# Basic examples

This vignette provides examples of various trial specifications using different combinations of settings, including various randomisation strategies including fixed randomisation, response-adaptive randomisation (RAR), and combinations.

The general-purpose function for specifying a trial is setup_trial(), but because trials with binary, binomially distributed and continuous, normally distributed outcomes are so common, the package comes with two convenience functions for specifying such trial designs: setup_trial_binom() and setup_trial_norm().

To keep things simple, this vignette uses only the setup_trial_binom() function and focuses on settings that apply to trial designs regardless of outcome type. The code is heavily annotated, but comments focus on settings not touched on earlier in the vignette (e.g. we do not keep annotating the arm and true_ys arguments).

For a general overview of how to use the adaptr package, please see vignette("Overview", "adaptr").

An advanced example on how to specify a trial design with setup_trial(), including the use of custom functions for generating outcomes and yielding posterior draws, is provided in vignette("Advanced-example", "adaptr").

library(adaptr)

## Trial designs without a common control arm

In this section, several examples for trials without a common control arm are provided. General settings applicable for all trial designs (including both trial specifications with and without a common control arm) are covered in this section.

### Example 1: general settings

setup_trial_binom(
# Four arms
arms = c("A", "B", "C", "D"),
# Set true outcomes (in this example event probabilities) for all arms
true_ys = c(0.3, 0.35, 0.31, 0.27), # 30%, 34%, 31% and 27%, respectively

# Set starting allocation probabilities
# Defaults to equal allocation if not specified
start_probs = c(0.3, 0.3, 0.2, 0.2),
# Set fixed allocation probability for first arm
# NA corresponds to no limits for specific arms
# Default (NULL) corresponds to no limits for all arms
fixed_probs = c(0.3, NA, NA, NA),
# Set minimum and maximum probability limits for some arms
# NA corresponds to no limits for specific arms
# Default (NULL) corresponds to no limits for all arms
# Must be NA for arms with fixed_probs (first arm in this example)
# sum(fixed_probs) + sum(min_probs) must not exceed 1
# sum(fixed_probs) + sum(max_probs) may be greater than 1, and must be at least
# 1 if specified for all arms
min_probs = c(NA, 0.2, NA, NA),
max_probs = c(NA, 0.7, NA, NA),

# Set looks - alternatively, specify both max_n AND look_after_every
data_looks = seq(from = 300, to = 1000, by = 100),

# No common control arm (as default, but explicitly specified in this example)
control = NULL,

# Set inferiority/superiority thresholds (different values than the defaults)
inferiority = 0.025,
superiority = 0.975,

# Define that the outcome is desirable (as opposed to the default setting)
highest_is_best = TRUE,

# No softening (the default setting, but made explicit here)
soften_power = 1,

# Use different simulation/summary settings than default
cri_width = 0.89, # 89% credible intervals
n_draws = 1000, # Only 1000 posterior draws in each arm
robust = TRUE, # Summarise posteriors using medians/MAD-SDs (as default)

# Trial description (used by print methods)
description = "example trial specification 1"
)
#> Trial specification: example trial specification 1
#> * Desirable outcome
#> * No common control arm
#> * Best arm: B
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.30         0.3         0.3        NA        NA
#>     B    0.35         0.3          NA       0.2       0.7
#>     C    0.31         0.2          NA        NA        NA
#>     D    0.27         0.2          NA        NA        NA
#>
#> Maximum sample size: 1000
#> Maximum number of data looks: 8
#> Planned data looks after:  300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  300, 400, 500, 600, 700, 800, 900, 1000
#>
#> Superiority threshold: 0.975 (all analyses)
#> Inferiority threshold: 0.025 (all analyses)
#> No equivalence threshold
#> No futility threshold (not relevant - no common control)
#> Soften power for all analyses: 1 (no softening)

### Example 2: equivalence testing, decreasing softening

• No common control arm
• Equivalence testing
• Different softening powers (decreasing softening as the trial progresses)
• Default settings for many unspecified arguments
setup_trial_binom(
# Specify arms and true outcome probabilities (undesirable outcome as default)
arms = c("A", "B", "C", "D"),
true_ys = c(0.2, 0.22, 0.24, 0.18),

# Specify adaptive analysis looks using max_n and look_after_every
# max_n does not need to be a multiple of look_after_every - a final look
# will be conducted at max_n regardless
max_n = 1250, # Maximum 1250 patients
look_after_every = 100, # Look after every 100 patients

# Assess equivalence between all arms: stop if >90 % probability that the
# absolute difference between the best and worst arms is < 5 %-points
# Note: equivalence_only_first must be NULL (default) in designs without a
# common control arm (such as this trial)
equivalence_prob = 0.9,
equivalence_diff = 0.05,

# Different softening powers at each look (13 possible looks in total)
# Starts at 0 (softens all allocation probabilities to be equal) and ends at
# 1 (no softening) for the final possible look in the trial
soften_power = seq(from = 0, to = 1, length.out = 13)
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * No common control arm
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20        0.25          NA        NA        NA
#>     B    0.22        0.25          NA        NA        NA
#>     C    0.24        0.25          NA        NA        NA
#>     D    0.18        0.25          NA        NA        NA
#>
#> Maximum sample size: 1250
#> Maximum number of data looks: 13
#> Planned looks after every 100
#>  patients have reached follow-up until final look after 1250 patients
#> Number of patients randomised at each look:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1250
#>
#> 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 each consequtive analysis: 0, 0.083, 0.167, 0.25, 0.333, 0.417, 0.5, 0.583, 0.667, 0.75, 0.833, 0.917, 1

## Trial designs with a common control arm

In this section, several examples for trials with a common control arm are provided and focus mostly on options specific to trial designs with a common control arm.

### Example 3: common control and sqrt-based fixed allocation

• A common control arm
• square-root-transformation-based fixed allocation probabilities (see description in setup_trial())
• Assessment of both equivalence and futility compared to the initial control only (not assessed for superior arms that become subsequent controls)
setup_trial_binom(
arms = c("A", "B", "C", "D"),
# Specify control arm
control = "A",

true_ys = c(0.2, 0.22, 0.24, 0.18),

data_looks = seq(from = 100, to = 1000, by = 100),

# Fixed, square-root-transformation-based allocation throughout
control_prob_fixed = "sqrt-based fixed",

# Assess equivalence: drop non-control arms if > 90% probability that they
# are equivalent to the common control, defined as an absolute difference of
# < 3 %-points
equivalence_prob = 0.9,
equivalence_diff = 0.03,
# Only assess against the initial control (i.e., not assessed if an arm is
# declared superior to the initial control and becomes the new control)
equivalence_only_first = TRUE,

# Assess futility: drop non-control arms if > 80% probability that they are
# < 10 %-points better (in this case lower because outcome is undesirable in
# this example) compared to the common control
futility_prob = 0.8,
futility_diff = 0.1,
# Only assessed for the initial control, as described above
futility_only_first = TRUE
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Common control arm: A
#> * Control arm probability fixed at 0.366 (for 4 arms), 0.414 (for 3 arms), 0.5 (for 2 arms)
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20       0.366       0.366        NA        NA
#>     B    0.22       0.211       0.211        NA        NA
#>     C    0.24       0.211       0.211        NA        NA
#>     D    0.18       0.211       0.211        NA        NA
#>
#> Maximum sample size: 1000
#> Maximum number of data looks: 10
#> Planned data looks after:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
#>
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> Equivalence threshold: 0.9 (all analyses) (only checked for first control)
#> Absolute equivalence difference: 0.03
#> Futility threshold: 0.8 (all analyses) (only checked for first control)
#> Absolute futility difference (in beneficial direction): 0.1
#> Soften power for all analyses: 1 (no softening - all arms fixed)

### Example 4: sqrt-based initial allocation and restricted RAR

• Square-root-transformation-based initial allocation probabilities
• Square-root-transformation-based allocation to the control arm (including subsequent controls, if a non-control arm is declared superior to the initial control)
• Restricted response-adaptive randomisation to the non-control arms
setup_trial_binom(
arms = c("A", "B", "C", "D"),
control = "A",

true_ys = c(0.2, 0.22, 0.24, 0.18),

data_looks = seq(from = 100, to = 1000, by = 100),

# Square-root-transformation-based control arm allocation including for
# subsequent controls and initial equal allocation to the non-control arms,
control_prob_fixed = "sqrt-based",

# Minimum probabilities of 20% for non-control arms, must be NA for the
# control arm with fixed allocation probability
# Limits are ignored for arms that become subsequent controls
min_probs = c(NA, 0.2, 0.2, 0.2),

# Constant softening of 0.5 (= square-root transformation)
soften_power = 0.5
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Common control arm: A
#> * Control arm probability fixed at 0.366 (for 4 arms), 0.414 (for 3 arms), 0.5 (for 2 arms)
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20       0.366       0.366        NA        NA
#>     B    0.22       0.211          NA       0.2        NA
#>     C    0.24       0.211          NA       0.2        NA
#>     D    0.18       0.211          NA       0.2        NA
#>
#> Maximum sample size: 1000
#> Maximum number of data looks: 10
#> Planned data looks after:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
#>
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> No equivalence threshold
#> No futility threshold
#> Soften power for all analyses: 0.5

### Example 5: sqrt-based allocation only to initial control arm

This example is similar to that above (with different restriction settings), but only use square-root-transformation-based allocation probabilities to the initial control arm. Hence, this will not apply if another arm is declared superior and becomes the new control.

setup_trial_binom(
arms = c("A", "B", "C", "D"),
control = "A",

true_ys = c(0.2, 0.22, 0.24, 0.18),

data_looks = seq(from = 100, to = 1000, by = 100),

# Square-root-transformation-based control arm allocation for the initial
# control only and initial equal allocation to the non-control arms, followed
control_prob_fixed = "sqrt-based start",

# Minimum probabilities of 20% for all non-control arms
# - must be NA for the initial control arm with fixed allocation probability
min_probs = c(NA, 0.2, 0.2, 0.2),
# Maximum probabilities of 65% for all non-control arms
# - must be NA for the initial control arm with fixed allocation probability
max_probs = c(NA, 0.65, 0.65, 0.65),

soften_power = 0.75
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Common control arm: A
#> * Control arm probability fixed at 0.366
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20       0.366       0.366        NA        NA
#>     B    0.22       0.211          NA       0.2      0.65
#>     C    0.24       0.211          NA       0.2      0.65
#>     D    0.18       0.211          NA       0.2      0.65
#>
#> Maximum sample size: 1000
#> Maximum number of data looks: 10
#> Planned data looks after:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
#>
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> No equivalence threshold
#> No futility threshold
#> Soften power for all analyses: 0.75

### Example 6: restricted RAR, matched control-arm allocation

• Control-arm allocation probability matched to that of the highest non-control arm (with re-scaling as necessary)
• Applies to both the initial and subsequent control arms
setup_trial_binom(
arms = c("A", "B", "C", "D"),
control = "A",

true_ys = c(0.2, 0.22, 0.24, 0.18),

data_looks = seq(from = 100, to = 1000, by = 100),

# Specify starting probabilities
# When "match" is specified below in control_prob_fixed, the initial control
# arm's initial allocation probability must match the highest initial
# non-control arm allocation probability
start_probs = c(0.3, 0.3, 0.2, 0.2),

control_prob_fixed = "match",

# - these are applied AFTER "matching" when calculating new allocation
#   probabilities
# - min_probs must be NA for the initial control arm when using matching
min_probs = c(NA, 0.2, 0.2, 0.2),

soften_power = 0.7
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Common control arm: A
#> * Control arm probability matched to best non-control arm
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20         0.3          NA        NA        NA
#>     B    0.22         0.3          NA       0.2        NA
#>     C    0.24         0.2          NA       0.2        NA
#>     D    0.18         0.2          NA       0.2        NA
#>
#> Maximum sample size: 1000
#> Maximum number of data looks: 10
#> Planned data looks after:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
#>
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> No equivalence threshold
#> No futility threshold
#> Soften power for all analyses: 0.7

### Example 7: follow-up (or data collection) lag

This example uses the randomised_at_looks argument to specify follow-up (or data collection) lag. In real use cases, this should usually be considered, as this may affect the relative performance of different trial designs and the extent to which the ‘final’ results after all patients have reached follow-up and are analysed may differ from the results when from the adaptive analyses with some randomised patients not included due to outcome data not being available yet.

setup_trial_binom(
arms = c("A", "B", "C", "D"),
control = "A",
true_ys = c(0.2, 0.22, 0.24, 0.18),

# Analyses conducted every time 100 patients have follow-up data available
data_looks = seq(from = 100, to = 1000, by = 100),
# Specify the number of patients randomised at each look - in this case, 200
# more patients are randomised than the number of patients that
# have follow-up data available at each look
randomised_at_looks = seq(from = 300, to = 1200, by = 100)
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Common control arm: A
#>
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20        0.25          NA        NA        NA
#>     B    0.22        0.25          NA        NA        NA
#>     C    0.24        0.25          NA        NA        NA
#>     D    0.18        0.25          NA        NA        NA
#>
#> Maximum sample size: 1200
#> Maximum number of data looks: 10
#> Planned data looks after:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200
#>
#> Superiority threshold: 0.99 (all analyses)
#> Inferiority threshold: 0.01 (all analyses)
#> No equivalence threshold
#> No futility threshold
#> Soften power for all analyses: 1 (no softening)

### Example 8: different probability thresholds over time

In this example, we specify different probability thresholds for superiority and inferiority stopping rules at different adaptive analyses. Varying probability thresholds may similarly be specified for stopping rules for equivalence and futility. Importantly, all probability thresholds must be specified such that each subsequent threshold is never stricter than the previous threshold

setup_trial_binom(
arms = c("A", "B", "C", "D"),
control = "A",
true_ys = c(0.2, 0.22, 0.24, 0.18),

# Analyses conducted every time 100 patients have follow-up data available
data_looks = seq(from = 100, to = 1000, by = 100),

# Specify varying inferiority/superiority thresholds
# When specifying varying thresholds, the number of thresholds must match
# the number of analyses, and thresholds may never be stricter than the
# threshold used in the previous analysis
# Superiority threshold decreasing from 0.99 to 0.95 during the first five
# analyses, and remains stationary at 0.95 after that
superiority = c(seq(from = 0.99, to = 0.95, by = -0.01), rep(0.95, 5)),
# Similarly for inferiority thresholds, but in the opposite direction
inferiority = c(seq(from = 0.01, to = 0.05, by = 0.01), rep(0.05, 5)),
)
#> Trial specification: generic binomially distributed outcome trial
#> * Undesirable outcome
#> * Common control arm: A
#>
#> * Best arm: D
#>
#> Arms, true outcomes, starting allocation probabilities
#> and allocation probability limits:
#>  arms true_ys start_probs fixed_probs min_probs max_probs
#>     A    0.20        0.25          NA        NA        NA
#>     B    0.22        0.25          NA        NA        NA
#>     C    0.24        0.25          NA        NA        NA
#>     D    0.18        0.25          NA        NA        NA
#>
#> Maximum sample size: 1000
#> Maximum number of data looks: 10
#> Planned data looks after:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 patients have reached follow-up
#> Number of patients randomised at each look:  100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
#>
#> Superiority thresholds:
#> 0.99, 0.98, 0.97, 0.96, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95
#> Inferiority thresholds:
#> 0.01, 0.02, 0.03, 0.04, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05
#> No equivalence threshold
#> No futility threshold
#> Soften power for all analyses: 1 (no softening)