generate_qualitative_data_rd {causalQual} | R Documentation |
Generate Qualitative Data (Regression Discontinuity)
Description
Generate a synthetic data set with qualitative outcomes under a regression discontinuity design. The data include a binary treatment indicator and a single covariate (the running variable). The conditional probability mass fuctions of potential outcomes are continuous in the running variable.
Usage
generate_qualitative_data_rd(n, outcome_type)
Arguments
n |
Sample size. |
outcome_type |
String controlling the outcome type. Must be either |
Details
Outcome type
Potential outcomes are generated differently according to outcome_type
. If outcome_type == "multinomial"
, generate_qualitative_data_rd
computes linear predictors for each class using the covariates:
\eta_{mi} (d) = \beta_{m1}^d X_{i1} + \beta_{m2}^d X_{i2} + \beta_{m3}^d X_{i3}, \quad d = 0, 1,
and then transforms \eta_{mi} (d)
into valid probability distributions using the softmax function:
P(Y_i(d) = m | X_i) = \frac{\exp(\eta_{mi} (d))}{\sum_{m'} \exp(\eta_{m'i}(d))}.
It then generates potential outcomes Y_i(1)
and Y_i(0)
by sampling from {1, 2, 3} using P(Y_i(d) = m | X_i), \, d = 0, 1
.
If instead outcome_type == "ordered"
, generate_qualitative_data_rd
first generates latent potential outcomes:
Y_i^* (d) = \tau d + X_{i1} + X_{i2} + X_{i3} + N (0, 1), \quad d = 0, 1,
with \tau = 2
. It then constructs Y_i (d)
by discretizing Y_i^* (d)
using threshold parameters \zeta_1 = 2
and \zeta_2 = 4
. Then,
P(Y_i(d) = m) = P(\zeta_{m-1} < Y_i^*(d) \leq \zeta_m) = \Phi (\zeta_m - \sum_j X_{ij} - \tau d) - \Phi (\zeta_{m-1} - \sum_j X_{ij} - \tau d), \quad d = 0, 1,
which allows us to analytically compute the probabilities of shift at the cutoff.
Treatment assignment
Treatment is always assigned as D_i = 1(X_i \geq 0.5)
.
Other details
The function always generates three independent covariates from U(0,1)
. Observed outcomes Y_i
are always constructed using the usual observational rule.
Value
A list storing a data frame with the observed data, and the true probabilities of shift at the cutoff.
Author(s)
Riccardo Di Francesco
See Also
generate_qualitative_data_soo
generate_qualitative_data_iv
generate_qualitative_data_did
Examples
## Generate synthetic data.
set.seed(1986)
data <- generate_qualitative_data_rd(100,
outcome_type = "ordered")
data$pshifts_cutoff