MPS/MSc in StatisticsAdaptive & Bayesian - Lect 41 Lecture 4 Sample size reviews 4.1A general approach to sample size reviews 4.2Binary data 4.3Normally.

MPS/MSc in Statistics Adaptive & Bayesian - Lect 4 1

Lecture 4

Sample size reviews

4.1 A general approach to sample size reviews

4.2 Binary data

4.3 Normally distributed data

4.4 Exact results: normally distributed responses


4.1 A general approach to sample size reviews

• Many sample size formulae depend on nuisance parameters, the values of which have to be guessed

• Part way through the trial we will have plenty of data on which to base a better guess

• So, do that, and recalculate the sample size• Now use the new sample size, perhaps within the limits of

minimum and maximum possible values• Assess the effect of this procedure on type I error: usually

it is very small


This is an adaptive design with a single interim analysiswhich may lead only to a reassessment of sample size

Its use is becoming widespread, and the regulatory authoritiesare generally well-disposed towards it


4.2 Sample size review for binary data

Treatments: Experimental (E) and Control (C)

Success probabilities: pE and pC

Hypotheses: H0: pE = pC H1: pE > pC

Type I error: a (one-sided)

Power: 1 – b, when pE = pER and pC = pCR

Sample sizes: nE and nC, where nE + nC = n

Allocation ratio: (1:1), that is nE = nC

pCR is the anticipated value of pC, and an improvement from

that value to pE = pER on E would be clinically worthwhile


Probability difference approach

Initial sample size calculation

Put q = pE - pC, and set power at q = qR = pER - pCR

Two popular formulae for n are:

(Machin et al., 1997)

and

(4.2) 2

1 1 1ER CR2

R

z zn 4p(1 p) phe pw re p

2

1 1 ER ER CR CR

R

z 2p(1 p) z p (1(4.1

p ) p (1 p )n )2


To use these formulae:

• use previous data and experience to guess pCR

• consider what difference qR would be clinically important

• deduce pER and

Using these values, find the required sample size n

Then, when data from about patients are available, a sample size review can be conducted

p

12 n


At the sample size review we do not change qR:

- this remains the clinically important difference

To recompute n based on (4.1), identify the control patients andfind an estimate, of pC as the success rate on C so far

Replace pCR by , pER by and

To recompute n based on (4.2), we need not break the blinding:just estimate as the overall success rate in the trial as a whole(over E and C)

The preservation of blindness makes the second option moreattractive

1C R2

ˆp by p

Cp̂

Cp̂ C Rp̂

p


Log-odds ratio approach

Initial sample size calculation

Put

and set power at q = qR computed from the values pER and pCR The resulting sample size formula is:

(4.3)

This formula can be updated at a sample size review in thesame way as (4.2), without breaking the blind

2

1 1

R

z z4n

p(1 p)

E C CE

C E E C

p 1 p pplog log log

p 1 p 1 p 1 p


Example

a = 0.025, 1 – b = 0.90: z1-a = 1.96, z1-b = 1.282

pCR = 0.3, pER = 0.5, = 0.4

(4.1) prob diff: qR = 0.2 n = 248

(4.2) prob diff: qR = 0.2 n = 252

(4.3) log-odds ratio: qR = 0.847 n = 244

p


After 120 observations, the sample size is reviewed

We find that = 0.2, rather than 0.4

Retaining probability difference: qR = 0.2, (4.2) n = 168

- sample size goes down

- qR = 0.2 consistent with pCR = 0.1, pER = 0.3

Retaining log-odds ratio: qR = 0.847, (4.3) n = 366

- sample size goes up

- qR = 0.847 consistent with pCR = 0.134, pER = 0.266

p


4.3 Normally distributed data

Treatments: Experimental (E) and Control (C)

Distributions: N(mE, s2) and N(mC, s2)

Hypotheses: H0: mE = mC H1: mE > mC

Type I error: a (one-sided)

Power: 1 – b, when mE = mER and mC = mCR

Sample sizes: nE and nC, where nE + nC = n

Allocation ratio: (1:1), that is nE = nC

Put q = mE - mC and qR = mER - mCR

Let denote the anticipated common variance

2R


The sample size is given by

see Slide 2.8

The actual values of mER and mCR have no effect on n otherthan through qR

The anticipated variance is very influential, and is replacedby an estimate at the sample size review

2

1 12R

R

z zn 4 ( 4. ) 4

2R


Estimating s2

(a) Use the conventional unbiased estimate

based on the n observations available so far

To use this requires breaking the blind, at least as far as

separating the two treatment groups: their identities need not

be revealed

22

hE E hC C2 2 x x x xˆ S

n 2

(4.5)

14MPS/MSc in Statistics Adaptive & Bayesian - Lect 4

(b) Avoid unblinding, using a simple adjustment (Gould, 1995)

For each term in (4.5)

Substitute in (4.5)

E E E

E

n n n2 2 2 2

hE E hE E hE E Eh 1 h 1 h 1

2n2 C

hE E C Eh 1

x x x x x x (x x) n (x x )

n(x x) n (x x )

n

22 2 2C EhE E hC C hj C E

j E,C h

n nx x x x (x x) (x x )

n

MPS/MSc in StatisticsAdaptive & Bayesian - Lect 4 15

If desired difference is present:

Can use estimate for sample size review without unblinding

(4.6)

: The estimate of total variance

E C Rx x

jCE

nnn22 2 2E C

hE E hC C hj Rh 1 h 1 j E,C h 1

n nx x x x (x x)

n

2%

jn

2 2E C 2 2E Chj R T Rj E,C h 12

n n n n(x x) n 1n nn 2 n 2

%%

2T


(c) Avoid unblinding, using an E-M algorithm (Gould and Shih, 1992; Gould, 1995)

DO NOT USE THIS METHOD!

See Friede and Kieser (2002) to see why you should not


Example

a = 0.025, 1 – b = 0.90: z1-a = 1.96, z1-b = 1.282

qR = 0.5, sR = 1.0

(4.4) n = 168

After 80 patients:

E E E C C Cn 40, x 5.6, S 1.45 n 40, x 5.3, S 1.26

2T 1.844


Unblinded approach

From (4.5)

So that

From (4.4), new sample size is

2 2 2E E C Cˆ(n 2) (n 1) S (n 1) S

143.9139

2ˆ 1.845

n 310


Blinded approach

From (4.6)

From (4.4), new sample size is

2

2

80145.7139 0.5

478

145.7139 5

781.804

n 304


Consider the unblinded approach to sample size review

An initial sample size is set

When n1 = patients have given responses, s2 isestimated by the usual pooled variance

The sample size is then recalculated (as n2) and that numberof subjects is taken, provided that n2 ≥ n1

Finally, a t-test is performed

4.4 Exact results: normally distributed responses

12 n


The final t-statistic is

where the subscript 2 identifies values computed for the finalsample

Now

E2 C22

2

x xt n

2S

2 22 E2 E2 C2 C22

2

(n 1)S (n 1)SS

n 2


With the subscript + relating to the extra patients recruited afterthe sample size review, it can be shown that

Dividing by the true value of s2, the quantities on the rhs aredistributed as independent c2 random variable on nE1 – 1, nE+ - 1 and 1 degrees of freedom respectively

A similar result holds for the control group

2E2 E2

22 2 E1 EE1 E1 E E E1 E

E2

n 1 S

n n n 1 S n 1 S x x

n


The final t-statistic, is a function of

and these quantities are mutually independent

From this result, the conditional distribution of t, given ,can be deduced

The revised sample size depends on the data only through

2 21E 1C E C 1x ,x ,x ,x an,S Sd

21S

21S

22 2 1n n S


The density of t is given by

where W represents the random variable

is a t density with degrees of freedom specified by

: it is a step function in w

is a c2 density with n1 - 2 degrees of freedom

In this way, for a given value of s2, the exact properties of thesample size review procedure can be determined

T WT Wf (t) f (t w)f (w)dw

2 21 1n 2 S /

21S

T Wf (t w)

Wf (w)


Example

a = 0.05, 1 – b = 0.90, = 1, qR = 0.414 n = 200

A sample size review is conducted after 100 responses

Suppose that the true value of s2 is 1.43

Then the true type I error rate will be 0.051

The true power will be 0.920

2R


References

General methods for sample size review:

Wittes and Brittain (1990)Gould (1992)Birkett and Day (1994)

Exact evaluations:

Kieser and Friede (2000)Friede and Kieser (2006)

Montague (2007)

MPS/MSc in StatisticsAdaptive & Bayesian - Lect 41 Lecture 4 Sample size reviews 4.1A general approach to sample size reviews 4.2Binary data 4.3Normally.

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