Overview

The adaptr package simulates adaptive (multi-arm, multi-stage) clinical trials using adaptive stopping, adaptive arm dropping and/or response-adaptive randomisation.

The package has been developed as part of the INCEPT (Intensive Care Platform Trial) project, primarily supported by a grant from Sygeforsikringen “danmark”.

Additional guidance on the key methodological considerations when planning and comparing adaptive clinical trials can be found in the open access article “An overview of methodological considerations regarding adaptive stopping, arm dropping and randomisation in clinical trials” available in Journal of Clinical Epidemiology.

Usage and workflow overview

The central functionality of adaptr and the typical workflow is illustrated here.

Setup

First, the package is loaded and a cluster of parallel workers is initiated by the setup_cluster() function to facilitate parallel computing:

library(adaptr)

setup_cluster(2)

Parallelisation is supported in many adaptr functions, and a cluster of parallel workers can be setup for the entire session using setup_cluster() early in the script as in this example. Alternatively, parallelisation can be controlled by the global "mc.cores" option (set by calling options(mc.cores = <number>)) or the cores argument of many functions.

Specify trial design

Setup a trial specification (defining the trial design and scenario) using the general setup_trial() function, or one of the special case variants using default priors setup_trial_binom() (for binary, binomially distributed outcomes; used in this example) or setup_trial_norm() (for continuous, normally distributed outcomes).

The example trial specification has the following characteristics:

See ?setup_trial() for details on all the arguments or vignette("Basic-examples", "adaptr") for basic example trial specifications and a thorough review of the general trial specification settings, and vignette("Advanced-example", "adaptr") for an advanced example including details on how to specify user-written functions for generating outcomes and posterior draws.

Below, the trial specification is setup and a human-readable overview printed:

binom_trial <- setup_trial_binom(
  arms = c("Arm A", "Arm B", "Arm C"),
  true_ys = c(0.25, 0.25, 0.25),
  min_probs = rep(0.20, 3),
  data_looks = seq(from = 300, to = 2000, by = 100),
  randomised_at_looks = c(seq(from = 400, to = 2000, by = 100), 2000),
  equivalence_prob = 0.9,
  equivalence_diff = 0.05,
  soften_power = 0.5
)

print(binom_trial, prob_digits = 3)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * No common control arm
#> * Best arms: Arm A and Arm B and Arm C
#> 
#> Arms, true outcomes, starting allocation probabilities 
#> and allocation probability limits:
#>   arms true_ys start_probs fixed_probs min_probs max_probs
#>  Arm A    0.25       0.333          NA       0.2        NA
#>  Arm B    0.25       0.333          NA       0.2        NA
#>  Arm C    0.25       0.333          NA       0.2        NA
#> 
#> Maximum sample size: 2000 
#> Maximum number of data looks: 18
#> Planned data looks after:  300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000 patients have reached follow-up
#> Number of patients randomised at each look:  400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2000
#> 
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> Equivalence threshold: 0.9 (all analyses) (no common control)
#> Absolute equivalence difference: 0.05
#> No futility threshold (not relevant - no common control)
#> Soften power for all analyses: 0.5

By default, (most) probabilities are shown with 3 decimals. This can be changed by explicitly print()ing the specification with the prob_digits arguments, for example:

print(binom_trial, prob_digits = 2)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * No common control arm
#> * Best arms: Arm A and Arm B and Arm C
#> 
#> Arms, true outcomes, starting allocation probabilities 
#> and allocation probability limits:
#>   arms true_ys start_probs fixed_probs min_probs max_probs
#>  Arm A    0.25        0.33          NA       0.2        NA
#>  Arm B    0.25        0.33          NA       0.2        NA
#>  Arm C    0.25        0.33          NA       0.2        NA
#> 
#> Maximum sample size: 2000 
#> Maximum number of data looks: 18
#> Planned data looks after:  300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000 patients have reached follow-up
#> Number of patients randomised at each look:  400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2000
#> 
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> Equivalence threshold: 0.9 (all analyses) (no common control)
#> Absolute equivalence difference: 0.05
#> No futility threshold (not relevant - no common control)
#> Soften power for all analyses: 0.5

Calibration

In the example trial specification, there are no true between-arm differences, and stopping rules for inferiority and superiority are not explicitly defined. This is intentional, as these stopping rules will be calibrated to obtain a desired probability of stopping for superiority in the scenario with no between-arm differences (corresponding to the Bayesian type 1 error rate). Trial specifications do not necessarily have to be calibrated. Instead,simulations can be run directly using the run_trials() function covered below (or run_trial() for a single simulation). This can be followed by assessment of performance metrics, and manually changing the specification (including the stopping rules) until performance metrics are considered acceptable. In this example, a full calibration procedure is performed.

Calibration of a trial specification is done using the calibrate_trial() function, which defaults to calibrate constant, symmetrical stopping rules for inferiority and superiority (expecting a trial specification with identical outcomes in each arm), but can be used to calibrate any parameter in a trial specification towards any performance metric if a user-defined calibration function (fun) is specified.

To perform the calibration, a target value, a search_range, a tolerance value (tol), and the allowed direction of the tolerance value (dir) must be specified (or alternatively, the defaults can be used). Of note, the number of simulations in each calibration step here is lower than generally recommended (to reduce the time required to build this vignette):

# Calibrate the trial specification
calibrated_binom_trial <- calibrate_trial(
  trial_spec = binom_trial,
  n_rep = 1000, # 1000 simulations for each step (more generally recommended)
  base_seed = 4131, # Base random seed (for reproducible results)
  target = 0.05, # Target value for calibrated metric (default value)
  search_range = c(0.9, 1), # Search range for superiority stopping threshold
  tol = 0.01, # Tolerance range
  dir = -1 # Tolerance range only applies below target
)

# Print result (to check if calibration is successful)
calibrated_binom_trial
#> Trial calibration:
#> * Result: calibration successful
#> * Best x: 0.9830921
#> * Best y: 0.045
#> 
#> Central settings:
#> * Target: 0.05
#> * Tolerance: 0.01 (at or below target, range: 0.04 to 0.05)
#> * Search range: 0.9 to 1
#> * Gaussian process controls:
#> * - resolution: 5000
#> * - kappa: 0.5
#> * - pow: 1.95
#> * - lengthscale: 1 (constant)
#> * - x scaled: yes
#> * Noisy: no
#> * Narrowing: yes
#> 
#> Calibration/simulation details:
#> * Total evaluations: 4 (previous + grid + iterations)
#> * Repetitions: 1000
#> * Calibration time: 1.71 mins
#> * Base random seed: 4131
#> 
#> See 'help("calibrate_trial")' for details.

The calibration is successful (if not, results should not be used, and the calibration settings should be changed and the calibration repeated). The calibrated, constant stopping threshold for superiority is printed with the results (0.9830921) and can be extracted using calibrated_binom_trial$best_x. Using the default calibration functionality, the calibrated, constant stopping threshold for inferiority is symmetrical, i.e., 1 - stopping threshold for superiority (0.0169079). The calibrated trial specification may be extracted using calibrated_binom_trial$best_trial_spec and, if printed, will also include the calibrated stopping thresholds.

Calibration results may be saved and reloaded by using the path argument, to avoid unnecessary repeated simulations.

Summarising results

The results of the simulations using the calibrated trial specification conducted during the calibration procedure may be extracted using calibrated_binom_trial$best_sims. These results can be summarised with several functions. Most of these functions support different ‘selection strategies’ for simulations not ending with superiority, i.e., performance metrics can be calculated assuming different arms would be used in clinical practice if no arm is ultimately superior.

The check_performance() function summarises performance metrics in a tidy data.frame, with uncertainty measures (bootstrapped confidence intervals) if requested. Here, performance metrics are calculated considering the ‘best’ arm (i.e., the one with the highest probability of being overall best) selected in simulations not ending with superiority:

# Calculate performance metrics with uncertainty measures
binom_trial_performance <- check_performance(
  calibrated_binom_trial$best_sims,
  select_strategy = "best",
  uncertainty = TRUE, # Calculate uncertainty measures
  n_boot = 1000, # 1000 bootstrap samples (more typically recommended)
  ci_width = 0.95, # 95% confidence intervals (default)
  boot_seed = "base" # Use same random seed for bootstrapping as for simulations
)

# Print results 
print(binom_trial_performance, digits = 2)
#>                   metric     est err_sd err_mad   lo_ci   hi_ci
#> 1           n_summarised 1000.00   0.00    0.00 1000.00 1000.00
#> 2              size_mean 1757.20  11.26   11.12 1736.20 1779.10
#> 3                size_sd  370.74   9.31    9.34  353.87  389.70
#> 4            size_median 2000.00   0.00    0.00 2000.00 2000.00
#> 5               size_p25 1500.00  47.25    0.00 1400.00 1500.00
#> 6               size_p75 2000.00   0.00    0.00 2000.00 2000.00
#> 7                size_p0  400.00     NA      NA      NA      NA
#> 8              size_p100 2000.00     NA      NA      NA      NA
#> 9            sum_ys_mean  440.16   2.90    2.91  434.50  445.89
#> 10             sum_ys_sd   95.56   2.34    2.41   91.15  100.14
#> 11         sum_ys_median  487.00   1.36    0.74  484.00  489.00
#> 12            sum_ys_p25  366.00   9.63    8.90  353.00  387.00
#> 13            sum_ys_p75  506.00   1.09    1.48  504.00  508.00
#> 14             sum_ys_p0   88.00     NA      NA      NA      NA
#> 15           sum_ys_p100  572.00     NA      NA      NA      NA
#> 16         ratio_ys_mean    0.25   0.00    0.00    0.25    0.25
#> 17           ratio_ys_sd    0.01   0.00    0.00    0.01    0.01
#> 18       ratio_ys_median    0.25   0.00    0.00    0.25    0.25
#> 19          ratio_ys_p25    0.24   0.00    0.00    0.24    0.24
#> 20          ratio_ys_p75    0.26   0.00    0.00    0.26    0.26
#> 21           ratio_ys_p0    0.19     NA      NA      NA      NA
#> 22         ratio_ys_p100    0.30     NA      NA      NA      NA
#> 23       prob_conclusive    0.42   0.02    0.01    0.39    0.45
#> 24         prob_superior    0.04   0.01    0.01    0.03    0.06
#> 25      prob_equivalence    0.38   0.02    0.01    0.35    0.41
#> 26         prob_futility    0.00   0.00    0.00    0.00    0.00
#> 27              prob_max    0.58   0.02    0.01    0.55    0.61
#> 28 prob_select_arm_Arm A    0.35   0.01    0.01    0.32    0.38
#> 29 prob_select_arm_Arm B    0.33   0.01    0.01    0.30    0.36
#> 30 prob_select_arm_Arm C    0.32   0.01    0.01    0.29    0.35
#> 31      prob_select_none    0.00   0.00    0.00    0.00    0.00
#> 32                  rmse    0.02   0.00    0.00    0.02    0.02
#> 33               rmse_te      NA     NA      NA      NA      NA
#> 34                   mae    0.01   0.00    0.00    0.01    0.01
#> 35                mae_te      NA     NA      NA      NA      NA
#> 36                   idp      NA     NA      NA      NA      NA

Similar results in list format (without uncertainty measures) can be obtained using the summary() method (as known from, e.g., regression models inR), which comes with a print() method providing formatted results. If the simulation results are printed directly, this function is called with the default arguments (all arguments, e.g., selection strategies may also be directly supplied to the print() method).

binom_trial_summary <- summary(
  calibrated_binom_trial$best_sims,
  select_strategy = "best"
)

print(binom_trial_summary, digits = 2)
#> Multiple simulation results: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Number of simulations: 1000
#> * Number of simulations summarised: 1000 (all trials)
#> * No common control arm
#> * Selection strategy: best remaining available
#> * Treatment effect compared to: no comparison
#> 
#> Performance metrics (using posterior estimates from final analysis [all patients]):
#> * Sample sizes: mean 1757.20 (SD: 370.74) | median 2000.00 (IQR: 1500.00 to 2000.00) [range: 400.00 to 2000.00]
#> * Total summarised outcomes: mean 440.16 (SD: 95.56) | median 487.00 (IQR: 366.00 to 506.00) [range: 88.00 to 572.00]
#> * Total summarised outcome rates: mean 0.2503 (SD: 0.0109) | median 0.2500 (IQR: 0.2435 to 0.2573) [range: 0.1900 to 0.2950]
#> * Conclusive: 42.50%
#> * Superiority: 4.50%
#> * Equivalence: 38.00%
#> * Futility: 0.00% [not assessed]
#> * Inconclusive at max sample size: 57.50%
#> * Selection probabilities: Arm A: 35.10% | Arm B: 32.90% | Arm C: 32.00% | None: 0.00%
#> * RMSE / MAE: 0.01767 / 0.01164
#> * RMSE / MAE treatment effect: not estimated / not estimated
#> * Ideal design percentage: not estimable
#> 
#> Simulation details:
#> * Simulation time: 36.4 secs
#> * Base random seed: 4131
#> * Credible interval width: 95%
#> * Number of posterior draws: 5000
#> * Estimation method: posterior medians with MAD-SDs

Individual simulation results can be extracted in a tidy data.frame using extract_results():

binom_trial_results <- extract_results(
  calibrated_binom_trial$best_sims,
  select_strategy = "best"
)

nrow(binom_trial_results) # Number of rows/simulations
#> [1] 1000

head(binom_trial_results) # Print the first rows
#>   sim final_n sum_ys ratio_ys final_status superior_arm selected_arm
#> 1   1    2000    478   0.2390  equivalence         <NA>        Arm A
#> 2   2    2000    488   0.2440          max         <NA>        Arm A
#> 3   3    2000    521   0.2605          max         <NA>        Arm C
#> 4   4    2000    500   0.2500          max         <NA>        Arm C
#> 5   5    2000    471   0.2355          max         <NA>        Arm A
#> 6   6    2000    503   0.2515          max         <NA>        Arm B
#>             err       sq_err err_te sq_err_te
#> 1 -0.0134029565 1.796392e-04     NA        NA
#> 2 -0.0118977741 1.415570e-04     NA        NA
#> 3  0.0004940695 2.441046e-07     NA        NA
#> 4 -0.0127647255 1.629382e-04     NA        NA
#> 5 -0.0232813002 5.420189e-04     NA        NA
#> 6 -0.0154278469 2.380185e-04     NA        NA

Finally, the probabilities of different remaining arms and their statuses (with uncertainty) at the last adaptive analysis can be summarised using the check_remaining_arms() function (dropped arms will be shown with an empty text string):

check_remaining_arms(
  calibrated_binom_trial$best_sims,
  ci_width = 0.95 # 95% confidence intervals (default)
)
#>     arm_Arm A   arm_Arm B   arm_Arm C   n  prop         se      lo_ci
#> 1      active      active      active 528 0.528 0.02172556 0.48541868
#> 2             equivalence equivalence 121 0.121 0.02964793 0.06289112
#> 3 equivalence equivalence             120 0.120 0.02966479 0.06185807
#> 4 equivalence             equivalence 108 0.108 0.02986637 0.04946299
#> 5 equivalence equivalence equivalence  31 0.031 0.03112876 0.00000000
#> 6                            superior  22 0.022 0.03127299 0.00000000
#> 7                superior              14 0.014 0.03140064 0.00000000
#> 8    superior                           9 0.009 0.03148015 0.00000000
#>        hi_ci
#> 1 0.57058132
#> 2 0.17910888
#> 3 0.17814193
#> 4 0.16653701
#> 5 0.09201126
#> 6 0.08329394
#> 7 0.07554412
#> 8 0.07069997

Visualising results

Several visualisation functions are included (all are optional, and all require the ggplot2 package installed).

Convergence and stability of one or more performance metrics may be visually assessed using plot_convergence() function:

plot_convergence(
  calibrated_binom_trial$best_sims,
  metrics = c("size mean", "prob_superior", "prob_equivalence"),
  # select_strategy can be specified, but does not affect the chosen metrics
)

Plotting other metrics is possible; see the plot_convergence() documentation. The simulation results may also be split into separate, consecutive batches when assessing convergence, to further assess the stability:

plot_convergence(
  calibrated_binom_trial$best_sims,
  metrics = c("size mean", "prob_superior", "prob_equivalence"),
  n_split = 4
)

The status probabilities for the overall trial according to trial progress can be visualised using the plot_status() function:

plot_status(
  calibrated_binom_trial$best_sims,
  x_value = "total n" # Total number of randomised patients at X-axis
)

Similarly, the status probabilities for one or more specific trial arms can be visualised:

plot_status(
  calibrated_binom_trial$best_sims,
  x_value = "total n",
  arm = NA # NA for all arms or character vector for specific arms
)

Finally, various metrics may be summarised over the progress of one or multiple trial simulations using the plot_history() function, which requires non-sparse results (the sparse argument must be FALSE in calibrate_trials(), run_trials(), or run_trial(), leading to additional results being saved - all other functions work with sparse results). This will be illustrated below.

Use calibrated stopping thresholds in another scenario

The calibrated stopping thresholds (calibrated in a scenario with no between-arm differences) may be used to run simulations with the same overall trial specification, but according to a different scenario (i.e., with between-arm differences present) to assess performance metrics (including the Bayesian analogue of power).

First, a new trial specification is setup using the same settings as before, except for between-arm differences and the calibrated stopping thresholds:

binom_trial_calib_diff <- setup_trial_binom(
  arms = c("Arm A", "Arm B", "Arm C"),
  true_ys = c(0.25, 0.20, 0.30), # Different outcomes in the arms
  min_probs = rep(0.20, 3),
  data_looks = seq(from = 300, to = 2000, by = 100),
  randomised_at_looks = c(seq(from = 400, to = 2000, by = 100), 2000),
  # Stopping rules for inferiority/superiority explicitly defined
  # using the calibration results
  inferiority = 1 - calibrated_binom_trial$best_x,
  superiority = calibrated_binom_trial$best_x,
  equivalence_prob = 0.9,
  equivalence_diff = 0.05,
  soften_power = 0.5
)

Simulations using the trial specification with calibrated stopping thresholds and differences present can then be conducted using the run_trials() function. Here, we specify that non-sparse results will be returned (to illustrate the plot_history() function).

binom_trial_diff_sims <- run_trials(
  binom_trial_calib_diff,
  n_rep = 1000, # 1000 simulations (more generally recommended)
  base_seed = 1234, # Reproducible results
  sparse = FALSE # Return additional results for visualisation
)

Again, simulations may be saved and reloaded using the path argument.

We then calculate performance metrics as above:

check_performance(
  binom_trial_diff_sims,
  select_strategy = "best",
  uncertainty = TRUE,
  n_boot = 1000, # 1000 bootstrap samples (more typically recommended)
  ci_width = 0.95,
  boot_seed = "base"
)
#>                   metric      est err_sd err_mad    lo_ci    hi_ci
#> 1           n_summarised 1000.000  0.000   0.000 1000.000 1000.000
#> 2              size_mean 1245.100 16.618  17.272 1215.185 1277.702
#> 3                size_sd  510.702  7.414   7.436  496.194  525.386
#> 4            size_median 1200.000 46.824   0.000 1200.000 1300.000
#> 5               size_p25  800.000 35.902   0.000  800.000  900.000
#> 6               size_p75 1700.000 46.345   0.000 1600.000 1700.000
#> 7                size_p0  400.000     NA      NA       NA       NA
#> 8              size_p100 2000.000     NA      NA       NA       NA
#> 9            sum_ys_mean  287.066  3.697   3.827  280.241  294.549
#> 10             sum_ys_sd  113.660  1.697   1.650  110.337  116.954
#> 11         sum_ys_median  286.000  5.981   7.413  274.500  295.000
#> 12            sum_ys_p25  194.750  7.147   7.413  180.731  207.756
#> 13            sum_ys_p75  382.250  6.961   7.042  370.000  395.250
#> 14             sum_ys_p0   85.000     NA      NA       NA       NA
#> 15           sum_ys_p100  518.000     NA      NA       NA       NA
#> 16         ratio_ys_mean    0.233  0.000   0.001    0.232    0.234
#> 17           ratio_ys_sd    0.016  0.000   0.000    0.015    0.016
#> 18       ratio_ys_median    0.232  0.001   0.001    0.231    0.233
#> 19          ratio_ys_p25    0.222  0.001   0.001    0.220    0.224
#> 20          ratio_ys_p75    0.243  0.001   0.001    0.241    0.244
#> 21           ratio_ys_p0    0.196     NA      NA       NA       NA
#> 22         ratio_ys_p100    0.298     NA      NA       NA       NA
#> 23       prob_conclusive    0.882  0.011   0.010    0.862    0.902
#> 24         prob_superior    0.719  0.015   0.015    0.690    0.747
#> 25      prob_equivalence    0.163  0.012   0.012    0.139    0.185
#> 26         prob_futility    0.000  0.000   0.000    0.000    0.000
#> 27              prob_max    0.118  0.011   0.010    0.098    0.138
#> 28 prob_select_arm_Arm A    0.033  0.005   0.004    0.023    0.043
#> 29 prob_select_arm_Arm B    0.967  0.005   0.004    0.957    0.977
#> 30 prob_select_arm_Arm C    0.000  0.000   0.000    0.000    0.000
#> 31      prob_select_none    0.000  0.000   0.000    0.000    0.000
#> 32                  rmse    0.020  0.001   0.001    0.019    0.022
#> 33               rmse_te       NA     NA      NA       NA       NA
#> 34                   mae    0.011  0.000   0.000    0.010    0.012
#> 35                mae_te       NA     NA      NA       NA       NA
#> 36                   idp   98.350  0.264   0.222   97.849   98.850

Similarly, overall trial statuses for the scenario with differences are visualised:

plot_status(binom_trial_diff_sims, x_value = "total n")

Statuses for each arm in this scenario are also visualised:

plot_status(binom_trial_diff_sims, x_value = "total n", arm = NA)

We can plot the median and interquartile ranges of allocation probabilities in each arm over time using the plot_history() function (requiring non-sparse results, leading to substantially larger objects and files if saved):

plot_history(
  binom_trial_diff_sims,
  x_value = "total n",
  y_value = "prob"
)

Similarly, the median (interquartile range) number of patients allocated to each arm as the trial progresses can be visualised:

plot_history(
  binom_trial_diff_sims,
  x_value = "total n",
  y_value = "n all"
)

Plotting other metrics is possible; see the plot_history() documentation.

Citation

If you use the package, please consider citing it:

citation(package = "adaptr")
#> 
#> To cite package 'adaptr' in publications use:
#> 
#>   Granholm A, Jensen AKG, Lange T, Kaas-Hansen BS (2022). adaptr: an R
#>   package for simulating and comparing adaptive clinical trials.
#>   Journal of Open Source Software, 7(72), 4284. URL
#>   https://doi.org/10.21105/joss.04284.
#> 
#> A BibTeX entry for LaTeX users is
#> 
#>   @Article{,
#>     title = {{adaptr}: an R package for simulating and comparing adaptive clinical trials},
#>     author = {Anders Granholm and Aksel Karl Georg Jensen and Theis Lange and Benjamin Skov Kaas-Hansen},
#>     journal = {Journal of Open Source Software},
#>     year = {2022},
#>     volume = {7},
#>     number = {72},
#>     pages = {4284},
#>     url = {https://doi.org/10.21105/joss.04284},
#>     doi = {10.21105/joss.04284},
#>   }