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title: "Methodological Background"
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## Introduction

Indirect treatment comparisons (ITCs) are vital when head-to-head clinical trials are absent. Unadjusted comparisons are biased when trial populations differ in effect modifiers; thus, population adjustment is required. `outstandR` provides a unified framework for these methods.

## Matching-Adjusted Indirect Comparison (MAIC)

MAIC is a method-of-moments weighting approach designed to match the aggregate characteristics of the comparator trial. Weights $w_i$ for each patient $i$ in the IPD trial are derived via a logistic regression formulation:

$$w_i=\exp(\beta^\top X_i)$$

The parameters $\beta$ are found by minimizing a convex objective function to match target covariate means:

$$\min_{\beta\in\mathbb{R}^p}\sum_{i=1}^{n_A}\exp(\beta^\top(X_i-\overline{X}_B))$$

## Outcome Regression Models

For an individual $i$ with outcome $Y_i$, treatment $T_i \in \{0,1\}$, and baseline covariates $X_i$, the general outcome model is:

$$g(\mathbb{E}[Y_i|X_i,T_i])=\alpha+X_i^\top\beta_0+(\beta_{trt}+X_i^\top\beta_1)\mathbb{I}(T_i=1)$$

## Parametric G-computation (Maximum Likelihood)

G-computation standardizes outcomes across treatment regimens. We estimate the marginal mean outcome by simulating a pseudo-population of size $N$ reflecting the target ALD trial, predicting outcomes, and averaging:

$$\hat{\mathbb{E}}[Y^T|\hat{\theta}]=\frac{1}{N}\sum_{i=1}^N\hat{Y_i}(T,\hat{\theta})$$

## Parametric G-computation (Bayesian Inference)

Bayesian G-computation estimates the full posterior distribution of the model parameters, offering robust uncertainty quantification. For $M$ posterior samples $\theta^{(m)}$, the marginal mean is computed as:

$$\hat{\mathbb{E}}[Y^T|\theta^{(m)}]=\frac{1}{N}\sum_{i=1}^N\hat{Y}_i(T,\theta^{(m)})$$

## Multiple Imputation Marginalization (MIM)

MIM conceptualizes unobserved potential outcomes as a missing data problem. Using $M$ posterior draws, MIM imputes counterfactuals and utilizes modified Rubin's rules to calculate total variance by subtracting within-imputation variance ($\overline{U}$) from between-imputation variance ($B$):

$$Var(\overline{Q})=B-\overline{U}$$