Theory: Causal objects (potential outcomes + estimands)

The causal extension of CausalMixGPD is built on the potential outcomes framework. The main inferential objects are treatment-specific conditional distributions, modeled with the same spliced DPM–GPD machinery as the one-arm setting.

Potential outcomes and treatment-specific conditional distributions

Let \(A\in\{0,1\}\) be a binary treatment indicator, \(X\) the observed pre-treatment covariates, and \(Y\) the observed outcome.

Each unit has two potential outcomes, \(Y(1)\) and \(Y(0)\), and the observed outcome satisfies

\[ Y = A\,Y(1) + (1-A)\,Y(0). \]

Define the treatment-specific conditional CDFs

\[ F_a(y\mid x)=\Pr\{Y(a)\le y \mid X=x\},\qquad a\in\{0,1\}. \]

The conditional density is \(f_a(y\mid x)=\frac{\partial}{\partial y}F_a(y\mid x)\), and the package models these densities using the same spliced DPM form for each treatment arm.

Conditional quantiles are defined by CDF inversion:

\[ Q_a(\tau\mid x)=\inf\{y:\,F_a(y\mid x)\ge \tau\},\qquad \tau\in(0,1). \]

Mean-type summaries are

\[ \mu_a(x)=\mathbb{E}\{Y(a)\mid X=x\}=\int y\,f_a(y\mid x)\,dy. \]

Marginal vs treated-standardized targets

Beyond conditional functions of \(x\), the package reports standardized (marginal) quantities by averaging conditional distributions over a covariate distribution.

Two common standards are:

\[ F_a^{m}(y)=\mathbb{E}\{F_a(y\mid X)\}, \qquad F_a^{t}(y)=\mathbb{E}\{F_a(y\mid X)\mid A=1\}. \]

Corresponding marginal quantiles are

\[ Q_a^{m}(\tau)=\inf\{y:\,F_a^{m}(y)\ge \tau\}, \qquad Q_a^{t}(\tau)=\inf\{y:\,F_a^{t}(y)\ge \tau\}. \]

Marginal means are

\[ \mu_a^{m}=\mathbb{E}\{\mu_a(X)\}, \qquad \mu_a^{t}=\mathbb{E}\{\mu_a(X)\mid A=1\}. \]

Identification assumptions and propensity score augmentation

Identification requires assumptions that connect potential outcomes to observed data.

• SUTVA (no interference and no hidden versions of treatment)

• Consistency: if \(A=a\) then \(Y=Y(a)\)

• Conditional ignorability:

  \[ (Y(0),Y(1))\perp A \mid X. \]

• Overlap / positivity:

  \[ 0<\Pr(A=1\mid X=x)<1 \]

In observational studies, CausalMixGPD supports propensity-score (PS) design-stage support via:

\[ \rho(x)=\Pr(A=1\mid X=x), \]

using the balancing property

\[ A\perp X \mid \rho(X). \]

With estimated scores \(\widehat{\rho}(x)\), the package augments the outcome model covariates to build a reduced confounding adjustment predictor \(r(x)\). Then it fits treatment-arm specific outcome models separately and computes causal contrasts as functionals of those fitted treatment-specific conditional distributions.

Reported causal estimands

Let \(\mu_a\) and \(Q_a\) denote the relevant conditional or standardized (marginal) quantities described above. The package reports:

• ATE (average treatment effect)

  \[ \mathrm{ATE}=\mu_1^{m}-\mu_0^{m}. \]

• ATT (average treatment effect on the treated)

  \[ \mathrm{ATT}=\mu_1^{t}-\mu_0^{t}. \]

• QTE (quantile treatment effect)

  \[ \mathrm{QTE}(\tau)=Q_1^{m}(\tau)-Q_0^{m}(\tau). \]

• QTT (quantile treatment effect on the treated)

  \[ \mathrm{QTT}(\tau)=Q_1^{t}(\tau)-Q_0^{t}(\tau). \]

• CATE (conditional average treatment effect)

  \[ \mathrm{CATE}(x)=\mu_1(x)-\mu_0(x). \]

• CQTE (conditional quantile treatment effect)

  \[ \mathrm{CQTE}(\tau\mid x)=Q_1(\tau\mid x)-Q_0(\tau\mid x). \]

What this means in the package

• One-arm posterior inference is “distribution first”; causal inference swaps in treatment-specific conditional distributions.
• Causal functions like ate(), att(), qte(), qtt(), cate(), and cqte() are contrast functionals of those fitted treatment-specific conditional distributions.

References (key)

• Rosenbaum & Rubin (1983), The Central Role of the Propensity Score in Observational Studies for Causal Effects — doi:10.1093/biomet/70.1.41
• Rubin (1974), Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies — doi:10.1037/h0037350
• Gelman et al. (2013), Bayesian Data Analysis — doi:10.1201/b16018

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