ferring pharmaceuticals the extended williams’ trend test - background and practical example...
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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