Sample size determination for cost-effectiveness trials

MPS Research Unit CHEBS Workshop - April 2003 1

Anne Whitehead

Medical and Pharmaceutical Statistics Research Unit

The University of Reading

Sample size determination for cost-effectiveness trials


Comparative study

• Parallel group design

• Control treatment (0)

New treatment (1)

• n0 subjects to receive control treatment

n1 subjects to receive new treatment


Measure of treatment difference

Let be the measure of the advantage of new over control

> 0 new better than control = 0 no difference < 0 new worse than control

Consider frequentist, Bayesian and decision-theoretic approaches


1. Frequentist approach

Focus on hypothesis testing and error rates- what might happen in repetitions of the trial

e.g. Test null hypothesis H0 : = 0against alternative H1

+: > 0

Obtain p-value, estimate and confidence interval

Conclude that new is better than control if the one-sided p-value is less than or equal to

Fix P(conclude new is better than control | = R) = 1–


Distribution of

= 0 = R

k

Fail to Reject H0 Reject H0


A general parametric approach

Assume

Reject H0 if > k

where is the standard normal distribution function andP(Z > z) = where Z ~ N(0, 1)

1ˆ ~ N ,w

ˆP( k | 0)

1 k w

k z w


Require

RˆP( k | ) 1

R1 (k ) w 1

R1 k w

R( k) w z

2

R

z zw


Application to cost-effectiveness trialsBriggs and Tambour (1998)

= k (E1 – E0) – (C1 – C0)

is the net benefit, where

E1, E0 are mean values for efficacy for new and control treatments

C1, C0 are mean costs for new andcontrol treatments

k is the amount that can be paid for aunit improvement in efficacy for asingle patient


E1 E0 C1 C0

ˆ k(x x ) (x x ) 22

EE 01E1 E0

1 0

22CC 01

C1 C01 0

var(x x )n n

var(x x )n n

E1 E0 C1 C0

E1 E0 C1 C0

cov (x x ),(x x )

var(x x )var(x x )

2

R

z z1w

ˆvar( )

Set

and solve for n0 and n1


2. Bayesian approach

Treat parameters as random variablesIncorporate prior informationInference via posterior distribution for parametersObtain estimate and credibility interval

Conclude that new is better than control if

P( > 0|data) > 1 –

Fix P0 (conclude new better than control) = 1 –


Likelihood function

Prior h0() is

Posterior h(|data)

i.e. h(|data) is

2ˆ ˆf ( | ) exp w( ) / 2

0 0N ,1/w

2 20 0

ˆexp{ w( ) / 2}exp{ w ( ) / 2}

20 0 0

ˆexp ( w w ) w w / 2

ˆ ~ N ,1/w

0 0

0 0

ˆw w 1N ,

w w w w


P ( > 0|data) > 1 –

if

i.e.

i.e.

0 0

0 0

ˆw w z

w w w w

0 0 0ˆw w z w w

0 0 0ˆw w z w w


Prior to conducting the study,

so

1ˆ ~ N ,w

00

1~ N ,

w

00

1 1ˆ ~ ,w w

2

00

wˆw ~ w, ww


Require

P0 0 0 0ˆw w z w w 1

0 0 0 0

20

w z w w w1 1

w w / w

0 0 0

20

(w w) z w w1

w w / w

20 0 0 0(w w) z w w z w w / w

20 0 0 0(w w) z w w z w w w

Express w in terms of n0 and n1, provide values for 0 and w0 and solve for n0 and n1


Application to cost-effectiveness trials O’Hagan and Stevens (2001)

= k (E1 – E0) – (C1 – C0)

Use multivariate normal distribution for- separate correlations between efficacy and cost for each treatment

Allow different prior distributions for the design stage (slide13) and the analysis stage (slide 11)

E1 E0 C1 C0ˆ k(x x ) (x x )

E1 E0 C1 C0x ,x ,x ,x


3. Decision-theoretic approach

Based on Bayesian paradigm

Appropriate when outcome is a decision

Explicitly model costs and benefits from possible actions

Incorporate prior information

Choose action which maximises expected gain


Actions

Undertake study and collect w units of information on ,

then one of the following actions is taken:

Action 0 : Abandon new treatment

Action 1 : Use new treatment thereafter


Table of gains (relative to continuing with control treatment)

Action 1 Action 0 0 – cw – b – cw

> 0 – cw – b + r1 – cw

c = cost of collecting 1 unit of information (w) b = further development cost of new treatment

r1 = reward if new treatment is better

G0,w() = – cw

1,w 1G ( ) cw b r I( 0)


Following collection of w units of information, the expected gain from action a is

Ga, w(x) = E {Ga,w()| x}

Action will be taken to maximise E {Ga,w()|x},

that is a*, w* where

Ga*, w*(x) = max {Ga, w(x)}

(Note: Action 1 will be taken if P ( > 0|data) > b/r1)


At design stage consider frequentist expectation:

E (Ga*, w(x))

and use this as the gain function Uw ()

a*,w

x

(x) f (x; ,w) dx G


Expected gain from collecting information w is

So optimal choice of w is w*, where

w 0 w w 0{U ( )} U ( )h ( )d

U E

w* ww

max { }U U


This is the prior expected utility or pre-posterior gain

w* 0 a,ww a

max E max (G ( ) | x) U E E

a,w 0xw amax max G ( )h( | x)d f (x; ,w)dx h ( )d


Note:

= E{– cw + max(r1 P ( > 0|data) – b, 0)}

a,wa

E max (G ( ) | x)E


Application to cost-effectiveness trials

Could apply the general decision-theoretic approach taking to be the net benefit

The decision-theoretic approach appears to be ideal for this setting, but does require the specification of an appropriate prior and gain function


Table of gains – ‘Simple Societal’(relative to continuing with control treatment)

Action 1 Action 0 0 – cw – b – cw

> 0 – cw – b + r1 – cw

c = cost of collecting 1 unit of information (w) b = further development cost of new treatment

r1 = reward if new treatment is more cost-effective

G0,w() = – cw

1,w 1G ( ) cw b r I( 0)


Gains – ‘Proportional Societal’(relative to continuing with control treatment)

c = cost of collecting 1 unit of information (w)

b = further development cost of new treatment


G0,w() = – cw

1,w 2G ( ) cw b r


Gains – ‘Pharmaceutical Company’

c = cost of collecting 1 unit of information (w)

b = further development cost of new treatment


where A is the set of outcomes which leads to Action 1,

e.g. for which P ( > 0|data) > 1 –

1,w 3G (x) cw b r I(x A)

0,wG (x) cw


References

Briggs, A. and Tambour, M. (1998). The design and analysis of stochastic cost-effectiveness studies for the evaluation of health care interventions (Working Paper series in Economics and Finance No. 234). Stockholm, Sweden: Stockholm School of Economics.

O’Hagan, A. and Stevens, J. W. (2001). Bayesian assessment of sample size for clinical trials of cost-effectiveness. Medical Decision Making, 21, 219-230.

Sample size determination for cost-effectiveness trials

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