Approaches to the Estimation of the Local Average Treatment Effect in a Regression Discontinuity Design

Abstract Regression discontinuity designs (RD designs) are used as a method for causal inference from observational data, where the decision to apply an intervention is made according to a ‘decision rule’ that is linked to some continuous variable. Such designs are being increasingly developed in medicine. The local average treatment effect (LATE) has been established as an estimator of the intervention effect in an RD design, particularly where a design's ‘decision rule’ is not adhered to strictly. Estimating the variance of the LATE is not necessarily straightforward. We consider three approaches to the estimation of the LATE: two‐stage least squares, likelihood‐based and a Bayesian approach. We compare these under a variety of simulated RD designs and a real example concerning the prescription of statins based on cardiovascular disease risk score.

The estimatorsβ 0a andβ 0b are those of the intercept term in a normal linear model and these have the following form, conditional on x = (x 1 , . . . , x n ) ⊤ , where the elements of x are centred about the threshold (x c notation is removed for convenience).
Furthermore, the maximum likelihood estimators for β 1a and β 1b are given bŷ We make the following definitions and, hence, we writeβ 0a andβ 0b aŝ Similarly, for i ∈ B we may writeβ Hence, we may writeβ as the maximum likelihood estimator of the LATE numerator. If Y i is normally distributed then the estimator forβ in (2) is the sum of independent normal random variables and thus is normally distributed. If not, standard asymptotic results apply andβ is approximately normally distributed for large samples.
Assuming that Var(Y i ) = σ 2 is constant for all i ∈ A ∪ B and that the outcomes Y are independent for each individual. Then In this work, we will assume that Var(Y i ) is the same for subjects above and below the threshold.
However, we could relax this assumption and, for example, set Var(Y i ) = σ 2 a above the threshold and Var(Y i ) = σ 2 b below the threshold. In this case We form similar linear models for T i , above and below the threshold, of the form with ω ai and ω bi normal mean-zero error terms such that Var(ω ai ) = φ 2 a and Var(ω bi ) = φ 2 b . In a similar manner to the linear models for Y i , the maximum likelihood estimates for π 0a and π 0b are and, for large samples,π is approximately normally distributed with Hence, we argue that (β,π) ⊤ will have, approximately, a bivariate normal distribution. We consider Assume that the covariance between Y i and T i may be written Rearranging, we see that

Hence, by substitution into Equation 3, we obtain
Similarly, Then, the distribution of (β,π) ⊤ is We write this as:  Our interest lies in determining Var(β/π). We have derived an approximate joint density for (β,π) ⊤ .
We set λ = g(β, π) = β π and consider a Taylor expansion (i.e. a multivariate delta method) to derive an approximation for Var(λ). The delta method approximation for the variance ofλ is given by: Since σ 2 , ρ a , ρ b are unknown, we estimate these parameters as followŝ Here, s 2 a and s 2 b denote the sample variance values from the linear models ((10) in Section 3.2 of the paper). The termsŷ i andt i denote fitted values from the corresponding models for the i th individual. Note thatρ a andρ b denote the sample covariance between Y i and T i for the populations above and below the threshold, respectively. Thus, our estimate for the variance of the LATE is

C Bayesian Priors for the Simulation Study
We assume the following prior distributions for the parameters of interest: