Are ‘Water Smart Landscapes’ Contagious? An Epidemic Approach on Networks to Study Peer Effects

Abstract

We test the existence of a neighborhood based peer effect around participation in an incentive based conservation program called ‘Water Smart Landscapes’ (WSL) in the city of Las Vegas, Nevada. We use 15 years of geo-coded daily records of WSL program applications and approvals compiled by the Southern Nevada Water Authority and Clark County Tax Assessors rolls for home characteristics. We use this data to test whether a spatially mediated peer effect can be observed in WSL participation likelihood at the household level. We show that epidemic spreading models provide more flexibility in modeling assumptions, and also provide one mechanism for addressing problems associated with correlated unobservables than hazards models which can also be applied to address the same questions. We build networks of neighborhood based peers for 16 randomly selected neighborhoods in Las Vegas and test for the existence of a peer based influence on WSL participation by using a Susceptible-Exposed-Infected-Recovered epidemic spreading model (SEIR), in which a home can become infected via autoinfection or through contagion from its infected neighbors. We show that this type of epidemic model can be directly recast to an additive-multiplicative hazard model, but not to purely multiplicative one. Using both inference and prediction approaches we find evidence of peer effects in several Las Vegas neighborhoods.

Appendices

Appendix A

1.1 A.1 Log Likelihood of the Epidemic Model

$$\begin{array}{@{}rcl@{}} \mathcal{L}(\bar{t}|\alpha,\mu)\!&=&\!\sum\limits_{i \in V | x_i^T=S} \left[ {\sum}_{t = 1}^{T} \log (1-\mu_i^t) + \sum\limits_{k \in \partial i | t_{kI}<T} ({T-t_{kiI}}) \log (1-\alpha)\right] \\ &&\!+\sum\limits_{i \in V | x_i^T\neq S} \left[ \sum\limits_{k \in \partial i | t_{kI}<t_{iE}-1} ({t_{iE}-\tau_{kiI}-1}) \log (1-\alpha) \right. \\ && \qquad~~~\qquad\left.\!+\log \left[1 - (1 - \mu_i^{t_{iE}})(1 - \alpha)^{n_{i}}\right] + \sum\limits_{t = 1}^{t_{iE}-1} \log (1-\mu_i^t) \right] \end{array} $$

(7)

1.2 A.2 Mapping Epidemic with Autoinfection to Additive-Multiplicative Hazard Model

Here we outline how the three models SI, additive-multiplicative and multiplicative hazard, describe the same quantity of interest, the hazard rate λ(t_i|Θ, D). In discrete time, this is the conditional probability that an event happens at time t_i, given it has not yet happened before. In general this quantity can depend on a set of parameters Θ and the data D. In our case Θ = {α, μ} include the transmission and autoinfection probabilities of all nodes, whereas D = {t_iE}, i.e. the data are the observed exposure times. In the SEIR model, (1 minus) the hazard rate of a susceptible node i at time t is:

$$ 1-\lambda_i (t | {\Theta}, D)= (1-\alpha)^{{n_{i}^{t}}} (1-{\mu_{i}^{t}}) \qquad \text{Epidemic model} $$

(8)

In discrete time (Kalbfleisch and Prentice 2011), the hazard rate in the additive-multiplicative hazard model (Höhle et al. 2005; Höhle 2008) can be written as:

$$ 1-\lambda_i(t|{\Theta},D) = \left( 1-\alpha\right)^{{n_{i}^{t}}} \left( e^{-\lambda_{0}^t}\right)^{e^{\bar{x}_{i} \cdot \bar{\beta}}} \qquad \text{Additive-Multiplicative} $$

(9)

whereas, in the purely multiplicative model used in Towe and Lawley (2013) we have:

$$ 1-\lambda_{i}(t|{\Theta},D) =\left( e^{-\lambda_{0}^t}\right)^{e^{ \bar{x}_{i} \cdot \bar{\beta}} e^{\alpha {n_{i}^{t}}} } \quad \qquad \qquad \text{Multiplicative} $$

(10)

where ${\lambda _{0}^{t}}$ is the baseline hazard at time t which represents a global contribution to the probability of getting infected which is the same for all houses; $\bar {x}_{i}$ is a vector of covariates and $\bar {\beta }$ is a vector of parameters coupling the covariates, in a similar flavor as in linear regression.

Comparing Eqs. 8 and 9 we obtain the mapping (4).

1.2.1 Proof

From the definition of hazard rate (Kalbfleisch and Prentice 2011) which, in discrete time, is a conditional probability, we have a relationship connecting the probability f(t_i) of an event happening at time t_i and the survival probability S(t_i), which is the probability that no event happens before time t_i (but it can happen at exactly t_i or later):

$$ \lambda(t)=\frac{f(t)}{S(t)} $$

(11)

Notice that for discrete time $S(t)={\sum }_{s=t}^{\infty } f(s)$, thus we can write:

$$ f(t)= S(t)-S(t + 1) =\sum\limits_{s=t}^{\infty} f(s)-\sum\limits_{s=t + 1}^{\infty} f(s)= f(t) $$

(12)

Substituting into Eq. 11:

$$\begin{array}{@{}rcl@{}} \lambda(t)&=&\frac{f(t)}{S(t)} \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} &=& 1-\frac{S(t + 1)}{S(t)} \end{array} $$

(14)

Using the equation valid in general (Kalbfleisch and Prentice 2011) relating the survival probability S(t) with the hazard rate $S_{i}(t)=e^{-{{\int }_{0}^{t}} \lambda _{i}(s) ds}$, the following relation valid for additive-multiplicative hazard model as in Scheike and Zhang (2002), where ${\lambda _{i}^{t}}=\alpha {n_{i}^{t}}+\lambda _{0} e^{\bar {x_{i}} \cdot \bar {\beta }}$ with covariates can be derived:

$$ S_i(t)=S_0(t)^{\exp(\bar{x_i} \cdot \bar{\beta})} S^i_{epi}(t) $$

(15)

where S₀(t) represents the baseline survival probability when $\bar {\beta }= 0$ and no epidemic effect is present (α = 0), whereas $S^{i}_{epi}(t)$ represents the epidemic contribution to the survival probability.

For discrete models we can write:

$$\begin{array}{@{}rcl@{}} S_{0}(t)&=&\prod\limits_{s = 0}^{t-1} (e^{-{\lambda_{0}^{s}}}) \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} S^{i}_{epi}(t) &=&\prod\limits_{s = 0}^{t-1} (1-\alpha)^{{n_{i}^{s}}} \end{array} $$

(17)

Substituting Eqs. 16 and 17 into Eq. 14, we get:

$$\begin{array}{@{}rcl@{}} \lambda_i^t&=&1-(e^{-\lambda_0^s})^{\exp(\bar{x_i} \cdot \bar{\beta})} (1-\alpha)^{n_i^t} \end{array} $$

(18)

For the purely Multiplicative hazard model as in Towe and Lawley (2013) where ${\lambda _{i}^{t}}={\lambda _{0}^{t}} e^{\bar {x_{i}} \cdot \bar {\beta }} e^{\alpha {n_{i}^{t}}}$ we cannot separate the baseline and epidemic contribution as above, instead we have:

$$ S_{i}(t)=S_{0}(t)^{\exp(\bar{x_{i}} \cdot \bar{\beta} +\alpha {n_{i}^{t}})} $$

(19)

which leads to:

$$\begin{array}{@{}rcl@{}} \lambda_i^t&=&1-(e^{-\lambda_0^s})^{\exp(\bar{x_i} \cdot \bar{\beta}+\alpha n_i^t)} \end{array} $$

(20)

Table 6 Maximum likelihood inference results for different recovery times τ_R (in months)

Appendix B: Estimate of WSL Rates’ Increase Due to the Presence of Active Neighbors

Here we estimate the increased number of participating houses due to the presence of active neighbors compared to what we would have observed in a counter-factual scenario where houses do not have any participating neighbors, in the model that allows for peer effects. In this way, the model is kept as faithful as possible to our best estimate models, but we prevent peer effects from occurring. The difference in WSL activation rates can then be attributed to peer effects, as described by our model.

In general, estimating this quantity analytically is not possible because the evolution of the activation dynamics depends on the entire history of the process, i.e. the dynamics are non Markovian; for instance, having 10 active neighbors at time T (and recovery τ_R > 10) and all of whom activated at T − 1 is different than having 10 active neighbors at time T but with one of them activating at T − 10, one at T − 9 and so on up to T − 1. In the former case the probability of activating will be a function of (1 − α)¹⁰, whereas in the latter it will be a function of (1 − α)(1 − α)²…(1 − α)¹⁰. In addition, the autoinfection probability varies from house to house based on covariates, and the epidemic term also varies from house to house based on the local network topology: the number of neighbors.

Therefore we run Markov Chain Monte Carlo simulations using the epidemic model with the parameters inferred in the prediction results of Table 5 but forcing the transmission probability α = 0. This is equivalent to the scenario where a house does not have any active neighbors, so that the final number of active houses obtained at the end of the 36-month period with this procedure will be a lower bound of the one we would get if each house were to have, at each time step in the observation period, a variable number of active neighbors $0 \leq {n_{i}^{t}} \leq k_{i}$, where k_i is the number of neighbors of node i (also known as the degree of a node in the network). We then calculate $r_{peer}=(\hat {I}^{Sim}_{T}-\hat {I}^{Sim}_{T,\alpha = 0})/{\Delta } \hat {I}_{\alpha = 0}$, the ratio between the increase in WSL rate at the end of the observation period between the case where houses have active neighbors, $\hat {I}^{Sim}_{T}$ (as in Table 5), and when we force α = 0, $\hat {I}^{Sim}_{T,\alpha = 0}$, over the increase in WSL rate during the last 36 months when houses do not have any participating neighbor ${\Delta } \hat {I}_{\alpha = 0}=\hat {I}^{Sim}_{T,\alpha = 0}-{I}^{obs}_{T-36,\alpha = 0}$, where ${I}^{obs}_{T-36,\alpha = 0}$ is the observed WSL rate at the beginning of the last 36-month period. We then consider the observed increase in number of participating houses during the last 36 months of the observation period ΔN_obs and calculate ΔN_peer = r_peerΔN_obs as an estimate of the number of participating houses in the 36 months due to peer effect. In Table 7 we show the results: the impact of peer effect varies across neighborhoods, from small impact as in neighborhood 11 of a 3% to a 27% increase of neighborhood 7.

Table 7 Activation increase estimate due to the presence of active neighbors