generate_obs_hte_26_rich()

The generate_obs_hte_26_rich() function provides a more complex and realistic observational dataset. It features 11 confounders, complex dependencies (including derived features), a low treatment rate, and a Tweedie (zero-inflated Gamma) outcome model.

1. Confounders ($X$)

The dataset contains 11 confounders $X = (X_1, \dots, X_{11})^T$:

• $X_1$: tenure_months
• $X_2$: avg_sessions_week
• $X_3$: spend_last_month
• $X_4$: age_years
• $X_5$: income_monthly
• $X_6$: prior_purchases_12m
• $X_7$: support_tickets_90d
• $X_8$: premium_user
• $X_9$: mobile_user
• $X_{10}$: urban_resident
• $X_{11}$: referred_user

Base Features Sampling: The base features $X_1, X_2, X_3, X_4, X_5, X_{10}$ are sampled using a Gaussian Copula with the following correlation matrix:

Derived Features: The remaining features are derived to mimic real-world behavioral dependencies:

• Premium User ($X_8$): $\text{logit}(P(X_8=1)) = -5.0 + 0.7 \ln(1+X_2) + 0.45 \ln(1+X_3) + 0.35 \ln(1+X_5) + 0.015 X_1$
• Mobile User ($X_9$): $\text{logit}(P(X_9=1)) = 1.5 - 0.03(X_4 - 35) + 0.30 X_{10} + 0.25 \ln(1+X_2)$
• Referred User ($X_{11}$): $\text{logit}(P(X_{11}=1)) = -1.2 - 0.02(X_1 - 12) + 0.45 X_9 + 0.20 X_{10}$
• Prior Purchases ($X_6$): $X_6 \sim \text{Poisson}(\lambda = \text{clip}(1.0 + 0.55 \ln(1+X_2) + 0.45 \ln(1+X_3) + 0.25 X_8, 0.1, 30.0))$
• Support Tickets ($X_7$): $X_7 \sim \text{Poisson}(\lambda = \text{clip}(0.6 + 0.25 \ln(1+X_2) + 0.30(1-X_8) + 0.15 \tanh(\frac{X_4 - 45}{12}), 0.05, 10.0))$

2. Treatment Assignment ($D$)

The treatment $D \in \{0, 1\}$ is assigned with a target rate of 5%: $P(D=1|X) = \sigma\left( \alpha_d + \text{bound}\left( f_d(X), 2.0 \right) \right)$ where $f_d(X) = \sum_{j=1}^{11} \beta_{d,j} X_j + g_d(X)$ with $\beta_d = (-0.004,\ 0,\ 0,\ -0.012,\ -0.00005,\ -0.04,\ 0.22,\ -0.45,\ -0.08,\ -0.12,\ 0.10),$ and

This design induces adverse selection (treated units can have lower observed outcomes than controls even when treatment helps on average).

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

The treatment effect on the link (log-mean) scale is: $\tau(X) = 0.08 + 0.08 \tanh\left(\ln\frac{1+X_2}{6}\right) + 0.11 X_8 + 0.03 X_9 + 0.03 X_{11} - 0.07 \tanh\frac{X_1}{48} - 0.05 \tanh\frac{X_4-40}{15} + 0.03 X_{10} \tanh\left(\ln\frac{1+X_3}{61}\right) - 0.04 \tanh\left(\ln\frac{1+X_7}{2.5}\right)$ and is clipped to $[0.005, 0.35]$.

Important: this clipping is for $\tau(X)$ on the link scale, not for CATE itself.

4. Outcome Model ($Y$)

The outcome follows a Tweedie (Two-part Hurdle) model:

1. Zero-Inflation (Participation): $Y > 0$ with probability $p_{pos} = \sigma(\alpha_{zi} + \sum \beta_{zi,j} X_j + g_{zi}(X) + D \cdot \tau_{zi}(X))$ where $\alpha_{zi} = -0.8$, and the nonlinear components are: $g_{zi}(X) = 0.45 \tanh\left(\ln\frac{1+X_2}{5}\right) + 0.25 \tanh\left(\ln\frac{1+X_3}{51}\right) + 0.20 \ln(1+X_6) - 0.30 \tanh\left(\ln\frac{1+X_7}{2}\right) + 0.15 X_8 + 0.10 X_9$ $\tau_{zi}(X) = 0.03 + 0.02 X_8 + 0.015 X_{11} - 0.02 \tanh\frac{X_1}{48}$

2. Positive Outcome (Magnitude): If $Y > 0$, then $Y \sim \text{Gamma}(\text{shape} = 2.2, \text{scale} = \mu_{pos} / 2.2)$ where $\ln(\mu_{pos}) = \sum \beta_{y,j} X_j + g_y(X) + D \cdot \tau(X)$. The baseline nonlinear part is: $g_y(X) = 1.4 \tanh\frac{X_1}{24} + 0.6 \left(\ln\frac{1+X_2}{6}\right)^2 + 0.25 \ln\frac{1+X_3}{61} \ln\frac{1+X_2}{6} + 0.35 \tanh\frac{\ln(1+X_5) - \ln 4001}{1.5} - 0.45 \ln(1+X_7) + 0.20 \ln(1+X_6) \tanh\frac{X_1}{18} + 0.30 X_9 \ln\frac{1+X_2}{6} + 0.25 X_8 (X_{10}-0.5) + 0.15 X_{11} \tanh(\frac{X_1}{12}-1) - 0.20 \tanh\frac{X_4-40}{15}$ The oracle natural-scale CATE is always computed as $\mathrm{CATE}(X)=g_1(X)-g_0(X),$ where $g_d(X)=\mathbb{E}[Y\mid X,D=d]$ under the two-part model.