Optimal Design of Experiments on Connected Units with Application to Social Networks

Abstract

When experiments are performed on social networks, it is difficult to justify the usual assumption of treatment-unit additivity, due to the connections between actors in the network. We investigate how connections between experimental units affect the design of experiments on those experimental units. Specifically, where we have unstructured treatments, whose effects propagate according to a linear network effects model which we introduce, we show that optimal designs are no longer necessarily balanced; we further demonstrate how experiments which do not take a network effect into account can lead to much higher variance than necessary and/or a large bias. We show the use of this methodology in a very wide range of experiments in agricultural trials, and crossover trials, as well as experiments on connected individuals in a social network.

Optimal block designs for experiments on networks

“…Important examples include the work of Druilhet (1999), Kunert and Martin (2000) and Kunert and Mersmann (2011), who provided models and efficient designs that concern experiments where units are arranged in a circle or a line and in which neighbours one unit apart can interfere with each other. More recently, Parker et al (2016) adopted the conceptual approach of Pearce (1957) and introduced a model, called the linear network effects model (LNM). The LNM is similar to the model of Kunert and Martin (2000).…”

“…However, it differs by relaxing the assumption of neighbour effects existing in only one direction and allows for a network setting in which units can be linked in a way that does not form a regular layout. Here, we extend the LNM of Parker et al (2016) to include the notion of blocks.…”

Optimal Block Designs for Experiments on Networks

“…We label these groups 1, …, κ and let group g have n (g) experimental units within it. We model the responses of experimental units by the block network model (BNM), which is an extension of the LNM (Parker et al, 2016) and is described by the equation where i = 1, 2, …, ; j = 1, 2, …, n (i) , y ij is the continuous response from unit j in the ith block receiving the treatment s = r(ij) ∈ {1, …, m}, μ represents the response for a baseline treatment or (unit) average, b i is the effect of block i, r(ij) is the (direct) treatment effect, A {ij,gh} / A = A {ij,gh} with j, h ∈ V is the adjacency matrix indicating the edge between units j and h belonging to blocks i and g ∈ {1, 2, …, κ} respectively, r(gh) is the network effect (neighbour or indirect treatment effect) and ij are the errors, which we assume to be independent and identically distributed with mean 0 and constant variance 2 . To overcome the model overparametrisation requires imposing some constraints, otherwise the normal equations have an infinite number of solutions and our parameters cannot be uniquely estimated.…”

“…Important examples include the work of Druilhet (1999), Kunert and Martin (2000) and Kunert and Mersmann (2011), who provided models and efficient designs that concern experiments where units are arranged in a circle or a line and in which neighbours one unit apart can interfere with each other. More recently, Parker et al (2016) adopted the conceptual approach of Pearce (1957) and introduced a model, called the linear network effects model (LNM). The LNM is similar to the model of Kunert and Martin (2000).…”

“…However, it differs by relaxing the assumption of neighbour effects existing in only one direction and allows for a network setting in which units can be linked in a way K E Y W O R D S clustering, connected experimental units, graphs, linear network models, treatment interference that does not form a regular layout. Here, we extend the LNM of Parker et al (2016) to include the notion of blocks.…”