deck_prjct.qmd

---
title: "Cross-trial evaluation of endTB regimens and 6BPaLM via data fusion"
format:
  clean-revealjs:
    self-contained: false
    slide-number: c
    preview-links: auto
    logo: figs/hsph.png
    # margin: 0.04
    footer: "TB Grupito Meeting"
    smaller: false
    incremental: false
    fig-width: 6
    fig-asp: 0.618
author:
  - name: Cong Jiang
    orcid: 0000-0002-4486-5423
    email: cjiang@hsph.harvard.edu
    affiliations: Harvard Biostatistics
  - name: Martha Boahene
    orcid: 0009-0005-7317-0304
    email: mboahene@g.harvard.edu
    affiliations: Harvard Biostatistics
  - name: Nima Hejazi
    orcid: 0000-0002-7127-2789
    email: nhejazi@hsph.harvard.edu
    affiliations: Harvard Biostatistics
date: last-modified
bibliography: refs.bib
csl: https://www.zotero.org/styles/journal-of-the-royal-statistical-society
---


## MDR/RR-TB and HIV

::: {style="font-size: 85%;"}
- **Drug-resistant TB (MDR/RR-TB)** remains a major global cause of preventable morbidity and mortality.
- People living with HIV (PLHIV) face substantially higher TB risk, and **TB is a leading cause of death among PLHIV**.
- Clinicians and programs need evidence to choose among **new all-oral regimens**. 
:::


## New hope: shorter, all-oral MDR/RR-TB regimens

::: {style="font-size: 85%;"}
Recent trials introduced **short-course, all-oral regimens**:

- **TB-PRACTECAL** [@nyangwa_24-week_2022;@nyangwa_short_2024]
  - 6-month regimen: **6BPaLM** (Bedaquiline + Pretomanid + Linezolid + Moxifloxacin)
  - Non-inferior vs standard care

- **endTB** [@guglielmetti_oral_2025] 
  - 9-month regimens: **9BLMZ, 9BCLLfxZ, 9BDLLfxZ**
  - Non-inferior vs standard care

**Standard care in each study was** 

- TB-PRACTECAL: 9–12 or 18–20 month all-oral regimens (region-specific) 
- endTB: 18–20 month all-oral regimens (region-specific) 
:::

## The decision problem: no head-to-head evidence 

::: {style="font-size: 85%;"}
- WHO guidelines recommend multiple **all-oral** MDR/RR-TB regimens; shorter regimens are appealing for feasibility and adherence.
- However, **comparative evidence between 6-month and 9-month regimens is limited**, and certainty of evidence is often low.

:::

. . .

::: {style="font-size: 85%; background-color: #f8f8f8; padding: 10px; border-left: 5px solid #2e8b57;"}
**Question**: Is **6BPaLM** truly preferable to **9-month regimens** from endTB?
:::


## What data do we have: two high-quality RCTs

::: {style="font-size: 85%;"}
We have individual patient data (IPD) from two non-inferiority RCTs:

| Feature | TB-PRACTECAL | endTB |
|---------------------|---------------------------------|------------------|
| **Duration** | 6-month regimens | 9-month regimens |
| **Standard care** | 9–12 or 18–20 month all-oral regimens (region-specific) | 18–20 month all-oral regimens (region-specific) |
| **Study regions** | Uzbekistan, Belarus, South Africa | Georgia, India, Kazakhstan, Lesotho, Pakistan, Peru, South Africa |
| **Follow-up time** | At least 72 weeks | At least 73 weeks |
:::

. . .

::: {style="font-size: 85%;"}

1) **No head-to-head regimen comparison** (no common arm)  
2) **Heterogeneous standard care** (region/time-specific)  
3) **Population and setting differences** across trial regions (affects transportability)
:::

. . .

::: {style="font-size: 85%;"}
Because a new head-to-head RCT is unlikely in the near term, we need to make the best possible use of existing evidence through cross-trial data fusion.
:::

## Aim and Approaches

::: {style="font-size: 85%;"}
**Primary aim**: Compare the efficacy of 6BPaLM versus endTB regimen (9BLMZ, 9BCLLfxZ, 9BDLLfxZ).
:::

. . .

::: {style="font-size: 85%;"}
**Approaches**: 

  - **Meta-analytic** approach: apply **meta analysis** to estimate direct contrasts.
  - **Target trial emulation** approach: use target trial emulation framework (Hernán & Robins, 2016), then analyze as an observational study. 
  - **Pooled-IPD “direct” strategy**: Pool individual patient data (IPD) from both trials and estimate the direct head-to-head contrast between the two treatments of intests.
  - **Pooled-IPD “indirect” strategy**: Pool IPD from both trials and estimate the indirect contrast between treatments <span style="background-color:#FFF2CC">despite non-common comparator (standard care)<span>.
:::

<!-- . . .

::: {style="font-size: 85%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
We will introduce four methodological strategies for data fusion in this cross-trial setting.
::: -->

## Related Literature and Contribution

::: {style="font-size: 85%;"}
- **Transportability**  
  - Prior work studies how to transport causal effects from multiple randomized trials to a **new target population** where experimental data may not be available  [@dahabreh2019extending; @dahabreh_toward_2020].
  - The core assumption of these methods lies in addressing the covariate distribution shift between the "experiment" and "target" groups.
:::

. . .

::: {style="font-size: 85%;"}
- **Common trial structure assumption**  
  Much of the causal transportability and IPD meta-analysis literature considers settings where trials evaluate the **same treatment set or comparator**, which simplifies pooling or transport of treatment effects (e.g., [@dahabreh_toward_2020]).
:::

## Our setting and contribution
::: {style="font-size: 85%;"}
- We define a **target population** as the **union of participants from endTB and TB-PRACTECAL** and focus on cross-trial causal comparison. 
:::

. . .

::: {style="font-size: 85%;"}
- We compare four strategies by combining ideas from **transportability, target trial emulation, and IPD meta-analysis** In particular, standard cares comparison can be solved in our framework by appropriately adjusting for differences.
:::

<!-- ## Data Structure and Notation
::: {style="font-size: 85%;"}
endTB trial $s = 1$ and TB-PRACTECAL trial $s = 2$ endTB trial $s = 1$ and TB-PRACTECAL trial and simple random sampling from the target population $\Omega$ to each trial population $\Omega_{s=1}$ and $\Omega_{s=2}$.
::: -->


## Data Structure and Notation
::: {style="font-size: 80%;"}

**Populations**: 

  - $S \in \{1,2\}$: trial indicator  
  - $S=1$: endTB  
  - $S=2$: TB-PRACTECAL  
  - $\Omega$: target population of interest  
:::

. . .

::: {style="font-size: 80%;"}
**Variables**:

  - $A \in \mathcal{A}$: treatment received  
  - $a$: 6BPaLM  
  - $b$: endTB regimen (9BLMZ, 9BCLLfxZ, 9BDLLfxZ)  
  - $c_s$: standard care in trial \(S=s\)  
  - $Y$: outcome (e.g., unfavorable outcome by end of follow-up)  
  - $L$: baseline covariates (measured pre-treatment)

 <!-- *Target Estimand*
$\psi(a,b) = \mathbb{E}\!\left[ Y(a) - Y(b) \mid \Omega \right]$ -->
:::




## Overview of four data fusion methods

::: {style="font-size: 80%; background-color: #fff0f0; padding: 2px; border-left: 2px solid #b22222;"}
Target population of interest $\Omega$: consisting of individuals at risk of TB recurrence in countries hosting enrollment sites for <span style="background-color:#FFF2CC">either the endTB or TB-PRACTECAL trials <span>.
:::

::: {style="font-size: 75%;"}
<!-- $\Omega = \Omega_{s=1} \cup \Omega_{s=2},$ where $\Omega_{s=1}$ and $\Omega_{s=2}$ denote the subpopulation of endTB and TB-PRACTECAL, respectively. -->
$\Omega = \Omega_{\text{TB-PRACTECAL}} \cup \Omega_{\text{endTB}}$
:::

. . .

::: {style="font-size: 75%;"}

| Method | Key Idea | Target Population |
|--------------|----------------------------------------------------|------------|
| **(1) Traditional Approach** | Compare each treatment to control in each trial, then take the difference of the estimates (possibly with weighting)| $\Omega$ (Pooled) |
| **(2) Target Trial Emulation Approach** | Combine treatment arms by developing target trial protocols, analyze as an observational study | $\Omega$ (Pooled) |
| **(3) Direct Method** | Estimate direct contrast by pooling data and adjusting for trial specific differences | $\Omega$ (Pooled) |
| **(4) Indirect Method** | Decompose contrast into treatment effect difference + control effect difference | $\Omega$ (Pooled) |
:::

. . .

::: {style="font-size: 80%;"}
**All methods aim to estimate the casual estimand**: $$\psi(a, b) = \mathbb{E}[Y(a) - Y(b) \mid \Omega].$$ 
:::
<!-- . . .

::: {style="font-size: 80%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
**Next**: We will examine each method in detail—their causal assumptions, statistical estimand, estimators, and implications for our cross-trial comparison.
::: -->

## Method 1: Traditional approach

::: {style="font-size: 85%;"}
**The Core Idea**: Use randomization within each trial, then take the difference of the estimated effects.

What the traditional approach does:

1.  **Analyze each trial separately**: Calculate within trial effect with data from TB-PRACTECAL and endTB separately.
2.  **Compare the two results**: Take the difference between the two trial-specific effects.

**Strength**: Uses within trial randomization

**Limitation**: Assumes negligible differences in standard-care efficacy between endTB and TB-PRACTECAL, as achieved by having a common comparator.


This directly reflects the common comparator: $\mathbb{E}[Y(c_1) - Y(c_2) \mid \Omega] = 0$

:::


## Method 2: Target Trial Emulation Approach

::: {style="font-size: 85%;"}
**Core idea**: Construct an emulated trial by pooling treated arms across trials and analyzing the data as an observational study, following a target trial emulation protocol while ignoring standard-care arms.

**Motivation**: Conducting a new head-to-head randomized trial is often infeasible.
By discarding control arms and comparing treated groups directly, we extract maximal information from existing trials under explicit causal assumptions.

**What this approach does**:

- Specify a target trial protocol (eligibility, treatment strategies, outcome, follow-up).
<!-- 
- Restrict data to treated participants. -->

- Pool treated arms across trials to form a single analytic dataset.

- Adjust for baseline covariates to control for  possible confounding.
:::

## Method 2: Target Trial Emulation Approach
::: {style="font-size: 85%;"}
**Identification**

- Exchangeability is assumed conditional on measured covariates.

- Randomization is no longer used once control arms are discarded.

**Limitation**

- Because control arms are ignored, identification relies entirely on covariate adjustment rather than randomization. 

- Lower sample size due to discarding control arms.
:::


## Methods 3 and 4: Direct and Indirect Approaches

::: {style="font-size: 85%;"}
<!-- What if we could combine data from both trials into one large dataset and then compare the treatments of interests? -->


Merge the data from TB-PRACTECAL and endTB, and <span style="background-color:#FFF2CC"> extend inferences from trial participants to a target population (i.e., $\Omega$).<span>
:::

. . .

::: {style="font-size: 85%;"}
Due to $\Omega = \Omega_{\text{TB-PRACTECAL}} \cup \Omega_{\text{endTB}},$ we  

- (1) use law of total expectation to decompose $\Omega$ into each trial's population;
:::

. . .

::: {style="font-size: 85%;"}
- (2) apply **transportability** to answer questions such as: *“What would be the average treatment effect if the endTB participants had received the TB-PRACTECAL intervention?”*
:::

. . .

::: {style="font-size: 85%;"}
Recall the definitions [@hernan2016; @dahabreh2019extending]:

- **Generalizability:** extension of inferences from the trial to a target population that *coincides, or is a subset of*, the trial-eligible population 

- **Transportability:** extension of inferences from the trial to
a target population that *includes individuals who are not part* of the trial-eligible population. 
:::

<!-- ## Method: Direct and Indirect Method

::: {style="font-size: 85%;"}
In contrast to the trial emulation strategy, both the Direct and Indirect methods leverage randomization: <span style="background-color:#FFF2CC">the within trial randomization implies the conditional ramdomizaiton.<span>
:::

::: {style="font-size: 85%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
In contrast to the trial emulation strategy, both the Direct and Indirect methods leverage randomization: the within trial randomization implies the conditional ramdomizaiton.
::: -->

## Method 3: Direct Method for $\psi(a, b)$

::: {style="font-size: 85%;"}
**Key Assumption 1: Consistency**
If an individual actually receives treatment $a$, then their observed outcome $Y$ is equal to their potential outcome $Y(a)$. This implicitly assumes that there are no multiple versions of the treatment, no interference.
:::

. . .

::: {style="font-size: 85%;"}
**Key Assumption 2: Exchangeability (unconfoundedness)**

$$Y(a)\perp\!\!\perp A\mid (L, S=1),\ \text{and}\ Y(b)\perp\!\!\perp A\mid (L, S=2)$$

If we do not have within trial randomization, we need to measure all important patient characteristics ($L$) to ensure **treatment assignment is unrelated to what outcomes would have been under different treatments**.
:::

. . .

::: {style="font-size: 85%;"}
<span style="background-color:#FFF2CC"> Fortunately, this is guaranteed by randomization within each trial, no matter how many covariates we measured, </span> i.e.,
marginal randomization implies this weaker assumption.
:::


## Key Assumptions of Method 3
::: {style="font-size: 85%;"}
**Key Assumption 3: Transportability (Conditional exchangeability over inclusion)**: 
$$Y(a),Y(b)\perp\!\!\perp S\mid L$$

What we learn about a treatment in one trial can be applied to patients in the other trial, given measured covariates.


*Information from patients on 6BPaLM in TB-PRACTECAL can tell us what would happen if endTB patients took 6BPaLM, and vice versa.*


:::

. . .

::: {style="font-size: 85%; background-color: #fff0f0; padding: 10px; border-left: 5px solid #b22222;"}
Any systematic differences between trials that are not captured in the measured covariates $L$ will break this assumption.
:::

## Key Assumptions of Method 3
::: {style="font-size: 85%;"}
**Key Assumption 4: Positivity — Treatment assignment**

Within each trial, everyone must have a non-zero probability to receive each treatment.
:::

. . .

::: {style="font-size: 85%;"}
**Key Assumption 5: Positivity — Trial participation/inclusion**

Across trials, everyone must have a non-zero probability to be in either trial. 

No group of patients should exist only in one trial and never the other.

If a certain kind of patient (say very sick patients) can only ever appear in trial 1 and never in trial 2, then we cannot use trial 2 to learn about that type of patient.
:::


## Identification Result (Statistical Estimand)

::: {style="font-size: 85%;"}
\begin{align}\label{psiid}
\psi(a, b) &:=  \mathbb{E}[Y(a) - Y(b) \mid \Omega] \nonumber \\
    &= \mathbb{E}_{L\mid S=1}[\mathbb{E}(Y \mid A = a, L, S =1)] \color{#E69F00}{\mathbb{P}_{\Omega}(S=1)} \\&\ \ \ \ - \mathbb{E}_{L\mid S=1}[\mathbb{E}(Y \mid A = b, L, S =2)] \color{#E69F00}{\mathbb{P}_{\Omega}(S=1)}  \\ &\ \ \ \ + \mathbb{E}_{L\mid S=2}[\mathbb{E}(Y \mid A = a, L, S =1)] \color{#6b72ff}{\mathbb{P}_{\Omega}(S=2)} \nonumber \\ &\ \ \ \ - \mathbb{E}_{L\mid S=2}[\mathbb{E}(Y \mid A = b, L, S =2)] \color{#6b72ff}{\mathbb{P}_{\Omega}(S=2)},
\end{align}
where $\color{#E69F00}{\mathbb{P}_{\Omega}(S=1)}$ is the marginal probability of being selected into trial 1 within the target population.
:::



. . .

::: {style="font-size: 85%;"}
Let's focus on the first two terms, and the last two terms follow the same pattern.
\begin{align}
\psi(a, b) &:=  \mathbb{E}[Y(a) - Y(b) \mid \Omega] \nonumber \\
    &= \color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ \mathbb{E}(Y \mid A = a, L, S =1)] \color{#E69F00}{\mathbb{P}_{\Omega}(S=1)} \\&\ \ \ \ - \color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ {\color{#FF6B6B}{\mathbb{E}(Y \mid A = b, L, S =2)}}] \color{#E69F00}{\mathbb{P}_{\Omega}(S=1)}  \\ &\ \ \ \ + \dots \nonumber \\ &\ \ \ \ - \dots,
\end{align}
:::


## Identification Result (G-computation)
::: {style="font-size: 85%;"}
The first term is the standard G-computation result of the trial 1,

\begin{align}
\psi(a, b) &:=  \mathbb{E}[Y(a) - Y(b) \mid \Omega] \nonumber \\
    &= \color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ \mathbb{E}(Y \mid A = a, L, S =1)] \color{#E69F00}{\mathbb{P}_{\Omega}(S=1)} \\&\ \ \ \ - \dots  \\ &\ \ \ \ + \dots \nonumber \\ &\ \ \ \ - \dots,
\end{align}
where subgroup-specific conditional means, i.e., $\color{#6b72ff}{\mathbb{E}(Y \mid A = a, L, S =1)}$, are used to estimate the treatment effect for patients in trial 1.

:::

. . .

::: {style="font-size: 85%;"}
$\color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ \mathbb{E}(Y \mid A = a, L, S =1)]$ is a sum over $l$ (in $S=1$) of 

\begin{align}
{\color{#6b72ff}{\mathbb{E}(Y \mid A = a, L = l, S =1)} * \mathbb{P}(L = l \mid S=1)}.
\end{align}
:::

## Identification Result (G-computation)
::: {style="font-size: 85%;"}
$\color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ \mathbb{E}(Y \mid A = a, L, S =1)]$ is

\begin{align}
{\color{#6b72ff}{\sum_{l: \mathbb{P}(l, S=1) \neq 0 }\mathbb{E}(Y \mid A = a, L = l, S =1)} * \mathbb{P}(L = l \mid S=1)}.
\end{align}

:::

. . .

::: {style="font-size: 85%;"}
- Fit a model using ${\color{#6b72ff}{\textbf{Trial 1}}}$ data to predict the outcome $Y$ given treatment $A=a$ and covariates $L$.

- Apply that same model to each patient in ${\color{#6b72ff}{\textbf{Trial 1}}}$ to get their predicted outcome under treatment ${\color{#6b72ff}{a}}$.

- Average those predictions across all ${\color{#6b72ff}{\textbf{Trial 1}}}$ patients.
:::


## Identification Result (G-computation)
::: {style="font-size: 85%;"}
The second term uses the **transportability** of the trial 2 for the trial 1,

\begin{align}
\psi(a, b) &:=  \mathbb{E}[Y(a) - Y(b) \mid \Omega] \nonumber \\
    &= \dots \\&\ \ \ \ - \color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ {\color{#FF6B6B}{\mathbb{E}(Y \mid A = b, L, S =2)}}] \color{#E69F00}{\mathbb{P}_{\Omega}(S=1)}  \\ &\ \ \ \ + \dots \nonumber \\ &\ \ \ \ - \dots,
\end{align}
where subgroup-specific conditional means, i.e., $\color{#FF6B6B}{\mathbb{E}(Y \mid A = b, L, S =2)}$, are fitted/estimated using data from trial 2.

:::

. . .

::: {style="font-size: 85%;"}
$\color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ {\color{#FF6B6B}{\mathbb{E}(Y \mid A = b, L, S =2)}}]$ is a sum over $l$ (in $S=1$) of 

\begin{align}
\color{#FF6B6B}{\mathbb{E}(Y \mid A = b, L = l, S =2)} * \color{#6b72ff}{\mathbb{P}(L = l \mid S=1)}.
\end{align}
:::

## Identification Result (G-computation)
::: {style="font-size: 85%;"}
$\color{#6b72ff}{\mathbb{E}_{L\mid S=1}}[ {\color{#FF6B6B}{\mathbb{E}(Y \mid A = b, L, S =2)}}]$ is 

\begin{align}
\color{#6b72ff}{\sum_{l: \mathbb{P}(l, S=1) \neq 0 } } \color{#FF6B6B}{ \mathbb{E}(Y \mid A = b, L = l, S =2)} * \color{#6b72ff}{\mathbb{P}(L = l \mid S=1)}.
\end{align}

:::

. . .

::: {style="font-size: 85%;"}
- Using ${\color{#FF6B6B}{\textbf{Trial 2}}}$ data, fit an outcome model for $Y$ given treatment $A=b$ and covariates $L$.

- Apply that same model to each patient in ${\color{#6b72ff}{\textbf{Trial 1}}}$ to get their predicted outcome under treatment ${\color{#FF6B6B}{b}}$.

- Average those predictions across all ${\color{#6b72ff}{\textbf{Trial 1}}}$ patients.
:::

## Estimation 

::: {style="font-size: 85%;"}
We consider three classes of estimators for the Direct contrast: 

1. **G-computation**; 
2. **Inverse Probability Weighting (IPW)**; 
3. **Causal machine learning estimator**.

The first two approaches rely heavily on correct specification of either the outcome model (g-computation) or the inclusion&treatment model (IPW). 
:::

. . .

::: {style="font-size: 85%;"}
To address their lack of robustness, we focus on a **causal machine learning estimator** in the main analysis. This estimator:

* Can flexibly incorporate modern machine learning methods for nuisance estimation.
* **Double Robustness**: remains consistent if **either** the outcome model **or** the *inclusion model* is correctly specified.
* Provides **valid** large-sample inference under standard regularity conditions.
:::


<!-- ## Summary: Direct Method

::: {style="font-size: 85%;"}
The Direct Method lets us “directly compare” treatments that were never tested head-to-head in the same trial, by combining trial data and applying advanced statistical adjustment.

**For Our Research**: This is **a straightforward and robust approach** to answer whether 6BPaLM is preferable to 9-month regimens for people living with HIV, provided the trial populations are sufficiently similar.
::: -->


## Method (4): Indirect Method

::: {style="font-size: 85%;"}

Instead of directly comparing 6BPaLM vs. 9-month regimens, we break the problem into **two parts**:

1. **The difference in treatment effects**  
   How much better is 6BPaLM vs. *its* standard care, compared to how much better a 9-month regimen is vs. *its* standard care?

2. **The difference in standard care**  
   How different are the two standard care treatments themselves?
:::

. . .

::: {style="font-size: 85%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
**Simple formula**:  Total difference = (Treatment effect difference) + (Standard care difference), i.e., 
$\psi(a, b)= \theta + \phi,$
where 
(1) $$\theta := \mathbb{E}[Y(a)-Y(c_1)\mid\Omega] - \mathbb{E}[Y(b)-Y(c_2)\mid\Omega],$$ difference in treatment effects,
and (2) $\phi := \mathbb{E}[Y(c_1)-Y(c_2)\mid\Omega],$ the difference between the two standard care arms.
:::


## Strengths & Limitations

::: {style="font-size: 85%;"}

| Method                      | Uses randomization | Uses all arms data| Intuitive | Statistically efficient | More Assumption | 
| --------------------------- | :----------------: | :-----------: | :---------------------: | :-----------: | :---------------------: | 
| **1: Traditional Approach**      |          ✔          |       ✔        |            ✔            |       △       |            Yes            |
| **2: Trial Emulation**      |          ✖         |       ✖       |            ✔            |       △       |            Yes            |
| **3: Pooled-IPD Direct**   |          ✔         |       ✔       |            ✔            |       ✔       |            Yes            |
| **4: Pooled-IPD Indirect** |          ✔         |       ✔       |            △            |       ✔       |            Yes            |
:::

::: {style="font-size: 85%; background-color: #f0f8ff; padding: 10px; border-left: 5px solid #4682b4;"}
✔ = fully satisfied; • △ = partially satisfied; ✖ = not applicable
:::

. . .

::: {style="font-size: 85%;"}
**Additional Targeting Conditions**

* **1 (Traditional):** requires a common control and equal contrasts across trials
* **2 (Trial Emulation):** exchangeability is assumed conditional on measured covariates.
* **3 (Direct):** requires transportability of treatment effects given $L$

* **4 (Indirect):** requires transportability of (1) standard care effects (e.g., given country $V$), and (2) treatment-control contrasts (given $L$)
:::

## References {.smaller footer="false"}

::: {#refs style="font-size: 80%;"}
:::

# Appendix {.appendix}


## Target Trial Emulation {.appendix}
### Design, Eligibility, and Treatment
 Criteria for emulating a target trial using endTB and TB-PRACTECAL RCTs.

::: {.small}
| Component | Target Trial (Ideal) | Emulated Trial (Real) |
|---|---|---|
| **Eligibility (baseline)** | | |
| Inclusion | Age ≥ 15; rifampin-resistant; fluoroquinolone-susceptible; baseline labs grade ≤ 4; HIV seropositive | Same |
| Exclusion | Pregnancy; elevated liver enzymes; QTcF ≥ 450 msec; uncorrectable electrolyte disorders; on MDR/RR-TB treatment ≥ 2 weeks; investigator discretion | Same |
| **Treatment strategies** | 6BPaLM; 9BLMZ; 9BCLLfxZ; 9BDLLfxZ | Same |
| **Assignment** | Randomized (1:1:1:1), stratified by country/region | IPTW using key baseline covariates |
|
:::
## Target Trial Emulation (cts) {.appendix}
### Follow-up, Estimands, and Analysis

**Table 1b.** Follow-up, outcomes, and causal contrasts for the emulated trial.

::: {.small}
| Component | Target Trial (Ideal) | Emulated Trial (Real) |
|---|---|---|
| **Time zero & follow-up** | Time zero: first dose initiation; follow-up ends at primary endpoint, censoring, or administrative study end | Same, with 1-week grace period (72/73 weeks) |
| **Outcome** | WHO composite outcome | Same |
| **Causal contrasts** | ITT: 6BPaLM vs 9BLMZ; 6BPaLM vs 9BCLLfxZ; 6BPaLM vs 9BDLLfxZ | Same, constructed as a single four-arm study in mITT population |
| **Statistical analysis** | ITT analysis | ITT with IPCW for censoring adjustment |
|
:::


## Comparison of Methods {.appendix}

![](figs/comparison.png){width=77% fig-align="center" .lightbox}


## Overview of source trials: TB-PRACTECAL vs. endTB {.appendix}

::: {style="font-size: 85%;"}
**TB-PRACTECAL** (Nyang’wa et al., 2022, 2024): 

-  **Design**: Phase 2/3 non-inferiority RCT 
-  **Countries**: South Africa, Belarus, Uzbekistan 
-  **Regimens tested**: 6-month all-oral regimens vs. standard care 
-  **Key finding**: 6BPaLM non-inferior to standard care 
-  **Primary endpoint**: Composite unfavorable outcome at 72 weeks

**endTB** (Guglielmetti et al., 2025): 

-  **Design**: Phase 3 non-inferiority RCT 
-  **Countries**: Georgia, India, Kazakhstan, Lesotho, Pakistan, Peru, South Africa 
-  **Regimens tested**: 9-month all-oral regimens vs. standard care 
-  **Key finding**: 9BLMZ, 9BCLLfxZ, 9BDLLfxZ non-inferior to standard care 
-  **Primary endpoint**: Favorable outcome at 73 weeks
:::


## Key differences between trials {.appendix}

::: {style="font-size: 85%;"}
| Feature | TB-PRACTECAL | endTB |
|---------------------|---------------------------------|------------------|
| **Duration** | 6-month regimens | 9-month regimens |
| **Standard care** | 9–12 or 18–20 month all-oral regimens (region-specific) | 18–20 month all-oral regimens (region-specific) |
| **Study regions** | Eastern Europe, Central Asia, Southern Africa | South Asia, Latin America, Eastern Europe, Africa |
| **Primary outcome** | Composite unfavorable outcome | Favorable outcome |
| **Assessment time** | 72 weeks | 73 weeks |
:::

. . .

::: {style="font-size: 85%;"}
**Critical implication**: 

- Standard care differs **both across regions and over time** 
- Direct comparison is **not straightforward** due to differing controls 
:::

## Challenges for cross-trial data fusion {.appendix}

::: {style="font-size: 85%;"}
**1. No head-to-head comparison** - Trials compared regimens to **different standard-of-care arms** - No common treatment arm between trials

**2. Heterogeneity in standard care** - Standard care varied by country and enrollment period - Changes in WHO guidelines during trial periods

**3. Regional and population differences** - Limited overlap in study regions - Different background epidemiology, healthcare systems
:::

. . .

::: {style="font-size: 85%; background-color: #fff0f0; padding: 10px; border-left: 5px solid #b22222;"}
**We need causal data fusion methods** that can: 

- Combine IPD from both trials 
- Account for differing standard care 
- Estimate direct contrasts between investigational regimens 
:::

## Identification Result (Statistical Estimand) {.appendix}

::: {style="font-size: 85%;"}
Identification of the parameter of interest $\psi$

\begin{align}
\psi(a, b) &:=  \mathbb{E}[Y(a) - Y(b) \mid \Omega] \nonumber \\
    &= \mathbb{E}_{L\mid S=1}[\mathbb{E}(Y \mid A = a, L, S =1)] \mathbb{P}_{\Omega}(S=1) \\ &\ \ \ \ + \mathbb{E}_{L\mid S=2}[\mathbb{E}(Y \mid A = a, L, S =1)] \mathbb{P}_{\Omega}(S=2) \nonumber \\ &\ \ \ \ - \mathbb{E}_{L\mid S=1}[\mathbb{E}(Y \mid A = b, L, S =2)] \mathbb{P}_{\Omega}(S=1)  \\ &\ \ \ \ - \mathbb{E}_{L\mid S=2}[\mathbb{E}(Y \mid A = b, L, S =2)] \mathbb{P}_{\Omega}(S=2),
\end{align}
where $S=1$ indicates TB-PRACTECAL and $S=2$ indicates endTB.

or equivalently, $\psi$ can be identified as the following IPW form
\begin{align}\label{psiid2}
\psi(a, b) &:=  \mathbb{E}[Y(a) - Y(b) \mid \Omega] \nonumber \\
&= 
\mathbb{E}\!\left[\frac{\mathbb{I}(A=a)\mathbb{I}(S=1)Y}{\mathbb{P}_{\Omega}(A=a \mid L, S=1)}\right]
+ \mathbb{E}\!\left[\frac{\mathbb{I}(A=a)\mathbb{I}(S=1)Y\, \delta(L)}{\mathbb{P}_{\Omega}(A=a \mid L, S=1)}\right] \nonumber \\
&\quad - \mathbb{E}\!\left[\frac{\mathbb{I}(A=b)\mathbb{I}(S=2)Y\, \delta^{-1}(L)}{\mathbb{P}_{\Omega}(A=b \mid L, S=2)}\right]- \mathbb{E}\!\left[\frac{\mathbb{I}(A=b)\mathbb{I}(S=2)Y}{\mathbb{P}_{\Omega}(A=b \mid L, S=2)}\right],
\end{align}
where $\delta(L):=
\frac{\mathbb{P}_{\Omega}(S = 2 \mid L)}{\mathbb{P}_{\Omega}(S = 1 \mid L)} = \frac{1 - \mathbb{P}_{\Omega}(S = 1 \mid L)}{\mathbb{P}_{\Omega}(S = 1 \mid L)}.$
:::


## Estimation {.appendix}

::: {style="font-size: 85%;"}
We implement the Direct Method using three estimation method (1) **g-computation** (2) **IPW** and (3) **one-step/doubly-robust estimation**:

**Doubly-Robust**: The one-step estimator is consistent if either the outcome model or the inclusion and treatment model is correctly specified.

1. **Outcome Model**: Predicts what the outcome    would be under each treatment, given patient characteristics, $\mu_s(z, L) = \E(Y\mid L,A=z,S=s)$ for any $(s, z) \in \{(1, a), (2, b)\}.$

2. **Inclusion and Treatment Model**: The trial inclusion (i.e., $g(L) := \mathbb{P}_{\Omega}(S=1\mid L)$) and treatment mechanism (i.e., $\mathbb{P}_{\Omega}(A=z\mid L,S=s)$) are consistently estimated.

Statistical properities of one-step estimator, such as consistency and asymptotic normality, are established under standard regularity conditions.
:::

. . .

::: {style="font-size: 85%; background-color: #f0f8ff; padding: 10px; border-left: 5px solid #4682b4;"}
Treatment mechanism are known in randomized trials, so only the <span style="background-color:#FFF2CC">inclusion model and outcome model </span> need to be estimated "well", and the doubly-robust property ensures consistency if either is correct.
:::


## Identification of $\theta$ {.appendix}

Decomposing $\theta := \underbrace{\mathbb{E}[Y(a) - Y(c_1) \mid \Omega]}_{\theta_1} - \underbrace{\mathbb{E}[Y(b) - Y(c_2) \mid \Omega]}_{\theta_2},$ we have: for $\theta_1$
\begin{align}\label{thetaid}
\mathbb{E}_{L \mid S = 1}& \left[ \mathbb{E}[Y \mid A = a, L, S = 1] - \mathbb{E}[Y \mid A = c_1, L, S = 1] \right] \mathbb{P}(S = 1)  \nonumber \\
+ \mathbb{E}_{L \mid S = 2} &\left[ \mathbb{E}[Y \mid A = a, L, S = 1] - \mathbb{E}[Y \mid A = c_1, L, S = 1] \right] \mathbb{P}(S = 2)
\end{align}

or equivalently, $\theta_1$ can be identified as the following IPW form,
\begin{align*}
&\mathbb{E}\!\left[\frac{\mathbb{I}(A=a)\mathbb{I}(S=1)Y}{\mathbb{P}(A=a \mid L, S=1)} - \frac{\mathbb{I}(A=c_1)\mathbb{I}(S=1)Y\, }{\mathbb{P}(A=c_1 \mid L, S=1)}\right] \\ +& \mathbb{E}\!\left[\frac{\mathbb{I}(A=a)\mathbb{I}(S=1)Y\, \delta(L)}{\mathbb{P}(A=a \mid L, S=1)} - \frac{\mathbb{I}(A=c_1)\mathbb{I}(S=1)Y\delta(L)}{\mathbb{P}(A=c_1 \mid L, S=1)}\right]
\end{align*}

where $\delta(L):=
\frac{\mathbb{P}_{\Omega}(S = 2 \mid L)}{\mathbb{P}_{\Omega}(S = 1 \mid L)} = \frac{1 - \mathbb{P}_{\Omega}(S = 1 \mid L)}{\mathbb{P}_{\Omega}(S = 1 \mid L)}.$

  <span style="background:#E6F7FF">
  There are 4 terms in the above identification formulas, corresponding to 2 components across 2 trials.
  </span>


## How It Works: Combining the Pieces {.appendix}
::: {style="font-size: 85%;"}
- **Step 1: Estimate Standard Care Difference**, $\hat{\phi}$
  - Use the Direct Method to compare SC1 v.s. SC2

- **Step 2: Estimate Treatment Effect Difference,** $\hat{\theta}$
  - Estimate 6BPaLM effect v.s. SC1 in TB-PRACTECAL, and 9-month effect vs. SC2 in endTB
  - Transport these effects to the combined population (4 terms in the identification formula)


- **Step 3: Add Them Up**
$\hat{\psi}(a, b)$ = $\hat{\theta}$ + $\hat{\phi}$
:::

. . .

::: {style="font-size: 85%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
**In Estimation**, we also proposed hree estimation method (1) **g-computation** (2) **IPW** and (3) **one-step/doubly robust estimation** to estimate both components and combine them for the final contrast. Statistical properities such as consistency and asymptotic normality are established under standard regularity conditions.
:::

## Strengths & Weaknesses {.appendix}

::: {style="font-size: 85%;"}

- **Strengths**
  - **Explicitly handles different standard cares** — a major advantage
  - **Uses randomization where possible** — more credible treatment effect estimates
  - **Flexible assumptions** — weaker assumptions for the standard care comparison

- **Challenges**
  - **More complex** — requires estimating two components (8 total key terms)
  - **Multiple assumptions** — each component has its own requirements

:::

. . .

::: {style="font-size: 85%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
When standard care differs substantially between trials, this method **explicitly accounts for that difference**, making it potentially more accurate than the traditional approach.
:::





















## Method (4): Indirect Method – decomposition for adjusting differing standard care {.appendix}

::: {style="font-size: 85%;"}
**The Core Idea**:
Instead of directly comparing 6BPaLM vs. 9-month regimens, we break the problem into **two parts**:

1. **The difference in treatment effects**  
   How much better is 6BPaLM vs. *its* standard care, compared to how much better a 9-month regimen is vs. *its* standard care?

2. **The difference in standard care**  
   How different are the two standard care treatments themselves?
:::

. . .

::: {style="font-size: 85%; background-color: #e6f7ff; padding: 10px; border-left: 5px solid #0077b6;"}
**Simple formula**:  
Total difference = (Treatment effect difference) + (Standard care difference), i.e., 
$$\psi(a, b)= \theta + \phi $$ 
where $\theta := \mathbb{E}[Y(a)-Y(c_1)\mid\Omega] - \mathbb{E}[Y(b)-Y(c_2)\mid\Omega]$, difference in treatment effects,
and $\phi := \mathbb{E}[Y(c_1)-Y(c_2)\mid\Omega],$ the difference between the two standard care arms.
:::

## The Two Components: A Closer Look
::: {style="font-size: 85%;"}

- **Component 1: Treatment Effect Difference ($\theta$)**
  - In **TB-PRACTECAL**: 6BPaLM v.s. Standard Care 1
  - In **endTB**: 9-month regimen v.s. Standard Care 2
  - Captures how much better each investigational regimen is compared to its own control, i.e., $\theta := \mathbb{E}[Y(a)-Y(c_1)\mid\Omega] - \mathbb{E}[Y(b)-Y(c_2)\mid\Omega]$

 - $(\text{Effect of 6BPaLM vs. SC1}) - (\text{Effect of 9-month regimen vs. SC2})$
:::

. . .

::: {style="font-size: 85%;"}
- **Component 2: Standard Care Difference ($\phi$)**
  - Direct comparison: Standard Care 1 vs. Standard Care 2
  - Captures differences in background treatment across trials

- $\phi := \mathbb{E}[Y(c_1)-Y(c_2)\mid\Omega]$
:::

. . .

::: {style="font-size: 85%; background-color: #fff0f0; padding: 10px; border-left: 5px solid #b22222;"}
**Fact of Component 2**:  
We can estimate $\phi$ using the **Direct Method** from before, since $\phi := \mathbb{E}\left[ Y(c_1) - Y(c_2) \mid \Omega \right]$ coincides with the pairwise contrast $\phi = \psi(c_1, c_2)$. 
:::

## Identification Assumptions (for $\theta$ and $\phi$) {.appendix}
::: {style="font-size: 85%;"}
- **(A1) Consistency**  

- **(A2) Exchangeability within each trial**  
  - Trial 1: $Y(a), Y(c_1)\perp\!\!\perp A \mid L,S=1$  
  - Trial 2: $Y(b), Y(c_2)\perp\!\!\perp A \mid L,S=2$

- **(A3) Positivity of treatment within trial**  

  Everyone in each trial has a non-zero probability to receive every treatment arm.

- **(A4) Transportability** : $Y(a),Y(c_1), Y(b), Y(c_2)\perp\!\!\perp S \mid L$

- **(A5) Positivity of trial inclusion**  

  For any covariate profile $L$, a subject must have a non-zeor probability to be in both trials.
:::




## Summary: Indirect Method {.appendix}

::: {style="font-size: 85%;"}

The Indirect Method breaks the comparison into "how much better the treatments are than their respective standard cares" plus "how different those standard cares are from each other."

**The Key Insight**:  
By separating treatment effects from background care differences, we get a clearer picture of what is really driving any observed differences between 6BPaLM and 9-month regimens.
:::

. . .

::: {style="font-size: 85%; background-color: #f0f8ff; padding: 10px; border-left: 5px solid #4682b4;"}
**For Our Study**:  
This method is particularly valuable because TB-PRACTECAL and endTB had different standard care regimens, something we can't ignore when comparing their results.
:::















## Comparison of Four Data-Fusion Methods {.appendix}

::: {style="font-size: 85%;"}
**Target Population**

| 1 Traditional    | 2 Trial Emulation  | 3 Indirect       | 4 Direct         |
| ----------------- | ----------------- | ----------------- | ----------------- |
| $\Omega$ (pooled) | $\Omega$ (pooled) | $\Omega$ (pooled) | $\Omega$ (pooled) |
:::

. . .

::: {style="font-size: 85%;"}

**Causal Estimand**

* **(1) Traditional:** contrast of trial-specific treatment effects.

* **(2) Trial Emulation:** contrast in the treat-only population.

* **(3) Direct:**
  $\psi=\E[Y(a)-Y(b)\mid\Omega]$

* **(4) Indirect:**
  $\psi=\theta+\phi$

  * $\phi$: difference between controls across trials
  * $\theta$: difference-in-differences of treatment effects
:::



## Comparison of Four Data-Fusion Methods {.appendix}

::: {style="font-size: 85%;"}

**Standard Assumptions**

Common to all methods:

* **Consistency**
* **Positivity within each trial**

What differs:

* **1, 3 & 4:** exchangeability holds *within each trial*
* **2 (Trial Emulation):** exchangeability in the treat-only population
:::


## Comparison of Four Data-Fusion Methods {.appendix}

::: {style="font-size: 85%;"}
**Additional Targeting Conditions**

* **1 (Traditional):** requires a common control and equal contrasts across trials
* **2 (Trial Emulation):** requires equality of treatment effect in $\Omega_{trt}$ and $\Omega$
* **3 (Direct):** requires transportability of treatment effects given $L$

* **4 (Indirect):** requires transportability for

  * controls (given country $V$), and
  * treatments (given $L$)
:::

. . .

::: {style="font-size: 85%;"}
**Uses Randomization**

| 1             | 2 | 3             | 4 |
| -------------- | -- | -------------- | -- |
| ✔ within-trial | ✖  | ✔ within-trial | ✔  within-trial|
:::






## Comparison of Four Data-Fusion Methods {.appendix}

::: {style="font-size: 85%;"}
**Sample Size Usage**

| Method               | How sample size is used                                                                          |
| -------------------- | ------------------------------------------------------------------------------------------------ |
| **1 Traditional**   | Uses **fully pooled IPD from all arms**, maximizing effective sample size.           |
| **2 Trial Emulation** | Uses **all treated units** across trials but may discard controls for the main contrast.         |
| **3 Indirect**      | Uses **fully pooled IPD from all arms**, maximizing effective sample size.   |
| **4 Direct**        | Uses **fully pooled IPD from all arms**, maximizing effective sample size.                       |
:::



## Summary: Indirect Method

::: {style="font-size: 85%;"}
The indirect method uses an identification strategy similar to the direct method, but must account for both **treatment effect difference** $\theta$ and **standard care difference** $\phi$.
:::

. . .

::: {style="font-size: 85%;"}
**Fact of Component 2**:  we can indentify and estimate $\phi$ using the **Direct Method** from before, since $\phi := \mathbb{E}\left[ Y(c_1) - Y(c_2) \mid \Omega \right]$ coincides with the pairwise contrast $$\phi = \psi(c_1, c_2).$$
:::