DGP generate_classic_rct_26

Math Explanation of the generate_classic_rct_26 DGP

The generate_classic_rct_26 function generates a synthetic dataset for a Classic Randomized Controlled Trial (RCT). In this scenario, treatment is assigned completely at random, and covariates $X$ affect the outcome (prognostic) but do not influence the treatment assignment (no confounding).

By default, it simulates a conversion experiment (binary 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 (conversion)

The outcome is a binary variable representing conversion. The probability of conversion for an individual is modeled using a logistic link function: $P(\text{conversion}=1 \mid D, X) = \sigma(L)$ where $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function.

The latent linear predictor $L$ is defined as: $L = \alpha_y + \sum_{j=1}^3 \beta_{y,j} x_j + g_y(X) + D \cdot \theta$

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).

• Baseline Intercept ($\alpha_y$): Derived from the target control conversion rate $p_A = 0.10$. $\alpha_y = \text{logit}(0.10) = \ln\left(\frac{0.10}{0.90}\right) \approx -2.197$ This sets the baseline rate at $X=0$; the marginal control rate can differ once $X$ shifts log-odds.
• Treatment Effect ($\theta$): Derived from the target treatment conversion rate $p_B = 0.11$. It represents the shift in log-odds. $\theta = \text{logit}(0.11) - \text{logit}(0.10) = \ln\left(\frac{0.11}{0.89}\right) - \ln\left(\frac{0.10}{0.90}\right) \approx 0.106$ This is a baseline log-odds shift; the marginal ATE on the probability scale is not exactly 1% once $X$ effects are present.
• Prognostic Coefficients ($\beta_y$): By default, $\beta_y = [0.6, 0.4, 0.8]$. These values determine how much each covariate shifts the log-odds of conversion.

The outcome is a binary variable representing conversion. The probability of conversion for an individual is modeled using a logistic link function: $P(\text{conversion}=1 \mid D, X) = \sigma(L)$ where $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function.

The latent linear predictor $L$ is defined as: $L = \alpha_y + \sum_{j=1}^3 \beta_{y,j} x_j + g_y(X) + D \cdot \theta$

• Baseline Intercept ($\alpha_y$): Derived from the target control conversion rate $p_A = 0.10$. $\alpha_y = \text{logit}(0.10) = \ln\left(\frac{0.10}{0.90}\right) \approx -2.197$
• Treatment Effect ($\theta$): Derived from the target treatment conversion rate $p_B = 0.11$. It represents the shift in log-odds. $\theta = \text{logit}(0.11) - \text{logit}(0.10) = \ln\left(\frac{0.11}{0.89}\right) - \ln\left(\frac{0.10}{0.90}\right) \approx 0.106$ On the probability scale, this corresponds to an Average Treatment Effect (ATE) of $\approx 1\%$.
• Prognostic Coefficients ($\beta_y$): By default, $\beta_y = [0.6, 0.4, 0.8]$. These values determine how much each covariate shifts the log-odds of conversion.

Summary of Default Parameters

• $N$: 10,000
• Control Conversion (baseline $p_A$): 10% at $X=0$ (marginal rate can differ with $X$)
• Treatment Conversion (baseline $p_B$): 11% at $X=0$ (marginal rate can differ with $X$)
• 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