generate_obs_hte_26()

Mathematical Specification of generate_obs_hte_26()

The generate_obs_hte_26() function generates an observational dataset with nonlinear outcome and treatment assignment mechanisms, along with heterogeneous treatment effects. The data generation process (DGP) is defined as follows:

1. Confounders ($X$)

The dataset contains five confounders $X = (X_1, X_2, X_3, X_4, X_5)^T$:

• $X_1$: tenure_months
• $X_2$: avg_sessions_week
• $X_3$: spend_last_month
• $X_4$: premium_user
• $X_5$: urban_resident

Base Features Sampling: The base features $X_1, X_2, X_3, X_5$ are sampled using a Gaussian Copula to introduce correlations while preserving specific marginal distributions. The correlation matrix for the underlying Gaussian variables is: $\Sigma = \begin{pmatrix} 1.0 & 0.3 & 0.2 & 0.0 \\ 0.3 & 1.0 & 0.4 & 0.0 \\ 0.2 & 0.4 & 1.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 1.0 \end{pmatrix}$

The marginal distributions are:

• $X_1 \sim \text{Lognormal}(\mu = \ln 24, \sigma = 0.6)$, clipped at $[0, 120]$.
• $X_2 \sim \text{NegativeBinomial}(\text{mean} = 5, \text{dispersion} = 0.5)$, clipped at $[0, 40]$.
• $X_3 \sim \text{Lognormal}(\mu = \ln 60, \sigma = 0.9)$, clipped at $[0, 500]$.
• $X_5 \sim \text{Bernoulli}(p = 0.60)$.

Derived Feature ($X_4$): The feature $X_4$ (premium_user) is generated based on a logistic model of the other features: $\text{logit}(P(X_4 = 1 | X)) = -5.0 + 0.7 \ln(1 + X_2) + 0.5 \ln(1 + X_3) + 0.01 X_1$ $X_4 \sim \text{Bernoulli}(P(X_4 = 1 | X))$

2. Treatment Assignment ($D$)

The treatment $D \in \{0, 1\}$ is assigned using a propensity score $e(X)$: $P(D=1|X) = \sigma\left( \alpha_d + \text{bound}\left( f_d(X), 2.0 \right) \right)$ where:

• $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function.
• $\text{bound}(z, c) = c \cdot \tanh(z/c)$ limits the propensity score range to ensure positivity (here $c=2.0$).
• $\alpha_d$ is a calibration constant chosen such that the overall treatment rate is approximately 35%.
• The score $f_d(X)$ is defined as: $f_d(X) = 0.005 X_1 + 0.8 X_4 + 0.25 X_5 + 0.8 \tanh\left(\ln\frac{1+X_3}{61}\right) + 0.2 \ln\frac{1+X_2}{6} \tanh\left(\frac{X_1}{24} - 1\right) + 0.4 X_4 (X_5 - 0.5)$

3. Heterogeneous Treatment Effect ($\tau(X)$)

The treatment effect (CATE) is nonlinear and depends on the confounders: $\tau(X) = \text{clip}\left( 1.2 + 0.6 \tanh\left(\ln\frac{1+X_2}{6}\right) + 0.4 X_4 - 0.5 \tanh\left(\frac{X_1}{48}\right) + 0.2 X_5 \tanh\left(\ln\frac{1+X_3}{61}\right), 0.1, 3.0 \right)$

4. Outcome Model ($Y$)

The outcome is a continuous variable generated as: $Y = f_y(X) + D \cdot \tau(X) + \epsilon, \quad \epsilon \sim \mathcal{N}(0, 3.5^2)$ The baseline outcome function $f_y(X)$ includes linear and nonlinear terms: $f_y(X) = 0.01 X_1 + 1.2 X_4 + 0.6 X_5 + g_y(X)$ where: $g_y(X) = 1.5 \tanh\left(\frac{X_1}{24}\right) + 0.5 \left(\ln\frac{1+X_2}{6}\right)^2 + 0.2 \ln\frac{1+X_3}{61} \ln\frac{1+X_2}{6} + 0.5 X_4 \ln\frac{1+X_2}{6} + 0.8 X_5 \tanh\left(\frac{1}{2} \ln\frac{1+X_3}{61}\right)$