Testing treatment combinations versus the corresponding monotherapies in clinical trials

Testing treatment combinations versus the corresponding monotherapies in clinical trials

Ekkekhard Glimm, Novartis Pharma AG

8th Tartu Conference on Multivariate Statistics

Tartu, Estonia, 29 June 2007

2 | Testing treatment combinations | Ekkehard Glimm | 29 June 2007

Setting the scene (I)

The problem

Two monotherapies available for the treatmentof a disease

Question: Does a combination / simultaneousadministration of the treatments („combination“) have a benefit?

Might be

• synergism ( positive interaction between monos)

• a way to overcome dose limitations of monos


Setting the scene (II)

Ultimate task is to find if the best combination therapydose is better than the best dose of any of the monos.

Frequent problem in clinical trials (e.g. hypertension treatment)

A lot of literature on the topic:

Laska and Meisner (1989)

Sarkar, Snapinn and Wang (1995)

Hung (2000)

Chuang-Stein, Stryszak, Dmitrienko and Offen (2007)


Setting the scene (III)

Limited goal in this talk:

Only two monotherapies

Optimal doses are known

Let A, B be the monotherapies, AB their combination.

Assume n individuals per treatment group with response

njABBAiNx iij ,,1,,,),,(~ 2

),max(:0 BAABH vs ),max(: BAABA


Min Test (I)

Laska and Meisner (1989):

/21 AABn xxZ

/22 BABn xxZ

Reject H0 if min(Z1, Z2)>u1- (with u1- N(0,1)-quantile).

Note: Assumption of known is just for convenience, min-t-test is also possible. Same with equal n’s.


Min Test (II)

Rejection probability of this test:

12125.0 22 , uur BAB

nAAB

nm

where .,. is the cdf of

1

1,

0

02

N

This test is uniformly most powerful

in the class of monotone tests (= tests whose teststatistic is a monotone function of Z1 and Z2).


Min Test (III)

The Min test is „conservative“:

Let AB = B > A and

Then the null rejection probability is

The „least favorable constellation“ under H0 is δ→∞

with

But at nominal =0.05!

/2 AB

n

115.00 , uur m

.lim 0

mr

0122.000 mr


Laska and Meisner Min Test (IV)

Min test rejection probability if AB=B>A

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

reje

cti

on

pro

b

Is there a way to alleviate this conservatism?


„Conditional“ tests

Tests uniformly more powerful (UMP) than the Min test can be derived, if we adjust the critical value based on the observed difference

/2 ABn xxV

In general such tests are of the form:

Reject if

To be UMP than the Min test, a sufficient condition is:

VcZZ ),min( 21

11 , uVcuVcV

and keeping is attainable. || somefor 1 VuVc


Sarkar et al. test

Suggestion by Sarkar et al. (1995):

Reject if

k, d such that -level is kept.

The null rejection prob. r0 can be written as a function of bivariate normal cdfs.

The derivative can be written as a function of bivariate normal cdfs and pdfs.

Using these two components, we can let the computer search for d corresponding to given k (or vice versa).

dVu

dVkZZ

if,

if ,),min(

121

0r


What can be inferred about the derivative

. As δ from 0, for all δ < some δ+.

For δ→∞: → 0.

There is either no or one δ* where .

If there is, for < * and for > *,

so r0 has a maximum in *.

0)0(0 r

0r

0*0 r

0)(0 r

0)(0 r

0r


Some remarks on computer implementation

k is fixed.

For given d, calculate at = 4.5

If this is <0, decrease d.

Stop if

Idea: If the conditions

hold, 0. = 4.5 is „close enough“ to .

This approach finds d within a few steps.

0r

60 100

r

11 , uVcuVcV

00r


Modified Suggestion: linearized conditional test

Reject if

With this, it is also possible to write down the rejection prob r0 and its derivative

Need to find k,c and d. To limit options, k=0 and c= u1/d were assumed, so just search for d.

Same search algorithm as before.

For non-linear c(|V|), I did not try to work out r0

(maybe possible for special functions).

.0

r

dVu

dVVckZZ

if,

if ,),min(

121


Rejection probability of conditional tests

Rejection probability is highest at max d which has k =

Once k > , power relatively quickly min test as d

0.01

0.02

0.03

0.04

0.05

0 0.5 1 1.5 2 2.5 3

reje

ctio

n p

rob

d=0.02

d=0.03

d=0.033, k=0

d=0.04

d=0.1

linear, d=0.047, k=0

min test


More Modifications

With both variants, we may want to allow

r0 approaches a value < as δ→∞.

Resulting test is no longer UMP than min test, but

we gain more power (in the vicinity of H0) for small δ.

Here, k, d, c are even easier to find:

The max r0 is at δ where

1ucVcV

.00 r


More Modifications: „Maximin test“

Idea: Find c(|V|) such that r0(0)=r0(∞).

This test maximizes the minimum rejection probability among all conditional tests.

Results:

Sarkar test with k=0: d=0.08025, c=1.767

r0(0)=r0(∞)=0.0386.

Linearized test with k=0: d=0.2125, c=8.539, c=1.81 r0(0)=r0(∞)=0.0348.


Rejection probability of „maximin“ test (k=0)

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

delta

reje

ctio

n pr

ob

min test

d=0.033

minimax

linear minimax

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

delta

reje

ctio

n pr

ob

min test

d=0.033

maximin

linear maximin


A remark on the power of conditional tests

Suppose

These tests do not dominate each other.

As and δ increase, the tests with large k, d overtake tests with small k, d.

Unfortunately, „real gains“ coincide with low power: Power gain over Min test (all at δ = 0):

- =0.8: k=0: 10.4%; Min test: 8.7% (max absolute gain, 1.73%)

- =2: k=0: 49.2%; Min test: 48.3%

- =2.8: k=1.3: 79.9%; Min test: 79.6%

- =3: k=1.5: 85.4%; Min test: 85.2%

./2

BABn


A few remarks on generalizations

Unequal n‘s: No problem. • The in the bivariate Normal distribution changes, so k, c and d

change, but approach remains the same.

Estimated s instead of known : In principle, same approach.

• Rejection prob a sum of bivariate t- rather than normal cdfs.

• Basic idea for constructing a UMP conditional test works the same.

• k, c and d can be found by a grid search.

> 2 monos: Again, in principle same approach, but gets messy: • more than one δ to be considered.

• Generalization of rule „if |V|< d“ to V1, ..., Vg not obvious.


Contentious issues about conditional tests

If we allow k<0, it is possible that we identify the combi as superior, although its observed average is lower than the better of the monos.

→ This can be avoided by requiring k 0.

Non-monotonicity: It can happen thatrejects, but does not, although

ABAB xxx ,,

**,, ABAB xxx

**, AABB xxxx

(However, we should keep in mind: The power never only depends on ).),max( BAAB


Conclusions

„Conditional“ non-monotone tests are UMP than the Laska-Meisner min test.

There is not that much to be gained:

The power depends on .

Even for modest n, the region where the min test‘s r0m<< is very small. E.g. n=8, (B A)/ =1 has r0m = 0.0471.

d is also very small. Only if the monotherapies arereally similar, this makes a difference.

/2 AB

n


Conclusions

Power profile is primarily driven by choice of k, irrespective of the variant of the conditional test.

Gains over the Min test are in the „wrong places“:• They are where power is low (10%). Here, small values of k

are best.

• At powers that matter to the pharma industry, „biggest“ gains are achieved for large k, but are generally very small (<<1%).

k and d are easy to obtain with a relatively simple search algorithm on a computer.

In practice, we‘ll rarely experience a difference from the Min test (with k=0, P( |V|<d |=0)=2.6%).


Literature

Laska, E.M. and Meisner, M. (1989): Testing whether an identified treatment is best. Biometrics 45, 1139-1151.

Hung, H.M.J. (2000): Evaluation of a combination drug with multiple doses in unbalanced factorial design clinical trials. Statistics in Medicine 19, 2079-2087.

Sarkar, S.K., Snapinn, S., and Wang, W. (1995): On improving the min test for the analysis of combination drug trials. Journal of Statistical Computation and Simulation 51, 197-213.

Chuang-Stein, C., Stryszak, P., Dmitrienko, A., Offen, W. (2007): Challenge of multiple co-primary endpoints: a new approach. Statistics in Medicine 26, 1181-1192.

Testing treatment combinations versus the corresponding monotherapies in clinical trials

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