# An Example of Fixed Design with Two Correlated Endpoints

In this vignette, we illustrate how to use `TrialSimulator` to simulate a trial of two correlated endpoints. Under a fixed design, only one milestone is specified for final analysis.

## Simulation Settings

- Trial consists of two active arms of high and low dose, and a standard-of-care arm.

- Patients are randomized into the trial with ratio `1:1:1`.

- 30 patients are randomized per month for the first 10 months, and 50 patients per month after then until 1000 patients.

- Dropout rate is 10% at month 18, which is modeled by an exponential distribution, with rate parameter `-log(1 - 0.1)/18`.

- Modeled by a three-state illness-death model, two endpoints `PFS` and `OS` are simulated.

  - Primary endpoint `OS` has a medians of 15, 18.5, 20 months in standard-of-care, low dose and high dose arms, respectively.
  - Key secondary endpoint `PFS` has a median of 7, 9, 10 months in the three arms.
  - Pearson’s correlation between `OS` and `PFS` are 0.68, 0.65, and 0.60 in the three arms.

- To ensures sufficient powers for both endpoints, final analysis is set when we have at least a total of 800 `PFS` events in the standard-of-care and high dose arm, meanwhile a total of 550 `OS` events are observed in three arms

  - `OS` is tested using one-sided logrank test.
  - `PFS` is tested using one-sided p-value from the Cox proportional hazard model as the PH assumption is assumed.
  - The Bonferroni correction is adopted to split overall \(\alpha = 5\%\), i.e., claiming significant effect for an endpoint when p-value is lower than \(0.05/4\).

## Transition Hazards of `PFS` and `OS`

We adopt the illness-death model to simulate the two endpoints. This ensures `PFS` \(\leq\) `OS` with probability one, and makes no assumption on latent variables or copula parameters. `TrialSimulator` offers a function `solveThreeStateModel()` to convert endpoints’ medians and correlation to the transition hazards, which are required by the built-in generator `CorrelatedPfsAndOs3()`. Refer to the vignette Simulate Correlated Progression-Free Survival and Overall Survival as Endpoints Under Illness-Death Model. Note that data generated under the illness-death model may induce a time-varying OS hazard ratio between treatment arms, violating the proportional hazards (PH) assumption in a Cox model. For a data model that is more suitable for PH Cox analysis, refer to the built-in generator `CorrelatedPfsAndOs2()` and vignette of the Gumbel copula method.

    pars_soc <- solveThreeStateModel(median_pfs = 7, median_os = 15, corr = .68,
                                     h12 = seq(.07, .10, length.out = 50))
    pars_soc
    #>   corr        h01       h02        h12        error
    #> 1 0.68 0.07498342 0.0240376 0.08959184 0.0001363832

    pars_low <- solveThreeStateModel(median_pfs = 9, median_os = 18.5, corr = .65,
                                     h12 = seq(.04, .07, length.out = 50))
    pars_low
    #>   corr        h01        h02        h12       error
    #> 1 0.65 0.05059501 0.02642134 0.06204082 0.001646964

    pars_high <- solveThreeStateModel(median_pfs = 10, median_os = 20, corr = .60,
                                      h12 = seq(.02, .06, length.out = 50))
    pars_high
    #>   corr        h01        h02        h12      error
    #> 1  0.6 0.03964373 0.02967099 0.04693878 0.00208503

Transition hazards in three treatment arms
arm | h01 | h02 | h12
--- | --- | --- | ---
soc | 0.075 | 0.024 | 0.090
low | 0.051 | 0.026 | 0.062
high | 0.040 | 0.030 | 0.047

## Define Treatment Arms

We use generator `CorrelatedPfsAndOs3()` with `endpoint()` and `arm()` to define the three treatment arms.

    #' define SoC
    pfs_os_in_soc <- endpoint(name = c('pfs', 'os'),
                              type = c('tte', 'tte'),
                              generator = CorrelatedPfsAndOs3,
                              h01 = 0.075, h02 = 0.024, h12 = 0.090)

    soc <- arm(name = 'soc')
    soc$add_endpoints(pfs_os_in_soc)

    #' define low dose arm
    pfs_os_in_low <- endpoint(name = c('pfs', 'os'),
                              type = c('tte', 'tte'),
                              generator = CorrelatedPfsAndOs3,
                              h01 = 0.051, h02 = 0.026, h12 = 0.062)

    low <- arm(name = 'low')
    low$add_endpoints(pfs_os_in_low)

    #' define high dose arm
    pfs_os_in_high <- endpoint(name = c('pfs', 'os'),
                               type = c('tte', 'tte'),
                               generator = CorrelatedPfsAndOs3,
                              h01 = 0.040, h02 = 0.030, h12 = 0.047)

    high <- arm(name = 'high')
    high$add_endpoints(pfs_os_in_high)

We can request for a summary report of, e.g., the high dose arm, by printing the arm object in R console. The medians of `PFS` and `OS` matches to the settings very well.

    high

## Define a Trial

With three arms, we can define a trial using the function `trial()`. Recruitment curve are specified through `enroller` with a built-in function `StaggeredRecruiter` of piecewise constant rate. We set `duration` to be an arbitrary large number (500) but controlling the end of trial through a pre-defined milestone later. Note that if `seed = NULL`, `TrialSimulator` will pick a seed for the purpose of reproducibility.

    accrual_rate <- data.frame(end_time = c(10, Inf),
                               piecewise_rate = c(30, 50))
    trial <- trial(
      name = 'Trial-3415', n_patients = 1000,
      seed = 1727811904, duration = 500,
      enroller = StaggeredRecruiter, accrual_rate = accrual_rate,
      dropout = rexp, rate = -log(1 - 0.1)/18, ## 10% by month 18
      silent = TRUE
    )

    trial$add_arms(sample_ratio = c(1, 1, 1), soc, low, high) ## 1:1:1
    trial
    #>  ⚕⚕         Trial Name:  Trial-3415
    #>  ⚕⚕        Description:  Trial-3415
    #>  ⚕⚕     Number of Arms:  3
    #>  ⚕⚕    Registered Arms:  soc, low, high
    #>  ⚕⚕       Sample Ratio:  1, 1, 1
    #>  ⚕⚕ Number of Patients:  1000
    #>  ⚕⚕   Planned Duration:  500
    #>  ⚕⚕            Regimen:  not set
    #>  ⚕⚕        Random Seed:  1727811904

## Define Milestone and Action for Final Analysis

To ensure sufficient powers of testing `PFS` and `OS`, final analysis is performed when we have at least 700 events for `OS` and 800 events for `PFS`. In the action function, we compute one-sided p-value of `PFS` using the proportional hazard Cox model, and one-sided p-value of `OS` using the logrank test. This is consistent with the assumption of illness-death model implemented in data generator `CorrelatedPfsAndOs3()`. Note that five columns are available in locked data: `arm`, `pfs`, ‘os’, ‘pfs_event’, and ‘os_event’, which are used to construct model formula. Estimates of hazard ratio are also computed. Built-in functions `fitCoxph` and `fitLogrank` return data frames. Refer to their help documants for more details.

    action <- function(trial){

      locked_data <- trial$get_locked_data('final')

      pfs <- fitCoxph((Surv(pfs, pfs_event) ~ arm), placebo = 'soc',
                      data = locked_data, alternative = 'less',
                      scale = 'hazard ratio')

      os <- fitLogrank((Surv(os, os_event) ~ arm), placebo = 'soc',
                       data = locked_data, alternative = 'less')

      ## Bonferroni test is applied to four hypotheses:
      ## PFS_low, PFS_high, OS_low, and OS_high
      pfs$decision <- ifelse(pfs$p < .05/4, 'reject', 'accept')
      os$decision <- ifelse(os$p < .05/4, 'reject', 'accept')

      trial$save(
        value = pfs %>% filter(arm == 'low') %>% select(estimate, decision, info),
        name = 'pfs_low')

      trial$save(
        value = pfs %>% filter(arm == 'high') %>% select(estimate, decision, info),
        name = 'pfs_high')

      trial$save(
        value = os %>% filter(arm == 'low') %>% select(decision, info),
        name = 'os_low')

      trial$save(
        value = os %>% filter(arm == 'high') %>% select(decision, info),
        name = 'os_high')

    }

Now we can define and register the milestone to a listener, which monitors the trial for us through a controller

    final <- milestone(name = 'final', action = action,
                       when = eventNumber(endpoint = 'pfs', n = 450,
                                          arms = c('soc', 'high')) &
                         eventNumber(endpoint = 'os', n = 550)
                       )

    listener <- listener()
    listener$add_milestones(final)
    #> A milestone <final> is registered.

    controller <- controller(trial, listener)

We can run a massive number of replicates in simulation to study operating characteristics of a trial design by specifying `n` in `Controller$run()`. The simulation results can be accessed by calling the member function `get_output()` of the controller.

    controller$run(n = 1000, plot_event = FALSE, silent = TRUE)
    output <- controller$get_output()

    output %>%
      head(5) %>%
      kable(escape = FALSE) %>%
      kable_styling(bootstrap_options = "striped",
                    full_width = FALSE,
                    position = "left") %>%
      scroll_box(width = "100%")

trial | seed | milestone_time_<final> | n_events_<final>_<pfs> | n_events_<final>_<os> | n_events_<final>_<patient_id> | n_events_<final>_<arms> | pfs_low_<estimate> | pfs_low_<decision> | pfs_low_<info> | pfs_high_<estimate> | pfs_high_<decision> | pfs_high_<info> | os_low_<decision> | os_low_<info> | os_high_<decision> | os_high_<info> | error_message
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
Trial-3415 | 1727811904 | 36.30115 | 769 | 550 | 1000 | c(“high”…. | 0.9455634 | accept | 524 | 0.7186429 | reject | 511 | accept | 381 | reject | 373 |
Trial-3415 | 1324384826 | 36.01193 | 768 | 550 | 1000 | c(“high”…. | 0.8100537 | reject | 517 | 0.7885803 | reject | 519 | accept | 380 | reject | 373 |
Trial-3415 | 1767496878 | 34.88384 | 775 | 550 | 1000 | c(“high”…. | 0.7731265 | reject | 522 | 0.8023016 | reject | 528 | reject | 381 | reject | 380 |
Trial-3415 | 295939270 | 35.21148 | 747 | 550 | 1000 | c(“high”…. | 0.8732924 | accept | 514 | 0.7772976 | reject | 496 | accept | 388 | reject | 369 |
Trial-3415 | 2110257267 | 34.37072 | 763 | 550 | 1000 | c(“high”…. | 0.8071533 | reject | 524 | 0.7257276 | reject | 511 | reject | 383 | reject | 372 |

For example, we can compute the powers and summarize the estimates of hazard ratio for `PFS`.

    output %>%
      summarise(
        across(matches('_<decision>$'), ~ mean(. == 'reject') * 100, .names = 'Power_{.col}'),
        across(matches('_<estimate>$'), ~ mean(.x), .names = 'HR_{.col}')
      ) %>%
      rename_with(~ sub('_<decision>$', '', .), starts_with('Power_')) %>%
      rename_with(~ sub('_<estimate>$', '', .), starts_with('HR_')) %>%
      kable(col.name = NULL, digits = 3, align = 'r') %>%
      add_header_above(c('Low', 'High', 'Low', 'High', 'Low', 'High'), align = 'r') %>%
      add_header_above(c('PFS' = 2, 'OS' = 2, 'PFS' = 2)) %>%
      add_header_above(c('Power (%)' = 4, 'Hazard Ratio' = 2)) %>%
      kable_styling(full_width = TRUE)

Power (%) | Hazard Ratio
--- | ---
PFS | OS | PFS
Low | High | Low | High | Low | High
73.5 | 95.5 | 64.5 | 82.7 | 0.78 | 0.71

Note that `OS` does not satisfy the proportional hazard assumption, and a composite condition on event numbers is used to trigger the final analysis. Thus, the powers in the table above would not match to the output from power calculation packages, which is as expected.