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![Page 1: Ferring Pharmaceuticals The extended Williams’ trend test - Background and practical example Anders Malmberg DSBS Generalforsamling May 26th, 2011.](https://reader037.fdocuments.in/reader037/viewer/2022110210/56649eaa5503460f94baec4b/html5/thumbnails/1.jpg)
Ferring Pharmaceuticals
The extended Williams’ trend test - Background and practical example
Anders Malmberg
DSBS Generalforsamling
May 26th, 2011
![Page 2: Ferring Pharmaceuticals The extended Williams’ trend test - Background and practical example Anders Malmberg DSBS Generalforsamling May 26th, 2011.](https://reader037.fdocuments.in/reader037/viewer/2022110210/56649eaa5503460f94baec4b/html5/thumbnails/2.jpg)
Outline
• BPH
• Degarelix in BPH
• Williams’ trend test
• Conclusion
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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
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Guidelines for Management
• Watchful waiting
• Medical management
• If medical therapy fails: surgery
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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
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Benign prostatic enlargement
The median lobe projects into the base of the bladder
The prostatic urethra narrows
The bladder shows thickening of the wall
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Symptoms of BPH
Storage symptoms
• Frequency
• Nocturia
• Urgency
Voiding symptoms
• Weak stream
• Intermittency
• Incomplete emptying
• Straining
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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
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Outline
• Benign Prostate Hyperplasia
• Degarelix in BPH
• Williams’ trend test
• Conclusion
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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
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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
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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...
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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...
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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
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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
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Outline
• Benign Prostate Hyperplasia
• Degarelix in BPH
• Williams’ trend test
• Conclusion
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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
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• 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
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How to find mean effect level of highest dose group under the restricted alternative...
Click to continue...
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X0
X1
X2 X3
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X0
X1
X2 X3X0<X1 ?
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X1<X2 ?
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M1 = M2<X3 ?
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M1 = M2 = M3
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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
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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 %
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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 %
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... 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
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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
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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
![Page 33: Ferring Pharmaceuticals The extended Williams’ trend test - Background and practical example Anders Malmberg DSBS Generalforsamling May 26th, 2011.](https://reader037.fdocuments.in/reader037/viewer/2022110210/56649eaa5503460f94baec4b/html5/thumbnails/33.jpg)
• 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
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Outline
• Benign Prostate Hyperplasia
• Degarelix in BPH
• Williams’ trend test
• Conclusion
![Page 35: Ferring Pharmaceuticals The extended Williams’ trend test - Background and practical example Anders Malmberg DSBS Generalforsamling May 26th, 2011.](https://reader037.fdocuments.in/reader037/viewer/2022110210/56649eaa5503460f94baec4b/html5/thumbnails/35.jpg)
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
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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