Table 4.1. Formulas for TS ρ Under RD Designs, by Score Distribution

List of Tables

Score Distribution	Formulas for ρ_TS	Parameter Definitions
Normal		; Φ^-1(.) is the inverse of the standard normal distribution
Uniform
Truncated Normal		k₁ and k₂ are the number of standard deviations from the mean of the full normal distribution that the left and right truncation points fall (see Figure 4.2).
Symmetric Bimodal Distribution: Mixture of Two Normal Distributions With Different Means but the Same Variances		w is the weight assigned to the first normal distribution (.5 in our case); l is the number of standard deviations to the right (left) of the mean of the overall bimodal distribution where the first (second) component normal distribution is centered (see Figure 4.2); d is the location of the cutoff score in the bimodal distribution.
Notes: The parameter p is the treatment group sampling rate and equals q—the percentage of the score distribution that lies to the right of the cutoff score K. The formulas for ρ_TS were calculated using the following relations: The moments for truncated normal random variables were obtained using results in Maddala (1983).