generate_multitreatment_binary_26()

generate_multitreatment_binary_26() defines a 3-arm multi-treatment observational DGP with correlated confounders and a binary outcome.

Treatments are one-hot columns (d_0, d_1, d_2) and are sampled from a multinomial-logit propensity model calibrated toward target class shares $0.50,0.25,0.25$.

1. Confounders and Copula Correlation

The confounder vector $X=(X_1,\dots,X_8)$ uses:

• $X_1$ (tenure_months) $\sim \mathcal{N}(24,12^2)$, clipped to $[0,120]$
• $X_2$ (weekly_active_days) $\sim \mathcal{N}(4.0,1.5^2)$, clipped to $[0,7]$
• $X_3$ (annual_income_k) $\sim \text{Gamma}(\text{shape}=4,\text{scale}=18)$, clipped at 300
• $X_4$ (premium_user) $\sim \text{Bernoulli}(0.22)$
• $X_5$ (family_plan) $\sim \text{Bernoulli}(0.38)$
• $X_6$ (recent_complaints) $\sim \text{Poisson}(0.8)$, clipped at 10
• $X_7$ (discount_eligible) $\sim \text{Bernoulli}(0.30)$
• $X_8$ (engagement_score) $\sim \text{Beta}(\alpha,\beta)$ with mean $0.60$ and concentration $\kappa=16$

Dependencies are induced with Gaussian copula correlation $\Sigma_{ij}=0.30^{|i-j|}.$

2. Treatment Assignment (Softmax)

Class scores are $s_k(X)=\alpha_k + X^\top\beta_{d,k}, \quad k\in\{0,1,2\},$ with propensities $m_k(X)=\frac{e^{s_k(X)}}{\sum_{j=0}^{2}e^{s_j(X)}}.$

Scenario coefficients:

• $\beta_{d,0}=\mathbf{0}$
• $\beta_{d,1}=[0.01,0.09,0.0018,0.45,0.20,0.08,0.30,0.28]$
• $\beta_{d,2}=[-0.004,0.07,0.0012,0.30,0.12,0.10,0.18,0.22]$

Treatment-score intercepts start at $0.0,0.0,0.0$ and are calibrated to match the target marginal rates.

3. Heterogeneous Effects on Logit Scale

Treatment shifts are additive on the link scale: $\tau^{link}_k(X)=\theta_k + \tau_k(X), \qquad \theta=(0.0,-0.18,0.26).$

For d_1 (enforced harmful relative to control): $\tau_1(X)=\min\Big(-0.16 -0.0008\,\text{tenure}-0.020\,\text{activeDays}-0.08\,\text{premium}-0.03\,\text{complaints}-0.10(\text{engagement}-0.60),\,-0.02\Big).$

For d_2 (enforced beneficial relative to control): $\tau_2(X)=\max\Big(0.14 +0.020\,\text{activeDays}+0.028\log(1+\text{income})+0.05\,\text{familyPlan}-0.010\,\text{complaints}+0.12(\text{engagement}-0.60),\,0.02\Big).$

4. Outcome Model (Binary Logistic)

Baseline logit: $\eta_0(X)=\alpha_y + X^\top\beta_y, \qquad \alpha_y=-1.1,$ where $\beta_y=[0.003,0.11,0.004,0.40,-0.25,-0.12,0.20,0.90].$

Observed logit under assigned treatment: $\eta(X,D)=\eta_0(X)+\sum_{k=0}^{2}D_k\,\tau^{link}_k(X).$

Then $P(Y=1\mid X,D)=\sigma(\eta)=\frac{1}{1+e^{-\eta}}, \qquad Y\sim\text{Bernoulli}(\sigma(\eta)).$

5. Oracle Outputs

With include_oracle=True, the generated frame includes:

• m_d_k: calibrated propensities
• tau_link_d_k: link-scale treatment shifts
• g_d_k: potential outcome probabilities under each arm
• cate_d_1, cate_d_2: contrasts vs control on probability scale