Handling Covariates in the Design Rosenberger of Clinical Trials - - PowerPoint PPT Presentation

▶
handling covariates in the design
SMART_READER_LITE
LIVE PREVIEW

Handling Covariates in the Design Rosenberger of Clinical Trials - - PowerPoint PPT Presentation

Aug 21, 2023 411 likes •667 views

Handling Covariates in Clinical Trials W. F. Handling Covariates in the Design Rosenberger of Clinical Trials I. Introduction Covariates and randomized phase III trials Two approaches for handling William F. Rosenberger covariates

slide-1
SLIDE 1

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Handling Covariates in the Design

  • f Clinical Trials

William F. Rosenberger Oleksandr Sverdlov

Department of Statistics, George Mason University

May 31, 2007

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 1 / 24

slide-2
SLIDE 2

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Outline

1

  • I. Introduction

Covariates and randomized phase III clinical trials Two approaches for handling covariates

2

  • II. Two approaches

Covariate-adaptive randomization The model-based approach

3

  • III. CARA Randomization (Hu and Rosenberger, 2006,

Chapter 9 Logistic regression Survival trials

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 2 / 24

slide-3
SLIDE 3

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Randomized phase III clinical trial

Two treatments: A and B n patients enter the trial sequentially and must be randomized immediately to either A or B Randomization sequence: Tn = (T1, ..., Tn)′, Tj =

  • 1,

if A; −1, if B. Patients’ covariate vectors: Z1, ..., ZM Patients’ responses: Yn = (Y1, ..., Yn)′ Statistical model: E(Yn) = f (θ|Tn, Z1, ..., ZM)

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 3 / 24

slide-4
SLIDE 4

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Treatment allocation procedures

Complete randomization: φj+1 = Pr(Tj+1 = 1) = 1/2 Restricted randomization: φj+1 = Pr(Tj+1 = 1|Tj) Permuted block design: B blocks of size m = n/B, where m is even; φj+1 = m/2 − NA|block(j) m − Nblock(j) Efron’s (1971) biased coin design: φj+1 =    1/2, if NA(j) = NB(j); p, if NA(j) < NB(j); 1 − p, if NA(j) > NB(j), where p ∈ (1/2, 1].

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 4 / 24

slide-5
SLIDE 5

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Philosophies regarding covariates

Design based: Use stratified blocks to ensure balance on a few known covariates; use covariate-adaptive randomization procedures if more than a few: “splitters” (Grizzle). Analysis based: Adjust for any covariates that are imbalanced using regression modeling of post-stratification after the trial: “lumpers” (Grizzle). Mantel, Whitehead are lumpers; Senn, Crowley, Harrell are splitters. In view of presenter, design should drive analysis; post-hoc adjustment should only be done if specified in design phase, not based on observed data imbalances.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 5 / 24

slide-6
SLIDE 6

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Two approaches for handling covariates

Let Tn = σ(T1, ..., Tn), Yn = σ(Y1, ..., Yn), Zn = σ(Z1, ..., Zn).

  • I. Covariate-adaptive
  • II. Covariate-adjusted

randomization response-adaptive (CARA) randomization

φn+1 = Pr(Tn+1 = 1|Tn, Zn+1) φn+1 = Pr(Tn+1 = 1|Tn, Yn, Zn+1) Goal Balance the treatments Target possibly across covariates unbalanced allocations Why?

  • 1. Increase credibility
  • 1. Allocate more patients
  • f the trial results

to the better treatment adjusting for covariates

  • 2. Balance ⇔ efficiency in
  • 2. Balance efficiency in

homoscedastic linear models heteroscedastic nonlinear models

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 6 / 24

slide-7
SLIDE 7

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Pocock and Simon’s (1975) procedure

Z1, ..., ZM: discrete covariates, Zi has levels 1, ..., li For patient (n + 1) observe (z1, ..., zM) Compute hypothetical imbalances within the levels: DiA(n) = (NiziA(n) + 1) − NiziB(n), DiB(n) = NiziA(n) − (NiziB(n) + 1). Compute overall covariate imbalances Gk(n) =

M

  • i=1

|Dik(n)|, k = A, B. Allocate the patient to treatment A with probability φn+1 =    1/2, if GA(n) = GB(n); p, if GA(n) < GB(n); 1 − p, if GA(n) > GB(n).

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 7 / 24

slide-8
SLIDE 8

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Taves (1974) was the first to propose a covariate-adaptive procedure, and Pocock-Simon reduces to Taves when p = 1, but the procedure is deterministic. Taves, now living in a retirement home, still favors his procedure, and does not believe in randomization:

I hope that the day is not too far distant when we look back on the current belief that randomization is essential to good clinical trial design and realize that it was... “credulous idolatry”. (Taves, 2004)

Countless simulations have been done on the Pocock-Simon method, but little theoretical work. Most simulations show that the method does achieve balance on a large number of

  • covariates. However, the use of covariate-adaptive

randomization is not without controversy. The CMPC states that the procedures should be strongly discouraged.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 8 / 24

slide-9
SLIDE 9

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Atkinson’s D− (or DA−) optimal approach

After n allocations one has Mn = diag{Z′

AWAZA, Z′ BWBZB},

where Zk is nk × p, and Wk = diag{pk(zi)qk(zi)}. Let Zn+1 = zn+1. Then det Mn+1 = det Mn(1 + z′

n+1(Z′ AWAZA)−1zn+1pAqA),

if A; det Mn(1 + z′

n+1(Z′ BWBZB)−1zn+1pBqB),

if B. Choose the treatment with the maximum value of d(k, θ, zn+1) = 1 + z′

n+1(Z′ kWkZk)−1zn+1pkqk

Randomized version: φn+1 = d(A, zn+1) B

k=A d(k, zn+1)

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 9 / 24

slide-10
SLIDE 10

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Non-linear models

There has been some controversy over the goals of efficiency and balance. Here is an exchange between John Whitehead and Stephen Senn: Whitehead: I think that one criterion is really to reduce the probability of some large imbalance rather than the variance of the estimates.... And to make sure that these unconvincing trials, because of the large imbalance, happen with very low probability, perhaps is more important.... I would always be wanting to adjust for these variables. Senn: I think we should avoid pandering to these foibles of physicians.... I think people worry far too much about imbalance from the inferrential (sic) point of view.... The way I usually describe it to physicians is as follows: if we have an unbalanced trial, you can usually show them that by throwing away some patients you can reduce it to a perfectly balanced

  • trial. So you can actually show that within it there is a perfectly balanced
  • trial. You can then say to them: ‘now, are you prepared to make an

inference on this balanced subset within the trial?’ and they nearly always say ‘yes’. And then I say to them, ‘well how can a little bit more information be worse than having just this balance trial within it?’

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 10 / 24

slide-11
SLIDE 11

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Non-linear models

For linear homoscedastic models, the most efficient design will induce balance over covariates. In non-linear models, such as logistic regression for binary response or covariate-adjusted survival models under exponential response with censoring, the most efficient design may be imbalanced. The most efficient design may also place more patients on the inferior treatment. A balanced design may be both inefficient and unethical. In non-linear models, the solution to the optimality problem is a function of unknown parameters which must be updated using the data, leading to response-adaptive randomization. Any

  • ptimization criterion must also take into account the ethical

considerations of minimizing the expected treatment failures or some other function of response.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 11 / 24

slide-12
SLIDE 12

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Multiple objectives of a phase III clinical trial

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 12 / 24

slide-13
SLIDE 13

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

CARA design for logistic regression

Model: log

  • pk

1 − pk

  • = θ′

kz, k = A, B,

where pk = Pr(Yk = 1|Z = z). πA = πA(θA, θB, z) = target proportion for A given z Use data from first n patients to compute ˆ θn = (ˆ θn,A, ˆ θn,B) If Zn+1 = zn+1, then allocate to A with probability φn+1 = πA(ˆ θn, zn+1)

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 13 / 24

slide-14
SLIDE 14

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Proposed target allocations for fixed z

Odds ratio (Rosenberger, Vidyashankar, Agarwal, 2001)

πRVA

A

= pA(z)/qA(z) pA(z)/qA(z) + pB(z)/qB(z) = eθ′

Az

eθ′

Az + eθ′ B z

For fixed power of simple difference test, minimize expected number of treatment failures (Rosenberger, S, I, H, R, 2001)

πRSIHR

A

= p pA(z) p pA(z) + p pB(z) = {1 + e−θ′

Az}−1/2

PB

k=A{1 + e−θ′

k z}−1/2

Minimize power:

πNeyman

A

= p pB(z)qB(z) p pA(z)qA(z) + p pB(z)qB(z)

For fixed power of log odds ratio test, minimize expected number of treatment failures (Ivanova and Rosenberger, 2002):

πOptimal

A

= p pB(z)qB(z) p pA(z)qA(z) + p pB(z)qB(z)

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 14 / 24

slide-15
SLIDE 15

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Atkinson, Biswas (2005) approach

Idea: Let ˆ θn be the estimated parameter based on the first n patients’ responses and treatment assignments. Since the D or DA-optimality criterion deals only with efficiency, skew the directional derivatives with some function that places more patients on the better treatment. Define f (A, ˆ θn)/ B

k=A f (k, ˆ

θn) to be that function. One can choose any of the previous allocation targets, for instance. Then randomize to A with probability φn+1 = f (A, ˆ θn)d(A, ˆ θn, zn+1) B

k=A f (k, ˆ

θn)d(k, ˆ θn, zn+1) . Note that there is nothing optimal about this allocation function.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 15 / 24

slide-16
SLIDE 16

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Zhang and Hu approach

Let πA = πA(θ, z). Then one can define φn+1 = g NA|zn+1(n) Nzn+1(n) , πA(ˆ θn, zn+1)

  • ,

(1) where g(x, y) =

  • y(y/x)γ

y(y/x)γ+(1−y)((1−y)/(1−x))γ ,

x, y ∈ [0, 1]; 1 − x, x=0,1, and γ ≥ 0 controls randomness. γ = 0 ⇔ CARA design; γ = ∞ ⇔ deterministic design (except when x = y). Hu and Rosenberger (2003) recommend γ = 2.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 16 / 24

slide-17
SLIDE 17

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Theorem (Zhang et al. (2007))

Let πA(θ, z) be continuous and differentiable under its expectation as a function of θ = (θA, θB). Define Ik = E{πk(θ, z)pkqkzz′}, k = A, B and V = diag(IA, IB). Then, under additional regularity conditions we have: Pr(Tn+1 = 1|ˆ θn, z) → πA a.s. (2) √n(ˆ θn − θ)

D

→ N(0, V); (3) √n(NA/n − vA)

D

→ N(0, σ2

A);

(4) √n(S/n − µs)

D

→ N(0, σ2

S),

(5) where S = #successes in the trial. If, in addition Pr(Z = z) > 0, then we have NA|z/Nz → πA a.s. and √ Nz(NA|z/Nz − πA)

D

→ N(0, σ2

A|z)

(6) The asymptotic variances σ2

A, σ2 A|z and σ2 S are known expressions.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 17 / 24

slide-18
SLIDE 18

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Preliminary simulation study

Model: logit(Pr(Yk = 1|z)) = αk + βk1z1 + βk2z2 + βk3z3, k = A, B. Z1 ∼Bernoulli(1/2), Z2 ∼DUnif[30,75], Z3 ∼Normal(200, 20). Allocation procedures: Complete randomization Pocock-Simon’s procedure with p = 0.75 Stratified PBD with m = 10 Four Zhang and Hu CARA randomization procedures, targeting the 4 π’s Atkinson-Biswas-type procedure with f the odds ratio function

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 18 / 24

slide-19
SLIDE 19

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Preliminary simulation study

Table: Model 1 with θA = θB and n = 200.

Procedure

NA(n) n

S.D.

NA0(n) N0(n)

S.D. dKS (z2) S.D. Err.Rate F(n) S.D. Complete 0.50 0.03 0.50 0.05 0.12 0.04 0.05 90 6 SPBD 0.50 0.03 0.50 0.04 0.12 0.03 0.05 90 6 P-S 0.50 0.00 0.50 0.01 0.10 0.03 0.05 90 6 πRSIHR 0.50 0.03 0.50 0.04 0.12 0.03 0.05 90 6 πRVA 0.50 0.03 0.50 0.04 0.11 0.03 0.06 90 6 πOptimal 0.50 0.02 0.50 0.04 0.12 0.03 0.06 90 6 πNeyman 0.50 0.02 0.50 0.04 0.11 0.03 0.06 90 6 A-B 0.50 0.02 0.50 0.04 0.12 0.04 0.05 90 6

Table: Model 2 with αA − αB = −1.0 and n = 200.

Procedure

NA(n) n

S.D.

NA0(n) N0(n)

S.D. dKS (z2) S.D. Power F(n) S.D. Complete 0.50 0.04 0.49 0.05 0.12 0.04 0.80 62 6 SPBD 0.50 0.03 0.50 0.04 0.12 0.03 0.81 62 6 P-S 0.50 0.01 0.50 0.01 0.10 0.03 0.81 62 6 πRSIHR 0.48 0.03 0.49 0.04 0.12 0.03 0.81 60 6 πRVA 0.40 0.03 0.45 0.04 0.12 0.03 0.76 56 6 πOptimal 0.45 0.03 0.48 0.04 0.12 0.03 0.80 58 6 πNeyman 0.48 0.03 0.49 0.04 0.12 0.03 0.81 60 6 A-B 0.47 0.03 0.50 0.04 0.12 0.04 0.81 60 6

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 19 / 24

slide-20
SLIDE 20

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

CARA designs for the exponential model

Exponential regression model: log T = β′

kz + σε, k = A, B,

where T ≥ 0 is survival time, σ = 1, and ε ∼ f (ε) = exp(ε − exp(ε)), ε ∈ R (extreme value distribution). Survival times are subject to the noninformative right censoring, i.e. a pair (ti, δi) is observed, where ti = min(Ti, Ci) and δi = 1 if ti = Ti if ti = Ci Information matrix for treatment k: Mk = (Z′DkZ)−1, where D = diag{Pr(δik = 1|zi)}.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 20 / 24

slide-21
SLIDE 21

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Difficulties: Censored observations Responses are survival/censoring times, which are not immediately observable Two-step approach (Zhang and Rosenberger, 2007): Establish the asymptotic properties of a given CARA procedure assuming that responses are immediate Argue that a moderate delay does not impact the asymptotics CARA procedure: πA = πA(βA, βB, z) target proportion for A given z Based on n patients, compute ˆ βn = (ˆ βn,A, ˆ βn,B). If Zn+1 = zn+1, then allocate to A with probability φn+1 = πA(ˆ βn, zn+1).

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 21 / 24

slide-22
SLIDE 22

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Proposed target allocations

Covariate-adjusted optimal allocations: πA =

  • λm

A (z)dB(z)

  • λm

A (z)dB(z) +

  • λm

B (z)dA(z), m = 0, 1, 2, 3.

where λk(z) = exp(β′

kz) and dk(z) = Pr(δk = 1|z).

Simple censoring scheme: tk = min(Tk, C) Tk ∼ Exp(λk(z)), C ∼ Exp(θ) Type I censoring scheme: tk = min(Tk, D − U) Tk ∼ Exp(λk(z)), U ∼ Unif(0, R), D = trial duration

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 22 / 24

slide-23
SLIDE 23

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

Theorem

Let β = (βA, βB), vk = E{πk(β, z)}, Vk = E{πk(β, z)dkzz′}, and V = diag(VA, VB). Then, for CARA procedures with the proposed targets and the two censoring schemes we have: Pr(Tn+1 = 1|ˆ βn, z) → πA a.s. (7) √n(ˆ βn − β)

D

→ N(0, V); (8) √n(NA/n − vA)

D

→ N(0, σ2

A).

(9) If, in addition Pr(Z = z) > 0, then we have NA|z/Nz → πA a.s. and √ Nz(NA|z/Nz − πA)

D

→ N(0, σ2

A|z),

(10) where the asymptotic variances σ2

A and σ2 A|z are known expressions.

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 23 / 24

slide-24
SLIDE 24

Handling Covariates in Clinical Trials

  • W. F.

Rosenberger

  • I. Introduction

Covariates and randomized phase III trials Two approaches for handling covariates

  • II. Two

approaches

Covariate- adaptive randomization Model-based approach

  • III. CARA

Randomiza- tion

Logistic regression Survival trials

The effect of delayed responses

Assume that: A1 Pr(Z = z) > 0 for every given z A2 For each z, Pr(τm(z) > tn+m − tm) = o(n−c), c > 0, where τm(z) is the delay in response for the m-th patient with zm = z. One can define a CARA procedure with DBCD(γ): φn+1 = g NA|zn+1(n) Nzn+1(n) , πA(ˆ βn, zn+1)

  • ,

By the result of Hu et al. (2006), the asymptotic properties of the DBCD still hold. If U ∼Unif(0, D), then tn+m − tm ∼Gamma(n, θ), and Pr(τm(z) > tn+m − tm) =

  • θ

λk(z) + θ n = o(n−1).

  • W. F. Rosenberger (GMU)

Handling Covariates in Clinical Trials May 31, 2007 24 / 24