generate_multitreatment_gamma_26()

generate_multitreatment_gamma_26() builds a 3-arm multi-treatment observational DGP with correlated confounders and a Gamma outcome.

Treatment columns are one-hot: $D=(D_0,D_1,D_2)$ with exactly one active class per row (d_0, d_1, d_2).

The scenario is configured to target marginal treatment shares close to $0.50, 0.25, 0.25$.

1. Confounders and Dependence

The confounder vector is $X=(X_1,\dots,X_8)$ with marginals:

• $X_1$ (tenure_months) $\sim \mathcal{N}(24, 12^2)$, clipped to $[0,120]$
• $X_2$ (avg_sessions_week) $\sim \mathcal{N}(5, 2^2)$, clipped to $[0,40]$
• $X_3$ (spend_last_month) $\sim \text{LogNormal}(\log 60, 0.9)$, clipped at 500
• $X_4$ (premium_user) $\sim \text{Bernoulli}(0.25)$
• $X_5$ (urban_resident) $\sim \text{Bernoulli}(0.60)$
• $X_6$ (support_tickets_q) $\sim \text{Poisson}(1.5)$, clipped at 15
• $X_7$ (discount_eligible) $\sim \text{Bernoulli}(0.35)$
• $X_8$ (credit_utilization) $\sim \text{Beta}(\alpha,\beta)$ with mean $0.45$ and concentration $\kappa=20$

Dependence is induced with a Gaussian copula whose correlation matrix is Toeplitz: $\Sigma_{ij} = \rho^{|i-j|}, \quad \rho=0.30.$

2. Treatment Assignment (Multinomial Logit)

For each class $k \in \{0,1,2\}$, define score $s_k(X) = \alpha_k + X^\top\beta_{d,k}.$ Then propensity is softmax: $m_k(X)=P(D_k=1\mid X)=\frac{\exp(s_k(X))}{\sum_{j=0}^2\exp(s_j(X))}.$

Coefficients used in this scenario:

• $\beta_{d,0}=\mathbf{0}$
• $\beta_{d,1}=[0.01, 0.10, 0.0015, 0.50, 0.20, 0.05, 0.35, 0.40]$
• $\beta_{d,2}=[-0.005, 0.07, 0.0010, 0.35, 0.10, 0.08, 0.20, 0.25]$

Treatment-score intercepts start at $0.0,0.0,0.0$, then are calibrated iteratively so mean class rates are close to target $0.50,0.25,0.25$.

3. Treatment Effects on Link Scale

The structural link shift for class $k$ is $\tau^{link}_k(X)=\theta_k+\tau_k(X), \qquad \theta=(0.0,-0.05,0.10).$

Control has no heterogeneous residual: $\tau_0(X)=0$.

For d_1 (forced harmful vs control): $\tau_1(X)=\min\Big(-0.22 -0.001\,\text{tenure}-0.006\,\text{sessions}-0.05\,\text{premium}-0.04\,\text{discount}-0.10(\text{credit}-0.45),\,-0.02\Big).$

For d_2 (forced beneficial vs control): $\tau_2(X)=\max\Big(0.16 +0.014\,\text{sessions}+0.030\log(1+\text{spend})+0.06\,\text{urban}-0.006\,\text{tickets}+0.12(\text{credit}-0.45),\,0.02\Big).$

4. Outcome Model (Gamma)

Baseline linear predictor: $\eta_0(X)=\alpha_y + X^\top\beta_y, \qquad \alpha_y=0,$ with $\beta_y=[0.01,0.08,0.0015,0.35,0.12,0.06,0.20,0.50].$

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

Gamma mean uses log link: $\mu(X,D)=\exp(\eta(X,D)).$ Given shape $a=2.0$, outcome is sampled as $Y\mid X,D \sim \text{Gamma}(\text{shape}=a,\,\text{scale}=\mu/a),$ so $\mathbb{E}[Y\mid X,D]=\mu$ and $\mathrm{Var}(Y\mid X,D)=\mu^2/a$.

5. Oracle Quantities

When include_oracle=True, the generator exposes:

• m_d_0, m_d_1, m_d_2: calibrated propensities $m_k(X)$
• tau_link_d_k: link-scale treatment shifts $\tau^{link}_k(X)$
• g_d_k: potential outcome means on natural scale under class $k$
• cate_d_1 = g_{d_1}-g_{d_0}, cate_d_2 = g_{d_2}-g_{d_0}