DGP classic_rct_gamma_26

Math Explanation of the classic_rct_gamma_26 DGP

The classic_rct_gamma_26 function generates a synthetic dataset for a Classic Randomized Controlled Trial (RCT). Treatment is assigned completely at random, and covariates $X$ affect the outcome (prognostic) but do not influence treatment assignment (no confounding).

By default, it simulates a revenue experiment (gamma outcome) with 10,000 samples and a 50/50 split.

Covariate Generation (Confounders)

Three binary covariates $X = [x_1, x_2, x_3]$ are generated independently:

• platform_ios ($x_1$): $x_1 \sim \text{Bernoulli}(0.5)$
• country_usa ($x_2$): $x_2 \sim \text{Bernoulli}(0.6)$
• source_paid ($x_3$): $x_3 \sim \text{Bernoulli}(0.3)$

Treatment Assignment ($D$)

Since it is an RCT, the treatment $D$ is independent of $X$. It is assigned with a probability $P(D=1) = 0.5$: $D \sim \text{Bernoulli}(0.5)$ The log-odds of treatment (intercept $\alpha_d$) is $0$.

Outcome Generation (revenue)

The outcome is a positive, skewed metric (e.g., revenue) modeled with a Gamma distribution using a log-mean link: $Y \mid D, X \sim \text{Gamma}(k, \theta)$ $\log \mathbb{E}[Y \mid D, X] = \log(k \cdot \theta_A) + \beta_y^\top X + g_y(X) + D \cdot \log(\theta_B/\theta_A)$

By default, $g_y(X)=0$ because add_pre=False. The nonlinear term is only included when add_pre=True (or when g_y/use_prognostic is provided).

• Shape ($k$): Default $k=2.0$.
• Scale (control $\theta_A$): Default $15.0$.
• Scale (treatment $\theta_B$): Default $16.5$.
• Prognostic Coefficients ($\beta_y$): Default $[0.25, 0.20, 0.45]$. These values shift the log-mean of revenue.

With the defaults, the group means are: $\mathbb{E}[Y \mid D=0] = k \cdot \theta_A = 2.0 \times 15.0 = 30.0$ $\mathbb{E}[Y \mid D=1] = k \cdot \theta_B = 2.0 \times 16.5 = 33.0$ This corresponds to a 10% uplift on the mean scale before covariate effects.

Summary of Default Parameters

• $N$: 10,000
• Control Mean (baseline): $30.0$ at $X=0$
• Treatment Mean (baseline): $33.0$ at $X=0$
• Treatment Split: 50%
• Confounders: 3 binary
• Nonlinear $g_y(X)$: only when add_pre=True or g_y/use_prognostic is provided

DGP

EDA

Balance check