Primary Analysis Workflow

Lei Shi

2026-05-06

Primary analysis

This vignette demonstrates the primary analysis workflow using the EC-IPW and EC-AIPW weighting estimators proposed in Zhou et al. (2024) for incorporating external controls in randomized trials with longitudinal outcomes.

1 load and visualize data

# load the simulated dataset
head(SyntheticData)

##   x1 x2 x3 x4       x5 A S T_cross         y1         y2         y3        y4
## 1  1  1  1  9 54.59836 1 1       2  3.4512377 -0.7642287 -2.4713591  3.935466
## 2  0  1  0  8 33.08006 1 1       2  0.4518106  6.3516296  4.5231869 -0.198674
## 3  1  1  1  7 48.51653 0 1       2  3.0532714 -2.0453190  5.9064870 -1.374919
## 4  1  1  1 15 31.68766 1 1       2 -9.1183948  0.2304339  4.7858172  8.490757
## 5  1  1  1 12 29.98495 0 1       2 -1.4270057  1.5878794  3.7006101  9.449632
## 6  1  1  0  7 46.08991 0 1       2 -2.6967072 -0.6130288  0.7482786 -2.413717

2 Estimation and inference:

2.1 Inverse probability weighting (IPW)

1. IPW with zero weight (wt = 0):

# test: within trial
## Data argument + column names (coxph, glm)
method_weighting_obj <- setup_method_weighting(
  method_name = "IPW",
  optimal_weight_flag = FALSE,
  wt = 0,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5"
)

analysis_primary_obj <- setup_analysis_primary(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  method_weighting_obj = method_weighting_obj
)

res <- run_analysis(analysis_primary_obj)
res

## $results
##      point_estimates standard_deviation lower_CI_normal upper_CI_normal
## tau1     -0.02808704          0.5367987      -1.0801931        1.024019
## tau2      0.40959558          0.5625763      -0.6930338        1.512225
## 
## $borrow_weight
## [1] 0

2. IPW with data-adaptive weight:

# test: within trial
method_weighting_obj <- setup_method_weighting(
  method_name = "IPW",
  optimal_weight_flag = TRUE,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5"
)

analysis_primary_obj <- setup_analysis_primary(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  method_weighting_obj = method_weighting_obj
)

res <- run_analysis(analysis_primary_obj)

res$borrow_weight

## [1] 0.1475196

res$results

##      point_estimates standard_deviation lower_CI_normal upper_CI_normal
## tau1      -0.1971969          0.5134018       -1.203446        0.809052
## tau2       0.4697209          0.5410007       -0.590621        1.530063

2.2 Augmented inverse probability weighting (AIPW)

The second approach is AIPW, which also accommodates two external borrowing strategies.

1. AIPW with zero weight (wt = 0):

# test: AIPW with 0 weight, should be same as IPW with 0 weight
method_weighting_obj <- setup_method_weighting(
  method_name = "AIPW",
  optimal_weight_flag = FALSE,
  wt = 0,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5",
  model_form_mu0_ext = c(
    "y1 ~ x1 + x2 + x3 + x4 + x5",
    "y2 ~ x1 + x2 + x3 + x4 + x5"
  )
)

analysis_primary_obj <- setup_analysis_primary(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  method_weighting_obj = method_weighting_obj
)

res <- run_analysis(analysis_primary_obj)

2. AIPW with data adaptive weight:

# test: AIPW with given weight
# bootstrap as part of the method and analysis
method_weighting_obj <- setup_method_weighting(
  method_name = "AIPW",
  optimal_weight_flag = TRUE,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5",
  model_form_mu0_ext = c(
    "y1 ~ x1 + x2 + x3 + x4 + x5",
    "y2 ~ x1 + x2 + x3 + x4 + x5"
  )
)

analysis_primary_obj <- setup_analysis_primary(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  method_weighting_obj = method_weighting_obj
)

res <- run_analysis(analysis_primary_obj)

3 Bootstrap inference

In this section we present Bootstrap inference results. We report Bootstrap confidence intervals with adjusted quantile ranges.

1. IPW with bootstrap CI

# test: within trial
## Data argument + column names (coxph, glm)
## having a bootstrap_flag outside the class
bootstrap_obj <- setup_bootstrap(
  replicates = 50,
  bootstrap_CI_type = "perc"
)

method_weighting_obj <- setup_method_weighting(
  method_name = "IPW",
  optimal_weight_flag = TRUE,
  bootstrap_flag = TRUE,
  bootstrap_obj = bootstrap_obj,
  wt = 0,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5"
)

analysis_primary_obj <- setup_analysis_primary(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  method_weighting_obj = method_weighting_obj
)

res <- run_analysis(analysis_primary_obj)
res

## $results
##      point_estimates standard_deviation lower_CI_boot upper_CI_boot
## tau1      -0.1971969          0.5134018    -1.1788096     0.8804926
## tau2       0.4697209          0.5410007    -0.6244215     1.8038458
## 
## $borrow_weight
## [1] 0.1475196

2. AIPW with bootstrap CI

# test: with optimal weight
## Data argument + column names (coxph, glm)
bootstrap_obj <- setup_bootstrap(
  replicates = 50,
  bootstrap_CI_type = "perc"
)

method_weighting_obj <- setup_method_weighting(
  method_name = "AIPW",
  optimal_weight_flag = TRUE,
  wt = 0,
  bootstrap_flag = TRUE,
  bootstrap_obj = bootstrap_obj,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5",
  model_form_mu0_ext = c(
    "y1 ~ x1 + x2 + x3 + x4 + x5",
    "y2 ~ x1 + x2 + x3 + x4 + x5"
  )
)

analysis_primary_obj <- setup_analysis_primary(
  data = SyntheticData,
  trial_status = "S",
  treatment = "A",
  outcome = c("y1", "y2"),
  covariates = c("x1", "x2", "x3", "x4", "x5"),
  method_weighting_obj = method_weighting_obj
)


res <- run_analysis(analysis_primary_obj)

## best to just do the last one, but also have options
## time dependent way of effects and modeling

4 Notes

1. When there are missing values in the data, the suggestion we have for now is to preprocess the dataset (such as deletion, imputing, etc.) to obtain a dataset without missingness, then apply the package. For general methodology development regarding missing values, we save it for future research work.

References

• Zhou X, Zhu J, Drake C, Pang H (2024). “Causal estimators for incorporating external controls in randomized trials with longitudinal outcomes.” Journal of the Royal Statistical Society Series A: Statistics in Society. doi: 10.1093/jrsssa/qnae075.
• Shi L, Pang H, Chen C, Zhu J (2025). “rdborrow: an R package for causal inference incorporating external controls in randomized controlled trials with longitudinal outcomes.” Journal of Biopharmaceutical Statistics, 35(6), 1043-1066. doi: 10.1080/10543406.2025.2489283.