Make Cuped Tweedie 26

The make_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 a pre-period covariate (y_pre) calibrated for CUPED benchmarks.

1. Confounders ($X$)

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 ($D$)

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, make_cuped_tweedie_26 defaults to a balanced random assignment).

3. Outcome Model ($Y$)

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

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

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

• $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function.
• $\alpha_{zi} = 0.0$ (resulting in ~50% baseline non-zero rate).
• $\text{u\_strength}_{zi} = 1.0$ (if add_pre=True).
• $U \sim \mathcal{N}(0, 1)$ is an unobserved confounder.

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 linear predictor on the log-mean scale: $\text{loc} = \alpha_y + D \cdot \tau(X) + \text{u\_strength}_y \cdot U$
• $\alpha_y = 2.0$.
• $\text{u\_strength}_y = 1.0$ (if add_pre=True).

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

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).
• $\theta_{log} = 0.12$ (Default log-uplift parameter).

5. Pre-period Covariate ($y_{pre}$)

The pre-period covariate is generated using the same two-part structure as the outcome but replaces the unobserved confounder $U$ with a shared latent driver $A \sim \mathcal{N}(0, 1)$: $y_{pre} = (I_{pre} \cdot Y_{pos, pre}) + \sigma_{noise} \cdot \epsilon$

• $\epsilon \sim \mathcal{N}(0, 1)$.
• The influence of the shared driver $A$ and the noise scale $\sigma_{noise}$ are calibrated (via numerical optimization) to ensure $y_{pre}$ achieves a target correlation (default $0.6$) with the post-period outcome $Y$ in the control group.