutes the consumption of five or more standard drinks in one evening for men and four or more drinks for women. When students are asked to estimate the proportion of students that engage in such behavior, the DAPA officers say, their estimates are strongly correlated with a student’s own drinking habits and those of his or her friends. It is here that a network effect will be explored. Does a student’s position in a network have an effect on his estimate of the proportion of students who binge drink? This paper will first explore the relationship between a student’s network centrality and his estimate of the proportion of students who binge drink. Next, we will analyze the effect on these individuals of joining a group with non‐normal drinking behaviors, such as a fraternity or DAPA. In this second case, we will distinguish between the effect on more and less central individuals. II. Position in Field Work in this field has already shown students’ perceptions of their friends’ binge drinking influences how much they actually drink, and that students who binge drink are more likely to over‐estimate the proportion of binge drinkers in the population (Wechsler & Kuo, 2000). It has also been shown quantitatively with real‐world data that a social

network effect exists in binge drinking within the cohort of Framingham Heart Study patients (Rosenquist, Christakis et Al., 2010). In colleges, proportions of drinkers and heavy drinkers have been found to be significantly positively related to respondents’ own drinking (Fondacaro & Heller, 1983). Clustering is often seen here, as over time, people have been found to be likely to restructure their social groups to be more homogeneous in terms of members’ drinking levels (Leonard & Mudar, 2003, Watts, 2003). Other work has shown that exposure to “wet” environments where alcohol is readily available and more heavily consumed than average is related to heavier binge drinking (Weitzman, Nelson, Wechsler 2003). Even more specifically, it has been found that having a social network of individuals who drink heavily relative to the population is a predictor of an individual’s own binge drinking, independent of high school binge drinking habits (Reifman & Watson, 2003). Such higher levels of drinking, in turn, have been found to lead to a higher risk of later substance abuse and dependency (Chassin, Pitts, Prost 2002). III. Model Basics and Assumptions In this model, the nodes are college students. The links between nodes are the undirected social tie of “going out” on weekend nights together on a regular basis. This may be alone or in groups, drinking or not. Undirected ties are used with the forethought that we will be discussing the observation of network neighbors’ regular drinking habits over a period of time. If person A goes out often enough with person B so as to have an idea of B’s drinking habits, it is reasonable to assume that B will have a similar idea of A’s habits. In building a random network of these individuals, we must consider variables that would affect the values of or cause clustering among the drinking habits of nodes. These

include but are not limited to day of the week, gender, proximity, interests, and age, particularly as over/under 21. To start, only weekend days will be considered. Although those individuals who drink more heavily and binge drink more frequently are likely to also drink and be observed drinking during the week, this behavior is very plausibly endogenous to drinking behavior on the weekends. As a result, we will only consider weekend observation for simplicity’s sake. Next we consider the effect of gender. It is common among professionals (Reifman, Watson, McCourt, 820) to consider a man’s drinking to be classified as binge drinking when he consumes more than 5 drinks in a night. Meanwhile, a woman’s drinking is considered binge drinking when she consumes more than 4 drinks in a night. Students commonly quantify binge drinking a bit differently, as 6 or more drinks for a man and 5 or more drinks for a woman. To reconcile these differences, this model normalizes the differences and considers 5 drinks to be binge drinking status for any node, ignoring gender. This is based on the assumptions that an individual’s estimate of the global proportion of binge drinkers (as opposed to the number of drinks) is gender‐neutral, and that women are just as likely to fall on either side of their binge drinking threshold as men are. The latter of these assumptions may be contested by some, and is worth further consideration in empirical tests and future models. Here we will simply assign equally probable random values di for each node, the average number of drinks consumed by an individual i in a given evening. By random assignment, this value will come out to be around 5, making half the respondents binge drinkers and half not. This is approximately consistent with the data observed in Wechsler, Davenport et Al. 1994.

∀i ∃ di ∈ rand(0,1,2,3…10)

€

Real‐world proximity, for example status as roommates, may be ignored by assuming that rooming is randomly assigned, or by the consideration that an individual is unlikely to have more than 3 roommates freshman year. Even a biased sample this small would not be strong enough to greatly sway our model, as long as the rooming is not entirely based on a variable that covaries highly with drinking habits. Most schools assign rooming randomly in freshman year. When this is not the case, roommates are likely to have known each other before, meaning that a connection was already present independent of proximity. This does not rule out the possibility of clustering based on rooming, but rather is to say that such clustering would be based on variables such as age, interests, and group membership that are otherwise accounted for in the model. It should be noted that in some cases, such as at Harvard, rooming is based on a survey that would cause a high covariance between lifestyle and proximity, leading to potential endogeneity. The best that this model can do is to not claim to fully account for those cases. What the model does claim to account for, however, is the interests of individuals in sports, music, style of upbringing, and geographic origin. None of these variables is endogenous in and of itself to higher binge drinking levels, though admittedly individuals may cluster based on interests and develop a preference for binge drinking. This model will assume that such clustering occurs such that a group would be formed such as a intramural sports team, a band, Greek society or other club. Variance across these clusters based on interests is precisely the variable we are able to analyze. This will be done by testing the effect of membership in a non‐normal drinking group on individuals.

With respect to age, this model operates under the soft assumption that nodes are under 21, the legal drinking age. However, we assume that all nodes are able to acquire alcohol, through whatever means. The under‐21 assumption is only critical to the extent that it implies the cohesion of groups in social settings. Specifically, to measure the effect of a team, fraternity, or other group on an individual’s perception of binge drinking, we must be able to assume that the group drinks (or does not) while together on weekend evenings. Individuals who are over 21 may fragment away from such structure, or conversely may be a cohesive force by supplying alcohol to the group. Thus this is a soft assumption. For the sake of tractability, we will say that the model is most applicable to college freshmen entering college randomly assigned to roommates and forming initial friendships randomly based on sorting of housing, introductory classes, interests, etc. We assume here that no set of introductory classes is particularly predisposed to attract any certain type of drinkers, with an eye to the idea that binge drinkers may take classes that are considered easier. This model puts forward that such information would likely be learned from the network that is developed after the student arrives to school, perhaps in the second or third semester. If we confine the analysis to the case of the student above, no issues of endogeneity will arise. IV. Formal Model As stated previously, the nodes in this model are individual college students, and the links are undirected indicators of who goes out together on weekend evenings. Each node is assigned a random value of di drawn from di = {0,1,2,3,…8,9,10}, the average number of drinks consumed on a weekend evening. This variable will be referred to as an individual’s

“drinking preference.” Further, each node will have a value bi, indicating whether or not the node’s drinking preference is considered to be binge or not. We define bi as:

∃ bi ∈ (0,1) ∍ ∀ di ≥ 5, bi = 1

The true global parameter B is the proportion of individuals in a network with bi=1: € # (bi = 1) B= n network The dependent variable in our analysis will be each node’s estimate of the global

€ parameter B in its own network component, estB.

estB =

# (bi = 1) all n1 i ∍ l(i, j )=1 + # (bi = 1) n1 + n 2

random ( n 2 5 )i ∍ l(i, j )= 2

Thus, a node’s estimate of the parameter B is formed based upon the known

€ drinking habits of its n 1 surrounding first‐step connections, as well as upon the drinking habits known from hearsay or stories about n2/5 of its second‐step connections. It is important to consider that an individual need not spend every weekend day with first‐step connections to observe their drinking preference, but rather merely enough to have an accurate estimate of all dj1. The model estimates that each node will hear stories about n2/5 of its path length=2 neighbors, and weight this information equally with the observations of its n1 path length=1 neighbors. In tests, n2 ranges from approximately 40‐200, and thus n2/5 from 8 to 40. The variable n1 ranges in tests from 5‐20. The specific number of path length=1 and path length=2 connections may be varied with network and group size, which will be discussed shortly. The key is the ratio of these values, which were constructed after speaking with individuals in various social groups who have varied drinking preferences.

In this analysis, the effect of membership in a group with abnormal drinking habits will be tested separately and compared for nodes with high and low centrality in the network. A node’s centrality will be determined by Jackson’s model of decay centrality [cite]: n network

ci =

∑δ

l(i, j )

i=1

where l(i,j) denotes the path length between node i and node j. Since the model of estB only

€ takes into account nodes one and two steps away from itself, so too the model of decay centrality will take into account only nodes one and two steps away. This will ensure the relevance of the centrality measure and its accuracy within the model. Further, this removes the usual issue of weighting distant neighbors that must be considered when using decay centrality to classify nodes. Decay will occur at δ=0.5, a value selected for the nice range of centrality values it gives us, from approximately 15‐50. When one decimal place is retained, this measure of centrality gives us few repeating values, and thus does a nice job of ordering the centrality of nodes without weighting too heavily the degree of its neighbors. In Part I of this analysis, a random undirected network of 100 nodes is created, and drinking preferences are randomized over the nodes with equal probability. Each node’s estimate of the parameter B will be calculated, and decay centrality will be used to rank the nodes. Then, the variance in estimates will be compared for the most, least, and median central nodes. This difference in variance will give an idea of the relative degree to which high and low centrality nodes will be influenced in their estimates of B given new information about other nodes’ drinking preferences.

In Part II of this analysis, the top 10 most central nodes and the top 10 least central nodes will each be randomly sorted into two groups. For both levels of centrality, one group will be assigned to be a control, making an estimate of the parameter B based on the model described above. The other group will be assigned membership to a group with abnormal drinking preferences, such that it is more likely for a node in this sub‐graph to binge drink. These second groups of individuals estimate B based on a similar function, but one that also incorporates the node’s observation of the drinking preferences of the new social group. This new function for estB is:

estBnew =

# (bi = 1) all n1 i ∍ l(i, j )=1 + # (bi = 1)

random ( n 2 5 )i ∍ l(i, j )= 2

+ # (bi = 1) all n soc i

n1 + n 2 5 + n soc

Two key assumptions are made here. First, we consider only those members of the

€ social group who are observed by node i on a regular basis such that a link exists between the two. Previous versions of the model were built on the assumption that the new social group was fully connected. That assumption can be relaxed, however, if the members of the new social group are interpreted as the totality of individuals within the new group whom node i observes. This may also account for an observation path length=2 away without any violation of the model’s assumptions, so long as

n soc = # (n soc ∍ l(i, j) = 1) +

# (n soc ∍ l(i, j) = 2)

Secondly, it is assumed that the nodes to which the members of the group are st step have a distribution of drinking preferences equal to that of the full connected at 1€

network excluding the group itself. Put plainly, this means that we assume that the friends of an individual’s new fraternity brothers who are not in the fraternity have a distribution

of drinking preferences equivalent to that of everyone the individual knew before being exposed to the fraternity. Since we assume that the distribution of 2nd step connections is the same, and since the model takes into account the number of observations, we need not sample from the 2nd step connections to which a node is exposed through the new 1st step connections within the new social group. This is one of the model’s strongest assumptions, yet it can help the model to explain clustering in drinking habits. The effect of clustering based on drinking preferences in this model is equivalent to exposure to a new group with distribution of drinking habits equal to the range and frequency of preferences that caused the cluster. Thus, in considering the effect of exposure to a heavy binge drinking group, we are also considering the potential effect of clustering based on heavy binge drinking, or in the terms of this paper, high drinking preferences. IV. Methods To model this estimation of B by all nodes in a network and to examine the effect of new group exposure, R was used to run simulations. The R code used to perform this simulation is attached in Appendix B. First, a random network was created between 100 individuals. To do so, links between nodes were propagated in a 100x100 adjacency matrix with probability 0.1. The simulation then checks to ensure that links are undirected by matching the connections above and below the diagonal and then clearing any links in the directional itself. This gives us an average degree of about 10 with a range of approximately 5‐20, which is close to what the assumptions call for.

A plot of a sample network formed in this manner may be found in Appendix A. Over such a network, we randomize the drinking preferences di from 0 to 10 by creating a vector of 100 random whole numbers in this range that are all equally probable. From this distribution we can find each individual’s bi, which equals 1 if di > 5. The distribution of bi allows us to find the true population value of B, the percentage of binge drinkers, that each node attempts to calculate using the model estB. Next, the simulation calculates the decay centrality of all 100 nodes. To do so, it first counts the number of 1st step connections by summing each of the rows in the adjacency matrix, and then counts the number of 2nd step connections by summing each of the rows in the square of the adjacency matrix1. Based on the ranking of this decay centrality, we pull out the 10 nodes with the highest centrality and the 10 with the lowest centrality, which we call highs and lows. Half of each of these groups of 10 will be randomly assigned new social group membership in a fraternity. The fraternity is a network of 30 individuals who are assigned drinking preferences on a scale of {4:10} such that the proportion of binge drinkers is systematically higher, though there is still chance variation. To estimate the parameter B, each node takes into account the binge drinking preferences (bis) of all its n1 connections that are path length 1 away, which we call his neighbors. Next each node takes into account a random sample of n2/5 of the connections of these neighbors, i.e. of those set of n2 nodes that are two steps away from the original node. The sampled nodes’ binge drinking preferences are combined with the set of preferences from the neighbors for all nodes.

1 Before summing, the original matrix is subtracted so that links from node i to i+1 and back to i are not

counted.

Those nodes that were randomly assigned to fraternity membership then also take into account the binge drinking preferences of these neighbors. Finally, the total set of considered preferences is divided by the number of preferences observed to give each node a value of estB, the estimated proportion of binge drinkers in the network. This process of randomly sorting the most‐connected and least‐connected nodes into two groups and the sampling of preferences is then conducted 100 times. This allows us to use the law of large numbers to find the population parameters mean value and variance of estB in the control and new social group‐exposed nodes. From the variances and the difference in the means between social group‐exposed and control nodes, a t‐test can be performed. The P‐values associated with the t‐tests allow us to test H0: µestB(control) = µestB(social group) vs. Ha: µestB(control) < µestB(social group) From the significance of these tests, conclusions can be made not only about the effect of social group exposure, but also about the difference in that effect across different values of network centrality. Separately, the variance of estimates estB in 100 trials made by the 10 most connected and 10 least connected individuals will be compared, to test H0: sdestB(low centrality) = sdestB (high centrality) vs. Ha: sdestB(low centrality) ≠ sdestB (high centrality)

Again, the law of large numbers will allow us to find the parameter of the true

variance of these estimates, from which we will test if there is a significant difference in the precision of the two estimators. In running this test we will also need the mean estimates

for each group, which will allow us to look at the accuracy of the two levels of centrality as well. V. Results Part I: Table 1. Effect of Centrality on Variance in Network’s Estimates of B Group µestB sdestB tdiff(sd) = 4.68 Low Centrality 0.5342 0.1091 df = 142 (use 100) High Centrality 0.5506 0.0786 P(T>t) < 0.001 Table 1 shows that a significant relationship was found between nodes’ centrality and the variance of their estimates of B. At a confidence level of α=0.0005, we can say that low centrality nodes’ estimates of B have greater variance and standard deviation. The magnitude of this effect is equivalent to a 30% change in the size of the standard deviation if a node moved from the periphery to the center of the network, a sizable real‐world effect. While the more peripheral individuals are less precise as a group, in this case they are actually more accurate than the high centrality group in estimating B, which in this trial happened to be B=0.5300. This is likely to merely be a chance effect of the randomization of di, however, and the difference in estimation is not large. Part II: Table 2. Effect of Exposure to New Social Group (high binge) across Centrality Centrality xestB(control)‐ sdpooled t P(T>t) < xestB(social group) Lows 0.0851 0.0072 11.869 0.10 Highs 0.0299 0.0054 5.518 0.10 Table 2 shows that a significant effect was found for both the high and low centrality groups at α=0.10. For low centrality nodes, observing the high binge drinking social group resulted in an 8.51% average increase in the node’s estimate of the network’s proportion of binge drinkers, holding all else equal. For high centrality nodes, observing the high binge

drinking social group resulted in a 2.99% average increase in the node’s estimate of the network’s proportion of binge drinkers, holding all else equal. While an 8.5% increase in average estimation of a proportion on the scale 0:1 is a sizable real‐world effect, a 3.0% increase in estimation is not. Thus, we conclude that exposure to a new network of drinkers with non‐normal drinking preferences can significantly sway an individual’s estimate of the percentage of binge drinkers in a network, an effect that is particularly noticeable in the peripheral nodes of a network. VI. Interpretation In real‐world terms, this conclusion means that individuals who have fewer friends to go out with on the weekends are more likely to mis‐estimate the average drinking behavior in the network, in either the positive or negative direction. When these peripheral individuals are exposed to a new group such as a fraternity that drinks heavily, therefore, their opinions of what is normal are easily skewed. This presents us with a social risk: individuals who have fewer friends with whom to go out on the weekend are at higher risk of thinking binge drinking is normal, a thought process that has been found to be associated with higher incidence of alcohol‐related problems themselves (Reifman & Watson, 2003). While there may be a degree of reverse causality present in that relationship, the association is certainly not good from either perspective, as it is certainly less healthy for a group to think that unsafe or unhealthy behaviors are normal. Alternatively, however, the results also indicate that if a peripheral individual were to join a group that exposed him or her to non‐normal drinking habits that are better than average, they could easily be skewed in their opinion of what is normal as well. This skew

would come just as frequently, but with a socially positive/quantitatively negative tilt as opposed to the example used in the results. Perhaps in such a group di would be distributed randomly along {0:6}, thereby giving the same probability to not binge drinking as the fraternity example used in the results did to binge drinking, and vice versa with the probability of binge drinking in this new sample. The results point to policy implications regarding the creation of safe‐drinking social groups for students to join, and the push for inclusion of peripheral individuals in these groups. If a individual who is less social and has few friends joins a heavy‐drinking fraternity, he is likely to develop unhealthy opinions and behaviors. On the other hand, if DAPA were highly publicized and more such groups would be available, then the peripheral individual’s new friends from this group would be likely to have a healthy effect on his opinions and behaviors. The prevalence, level of participation, and cohesion of safe‐ drinking groups is important, but this model’s conclusion is that the key is to target individuals based on centrality in a network, as the effect of groups positive and negative is smaller for central individuals. Such targeting might occur via a survey or by analyzing online social network data. VII. Empirical Verification Although this paper is somewhat of an empirical verification of the model by simulation, further possibilities abound. The ideal data for testing this in the real world would be a study over time in which individuals fill out a survey at the beginning of freshman year listing their friends, and their estimate of the parameter B. Subsequent surveys that include information on group membership and close friends at each time period would allow the analysis of clustering over time and any changes in the estimate of

B. Each individual’s drinking could be estimated by what his or her friends say, and the groups’ members and influence would be calculated. Such a study could occur in a similar fashion to Prof. Christakis’ work on the spread of the flu at Harvard in 2010 (Christakis & Fowler 2010). The success of this study indicates that there is a huge opportunity to gather such simple data from college students by merely motivating them with $5 ice cream gift certificates (the author of this paper was one of the participants). Limitations with such data include the typical biases in listing friends, such as the instinct to list friends who you see together, though analysis of who is listed over time would help remove such bias. Though the study would avoid the precarious self‐reporting bias of how much the individual drinks, participants might be hesitant to list the drinking habits of their friends, and might therefore underestimate their average number of drinks. This could be accounted for by normalizing the distribution of drinks onto the scale used in this model of 0‐10. Cumulative effects would be expected for positive and negative group influences That is to say, membership in a fraternity and DAPA would be expected even out and have no effect on estimation of B, assuming that the two organizations were composed of individuals with drinking habits equally divergent from average in opposite directions. It is expected that such empirical verification would show results similar to those estimated by this model.

Bibliography Chassin, Laurie, Justin Prost, and Steven C. Pitts. "Binge Drinking Trajectories From Adolescence to Emerging Adulthood in a High-Risk Sample: Predictors and Substance Abuse Outcomes." Journal of Consulting and Clinical Psychology 70.1 (2002): 67-78. Web. Reifman, Alan, and Wendy K. Watson. "Binge Drinking During the First Semester of College: Continuation and Desistance From High School Patterns." Journal of American College Health 52.2 (2003): 73-81. Web. Reifman, Alan, Wendy K. Watson, and Andrea McCourt. "Social Networks and College Drinking: Probing Processes of Social Influence and Selection." Society for Personality and Social Psychology 32.6: 820-32. Web. Seeman, Melvin, and Carolyn Anderson. "Alienation and Alcohol: The Role of Work, Mastery, and Community in Drinking Behavior." American Sociological Review 48 (1983): 60-77. Web. Wechsler, Henry, and Meichun Kuo. "College Students Define Binge Drinking and Estimate Its Prevalence: Results of a National Survey." Journal of American College Health 49 (2000): 5764. Web. Weitzman, Elissa R., Toben F. Nelson, and Henry Wechsler. "Taking Up Binge Drinking in College: The Influences of Person, Social Group, and Environment." Journal of Adolescent Health 32 (2003): 26-35. Print.

Appendix A. Figure 1. Sample network generated in R. This is the network used for this estimate. The group of low centrality nodes was (97,23,74,40,47,82,90,48,52,38) and the high centrality group was (61,49,88,78,84,32,92,83,7,100). Some of those nodes can be seen here.