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
Prevalence of BPH with age
100
87%
77%
80
60
48%
40
20
92%
29%
11%
0
31–40
41–50
51–60
61–70
71–80
80+
Berry SJ et al. J Urol 1984; 132: 474–9
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
Voiding symptoms
• Frequency
• Weak stream
• Nocturia
• Intermittency
• Urgency
• 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
Primary endpoint: Reduction in IPSS at Month 3
End at Month 12
Screening Dose
Follow-up Period
A: Placebo, mannitol
B: 10 mg degarelix
C: 20 mg degarelix
D: 30 mg degarelix
Trial design
Primary endpoint: Reduction in IPSS at Month 3
Interim analysis at Month 6
Screening Dose
Follow-up Period
A: Placebo, mannitol
B: 10 mg degarelix
C: 20 mg degarelix
D: 30 mg degarelix
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
Williams’ trend test – background (2)
• Bartholomew (1961) used the following test statistic:
3
   2 ( M i  X ) 2
2
3
i 0
• van Eeden (1958) derived method for computing mean effect levels under
restrictive alternative hypothesis
• Williams (1971) tested highest dose versus control:
W3  ( M 3  X 0 ) / 2 s 2 / n
How to find mean effect level of highest dose
group under the restricted alternative...
Click to continue...
X1
X2
X0
X3
X1
X0<X1 ?
X0
X2
X3
X1<X2 ?
M1 = M2<X3 ?
M 1 = M2 = M3
Williams’ trend test – background (4)
-
Williams (1971) tested highest dose versus control
W3  ( M 3  X 0 ) / 2 s 2 / n
-
In SAS: probmc("williams",W 3,.,3*(n-1),3)
-
For step 2 the procedure is repeated with W 2
Where’s the gain? (1)
Assuming mean differences versus placebo of 1, 3, 3
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 %
N=95 per arm and SD=6
power using Williams test = 90 % power with t-test = 88 %
Where’s the gain? (2)
Assuming mean differences versus placebo of 1, 2.5, 3
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 %
N=130 per arm and SD=6
power using Williams test = 90 % power with t-test = 90 %
... 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
How Williams’ test is extended
Numerator of W3 can be written as

0
  1

M 3  X 1  max  1
0
  1 n / n
2
234


0
n3 / n34
n3 / n234
Which gives three studentized variables
Ti 
 X 1 
1  
 X 2 
n4 / n34  
X
n4 / n234  3 
 X 4 
d it X

t
i
var d X
Where the extended test statistic W3 = max(T1, T2, T3)

, i  1,2,3
Linear fixed effects model
Y  X  
• Interested in differences between the adjusted means
• Use the following standardized quantities
T
2
ˆ
ˆ
T j  d j  /  d j ( X T X ) 1 d j
T
• Where Tj j=1,..., 3 are multivariate t with known correlation matrix
Extensions that Bretz made
• 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
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, 17701774
• 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