generate_cuped_tweedie_26

The generate_cuped_tweedie_26 data generating process (DGP) creates a synthetic dataset characterized by a Tweedie-like outcome (zero-inflated with a heavy right tail), correlated confounders, and structured heterogeneous treatment effects (HTE). It also includes pre-period covariates (y_pre and y_pre_2) calibrated for CUPED benchmarks.

1. Confounders

Five confounders are generated using a Gaussian Copula to induce specific correlations:

• tenure_months ($X_1$): $\text{Lognormal}(\mu=2.5, \sigma=0.5)$
• avg_sessions_week ($X_2$): $\text{NegativeBinomial}(\mu=5, \alpha=0.5)$
• spend_last_month ($X_3$): $\text{Lognormal}(\mu=4.0, \sigma=0.8)$
• discount_rate ($X_4$): $\text{Beta}(\text{mean}=0.1, \kappa=20)$
• platform ($X_5$): Categorical with levels android (65%), ios (30%), web (5%).

Correlations: $\text{corr}(X_1, X_2) = 0.4$ and $\text{corr}(X_2, X_3) = 0.5$.

2. Treatment Assignment

The treatment $D$ is assigned via a Bernoulli trial with a constant propensity score (RCT): $D \sim \text{Bernoulli}(0.5)$ (Note: While the generator supports complex propensity models, generate_cuped_tweedie_26 defaults to a balanced random assignment).

3. Outcome Model

The outcome is generated as a two-part (hurdle) process: $Y = I \cdot Y_{pos}$

A. Binary Indicator of Non-zero Outcome ($I$)

$I \sim \text{Bernoulli}(\sigma(\alpha_{zi} + u_{zi} \cdot U))$

• $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function.
• $\alpha_{zi} = 0.0$ (about 50% baseline non-zero rate).
• $u_{zi} = 1.0$ when add_pre=True, and $0$ otherwise.

B. Positive Outcome Value ($Y_{pos}$)

If $I=1$, the value is drawn from a Gamma distribution: $Y_{pos} \sim \text{Gamma}(\text{shape}=k, \text{scale}=\mu_{pos} / k)$

• $k = 2.0$ (shape parameter).
• $\mu_{pos} = \exp(\text{loc})$, where $\text{loc}$ is the log-mean linear predictor: $\text{loc} = \alpha_y + D \cdot \tau(X) + u_y \cdot U$
• $\alpha_y = 2.0$.
• $u_y = 1.0$ when add_pre=True, and $0$ otherwise.
• $U \sim \mathcal{N}(0,1)$ is a shared latent prognostic driver used in the outcome/pre construction (treatment remains randomized because $u_d=0$).

4. Heterogeneous Treatment Effect

The treatment effect $\tau(X)$ is defined on the log-mean scale and incorporates monotone effects, diminishing returns, and categorical modifiers: $\tau(X) = \text{clip}\left(\theta_{log} \cdot g_t(X_1) \cdot g_s(X_2) \cdot \text{seg}(X_5), 0, 2.5 \cdot \theta_{log}\right)$ Where:

• $g_t(X_1) = \frac{1}{1 + \exp(-0.25 \cdot (X_1 - 12))}$ (saturating effect of tenure).
• $g_s(X_2) = \frac{1}{1 + \exp(-0.35 \cdot (X_2 - 4))}$ (diminishing returns of sessions).
• $\text{seg}(X_5) = 1 + 0.3 \cdot \mathbb{1}_{\text{platform}=\text{ios}}$ (premium segment modifier).
• generate_cuped_tweedie_26 uses $\theta_{log}=0.38$ by default.

5. Pre-period Covariate

The first pre-period covariate uses the same latent driver $A \sim \mathcal{N}(0,1)$ that also drives outcome stochasticity when add_pre=True: $y_{pre} = base_1 + \sigma_{noise} \cdot \epsilon, \quad \epsilon \sim \mathcal{N}(0,1)$ where $base_1$ is built from the Tweedie two-part mechanism, and $\sigma_{noise}$ is numerically calibrated so that \n$\mathrm{corr}(y_{pre}, y \mid d=0) \approx \rho_{target,1}$ with default $\rho_{target,1}=0.82$ in generate_cuped_tweedie_26.

6. Second Pre-period Covariate

The second pre-period covariate is constructed from a second noisy view of the same latent driver $A$ (not an orthogonalized independent latent):

A. Latent-driven two-part base

With standardized covariates $tenure_z, sessions_z, spend_z, discount_z$ and $a=z(A)$: $zi\_score = 0.20 + 1.05a + 0.10\,tenure_z - 0.08\,discount_z$ $p_{pos,2} = \sigma(zi\_score), \quad I_2 \sim \text{Bernoulli}(p_{pos,2})$ $loc_2 = 1.90 + 0.95a + 0.10\,sessions_z + 0.08\,spend_z$ $\mu_2 = e^{loc_2}, \quad Y_{2,+} \sim \text{Gamma}(shape=2.3, scale=\mu_2/2.3)$ $base_2 = I_2 \cdot Y_{2,+}$

B. Feasibility blending and calibration

If $\mathrm{corr}(base_2, y \mid d=0)$ is below the second target, the base is blended with $y_{pre}$: $base_2^{mix} = (1-w)\,base_2 + w\,y_{pre}, \quad w \in [0.10,1.00]$ using the first $w$ on a grid that reaches the target (or the best achievable one). Then Gaussian noise is calibrated as: $y_{pre,2} = base_2^{mix} + \sigma_2 \cdot \epsilon_2, \quad \epsilon_2 \sim \mathcal{N}(0,1)$ to match the requested second control-group correlation. If pre_target_corr_2 is omitted, the default target is $\rho_{target,2}=\min(\rho_{target,1},\,\min(0.72,\rho_{target,1}-0.10)).$