# Selection Effects in Online Sharing: Consequences for Peer Adoption

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7 (a) (b) (c) Fig. 1: A story for (a) a Page posting a new Offer on a mobile web browser and (b) a friend claiming (adopting) an Offer on the Facebook iphone application (c) a friend claiming an Offer on the dekstop interface. (a) Fig. 2: Mobile interface presented to subjects after adoption for the (a) active and (b) passive sharing conditions. (b) We examine downstream effects of the sharing interface by measuring the subsequent behavior of peers who were exposed to subjects adoption activity. It is important to note that peers who see the activity of subjects who share under the passive sharing condition are different than those who share in the active condition. This introduces a selection effect that shapes the population of exposed peers, and this effect is what we intend to measure.

10 0.75 cumulative shares active passive number of exposed peers Fig. 4: The empirical cumulative distribution function for the number of exposed peers for each sharing subject by treatment condition. For clarity in comparison, the x-axis is truncated at the 90% percentile of the distribution. The empirical distributions show sharers in the passive condition usually expose more of their peers. (0,18] (18,35] (35,58] (58,98] (98,max] peer adoption rate active passive active passive active passive active passive active passive Fig. 5: Average adoption rate of sharing subjects peers, broken down by quintiles of the total number of exposed peers. Error bars show the 95% confidence intervals. We fit a random effects probit model, P r(s ik = 1 µ k ) = Φ(µ k + ɛ ik ), in order to estimate the variation in product sharing utilities, σµ. 2 To identify the parameters in the probit model, we let σɛ 2 = 1. Table III contains the model parameter estimates we obtain. The estimated intraclass correlation coefficient is σ = 0.064, indicating that µ 2 +σ2 ɛ the Offer-level random effects do not explain much additional variance in the sharing model. This implies that the product selection mechanism is likely to be weak. σ 2 µ

12 Parameter γ β Estimate (0.004) (0.003) σ λ Log-likelihood -151,521 Groups 25,726 N 702,090 Table IV: Maximum likelihood parameter estimates for binomial regression predicting the number of adopting alters with random effects at the Offer level. γ is the mean of the random effect λ k, while β represents a reduced form measure of total selection effect. * denotes significance at the level. Our setup is similar to the simultaneous discrete-choice models with interdependent preferences considered in Yang et al. [2006], motivating a similar estimation procedure using Bayesian methods. Bayesian estimation is ideal for this setting because it allows us to flexibly perform inference on the correlation parameters ρ and ψ at the cost of parametric assumptions. We use non-informative priors on all parameters and run Markov-chain Monte Carlo simulations to estimate their posterior distributions given the observed data. Due to the scale of our data, we used an efficient Hamiltonian Monte Carlo sampler [Hoffman and Gelman 2012] and computed our results using a state-of-the-art Bayesian model compiler [Stan Development Team 2013]. We simulated three Markov chains for 2,000 iterations, discarding the first 1,000 iterations for burn-in. We then used the last 1,000 draws for estimation. We evaluated convergence by computing a potential scale reduction factor for each estimated parameter in the model [Gelman and Rubin 1992]. Parameter Mean 2.5% Median 97.5% Mean product sharing utility α Mean product adoption utility γ Std. dev. of product sharing utility σ µ Std. dev. of product adoption utility σ λ Product-level correlation coefficient ρ Dyad-selection correlation coefficient ψ Table V: HMC posterior mean, median, and 95% credible interval estimates for the parameters of the joint structural model of sharing and adoption specified in Section 3.2. Estimates for ρ and ψ are positive and significant, providing evidence for both product- and dyad-selection effects. We estimate three main types of parameters in the model (Table V). First, there are mean sharing and adoption utilities, α and γ, which rationalize the average rates of sharing and adoption. Second, there are correlations between the unobserved utilities at the product level, ρ, and at the dyad level, ψ. Third, we estimate the standard deviations of the zeromean product-level utilities, σ µ and σ λ. Note that as in the regression models, σ ɛ and σ ν are fixed at 1 in order to identify the other parameters of the model. This is a typical assumption in this modeling situation where we have no absolute utility scale.

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