Dropping a Treatment Arm in A Clinical Trial: "Re-distributing Remaining
Sample Size" Scheme\*

**Yi Tsong, H.M. James Hung,**

**Sue-Jane Wang, Lu Cui**

Office of Biometrics/CDER/FDA

\*Funded by CDER Reviewer Science Research Fund RSR 99/00-008

\*The views expressed in this article are of the authors' professional
opinions and do not represent the position of the U.S. Food and Drug
Administration

I.  ***INTRODUCTION***

**Phase III trials with multiple treatment arms (e.g. low dose and high
dose and control).**

**Study Objective - show efficacy of both or either one of the two
treatment arms.**

**Group sequential analysis plan.**

**Research Goal --**

**Evaluate adaptive designs of dropping one arm at interim look when
lack of potential to show efficacy**

**Earlier Work:**

**Tsong, Hung, Wang and Cui (JSM 1997)**

Research Goal: Evaluate the adaptive design of reallocate unused type I
error rate of the dropped arm based on result at interim look

Results:

+-------------------+----------------+------------+------------------+
| #### *Anal        | ####           | #### *α*   | #### *Power*     |
| ytical Procedure* |  *Difference b |            |                  |
|                   | etween 2 arms* |            |                  |
+-------------------+----------------+------------+------------------+
| ##                | #### NO        | ###        | #### Improved    |
| ## *Single Stage* |                | # Inflated |                  |
|                   |                |            |                  |
| ####              |                |            |                  |
+-------------------+----------------+------------+------------------+
|                   | #### SMALL     | ###        | #### Improved    |
|                   |                | # Inflated |                  |
+-------------------+----------------+------------+------------------+
|                   | #### LARGE     | ###        | #### Improved    |
|                   |                | # Inflated |                  |
+-------------------+----------------+------------+------------------+
| #### *2-Stage*    | #### NO        | ###        | #### Same        |
|                   |                | # Inflated |                  |
+-------------------+----------------+------------+------------------+
|                   | #### SMALL     | ###        | #### Improved    |
|                   |                | # Inflated |                  |
+-------------------+----------------+------------+------------------+
|                   | #### LARGE     | ###        | ##               |
|                   |                | # Inflated | ## More Improved |
+-------------------+----------------+------------+------------------+

#### 

Yr 2000 Research Goal:

Evaluate the adaptive design of reassign unused sample size of the
dropped arm

**&**

**Minimize the error of dropping an efficacy arm**

**II. ANALYTICAL PROCEDURE**

**μ~0~, μ~1~, μ~2~ = mean of reference, test arm 1, test arm 2,**

**H~01~: ∆~1~ = 0 vs. H~a1~: ∆~1~ \> 0**

**H~02~: ∆~2~ = 0 vs. H~a2~: ∆~2~ \> 0 with ∆~i~ = μ~i~ -μ~0~.**

**At each interim analysis, test H~0i~ and terminate [the
]{.underline}**

**[i-th test treatment arm]{.underline} once H~0i~ is rejected.**

-   **Use 0.54α for testing H~01~ and H~02~.**

> **Corr(Z~1~, Z~2~) = 0.5, Prob(Z~1~ \> C or Z~2~ \> C) = 0.25**
>
> **⇒ C = 2.21 =z~0.0135~**

-   **α~ik,0.54α~ = α~k,0.54α~ = portion of 0.54α**

**spent on the k-th interim look.**

-   **T~ik~ = (Y~ik~ - Y~0k~)/v~ik~,**

**v~ik~ - standard error of (Y~ik~ - Y~0k~).**

-   **H~0i~ is rejected if T~ik~ \>C~k,0.54α~ for either i=1 or 2.**

**III. SCHEMES TO DROP A TREATMENT ARM**

# 

# Conditional Power 

**CP~ik~(θ) = P\[T~i,K+1~\> C~K+1~\| T~ik~, θ\] ,**

**θ = the value assumed for ∆~i~**

**[Drop the i-th test treatment arm]{.underline} at the k-th look if**

**CP~ik~(∆~ik~) \< CP~ik~(εδ~a~)**

**where ∆~ik~ is the estimate of difference at the k-th look, δ~a~ is
the target difference specified in design**

**IV. SCREENING TO PERFORM "DROP-ARM" TEST**

**Two options for performing "drop-arm" test,**

**1. At any interim look**

2.  **Only if**

```{=html}
<!-- -->
```
a.  **Large difference between the two statistics**

> **\|T~1k~ - T~2k~ \| \> δ~T,~ δ~T~ = pre-specified difference**

**and**

**b. Small conditional powers**

> **CP~1k~(∆~1k~) + CP~2k~(∆~2k~) \< δ~CP~, δ~CP~ = pre-specified**

V.  **SCHEMES TO DEAL WITH UNUSED SAMPLE SIZE AFTER DROP/TERMINATE AN
    ARM**

## No re-distributing (NR)

**Re-distributing the unused sample size (RD)**

**if drop arm 1 at the k\*-the look**

> **T~2k~ for k ≤ k\***

**T\*~2k~ =**

> **√(k\*/k) T~2k\*~ + √(1-k\*/k) W~2k,~ k\>k\***
>
> **(if change critical limits)**
>
> **or**

**√(nk\*/m~k~) T~2k\*~ + √(1-nk\*/m~k~) W~2k,~ k\>k\***

> **(if use same critical limits)**
>
> **m~k~ = (3k/2 --k\*/2), T~ik~ ∼ N(√(nk) ∆~i~, 1),**
>
> **W~2k~ ∼ N(√(m~k~ -- nk\*)∆~2~, 1)**

### VI. SIMULATION PLAN

-   **3 interim looks (0.25/0.50/0.75) and final**

2.  **interim looks (0.50/0.75) and a final test**

-   **One sided overall type I error rate =0.025**

-   **O'Brien-Fleming α spending functions**

-   **Single stage testing procedure**

-   **RD (sample size reallocation) vs. ND (no reallocation)**

-   **N/group = 60/156**

-   **Simulation size = 100,000**

-   **δ~a~ = 0.3 (effect size) for CPA computation**

-   **Screen for drop (δ~T~ = 2/1.5, δ~CP~ = ½ δ~T~) vs. no screen**

######## Sample size re-distribution scheme 

  -------------- ------------- ---------------- --------------- -------- ------- ---------
  ***Info Time   ***Critical   ***Unused S.     ***Cumulative                    
  When Drops One Value***      Size To Be       Sample Size                      
  Arm***                       Distributed***   (Remaining Arm)                  
                                                At Inf. Time***                  

                                                .25             .50      .75     1.0

  ***.25***      4.852         3n/4             n/4             5n/8     n       11n/8

  ***.50***      3.345         2n/4             n/4             2n/4     7n/8    10n/8

  ***.75***      2.671         n/4              n/4             2n/4     3n/4    9n/8

  ***1.0***      2.282         0                n/4             2n/4     3n/4    n
  -------------- ------------- ---------------- --------------- -------- ------- ---------

**Adaptive scheme with sample size change**

######### Treatment effect configuration

  ------------------------------- ---------------------------------------
  ***Max(∆~i~) = Max(μ~i~         ***Means of the three arms***
  -μ~0~)***                       

  ***0.0***                       N(0,0), N(0,0), N(0,0)

  ***0.3***                       N(0,1), N(0,1), N(0.3,1)

                                  N(,1), N(0.15,1), N(0.3,1)

                                  N(0,1), N(0.3,1), N(0.3,1)

  ***0.5***                       N(0.1), N(0,1), N(0.5,1)

                                  N(0.1), N(0.25,1), N(0.5,1)

                                  N(0,1), N(0.5,1), N(0.5,1)

  ***0.7***                       N(0,1), N(0,1), N(0.7,1)

                                  N(0,1), N(0.35.1), N(0.7,1)

                                  N(0,1), N(0.7,1) N(0.7,1)
  ------------------------------- ---------------------------------------

#  

# VII. SIMULATION RESULTS

#### Type I Error Rate

  ----------------- ------------ ------------ ------------- ------------- --------- ------------- ---------------- --
  ***Interim        ***ε***      ***δ~T~***                 ***RA/NA***                           ***Type I        
  looks***                                                                                        error/arm***     

  ***¼, ½, ¾, 1***  Reg. Seq                                                                      0.012            

                    0.2          no screen                  ND                                    0.010            

                                                            RD                                    0.0102           

                                 1.5                        ND                                    0.012            

                                                            RD                                    0.012            

                                 2.0                        ND                                    0.012            

                                                            RD                                    0.012            

                    0.1          no screen                  ND                                    0.011            

                                                            RD                                    0.011            

                                 1.5                        ND                                    0.012            

                                                            RD                                    0.012            

                                 2.0                        ND                                    0.012            

                                                            RD                                    0.012            

  ***½, ¾, 1***     ***Reg.                                                         ***0.012***                    
                    Seq***                                                                                         

                    0.2                       no screen                   ND        0.009-0.011                    

                                                                          RD        0.010-0.011                    

                                              1.5                         ND        0.012                          

                                                                          RD        0.012                          

                                              2.0                         ND        0.012                          

                                                                          RD        0.012                          

                    0.1                       no screen                   ND        0.010-0.011                    

                                                                          RD        0.010-0.011                    

                                              1.5                         ND        0.012                          

                                                                          RD        0.012                          

                                              2.0                         ND        0.012                          

                                                                          RD        0.012                          
  ----------------- ------------ ------------ ------------- ------------- --------- ------------- ---------------- --

#### Power

+-------+----+---------+---+------+-------+-------+-------+-------+
| ***   | *  | ***size | * | **   | *     | *     |       | ***Pr |
| Inter | ** | /arm*** | * | *δ~T | **RA/ | **Pow |       | ob*** |
| im*** | μ~ |         | * | ~*** | NA*** | er*** |       |       |
|       | 0~ |         | ε |      |       |       |       | *     |
|       | /* |         | * |      |       |       |       | **(dr |
|       | ** |         | * |      |       |       |       | op*** |
|       |    |         | * |      |       |       |       |       |
|       | *  |         |   |      |       |       |       | *     |
|       | ** |         |   |      |       |       |       | **arm |
|       | μ~ |         |   |      |       |       |       | 2)*** |
|       | 1~ |         |   |      |       |       |       |       |
|       | /* |         |   |      |       |       |       |       |
|       | ** |         |   |      |       |       |       |       |
|       |    |         |   |      |       |       |       |       |
|       | ** |         |   |      |       |       |       |       |
|       | *μ |         |   |      |       |       |       |       |
|       | ~2 |         |   |      |       |       |       |       |
|       | ~* |         |   |      |       |       |       |       |
|       | ** |         |   |      |       |       |       |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   |      |       | Arm 1 | Arm 2 |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
| ***½, | 0  | 60      | 0 | reg. |       | 0.013 | 0.270 |       |
| ¾***  | .0 |         | . | Seq  |       |       |       |       |
|       | /0 |         | 1 |      |       |       |       |       |
|       | .0 |         |   |      |       |       |       |       |
|       | /0 |         |   |      |       |       |       |       |
|       | .3 |         |   |      |       |       |       |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   | No   | ND    | 0.006 | 0.236 | 0.123 |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   |      | RD    | 0.006 | 0.318 |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   | 1.75 | ND    | 0.013 | 0.270 | 0.001 |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   |      | RD    | 0.013 | 0.299 |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   | 2.0  | ND    | 0.013 | 0.270 | 0.001 |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   |      | RD    | 0.013 | 0.286 |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    | 156     | 0 | Reg. |       | 0.012 | 0.650 |       |
|       |    |         | . | Seq  |       |       |       |       |
|       |    |         | 1 |      |       |       |       |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   | 1.75 | ND    | 0.011 | 0.650 | 0.000 |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   |      | RD    | 0.012 | 0.696 |       |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   | 2.0  | ND    | 0.012 | 0.650 | 0.000 |
+-------+----+---------+---+------+-------+-------+-------+-------+
|       |    |         |   |      | RD    | 0.012 | 0.696 |       |
+-------+----+---------+---+------+-------+-------+-------+-------+

+-------+----+---------+----+-----+--------+------+-------+-------+
| ***   | *  | ***size | *  | *   | ***RA  | ***  |       | ***Pr |
| Inter | ** | /arm*** | ** | **δ | /NA*** | Powe |       | ob*** |
| im*** | μ~ |         | ε* | ~T~ |        | r*** |       |       |
|       | 0~ |         | ** | *** |        |      |       | ***   |
|       | /* |         |    |     |        |      |       | (Drop |
|       | ** |         |    |     |        |      |       | arm   |
|       |    |         |    |     |        |      |       | 2)*** |
|       | *  |         |    |     |        |      |       |       |
|       | ** |         |    |     |        |      |       |       |
|       | μ~ |         |    |     |        |      |       |       |
|       | 1~ |         |    |     |        |      |       |       |
|       | /* |         |    |     |        |      |       |       |
|       | ** |         |    |     |        |      |       |       |
|       |    |         |    |     |        |      |       |       |
|       | ** |         |    |     |        |      |       |       |
|       | *μ |         |    |     |        |      |       |       |
|       | ~2 |         |    |     |        |      |       |       |
|       | ~* |         |    |     |        |      |       |       |
|       | ** |         |    |     |        |      |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    |     |        | Arm  | Arm 2 |       |
|       |    |         |    |     |        | 1    |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
| ***½, | 0  | 60      | 0  | r   |        | 0    | 0.683 |       |
| ¾***  | .0 |         | .1 | eg. |        | .012 |       |       |
|       | /0 |         |    | Seq |        |      |       |       |
|       | .0 |         |    |     |        |      |       |       |
|       | /0 |         |    |     |        |      |       |       |
|       | .5 |         |    |     |        |      |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    | No  | NA     | 0    | 0.673 | 0.027 |
|       |    |         |    |     |        | .006 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    |     | RA     | 0    | 0.774 |       |
|       |    |         |    |     |        | .005 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    | 1   | NA     | 0    | 0.683 | 0.000 |
|       |    |         |    | .75 |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    |     | RA     | 0    | 0.751 |       |
|       |    |         |    |     |        | .011 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    | 2.0 | NA     | 0    | 0.683 | 0.000 |
|       |    |         |    |     |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    |     | RA     | 0    | 0.729 |       |
|       |    |         |    |     |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    | 156     | 0  | R   |        | 0    | 0.984 |       |
|       |    |         | .1 | eg. |        | .012 |       |       |
|       |    |         |    | Seq |        |      |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    | 1   | ND     | 0    | 0.984 | 0.000 |
|       |    |         |    | .75 |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    |     | RD     | 0    | 0.993 |       |
|       |    |         |    |     |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    | 2.0 | ND     | 0    | 0.984 | 0.000 |
|       |    |         |    |     |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+
|       |    |         |    |     | RD     | 0    | 0.993 |       |
|       |    |         |    |     |        | .012 |       |       |
+-------+----+---------+----+-----+--------+------+-------+-------+

+-------+----+---------+---+--------+--------+-----------+---------+
| ***   | *  | ***size | * | ***δ   | ***RA  | **        | ***     |
| Inter | ** | /arm*** | * | ~T~*** | /NA*** | *Power*** | Prob*** |
| im*** | μ~ |         | * |        |        |           |         |
|       | 0~ |         | ε |        |        |           | ***(    |
|       | /* |         | * |        |        |           | drop*** |
|       | ** |         | * |        |        |           |         |
|       |    |         | * |        |        |           | ***     |
|       | *  |         |   |        |        |           | arm)*** |
|       | ** |         |   |        |        |           |         |
|       | μ~ |         |   |        |        |           |         |
|       | 1~ |         |   |        |        |           |         |
|       | /* |         |   |        |        |           |         |
|       | ** |         |   |        |        |           |         |
|       |    |         |   |        |        |           |         |
|       | ** |         |   |        |        |           |         |
|       | *μ |         |   |        |        |           |         |
|       | ~2 |         |   |        |        |           |         |
|       | ~* |         |   |        |        |           |         |
|       | ** |         |   |        |        |           |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   |        |        | Each Arm  |         |
+-------+----+---------+---+--------+--------+-----------+---------+
| ***½, | 0  | 60      | 0 | reg.   |        | 0.270     |         |
| ¾***  | .0 |         | . | Seq    |        |           |         |
|       | /0 |         | 1 |        |        |           |         |
|       | .3 |         |   |        |        |           |         |
|       | /0 |         |   |        |        |           |         |
|       | .3 |         |   |        |        |           |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   | No     | ND     | 0.193     | 0.485   |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   |        | RD     | 0.227     |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   | 1.75   | ND     | 0.270     | 0.026   |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   |        | RD     | 0.274     |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   | 2.0    | ND     | 0.270     | 0.026   |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   |        | RD     | 0.275     |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    | 156     | 0 | Reg.   |        | 0.644     |         |
|       |    |         | . | Seq    |        |           |         |
|       |    |         | 1 |        |        |           |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   | 1.75   | NA     | 0.640     |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   |        | RD     | 0.664     |         |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   | 2.0    | ND     | 0.650     | 0.017   |
+-------+----+---------+---+--------+--------+-----------+---------+
|       |    |         |   |        | RD     | 0.664     |         |
+-------+----+---------+---+--------+--------+-----------+---------+

+-------+----+---------+----+-----+--------+-------------+---------+
| ***   | *  | ***size | *  | *   | ***RA  | ***Power*** | ***     |
| Inter | ** | /arm*** | ** | **δ | /NA*** |             | Prob*** |
| im*** | μ~ |         | ε* | ~T~ |        |             |         |
|       | 0~ |         | ** | *** |        |             | *       |
|       | /* |         |    |     |        |             | **(Drop |
|       | ** |         |    |     |        |             | arm     |
|       |    |         |    |     |        |             | 2)***   |
|       | *  |         |    |     |        |             |         |
|       | ** |         |    |     |        |             |         |
|       | μ~ |         |    |     |        |             |         |
|       | 1~ |         |    |     |        |             |         |
|       | /* |         |    |     |        |             |         |
|       | ** |         |    |     |        |             |         |
|       |    |         |    |     |        |             |         |
|       | ** |         |    |     |        |             |         |
|       | *μ |         |    |     |        |             |         |
|       | ~2 |         |    |     |        |             |         |
|       | ~* |         |    |     |        |             |         |
|       | ** |         |    |     |        |             |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    |     |        | Each Arm    |         |
+-------+----+---------+----+-----+--------+-------------+---------+
| ***½, | 0  | 60      | 0  | r   |        | 0.611       |         |
| ¾***  | .0 |         | .1 | eg. |        |             |         |
|       | /0 |         |    | Seq |        |             |         |
|       | .5 |         |    |     |        |             |         |
|       | /0 |         |    |     |        |             |         |
|       | .5 |         |    |     |        |             |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    | No  | NA     | 0.451       | 0.433   |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    |     | RA     | 0.492       |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    | 1   | NA     | 0.678       | 0.016   |
|       |    |         |    | .75 |        |             |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    |     | RA     | 0.683       |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    | 2.0 | NA     | 0.678       | 0.016   |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    |     | RA     | 0.693       |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    | 156     | 0  | R   |        | 0.976       |         |
|       |    |         | .1 | eg. |        |             |         |
|       |    |         |    | Seq |        |             |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    | 1   | NA     | 0.973       | 0.004   |
|       |    |         |    | .75 |        |             |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    |     | RA     | 0.986       |         |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    | 2.0 | NA     | 0.981       | 0.003   |
+-------+----+---------+----+-----+--------+-------------+---------+
|       |    |         |    |     | RA     | 0.986       |         |
+-------+----+---------+----+-----+--------+-------------+---------+

**VIII. SUMMARY**

**Type I error rate:**

**RA ⇒ type I error rate under control**

**RA + screening ⇒**

> **Type I error rate near nominal level**

**Power:**

**With unequal treatment effect --**

####### RA ⇒ power increase 

**RA + screening ⇒**

> **More power increases and reduces the probability of dropping an
> effective arm**

**With equal treatment effect --**

**Drop arm may lead to lose power**

**RA + screening ⇒**

> **More power increases and reduces the probability of dropping an
> effective arm**

**Results can be generalized to more than two treatment arms.**