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Typically social contagion models in the previous studies have been performed using GLMs (Christakis and Fowler, 2013), with a form like

g`E“Y<sub>t</sub>i<sub>`1</sub>‰˘“β0`β1yit`β2yjt`1`β3yjt` M

ÿ

k“4

βkxk (3.1)

wheregp¨qis the link function of the particular generalised regression,y_tiis the state of individual i at time t, xk are any other covariates considered in the regression,

βkare the regression parameters, and the model is considering effects between pairs

of friends. The claim is then that if the coefficientβ2 is significantly positive this is

evidence of a causal influence on the state of individual iby the state of individual

j.

The model protects against homophily due to the inclusion of theyj_t term. Modelling

yi_t`1 as dependent on yi_t leads to the model acting as a Markov chain, resulting in

yi_t`1 being independent of both the state of individual i and that of individual j

at the time they became friends. It protects against shared context by a found asymmetry in the impact of individual j oni from that of individualion j.

As mentioned, this method has come under a lot of extreme (in some cases perhaps too extreme) criticism. One such criticism is that consideringβ2 to be the effect of

interest is a strange choice as causality should occur over time, so ratherβ3 should

contain the contagion effect (Lyons, 2011). Another argued that the model can be used to provide evidence for contagion of nonsensical things such as height (Cohen-

Cole and Fletcher, 2008a). Yet another argued that whilst the lagged friend term may control for homophily causing the initial creation of the friendship, it does not control for homophily causing the retention of the friendship (Noel and Nyhan, 2011). There were also criticisms as to the size and directions of the effects found, such as that commonlyβ2 and β3 were found to be of opposing sign (which would

produce a negative homophily effect insinuating that any possible homophily leads to dissimilarity rather than similarity) and that the differences in the impact of individualj onifrom that of individualionj were commonly found to be insignif- icant (Lyons, 2011). Despite all these criticisms it has been shown that the model can successfully support the null hypothesis of no contagion (VanderWeele et al., 2012).

However, the sheer amount of criticism, and the lack of support for the models ability to show a significant result of actual social contagion, shows the difficulty in using such linear models when attempting the difficult task of inferring evidence for causal effects that are highly prone to confounding. It is therefore here that we present our first application of a more sophisticated statistical model to what is arguably a complex underlying system involving socio-economic variables by way of the social connections. This is done in the form of a non-linear parametric model, specifically that initially developed by Hill et al. (2015), where we also show how the model must be generalised to improve its applicability.

If we let a component of mood for an individual at time t with k` _{friends with} better mood andk´ _{friends with worse mood be represented by an integer random} variableYptq, we can imagine a very general probabilistic model for mood in which

PrpYpt`1q “y1

|yptq “yq “fpy1_{, y, k}`_{, k}´

q . (3.2)

In practice, finding an appropriate functionf for such a general model becomes too difficult and so we will normally need to consider special cases of this general model. In the work of Hill et al. (2015) they considered only binary statesYptq “D for an individual with depressive symptoms at time t and Yptq “ N for a non-depressed (healthy) individual, and sought to distinguish between sigmoidal dependence on the number of friends in a given state and no such dependence.

Such an approach is robust to confounding from homophily and shared context, as shown in Hill et al. (2015). In simplified terms, this comes from the fact that this model considers transition probabilities between states, which are distinguishable for contagion and the other basic phenomena that could confound it, rather than

stationary distributions, which are not. If it can be shown that the transition prob- ability for an individual of going from being depressed to not depressed is stratified by the number of friends they have who are depressed at the initial time point, i.e. that the probability PrpYpt`1q “ D|Yptq “ Nq is higher, and the probability PrpYpt`1q “N|Yptq “ Dq is lower, for individuals with more friends in state D

than the baseline values (with zero friends in stateD), then we can infer evidence of social contagion that is not confounded by the typical confounding phenomena. If homophily or shared context were occurring within the data, rather than contagion, then we would simply expect to see more clusters of same state individuals in the dataset and we would expect individuals within these clusters to transition together. We would not expect either of these phenomena to result in the stratification of the transition probability by the number of contagious state friends.

It is possible that some more complicated phenomena could be confounding the method, but that does not negate the improvement this method presents over pre- vious methods. There is also the argument that the simplest possible explanation is the most likely. If this is true here, then ruling out shared context and homophily leaves contagion as the simplest explanation.

Despite the robustness this method does not account for the possibility of non-binary states, such as the different continuous numerical scores for the weights individuals have. To relax this assumption we now let Yptq be an integer (for discrete states) or a continuous number (for continuous states), and consider a trinomial model specified by three probabilities: the probability of increasing state, the probability of decreasing state, and the probability of remaining in the same state

PrpYipt`1q ąYiptqq “p ,

PrpYipt`1q ăYiptqq “q ,

PrpYipt`1q “Yiptqq “1´p´q .

(3.3)

We can then examine whether these probabilities were dependent on the states of an individuals friends relative to their own at the first time point by comparing two different functional forms for p and q. The first is conditioned on the number of friends an individual had who had a higher/lower score at the first time point,

k. This takes the form of a discrete S-shaped (sigmoidal) function, appropriate for behavioural contagion being a type of complex contagion (Centola and Macy, 2007;

Centola, 2010; Valente, 1996), with the following mathematical formulation: pk“α`β k ÿ l“0 ˆ 10 l ˙ γlp1´γq1´l , qk“δ` k ÿ l“0 ˆ 10 l ˙ ζlp1´ζq1´l . (3.4)

Here the parameters α and δ correspond to the baseline transition probabilities. The contribution to the transition probabilities by each additional friend is given by the binomial terms dependent on the parametersγ and ζ with an overall amplitude given by β and , which leads to the sigmoidal complex contagion form required. For each individual with k higher or lower scoring friends,k binomial terms (from a binomial distribution with up to 10 possible successes, reflecting the maximum number of friends an individual is allowed to list in the data set) are added to the baseline transition probability to give the individual’s transition probability given their friends’ states. The second functional form forp and q is independent of the states of the friends:

pk “α , qk “δ . (3.5)

Each one is dependent on only the baseline transition probability. Using each pos- sible combination of these two functional forms gives us four models to compare. Model 1, where pk and qk are given by (3.4), has both increasing and decreasing

state being dependent on friend states. Model 2, wherepkandqkare given by (3.5),

has neither increasing nor decreasing state being dependent on friend states. Model 3, where pk is given by (3.4) and qk by (3.5), has increasing state alone being de-

pendent on friend states. Model 4, wherepk is given by (3.5) and qk by (3.4), has

decreasing state alone being dependent on friend states.

These models can each be fitted to data using maximum-likelihood estimation (MLE), with separate model variants conditioned either on higher scoring friends or lower scoring friends. The likelihood takes the form

Lpn,m|p,q,Nq “ź k ˆ Nk nk, mk, Nk´nk´mk ˙ pnk k q mk k p1´pk´qkq Nk´nk´mk (3.6) wherenk was the number of individuals with k higher / lower scoring friends who

worsened, mk was the number of individuals with k higher / lower scoring friends

friends, and the multinomial coefficient is ˆ Nk nk, mk, Nk´nk´mk ˙ “ Nk! nk!mk!pNk´nk´mkq! . (3.7)

Competing models are then compared using their Akaike Information Criterion (AIC) values in order to find the preferred model in each case

AIC“2ν´2 logLpn,m|pˆ,qˆ,Nq (3.8)

whereν is the number of parameters in the model, and ˆpand ˆqare the values ofp andq dependent on the fitted parameters (Akaike, 1974).

We initially used this method to model social contagion for discrete mood states (Eyre, 2014). We now apply it to continuous weight states in order to examine the possi- bility of contagion of mood changes in an adolescent population.