OLE Analysis Workflow

OLE phase

This vignette demonstrates the open-label extension (OLE) phase analysis workflow using the difference-in-differences (DID) and synthetic control method (SCM) estimators proposed in Zhou et al. (2024) for estimating long-term treatment effects when the control group switches to treatment.

The SyntheticData dataset has outcomes y1, y2, y3, y4 measured at four time points, and T_cross = 2. This means y1 and y2 are from the placebo-controlled phase (Period I) and y3 and y4 are from the open-label extension (Period II). T_cross is the last column index of Period I in outcome_col_name.

head(SyntheticData[, c("A", "S", "y1", "y2", "y3", "y4")])

##   A S         y1         y2         y3        y4
## 1 1 1  3.4512377 -0.7642287 -2.4713591  3.935466
## 2 1 1  0.4518106  6.3516296  4.5231869 -0.198674
## 3 0 1  3.0532714 -2.0453190  5.9064870 -1.374919
## 4 1 1 -9.1183948  0.2304339  4.7858172  8.490757
## 5 0 1 -1.4270057  1.5878794  3.7006101  9.449632
## 6 0 1 -2.6967072 -0.6130288  0.7482786 -2.413717

1 DID methods

1.1 DID-EC-IPW

method <- did_ec_ipw(
  ps_formula = "S ~ x1 + x2 + x3 + x4 + x5",
  trt_formula = "A ~ x1 + x2 + x3 + x4 + x5",
  bootstrap = 50
)

analysis <- setup_analysis_OLE(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2", "y3", "y4"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  T_cross = 2,
  method_OLE_obj = method
)

run_analysis(analysis)

##      point_estimates lower_CI_boot upper_CI_boot
## tau3        2.075926    -0.2531514      3.994431
## tau4        4.389438     0.5095401      7.982780

1.2 DID-EC-AIPW

model_forms <- c(
  "y1 ~ x1 + x2 + x3 + x4 + x5",
  "y2 ~ x1 + x2 + x3 + x4 + x5",
  "y3 ~ x1 + x2 + x3 + x4 + x5",
  "y4 ~ x1 + x2 + x3 + x4 + x5"
)

method <- did_ec_aipw(
  ps_formula = "S ~ x1 + x2 + x3 + x4 + x5",
  trt_formula = "A ~ x1 + x2 + x3 + x4 + x5",
  outcome_formula = model_forms,
  bootstrap = 50
)

analysis <- setup_analysis_OLE(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2", "y3", "y4"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  T_cross = 2,
  method_OLE_obj = method
)

run_analysis(analysis)

##      point_estimates lower_CI_boot upper_CI_boot
## tau3        2.041727     -1.232759      4.558105
## tau4        4.036118      1.492481      7.712051

1.3 DID-EC-OR

model_forms <- c(
  "y1 ~ x1 + x2 + x3 + x4 + x5",
  "y2 ~ x1 + x2 + x3 + x4 + x5",
  "y3 ~ x1 + x2 + x3 + x4 + x5",
  "y4 ~ x1 + x2 + x3 + x4 + x5"
)

method <- did_ec_or(
  outcome_formula_ext = model_forms,
  outcome_formula_rct_ctrl = model_forms,
  outcome_formula_rct_trt = model_forms,
  bootstrap = 50
)

analysis <- setup_analysis_OLE(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2", "y3", "y4"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  T_cross = 2,
  method_OLE_obj = method
)

run_analysis(analysis)

##      point_estimates lower_CI_boot upper_CI_boot
## tau3        1.568947    -0.7157897      3.610432
## tau4        4.407834     1.4696074      6.400508

2 Synthetic control method

method <- scm(
  lambda_min = 0,
  lambda_max = 1e-3,
  nlambda = 2,
  bootstrap = 3,
  bootstrap_ci_type = "perc"
)

analysis <- setup_analysis_OLE(
  data = SyntheticData,
  trial_status_col_name = "S",
  treatment_col_name = "A",
  outcome_col_name = c("y1", "y2", "y3", "y4"),
  covariates_col_name = c("x1", "x2", "x3", "x4", "x5"),
  T_cross = 2,
  method_OLE_obj = method
)

run_analysis(analysis)

##      point_estimates lower_CI_boot upper_CI_boot
## tau3        2.064756     0.8669477      4.262813
## tau4        3.943234     3.0478559      7.319810