SCT_GSD.Rmd

---
title: "Covariate Adjustment and Group Sequential Designs"
subtitle: "Worked Examples using Simulated Trial Data"
author: "Kelly Van Lancker (kvanlan3@jhu.edu), Josh Betz (jbetz@jhu.edu), and Michael Rosenblum (mrosen@jhu.edu)"
date: "`r format(Sys.time(), '%Y-%m-%d %I:%M')`"
output:
  html_document:
    toc: true # table of content true
    # toc_depth: 3
    # number_sections: true
    theme: united
    highlight: tango
    # css: my.css   # For custom CSS - In same folder
---

```{r setup, include=FALSE}
options(
  knitr.kable.NA = ""
)

knitr::opts_chunk$set(
  echo = TRUE,
  message = FALSE,
  # warning = FALSE,
  results = "markup"
) 
```

## Installing R Packages

We first install the required packages
```{r install-packages, eval = FALSE}
required_packages <-
  c("rpact", "tidyr", "dplyr", "parallel", "ldbounds")

install.packages(required_packages)
```

Once the required packages are installed, they can be loaded using `library()`

```{r load-packages}
library(rpact)
library(tidyr)
library(dplyr)
library(parallel)
library(ldbounds)
```

## Simulated Dataset

### Design Parameters
We first determine the maximum/total information and sample size based on a design with 2 analyses: 1 interim analyses when 50\% of the information is available in addition to the final analysis. The two-sided significance level equals 5\% and the power to detect a 5\% difference in the success rate (55\% in treatment arm versus 50\% in control) equals 90\%. An alpha-spending function that approximates the Pocock boundaries are used for efficacy stopping (futility stopping is not considered).
```{r design-parameters}
design_par = getDesignGroupSequential(sided = 2, alpha = 0.05, beta=0.1,
                                      informationRates = c(0.50, 1),
                                      typeOfDesign = "asP")
# Determine Inflation Factor
#getDesignCharacteristics(design_par) #1.1110  

inf_total = 1.1110*(qnorm(0.975)+qnorm(0.90))^2/0.05^2

n_total = round(1.1110*(power.prop.test(power=0.90,p1=0.55,p2=0.5, 
                                         alternative="two.sided", 
                                         sig.level=0.05)$n)*2
)
```

We load the code to create the data (we need function `expit()`) and conduct the data analysis.
```{r load-code-github, warning=FALSE}
data_url_code <-
  "https://github.com/jbetz-jhu/CovariateAdjustmentTutorial/raw/main/codeGeneral.R"

source(file = url(data_url_code))

```


### Creating Simulated Dataset
Create data frame 'study_data' with baseline covariates `x_1`, `x_2`, `x_3` and `x_4`, a patient identifier (`id`), a treatment indicator (`treatment`), the primary outcome of interest (`y_1`), the time of enrollment (`enrollment_times`), and the time a patient's outcome is measured (`outcome_times`).

```{r data-generation, eval=FALSE}
set.seed(12345)
# Generate baseline covariates
baseline_covariates_orig <-
  matrix(
    data =
      mvtnorm::rmvnorm(
        n = n_total,
        mean = matrix(data = 0, nrow = 4),
        sigma = diag(x = 1, nrow = 4)
      ),
    nrow = n_total,
    ncol = 4
  ) %>% 
  data.frame() %>% 
  setNames(
    object = .,
    nm = paste0("x_", 1:4)
  )

# Generate treatment indicator
treatment <-
      rbinom(
        n = n_total,
        size = 1,
        prob = 0.5
      )

# (Counterfactual) outcomes under new experimental treatment
n_outcomes = 1
outcomes1 <-
      matrix(
        data =
          rbinom(
            n = n_total*n_outcomes,
            size = 1,
            prob = expit(-2.5*baseline_covariates_orig$x_1+
                           2*baseline_covariates_orig$x_2-
                           2.5*baseline_covariates_orig$x_3+
                           2.1*baseline_covariates_orig$x_4)
          ),
        nrow = n_total,
        ncol = n_outcomes
      ) %>% 
      data.frame() %>% 
      setNames(
        object = .,
        nm = paste0("y1_", 1:n_outcomes)
      )

# (Counterfactual) outcomes under control
outcomes0 <-
      matrix(
        data =
          rbinom(
            n = n_total*n_outcomes,
            size = 1,
            prob = expit(-2*baseline_covariates_orig$x_1+
                           2.5*baseline_covariates_orig$x_2-
                           2.25*baseline_covariates_orig$x_3-
                           2.1*baseline_covariates_orig$x_4)
          ),
        nrow = n_total,
        ncol = n_outcomes
      ) %>% 
      data.frame() %>% 
      setNames(
        object = .,
        nm = paste0("y0_", 1:n_outcomes)
      )

# Enrollment times
daily_enrollment = 5
enrollment_times <-
      round(
        runif(
          n = n_total, min = 0, max=n_total/daily_enrollment
        )
      )

# Outcome times
outcome_times <-
      setNames(
        object = 365 + enrollment_times,
        nm = paste0("outcome_time_", 1)
      )

# Measured baseline covariates   
baseline_covariates_meas = as.data.frame(cbind(
  x_1 = exp(baseline_covariates_orig$x_1/2),
  x_2 = baseline_covariates_orig$x_2/
  (1+exp(baseline_covariates_orig$x_1))+10,
  x_3 = (baseline_covariates_orig$x_1*
                                  baseline_covariates_orig$x_3/25+0.6)^3,
  x_4 = (baseline_covariates_orig$x_2
                                +baseline_covariates_orig$x_4+20)^2
  ))

# Make dataset    
study.data <-
      tibble(
        id = 1:n_total,
        baseline_covariates_meas,
        enrollment_times,
        treatment,
        outcome_times,
        y_1 = outcomes1$y1_1*treatment+outcomes0$y0_1*(1-treatment)
      )
```

The data can also be directly loaded from Github

```{r load-simulated-data-github}
data_url_gsd_data <-
  "https://github.com/jbetz-jhu/CovariateAdjustmentTutorial/raw/main/simData.RData"

load(file = url(data_url_gsd_data))

```

The complete simulated trial data without any missing values are in a `data.frame` named `study.data`.

  - Randomization Information
    - `treatment`: Treatment Arm
  - Baseline Information
    - `x_1`, `x_2`, `x_3` and `x_4`: 4 continuous baseline covariates
    - `id`: patient identifier
    - `enrollment_times`: time of enrollment
  - Outcome information:
    - `y_1`: Binary outcome (1: success; 0: otherwise) 
    - `outcome_times`: time someone outcome is measured (365 days after enrollment)

## Combining Group Sequential Designs and Covariate Adjustment

### Interim analysis

Via the function `data_at_time_t()` we construct a dataframe of the data available at the interim analysis, which is conducted when 50\% of the patients have their primary endpoint available.
```{r interim-data}
n_total_ia = round(0.5*n_total)
time_ia = sort(study.data$outcome_times)[n_total_ia]
n_recr_ia = length(which(study.data$enrollment_times<=time_ia))

data_ia = data_at_time_t(
  data = study.data,
  id_column = "id",
  analysis_time = time_ia,
  enrollment_time = "enrollment_times",
  treatment_column = "treatment",
  covariate_columns = c("x_1", "x_2", "x_3", "x_4"),
  outcome_columns = c("y_1"),
  outcome_times = c("outcome_times")
      ) 
    
```

We can then call the function `interimAnalysis()` to conduct the interim analysis. This gives us a decision based on the original estimator and updated estimator (which is the same as the original one at the first analysis). In addition, we get the estimates(both the original and updated estimate), the corresponding standard errors, test statistics, and information fractions. Finally, we also get the current covariance matrix based on the original estimates, which we will need at the final analysis to do the orthogonalization (in order to obtain the updated estimate).

```{r interim-analysis, warning=FALSE}
results_ia = interimAnalysis(data = data_ia,
                             totalInformation = inf_total, 
                             estimationMethod= standardization,
                             null.value = 0,
                             alpha = 0.05, 
                             alternative = "two.sided", 
                             boundaries = 2,
                             plannedAnalyses=2,
                             y0_formula=y_1 ~ x_1, 
                             y1_formula=y_1 ~ x_1,
                             family="binomial"
                             )
```

#### Results
Decision based on original interim estimator
```{r ia-original-decision}
results_ia$decisionOriginal
```
So we do not stop the trial early for efficacy based on the original interim estimator.

Decision based on updated interim estimator
```{r ia-updated-decision}
results_ia$decisionUpdated
```
So we do not stop the trial early for efficacy based on the updated interim estimator.

Original interim estimate
```{r ia-original-estimate}
results_ia$estimateOriginal
previousEstimates = results_ia$estimateOriginal
```

Updated interim estimate
```{r ia-updated-estimate}
results_ia$estimateUpdated
```

Covariance matrix (i.e., variance) of the original interim estimate
```{r ia-original-covariance}
results_ia$covMatrixOriginal
covMatrix = results_ia$covMatrixOriginal
```  

Information time (current information divided by expected information at end of trial) corresonding with original interim estimate
```{r ia-original-informationtime}
results_ia$informationTimeOriginal
previousTimes = results_ia$informationTimeOriginal
```  

Information time (current information divided by expected information at end of trial) corresonding with updated interim estimate
```{r ia-updated-informationtime}
results_ia$informationTimeUpdated
previousTimesUpd = results_ia$informationTimeUpdated
```

### Final analysis

Via the function `data_at_time_t()` we construct a dataframe of the data available at the final analysis (which equals the full dataset!).
```{r final-data}
time_fin = sort(study.data$outcome_times)[n_total]
n_recr_fin = length(which(study.data$enrollment_times<=time_fin))

data_fin = data_at_time_t(
  data = study.data,
  id_column = "id",
  analysis_time = time_fin,
  enrollment_time = "enrollment_times",
  treatment_column = "treatment",
  covariate_columns = c("x_1", "x_2", "x_3", "x_4"),
  outcome_columns = c("y_1"),
  outcome_times = c("outcome_times")
      ) 
    
```

We can then call the function `interimAnalysis()` to conduct the final analysis. This gives us a decision based on the original estimator and updated estimator (which is the same as the original one at the first analysis). In addition, we get the estimates(both the original and updated estimate), the corresponding standard errors, test statistics, and information fractions. Finally, we also get the current covariance matrix based on the original estimates, which we will need at the final analysis to do the orthogonalization (in order to obtain the updated estimate).

```{r final-analysis}
results_fin = interimAnalysis(data = data_fin, 
                            totalInformation = inf_total, 
                            estimationMethod= standardization,
                            previousEstimatesOriginal=previousEstimates,
                            previousCovMatrixOriginal=covMatrix,
                            previousInformationTimesOriginal=previousTimes,
                            previousInformationTimesUpdated=previousTimesUpd,
                            previousDatasets = list(data_ia),
                            null.value = 0,
                            alpha = 0.05, 
                            alternative = "two.sided", 
                            boundaries = 2,
                            plannedAnalyses=2,
                            y0_formula=y_1 ~ x_2, 
                            y1_formula=y_1 ~ x_2,
                            family="binomial",
                            parametersPreviousEstimators = list(list(
                              y0_formula=y_1 ~ x_1,
                              y1_formula=y_1 ~ x_1,
                              family="binomial"
                              )) 
)

```

#### Results
Decision based on original final estimator
```{r final-original-decision}
results_fin$decisionOriginal
```
So we do not reject the null hypothesis based on the original final estimator.

Decision based on updated final estimator
```{r final-updated-decision}
results_fin$decisionUpdated
```
So we do not reject the null hypothesis based on the updated final estimator.

Original final estimate
```{r final-original-estimate}
results_fin$estimateOriginal
```

Updated final estimate
```{r final-updated-estimate}
results_fin$estimateUpdated
```