Response-Adaptive Randomization: Stage 1 and Stage 2

Overview

This example demonstrates a simple response-adaptive randomization design with two stages.

• Stage 1: The first 100 patients are randomized 1:1 to Treatment or Control.
• Observed response rates are calculated after Stage 1.
• The Stage 2 allocation probability is updated according to the observed response rates.
• Stage 2: The next 100 patients are randomized using the updated allocation probability.
• A randomization dataset is created for possible use in an EDC or IRT/RTSM workflow.

The adaptive allocation probability is defined as:

\[ P(\text{Treatment}) = \frac{\hat p_{\text{Treatment}}} {\hat p_{\text{Treatment}}+\hat p_{\text{Control}}} \]

To avoid extreme allocation, the probability is restricted to:

\[ 0.30 \leq P(\text{Treatment}) \leq 0.70 \]

1. Set Seed

set.seed(123)

2. Stage 1: First 100 Patients

The first 100 patients are randomized equally between Treatment and Control.

Treatment coding is:

• 1 = Treatment
• 0 = Control

n_stage1 <- 100

trt1 <- rbinom(
  n_stage1,
  1,
  0.50
)

3. Simulate Stage 1 Responses

For illustration, assume the true response rates are:

\[ P(\text{Response}|\text{Treatment}) = 0.70 \]

and

\[ P(\text{Response}|\text{Control}) = 0.50 \]

response1 <- rbinom(
  n_stage1,
  1,
  prob = ifelse(
    trt1 == 1,
    0.70,
    0.50
  )
)

4. Calculate Observed Response Rates

rate_trt <- mean(
  response1[trt1 == 1]
)

rate_ctl <- mean(
  response1[trt1 == 0]
)

cat(
  "Observed Treatment response rate =",
  round(rate_trt, 3),
  "\n"
)

## Observed Treatment response rate = 0.745

cat(
  "Observed Control response rate =",
  round(rate_ctl, 3),
  "\n"
)

## Observed Control response rate = 0.528

5. Calculate Adaptive Allocation Probability

The Stage 2 Treatment allocation probability is calculated as:

\[ P(\text{Treatment}) = \frac{\hat p_T} {\hat p_T+\hat p_C} \]

where:

• (p_T) is the observed Treatment response rate;
• (p_C) is the observed Control response rate.

prob_trt_next <- rate_trt /
  (rate_trt + rate_ctl)

To avoid extreme allocation, restrict the Treatment probability to the interval 0.30 to 0.70.

prob_trt_next <- max(
  0.30,
  min(
    0.70,
    prob_trt_next
  )
)

prob_ctl_next <- 1 -
  prob_trt_next

Display the Stage 2 allocation probabilities.

cat(
  "Stage 2 Treatment allocation probability =",
  round(prob_trt_next, 3),
  "\n"
)

## Stage 2 Treatment allocation probability = 0.585

cat(
  "Stage 2 Control allocation probability =",
  round(prob_ctl_next, 3),
  "\n"
)

## Stage 2 Control allocation probability = 0.415

6. Stage 2: Randomize the Next 100 Patients

n_stage2 <- 100

7. Create Subject IDs

The next 100 patients are assigned Subject IDs from SUBJ-101 through SUBJ-200.

subject_id <- sprintf(
  "SUBJ-%03d",
  101:200
)

8. Create Randomization Numbers

randomization_number <- sprintf(
  "R%04d",
  101:200
)

9. Generate Random Numbers

For each Stage 2 patient, generate one random number from a Uniform(0,1) distribution.

\[ U_i \sim Uniform(0,1) \]

u_random <- runif(
  n_stage2
)

10. Assign Treatment

The randomization rule is:

\[ \text{Treatment}_i = \begin{cases} \text{Treatment}, & U_i \leq P(\text{Treatment}) \ \text{Control}, & U_i > P(\text{Treatment}) \end{cases} \]

trt2 <- ifelse(
  u_random <= prob_trt_next,
  1,
  0
)

11. Create Treatment Labels

treatment_assignment <- ifelse(
  trt2 == 1,
  "Treatment",
  "Control"
)

12. Create Blinded Treatment Codes

For illustration:

• A = Treatment
• B = Control

In a real double-blind trial, the actual mapping between blinded code and treatment would normally be restricted.

treatment_code <- ifelse(
  trt2 == 1,
  "A",
  "B"
)

13. Create the Stage 2 Randomization Dataset

stage2_randomization <- data.frame(
  
  Subject_ID =
    subject_id,
  
  Randomization_Number =
    randomization_number,
  
  Stage =
    2,
  
  Allocation_Probability_Treatment =
    rep(
      prob_trt_next,
      n_stage2
    ),
  
  Allocation_Probability_Control =
    rep(
      prob_ctl_next,
      n_stage2
    ),
  
  Random_Number =
    round(
      u_random,
      6
    ),
  
  Treatment_Code =
    treatment_code,
  
  Treatment_Assignment =
    treatment_assignment,
  
  stringsAsFactors = FALSE
)

14. Display the First 10 Randomized Patients

head(
  stage2_randomization,
  10
)

##    Subject_ID Randomization_Number Stage Allocation_Probability_Treatment
## 1    SUBJ-101                R0101     2                         0.584989
## 2    SUBJ-102                R0102     2                         0.584989
## 3    SUBJ-103                R0103     2                         0.584989
## 4    SUBJ-104                R0104     2                         0.584989
## 5    SUBJ-105                R0105     2                         0.584989
## 6    SUBJ-106                R0106     2                         0.584989
## 7    SUBJ-107                R0107     2                         0.584989
## 8    SUBJ-108                R0108     2                         0.584989
## 9    SUBJ-109                R0109     2                         0.584989
## 10   SUBJ-110                R0110     2                         0.584989
##    Allocation_Probability_Control Random_Number Treatment_Code
## 1                        0.415011      0.238726              A
## 2                        0.415011      0.962359              B
## 3                        0.415011      0.601366              B
## 4                        0.415011      0.515030              A
## 5                        0.415011      0.402573              A
## 6                        0.415011      0.880247              B
## 7                        0.415011      0.364092              A
## 8                        0.415011      0.288239              A
## 9                        0.415011      0.170645              A
## 10                       0.415011      0.172172              A
##    Treatment_Assignment
## 1             Treatment
## 2               Control
## 3               Control
## 4             Treatment
## 5             Treatment
## 6               Control
## 7             Treatment
## 8             Treatment
## 9             Treatment
## 10            Treatment

15. Check Stage 2 Treatment Allocation

table(
  stage2_randomization$
    Treatment_Assignment
)

## 
##   Control Treatment 
##        40        60

16. Calculate Observed Stage 2 Allocation Proportions

prop.table(
  table(
    stage2_randomization$
      Treatment_Assignment
  )
)

## 
##   Control Treatment 
##       0.4       0.6

Interpretation

The overall adaptive randomization process is:

\[ \text{Stage 1: 1:1 Randomization} \]

\[ \downarrow \]

\[ \text{Observe Stage 1 Responses} \]

\[ \downarrow \]

\[ \text{Calculate New Allocation Probability} \]

\[ \downarrow \]

\[ \text{Stage 2 Adaptive Randomization} \]

For example, if the observed Stage 1 response rates are:

\[ \hat p_T = 0.70 \]

and

\[ \hat p_C = 0.50 \]

then:

\[ P(\text{Treatment}) = \frac{0.70}{0.70+0.50} = 0.583 \]

Therefore, approximately 58.3% of Stage 2 patients would be randomized to Treatment and 41.7% to Control.

Importantly, the exact number of patients assigned to each group will not necessarily equal these proportions because each patient is still randomized individually.

Practical Clinical Trial Note

This example uses one adaptation after the first 100 patients.

Therefore, all Stage 2 patients use the same updated allocation probability.

The design is:

\[ 100\text{ patients} \rightarrow \text{Interim Analysis} \rightarrow \text{Update Allocation Probability} \rightarrow 100\text{ additional patients} \]

This is operationally simpler than updating the allocation probability after every individual patient.

In an actual clinical trial, the randomization algorithm, allocation restrictions, timing of adaptation, treatment-code mapping, and IRT/RTSM implementation should be prespecified and validated.