# Introduction to MRStdCRT

# Introduction

The `MRStdCRT`` package provides tools for computing the model-robust standardization estimator with jackknife variance estimator for the cluster average treatment effect(c-ATE) and individual average treatment effect(i-ATE) in clustered randomized trials (CRTs).

# Model robust standardization

In CRT, assume the cluster size \(N_{i}\) is the natural cluster panel size. The total sample size of the study is \(N=\sum_{i=1}^m N_{i}\). Let \(A_i\in\{0,1\}\) be the randomized cluster-level treatment indicator, with \(A_i=1\) indicating the assignment to the treatment condition and \(A_i=0\) to usual care. The potential outcomes framework and define \(\{Y_{ij}(1),Y_{ij}(0)\}\) as a pair of potential outcomes for each individual \(j \in \{1,\dots, N_i\}\) under the treatment and usual care conditions, respectively. Denote \(\boldsymbol{X}_i = {\boldsymbol{X}_{i1},\dots,\boldsymbol{X}_{iN_i}}^\top\) as the collection of baseline covariates across all individual, and \(\boldsymbol{H}_i\) as the collection of cluster-level covariates. Writing \(f(a,b)\) as a pre-specified contrast function, a general class of weighted average treatment effect in CRTs is defined as \[\Delta_{\omega}=f(\mu_\omega(1),\mu_\omega(0)),\] where the weighted average potential outcome under treatment condition \(A_i=a\) is \[\begin{align*} \mu_\omega(a)=\frac{E\left((\omega_i/N_i)\sum_{j=1}^{N_i}Y_{ij}(a)\right)}{E(\omega_i)}. \end{align*}\]

In this formulation, \(\omega_i\) is a pre-specified cluster-specific weight determining the contribution of each cluster to the target estimand, and can be at most a function of the cluster size \(N_i\), or additional cluster-level covariates \(\boldsymbol{H}_i\). In CRTs, two typical estimands of interest arise from different specifications of \(\omega_i\). First, setting \(\omega_i=1\) gives each cluster equal weight and leads to the , \(\Delta_C=f(\mu_C(1),\mu_C(0))\), with \[\begin{align*} \mu_C(a)=E\left(\frac{\sum_{j=1}^{N_i}Y_{ij}(a)}{N_i}\right). \end{align*}\] Second, setting \(\omega_i=N_i\) gives equal weight to each individual in the study regardless of their cluster membership and leads to the , \(\Delta_I=f(\mu_I(1),\mu_I(0))\), with \[\begin{align*} \mu_I(a)=\frac{E\left(\sum_{j=1}^{N_i}Y_{ij}(a)\right)}{E(N_i)}. \end{align*}\] Under the general setup, the average potential outcomes can be estimated by \[\begin{align*} \widehat{\mu}_\omega(a)=\sum_{i=1}^m \frac{\omega_i}{\omega_{+}}\left\{\underbrace{\widehat{E}(\overline{Y}_{i}|A_i=a,\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)}_{\text{regression prediction}}+\underbrace{\frac{I(A_i=a)\left(\overline{Y}_i-\widehat{E}(\overline{Y}_{i}|A_i=a,\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)\right)}{\pi(\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)^a\left(1-\pi(\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)\right)^{1-a}}}_{\text{weighted cluster-level residual}}\right\}, \end{align*}\] where \(\omega_{+}=\sum_{i=1}^m \omega_i\) is the sum of weights across clusters, \(\pi(\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)=P(A_i=a|\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)\) is the conditional randomization probability of each cluster given baseline information, and \(\widehat{E}(\overline{Y}_{i}|A_i=a,\boldsymbol{X}_i,\boldsymbol{H}_i,N_i)\) is the conditional mean of the cluster average outcome, \(\overline{Y_i}=N_i^{-1}\sum_{i=1}^{N_i}Y_{ij}\), given baseline covariates and cluster size, which could be estimated via any sensible outcome regression model.

# Data Structure and Description

In the context of cluster-randomized trials (CRT), we observe the following data vector for each subject \(j\) in cluster \(i\): \(\{Y_{ij}, A_{i}, \boldsymbol{X}_{ij}, \boldsymbol{H}_i, N_i\}\), where:

- \(Y_{ij}\) represents the observed outcome for individual \(j\) in cluster \(i\),
- \(A_{i}\) is the treatment assignment for cluster \(i\),
- \(\boldsymbol{X}_{ij}\) denotes the individual-level covariates,
- \(\boldsymbol{H}_i\) denotes the cluster-level covariates,
- \(N_i\) is the number of individuals in cluster \(i\).

Different working outcome mean models are proposed based on either individual-level observations or cluster-level summarizes (means), which can be further used to construct the model-robust standardization estimator for either the weighted cluster-averaged treatment effect (c-ATE) or weighted individual-averaged treatment effect (i-ATE).

## Syntax

The primary data fitting function is `MRStdCRT_fit`, which generates a summary for the target estimands, including both c-ATE and i-ATE. In particular, the output provides:

- The test statistics for testing non-informative cluster sizes,
- The number of clusters,
- The cluster sizes,
- The outcome regression model used.

User can call the `MRStdCRT_fit` function as follows:

`MRStdCRT_fit <- function(formula, data, clus_id, trt, prob, method,family,corstr,scale, jack, alpha)`

with the following arguments:

- `formula`: The model formula for the outcome regression.
- `data`: The dataset being analyzed.
- `cluster`: The identifier for the clusters.
- `trt`: The treatment variable.
- `trtprob`: The vector of treatment probabilities for each cluster. If NULL, the probability will be calculated.
- `method`: The method used for outcome regression model fitting, i.e. “GLM”, “LMM”, “GEE”, “GLMM”.
- `family`: The distributional family for the outcome model (default is `gaussian(link="identity")`).
- `corstr`: The correlation structure used in GEE or GLMM.
- `scale`: The scale of estimand including risk difference (RD), risk ratio (RR), and odds ratio (RR).
- `jack`: Categorical variable for jackknife variance estimator (default is 1, i.e., the standard jackknife variance estimator).
- `alpha`: The significance level (default is 0.05).

# Illustrative example with PPACT data sets

## Illustrative Example

In this example, we demonstrate how to use the `MRStdCRT_fit` function to estimate treatment effects in a CRT using the `ppact` dataset. The goal is to estimate the cluster-averaged treatment effect (c-ATE) and the individual-averaged treatment effect (i-ATE) using the marginal model fitted by generalized estimating equation (GEE).

### Step 1: Prepare the Treatment Assignment Probabilities

Before fitting the model, user needs to calculate the probabilities of treatment assignment for each cluster, which is either known by design (e.g., simple randomization and pair-matched randomization) or estimated properly under covariate-constrained randomization. See Section 3.2 in the main manuscript for more information.

For example, one can estimate the treatment assignment probabilities as follows:

    library(MRStdCRT)
    data(ppact)

    ppact_prob <- ppact %>%
      group_by(CLUST) %>%
      mutate(first_trt = first(INTERVENTION)) %>%
      ungroup() %>%
      mutate(prob_A_1 = mean(first_trt == 1, na.rm = TRUE),  # Proportion trt = 1
             prob_A_0 = mean(first_trt == 0, na.rm = TRUE)) %>%
      mutate(assigned_value = ifelse(INTERVENTION == 1, prob_A_1, prob_A_0))

    prob <- ppact_prob$assigned_value

As another example, the following hypothetical treatment probability vector represents a blocked CRT with three blocks, assigned probabilities of 0.3, 0.5, and 0.6, respectively.

    clusters <- sort(unique(ppact$CLUST))
    n_clusters <- length(clusters)
    block_info <- data.frame(
      CLUST = clusters
    ) %>%
      mutate(
        # Create a 'block' identifier for each cluster
        block = case_when(
          row_number() <= 25 ~ 1,
          row_number() <= 66 ~ 2,
          TRUE ~ 3
        )
      ) %>%
      mutate(
        # Assign the desired hypothetical treatment probability to each block
        hypothetical_prob = case_when(
          block == 1 ~ 0.3,
          block == 2 ~ 0.5,
          block == 3 ~ 0.6
        )
      )
    prob_lookup <- setNames(block_info$hypothetical_prob, block_info$CLUST)
    probs_vector <- prob_lookup[as.character(ppact$CLUST)]

### Step 2: Fit the `robust_CRT` Model

Specifically, the argument trtprob specifies the treatment probability vector for all clusters. For example, if the CRT follows a simple Bernoulli trial with probability of assignment 0.5, one can simply set trtprob = 0.5; otherwise, one needs to provide a vector that contains assignment probabilities for all individuals, where the individuals within the same cluster share the same value of assignment probability.

    example <- MRStdCRT_fit(
      formula = PEGS ~ AGE + FEMALE + comorbid + Dep_OR_Anx + pain_count + PEGS_bl +
        BL_benzo_flag + BL_avg_daily + satisfied_primary + n,
      data = ppact,
      cluster = "CLUST",
      trt = "INTERVENTION",
      trtprob = prob,
      method = "GEE",
      corstr = "independence",
      scale = "RR"
    )
    ## To view the summary, use the following command
    summary(example)
    ## One may also extract specific statistical feature, such as table for point and interval estimates, and standard errors
    example$estimate