## Subject: Statistics

## Paper: Biostatistics

## Module 20: Sequential Clinical Trials I

Development Team

Principal investigator: Dr. Bhaswati Ganguli,Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Sugata SenRoy,Professor, Department of Statistics, University of Calcutta

Content writer: Dr.Atanu Bhattacharjee, Assistant Professor, Centre for Cancer Epidemiology, The Advanced Centre for Treatment, Research and Education in Cancer (ACTREC) Tata Memorial Centre

Content reviewer: Department of Statistics, University of Calcutta

Sequential Clinical Trials

Sequential Clinical Trials

There is an increasing need for innovative clinical trial designs that allow effective agents to be identified more efficiently compared with the standard designs.

Group sequential designs

The flexibility to use cumulated information to modify the subsequent course of the trial is supported by Group sequential designs.

Sequential Clinical Trials

Monitoring in a sequential trial

Monitoring a group sequential clinical trial is a dynamic process.

The time of performing an interim analysis in a group sequential trial is a natural time to update and combine the information from prior experience and accumulated data.

The decision-theoretic approach has the potential ability to concurrently consider efficacy, futility and cost (Berry,1994).

Sequential Clinical Trials

Monitoring in a sequential trial

Monitoring a group sequential clinical trial is a dynamic process.

The time of performing an interim analysis in a group sequential trial is a natural time to update and combine the information from prior experience and accumulated data.

The decision-theoretic approach has the potential ability to concurrently consider efficacy, futility and cost (Berry,1994).

Sequential Clinical Trials

Monitoring in a sequential trial

Monitoring a group sequential clinical trial is a dynamic process.

The time of performing an interim analysis in a group sequential trial is a natural time to update and combine the information from prior experience and accumulated data.

The decision-theoretic approach has the potential ability to concurrently consider efficacy, futility and cost (Berry,1994).

Introduction

Monitoring in a sequential trial

During the monitoring in clinical trial it is obvious that multiple testing problem will occur. It generates the false positive and false negative conclusion.

Adaptive Design and random sample size

However, this issue may be avoided by selecting the sample size in random manner through adaptive design.

Introduction

Monitoring in a sequential trial

During the monitoring in clinical trial it is obvious that multiple testing problem will occur. It generates the false positive and false negative conclusion.

Adaptive Design and random sample size

However, this issue may be avoided by selecting the sample size in random manner through adaptive design.

Introduction

Stopping Rule in Sequential Trial

The stopping rule in any clinical trial is based on the loss function that reflects the evidence of operation characteristics at interim analysis.

The maximum sample size is sequentially determined using interim data.

The overall type I error rate can be controlled by choosing related designing parameters.

Introduction

Stopping Rule in Sequential Trial

The stopping rule in any clinical trial is based on the loss function that reflects the evidence of operation characteristics at interim analysis.

The maximum sample size is sequentially determined using interim data.

The overall type I error rate can be controlled by choosing related designing parameters.

Introduction

Stopping Rule in Sequential Trial

The stopping rule in any clinical trial is based on the loss function that reflects the evidence of operation characteristics at interim analysis.

The maximum sample size is sequentially determined using interim data.

The overall type I error rate can be controlled by choosing related designing parameters.

Introduction

False Postive and False Negative

In Sequential trial testing the decision about the false-positive and false negative can be measured by loss structure.

But it can be concluded that in sequential setting the type I error rate is often smaller than the specified significance level.

Moreover, the procedures that are based on the non-sufficient statistics may increase the type II error rate with or without the decreased type I error rate.

Challenge in Sequential Trial

Challenge in Sequential Trial

It is difficult to get unbiased estimator through sequential trial because the sample size is random in nature.

Example in Sequential Trial

Example

Consider a clinical trial that compares a new treatmentT with a
controlC, where the individual treatment response isX_{T} and the
individual control response isXC. For notation simplicity, we
assume a design with equal randomization, which can be easily
extended to more general two-arm trials.

Example in Sequential Trial

Example

The block size at each stage is2B_{i}, i= 1,2, ...., whereB_{i} for each
treatment arm is fixed before the trial starts, but the maximum
number of blocks is not predetermined and is determined using
interim data.

Example in Sequential Trial

Example

LetX¯_{T}_{i} andX¯_{C}_{1} be the observed mean responses in the ith block.

Given the parameter of interest,θ, which is a measure of the treatment difference, let

Xi = ¯XTi−X¯Ci ∼Fi(.|θ),andθ= Z ∞

−∞

xdFi(x|θ) (1)

Introduction

Hypothesis Test

The one-sided hypothesis to be tested is

H_{0} ≤0 versusH_{1} :θ >0 (2)
The termθ can be assumed with prior distributionπ with

E(θ|π) =δ.

Decision Rule

It is expected that a loss structure consisting of the respective cost of making a false-positive conclusion, a false-negative conclusion, and the cost of the total sample, in which the risk and benefit of the new agent under investigation should be balanced via these cost parameters.

Introduction

Decision Rule

LetDbe the decision of either accepting (A) or rejecting(R) null hypothesis. At the jth interim analysis, the loss function is defined by

L(θ, D, j) =L(θ, D) + 2K_{2}

j

X

i=1

B_{i} (3)

where

L(θ, D) =K0h0(θ)I{≤0, D=R}+K1h1(θ)I{θ >0, D=A}

(4)
K_{0} is a constant and positive penalty for each unit of h_{0}(θ) when
H_{0} is incorrectly rejected, andK_{1} is a positive penalty for each
unit ofh1(θ) when H0 is incorrectly accepted.

Introduction

Decision Rule

Here,h0(.)and h1(.) are positive and continuous functions, and
the positive loss may change with the actual value ofθ. One
sensible choice is to allow the loss function to change with the
magnitude ofθ, such as h0(θ) =h1(θ) =|θ|^{w}+c with c≥0and
w >0.

Introduction

Decision Rule

The loss to accept the null hypothesis increases as the distance of a positiveθ to zero gets larger. Similarly, the loss to rejectH0

increases as a negativeθdeviates from zero further. The unit cost
of each sample,K_{2}, is relative to K_{0} and K_{1}, which should be
much smaller thanK_{0} andK_{1}.

Another special case for the type of loss functions is the commonly
used0−K_{i} loss, withh_{0}(θ) =h_{1}(θ) = 1. Following a common
convention that making a false-positive conclusion leads to a more
severe penalty than making a false-negative conclusion, we let
K_{0} > K_{1}.

Decision Rule

Decision Rule

LetXj ={X_{1}, ...., Xj}be the accumulated data up to step j,
whereX_{j} represents data from thej^{th} block. At each interim
analysis the study will be terminated for futility or for efficacy if
the evidence is strong enough. Otherwise, the study will continue
to(j+ 1)^{th} block.

The loss function related to the above two decisions at the jth interim analysis is defined as the expected loss of stopping the trial and accepting the null hypothesis, that is,

L_{A}(K_{j}) =E{L(θ, A, j)|K_{j}}, (5)

Decision Rule

Decision Rule

L_{cont}(K_{j}) =E[min_{D+A}orRE()|K_{j+1}}|K_{j}] (6)
It is possible to achieve the targeted power while minimizing the
total sample sizeas monitoring rules. This type of the group
sequential designs motivated by internal reasons in view of the
interim data towards treatment effect comparison.