generate_obs_hte_binary_26()

The generate_obs_hte_binary_26() function provides a richer observational dataset with binary outcomes, nonlinear confounding, and heterogeneous treatment effects. It uses 11 confounders with behavior-driven derived features and a moderate target treatment rate.

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
• $X_5$: prior_purchases_12m
• $X_6$: support_tickets_90d
• $X_7$: premium_user
• $X_8$: mobile_user
• $X_9$: weekend_user
• $X_{10}$: email_opt_in
• $X_{11}$: referred_user

Base Features Sampling: The base features $X_1, X_2, X_3, X_4, X_9$ are sampled using a Gaussian Copula with correlation matrix:

Derived Features: The remaining confounders are generated from behavior-linked models:

• Premium User ($X_7$): $\text{logit}(P(X_7=1)) = -4.6 + 0.70 \ln(1+X_2) + 0.45 \ln(1+X_3) + 0.012 X_1 + 0.20 X_9$
• Mobile User ($X_8$): $\text{logit}(P(X_8=1)) = 1.4 - 0.03(X_4 - 35) + 0.25 X_9 + 0.22 \ln(1+X_2)$
• Referred User ($X_{11}$): $\text{logit}(P(X_{11}=1)) = -1.3 - 0.02(X_1 - 12) + 0.42 X_8 + 0.25 X_9$
• Email Opt-in ($X_{10}$): $\text{logit}(P(X_{10}=1)) = -0.6 + 0.50 X_8 + 0.35 X_7 + 0.20 X_9 + 0.12 \ln(1+X_2)$
• Prior Purchases ($X_5$): $X_5 \sim \text{Poisson}(\lambda = \text{clip}(0.9 + 0.52 \ln(1+X_2) + 0.40 \ln(1+X_3) + 0.24 X_7 + 0.12 X_{10}, 0.05, 25.0))$
• Support Tickets ($X_6$): $X_6 \sim \text{Poisson}(\lambda = \text{clip}(0.55 + 0.22 \ln(1+X_2) + 0.32(1-X_7) + 0.18(1-X_{10}) + 0.12 \tanh(\frac{X_4 - 45}{12}), 0.05, 10.0))$

2. Treatment Assignment ($D$)

The treatment $D \in \{0, 1\}$ is assigned with a target rate of 15%: $P(D=1|X) = \sigma\left( \alpha_d + \text{bound}(f_d(X), 2.0) \right)$ where $f_d(X) = \sum_{j=1}^{11} \beta_{d,j} X_j + g_d(X)$ and $g_d(X) = 0.85\tanh\left(\ln\frac{1+X_3}{61}\right) + 0.24\ln\frac{1+X_2}{6}\tanh\left(\frac{X_1}{24}-1\right) + 0.30X_7(X_9-0.5) + 0.22X_{10}\tanh\left(\ln\frac{1+X_2}{4}\right) + 0.25X_{11}\left(1-\tanh\frac{X_1}{36}\right) + 0.14\tanh\left(\ln\frac{1+X_6}{2.5}\right) - 0.10\tanh\left(\frac{X_4-45}{12}\right).$

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

For this binary-outcome scenario, treatment effect is on the log-odds scale: $\tau(X) = 0.30 + 0.40\tanh\left(\ln\frac{1+X_2}{6}\right) + 0.45X_7 + 0.12X_8 + 0.10X_{10} + 0.12X_{11} - 0.32\tanh\frac{X_1}{48} - 0.18\tanh\frac{X_4-40}{15} + 0.10X_9\tanh\left(\ln\frac{1+X_3}{61}\right) - 0.12\tanh\left(\ln\frac{1+X_6}{2.5}\right).$ The treatment effect is clipped to $[-0.35, 1.4]$.

4. Outcome Model ($Y$)

The outcome is binary with logistic link: $P(Y=1\mid X,D) = \sigma\left(\alpha_y + \sum_{j=1}^{11}\beta_{y,j}X_j + g_y(X) + D\cdot\tau(X)\right),$ with $\alpha_y = -1.7$.

The nonlinear baseline component is: $g_y(X) = 1.1\tanh\frac{X_1}{24} + 0.50\left(\ln\frac{1+X_2}{6}\right)^2 + 0.22\ln\frac{1+X_3}{61}\ln\frac{1+X_2}{6} - 0.35\ln(1+X_6) + 0.18\ln(1+X_5)\tanh\frac{X_1}{18} + 0.22X_8\ln\frac{1+X_2}{6} + 0.20X_7(X_9-0.5) + 0.18X_{10}\tanh\left(\frac{1}{2}\ln\frac{1+X_3}{61}\right) + 0.14X_{11}\tanh\left(\frac{X_1}{12}-1\right) - 0.20\tanh\frac{X_4-40}{15}.$

Oracle CATE is reported on the natural scale as a risk difference (g1 - g0).