dgp

Pre-configured multi-treatment dataset with Gamma-distributed outcome.

• 3 treatment classes: d_0 (control), d_1, d_2

• 8 confounders with realistic marginals sampled through a Gaussian copula

• Gamma outcome with log-link confounding and heterogeneous arm effects

Notes

Let $X = (\text{tenure}, \text{sessions}, \text{spend}, \text{premium}, \text{urban}, \text{tickets}, \text{discount}, \text{credit})$ denote the 8 observed confounders. The treatment assignment mechanism is a multinomial logit with calibrated marginal arm rates near $(0.50, 0.25, 0.25)$:

The confounders are jointly sampled through a Toeplitz copula with $\mathrm{Corr}(X_i, X_j) = 0.3^{|i-j|}$.

The outcome uses a log link. For arm $k$,

This scenario fixes $\theta = (0, -0.05, 0.10)$ and uses the heterogeneous shifts

So d_1 is always weakly worse than control on the log-mean scale, while d_2 is always weakly better than control.

Pre-configured multi-treatment dataset with Binary outcome.

• 3 treatment classes: d_0 (control), d_1, d_2

• 8 confounders with realistic marginals sampled through a Gaussian copula

• Binary outcome with a logistic baseline and heterogeneous arm effects

Notes

Let $X = (\text{tenure}, \text{active days}, \text{income}, \text{premium}, \text{family}, \text{complaints}, \text{discount}, \text{engagement})$ denote the 8 confounders. Treatment assignment again follows a calibrated multinomial logit with target arm rates near $(0.50, 0.25, 0.25)$:

The outcome uses a logistic link with alpha_y = -1.1:

This scenario fixes $\theta = (0, -0.18, 0.26)$ and uses

The clipping keeps d_1 uniformly below control and d_2 uniformly above control on the log-odds scale, while the Gaussian copula with $\mathrm{Corr}(X_i, X_j) = 0.3^{|i-j|}$ induces cross-feature dependence.

The notebook simulates overlapping contact and repeat actions. This packaged DGP resolves them into a mutually exclusive one-hot treatment:

• control

• neg_contact_flg

• error_flg

• neg_contact_flg_error_flg

Treatment assignment matches the notebook’s independent Bernoulli contact and repeat mechanisms exactly after overlap-resolution, but is exposed through the shared multi-treatment generator so it integrates with MultiCausalData and the scenario tooling.

Notes

Write $a(X)$ for the contact logit and $b(X)$ for the repeat logit. The notebook first draws two conditionally independent Bernoulli actions,

where $\sigma(z) = 1 / (1 + e^{-z})$. In this packaged benchmark the pair $(C, R)$ is re-encoded as a one-hot treatment:

Let $p_c = \sigma(a(X))$ and $p_r = \sigma(b(X))$. Then the arm probabilities are

Equivalently, this is exactly the softmax model with class scores $(0, a(X), b(X), a(X)+b(X))$, which is why the implementation passes g_d=[None, _cx_contact_logit, _cx_repeat_logit, lambda x: _cx_contact_logit(x) + _cx_repeat_logit(x)].

The observed outcome uses a binary logit baseline $g_y(X)$ plus a class effect

with $\theta(\text{control}) = \theta(\text{neg\_contact\_flg}) = 0$ and $\theta(\text{error\_flg}) = \theta(\text{neg\_contact\_flg\_error\_flg}) = -0.65$.

Worked overlap example: if $a(X)=0.8$ and $b(X)=-0.2$, then $p_c \approx 0.690$ and $p_r \approx 0.450$, giving arm probabilities approximately (0.170, 0.379, 0.140, 0.311) for (control, neg_contact_flg, error_flg, neg_contact_flg_error_flg).

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