generate_cuped_binary_26

The make_cuped_binary_26 data generating process (DGP) creates a synthetic binary-outcome dataset with randomized treatment, richer confounders, structured heterogeneous treatment effects (HTE), and a calibrated pre-period covariate (y_pre) for CUPED benchmarking.

1. Confounders

The DGP uses mixed-type confounders sampled from independent marginals:

• tenure_months ($X_1$): $\text{Lognormal}(\mu=2.5, \sigma=0.5)$
• spend_last_month ($X_2$): $\text{Lognormal}(\mu=4.0, \sigma=0.8)$
• discount_rate ($X_3$): $\text{Beta}(\text{mean}=0.1, \kappa=20)$
• support_tickets ($X_4$): $\text{Poisson}(\lambda=1.8)$
• email_open_rate ($X_5$): $\text{Beta}(\text{mean}=0.45, \kappa=14)$
• referral_count ($X_6$): $\text{Poisson}(\lambda=0.8)$
• plan_tier ($X_7$): Categorical with levels free (55%), plus (30%), pro (15%), encoded as plan_tier_plus and plan_tier_pro.
• region ($X_8$): Categorical with levels na (80%), eu (20%), encoded as region_eu.

2. Treatment Assignment

Treatment is randomized with constant propensity: $D \sim \text{Bernoulli}(0.5)$ This is implemented with $\alpha_d = \text{logit}(0.5)$ and no $X$- or latent-driven treatment terms.

3. Outcome Model

The outcome is binary with a logistic link: $Y \sim \text{Bernoulli}(p), \quad \text{logit}(p)=\alpha_y + g_y(X) + \lambda U + D \cdot \tau(X)$

• $\alpha_y=-1.4$.
• $U \sim \mathcal{N}(0,1)$ is a shared latent prognostic signal used only in the outcome equation (so treatment remains randomized/unconfounded).
• $\lambda=1.2$ when add_pre=True, and $\lambda=0$ otherwise.
• Baseline score $g_y(X)$ combines monotone transformations of confounders and segment indicators.

4. Heterogeneous Treatment Effect

Treatment effect is defined on the log-odds scale: $\tau(X)=\text{clip}(\theta_{logit} \cdot g_t \cdot g_{sp} \cdot g_o \cdot g_{ticket} \cdot g_{ref} \cdot seg, 0, 2\theta_{logit})$ Where:

• $g_t = \sigma(0.20 \cdot (\text{tenure} - 10))$
• $g_{sp} = \sigma(0.55 \cdot (\log(1+\text{spend}) - 4))$
• $g_o = 0.85 + 0.35 \cdot \text{email\_open\_rate}$
• $g_{ticket} = \frac{1}{1+\exp(0.45 \cdot (\text{support\_tickets}-2.5))}$
• $g_{ref} = 0.90 + 0.18 \cdot \tanh(0.8 \cdot \text{referral\_count})$
• $seg = 1 + 0.12\,\mathbb{1}_{plan=plus} + 0.26\,\mathbb{1}_{plan=pro} + 0.06\,\mathbb{1}_{region=eu}$
• $\theta_{logit}=0.38$ by default.

5. Pre-period Covariate

If add_pre=True, first define a centered probability-scale baseline signal: $s(X,U)=\sigma\!\left(\alpha_y + g_y(X) + \lambda U\right)-\mathbb{E}\left[\sigma\!\left(\alpha_y + g_y(X) + \lambda U\right)\right]$ Then construct: $y_{pre}=s(X,U)+\sigma_{noise}\epsilon, \quad \epsilon \sim \mathcal{N}(0,1)$ The noise scale is calibrated numerically so that $\text{corr}(y_{pre}, y)$ in the control group matches the target correlation (default $0.65$).