Ferring Pharmaceuticals

The extended Williams’ trend test - Background and practical example

Anders Malmberg

DSBS Generalforsamling

May 26th, 2011

Outline

• BPH

• Degarelix in BPH

• Williams’ trend test

• Conclusion

11%

29%

48%

77%87%

92%

0

20

40

60

80

100

31–40 41–50 51–60 61–70 71–80 80+

Berry SJ et al. J Urol 1984; 132: 474–9

Prevalence of BPH with age

Guidelines for Management

• Watchful waiting

• Medical management

• If medical therapy fails: surgery

Normal bladder and prostate

BPH is the most common benign condition in man

The cause of BPH is multifactorial but there are two essential pre-requisites: the presence of testes and ageing

Benign prostatic enlargement

The median lobe projects into the base of the bladder

The prostatic urethra narrows

The bladder shows thickening of the wall

Symptoms of BPH

Storage symptoms

• Frequency

• Nocturia

• Urgency

Voiding symptoms

• Weak stream

• Intermittency

• Incomplete emptying

• Straining

Symptom Score - IPSS

• Each one of the symptoms is rated on a 0 – 5 scale (0 = not bothersome; 5 = very bothersome)

• Total IPSS = sum of the symptom scores

• Mild patients score 0 – 7

• Moderate patients score 8 – 19

• Severe patients score 20 – 35

• Primary objective of BPH trials is to reduce IPSS score

Outline

• Benign Prostate Hyperplasia

• Degarelix in BPH

• Williams’ trend test

• Conclusion

Degarelix in BPH

Prostate cancer

• Degarelix is marketed for the treatment of prostate cancer in the U.S.A. and EU

• Patients are castrated and growth of prostate is arrested

BPH

• In an earlier phase IIa study, it was found that degarelix can induce a marked but transient testosterone suppression resulting in sustained symptom relief in patients with BPH

• Primary objective of the study was to find a dosing regimen that provides a clinical effect defined as reduction in IPSS at Month 3

Trial design

B: 10 mg degarelix

C: 20 mg degarelix

D: 30 mg degarelix

A: Placebo, mannitol

Screening Follow-up Period Dose

Primary endpoint: Reduction in IPSS at Month 3

End at Month 12

Trial design

B: 10 mg degarelix

C: 20 mg degarelix

D: 30 mg degarelix

A: Placebo, mannitol

Screening Follow-up PeriodDose

Primary endpoint: Reduction in IPSS at Month 3

Interim analysis at Month 6

End of Phase II meeting...

Power calculation (1)

• Expected mean differences in reduction from baseline in IPSS vs placebo at Month 3 is assumed to be 1, 3, and 3 points for the 10, 20 and 30 mg dose group

• Between-subject standard deviation of change from baseline 6 points

• Type I error 5% (two-sided)

• Power of 90% to declare mean IPSS response in both 20 and 30 mg to be significantly different from placebo...

Power calculation: Multiple testing

• Dunnetts’ Type-I error correction for many to one comparison

• Step-down (30 mg vs placebo then 20 mg vs placebo) t-test Williams’ test

Power calculation (2)

• Expected mean differences in reduction from baseline in IPSS vs placebo at Month 3 is assumed to be 1, 3, and 3 points for the 10, 20 and 30 mg dose group

• Between-subject standard deviation of change from baseline 6 points

• Type I error 5% (two-sided)

• Power of 90% to declare mean IPSS response in both 20 and 30 mg to be significantly different from placebo

• The number of patients saved using Williams’ trend instead if t-test is about 36 patients (8 %)

• For our phase II b study this translated to ~ 1.000.000 EUR

Outline

• Benign Prostate Hyperplasia

• Degarelix in BPH

• Williams’ trend test

• Conclusion

Williams’ trend test – background (1)

• Useful when an overall dose related trend is to be expected

• An estimate of target dose is of interest

• Null hypotesis: all means are equal

μ0= μ1= μ2= μ3

• Restrictive alternative hypothesis

μ0<= μ1<= μ2<= μ3, μ0< μ3

• Bartholomew (1961) used the following test statistic:

• van Eeden (1958) derived method for computing mean effect levels under restrictive alternative hypothesis

• Williams (1971) tested highest dose versus control:

Williams’ trend test – background (2)

nsXMW /2/)( 2033

3

0

2223 )(

ii XM

How to find mean effect level of highest dose group under the restricted alternative...

Click to continue...

X0

X1

X2 X3

X0

X1

X2 X3X0<X1 ?

X1<X2 ?

M1 = M2<X3 ?

M1 = M2 = M3

Williams’ trend test – background (4)

- Williams (1971) tested highest dose versus control

- In SAS: probmc("williams",W3,.,3*(n-1),3)

- For step 2 the procedure is repeated with W2

nsXMW /2/)( 2033

Where’s the gain? (1)

Assuming mean differences versus placebo of 1, 3, 3

N=95 per arm and SD=6

power using Williams test = 90 % power with t-test = 88 %

Conditional power, given the estimated shape under the

isotonic restriction

Relative frequency

Williams power

power of t-test

M0 <= M1 <= M2 < M3 50 % 87 % 88 %

M0 <= M1 < M2 = M3 49 % 94 % 87 %

M0 < M1 = M2 = M3 1 % 79 % 79 %

Where’s the gain? (2)

Assuming mean differences versus placebo of 1, 2.5, 3

N=130 per arm and SD=6

power using Williams test = 90 % power with t-test = 90 %

Conditional power, given the estimated shape under the

isotonic restriction

Relative frequency

Williams power

power of t-test

M0 <= M1 <= M2 < M3 74 % 88 % 89 %

M0 <= M1 < M2 = M3 25 % 97 % 94 %

M0 < M1 = M2 = M3 1 % 89 % 79 %

... but

• Williams’ test works only for balanced one-way layouts

• Instead, use the extended Williams’ test (Bretz, 2006)

– General unbalanced linear models

– Accurate computation of p-values using multivariate t-distribution

Numerator of W3 can be written as

Which gives three studentized variables

Where the extended test statistic W3 = max(T1, T2, T3)

4

3

2

1

234423432342

34434313

///1

//01

1001

max

X

X

X

X

nnnnnn

nnnnXM

How Williams’ test is extended

3,2,1,var

iXd

XdT

ti

ti

i

Linear fixed effects model

jTT

jTjj dXXddT

XY

12 )(ˆ/ˆ

• Interested in differences between the adjusted means

• Use the following standardized quantities

• Where Tj j=1,..., 3 are multivariate t with known correlation matrix

• Wrote the solution using matrices

• Considered the multivariate t-disribution of (T1, T2, T3)

– Remember

Prob(max (T1, T2, T3) <= W3) = Prob(T1<=W3, T1<=W3, T1<=W3)

• Used numerical integrations of Genz and Bretz (2002) to calculate the p-value

• SAS code for computing p-values is available for downloading from Bretz’ homepage

Extensions that Bretz made

Outline

• Benign Prostate Hyperplasia

• Degarelix in BPH

• Williams’ trend test

• Conclusion

Conclusions

• Consider to use the extended Williams’ trend test if an overall dose related trend is expected

• The modified version (smoothing also the control grop) is more powerful in concave cases (but will increase p-value since number of dimensions in joint test statistic will increase)

To think of

- Algorithm to estimate target dose

- Confidence interval estimation is not available

References

• Bartholomew, D.J., 1961, A test of homogeneity of means under restriced alternatives J. R. Statisti. Soc. B 23, 239-281

• Barry, M. J., et al., 1995, Benign prostatic hyperplasia specific health status measures in clinical research: How much change in the American Urological Association Symptom index and the Benign prostatic hyperplasia impact index is perceptible to patients? J. Urol., 154, 1770-1774

• Berry S. J., et al., 1984, The Development of human benign prostatic hyperplasia with age J. Urol., 132, 474–9

• Bretz, F., 2006, An extention of the Williams trend test to general unbalanced linear models Comp. Stat. & Data Ana. 50, 1735-1748

• Genz, A., Bretz, F., 2002, Methods for the computation of multivariate t-probabilities J. Comp. Graph. Statist. 11, 950-971

• Marcus, R. 1976, The power of some tests of the equality of normal means against an ordered alternative, Biometrika 63, 177-183

• Williams, D.A., 1971, A test for differences between treatment means when several dose levels are compared with a zero dose control Biometrics 27, 103-117