1. 臨床試驗 Analysis of Clinical Data (III) Sample Size, Power, and Interim Analysis
授課老師: 劉仁沛教授
國立台灣大學
與 國家衛生研究院
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2. Outlines Sample Size Estimation Interim Analyses Continuous
Categorical
Censored
Interim Analyses
Group sequential methods
Data monitoring

3. Reading Sample Size Estimation: Chapter 11 (p.441- p.518)
Chow, S. C., and Liu, J.P. (2014) Design and Analysis of Clinical Trials: Concepts and Methodologies, 3rd Ed., Wiley, New York, New York.
Interim Analysis: Chapter 10 (p.419 – p.437)

4. Sample Size Estimation
Introduction
E = experimental treatment group
C = control treatment group
We consider
Mean difference
Difference in proportions
Time to event

5. Notation  = type I error  = type II error
Zα= the th normal quartile: Level of Significance
P{ Z > Z} = , one sided
P{|Z| > Z /2} = , two sided
Zβ= the th normal quartile: Level of Power
P{Z < Z} = 1 - 

6. Typical Design Assumptions
Two Sided
Significance Level
Power

7. Basic Formula

8. Basic Formula

9. Basic Formula

10. Basic Formula

11. Assume Equal Allocation
Mean difference: E – C

12. Example (Continuous Endpoints)

13. Risk difference: PE - PC

14. Example Primary efficacy endpoint: 3-year mortality rate
Control group: 60%
Expected mortality rate of the experiment group: 40%
2-tailed test at 5% significance level
Desired power: 80%

15. Example

16. Unequal Allocation

17. Relative Risk: PE / PC

18. Example

19. McNemar test for paired design
Diagnosis for test marker Diagnosis for reference marker Total
Yes (1) No (0)
Yes (1) x11(p11) x10(p10) xt(pt)
No (0) x01(p01) x00(p00) n-xt(1-pt)
Total xr(pr) n-xr(1-pr) n

20. McNemar test for paired design

21. Example
A new diagnostic device is expected to improve the accuracy for detection of positive rate by 0.15, i.e., pt – pr =0.15
Assume that p01 = 0.05  p10 = 0.20  (p01 + p10)/2 = 0.125
For = 0.05 and =0.10, n=80

22. Time to Event --- Assume exponential Distribution

23. Time to Event --- Assume exponential Distribution
Patient’s enter the trial at a uniform rate over a T-year period. If the trial terminates at a time T, then

24. Time to Event --- Assume exponential Distribution
Patients are recruited over the interval (0, T0), but with a follow up until T, then

25. Example (Censored Data)

26. Example (Censored Data)

27. Non-inferiority Trials

28. Non-inferiority Trials

29. Example
Experiment: lifestyle modification package for treatment of hypertension: relaxation therapy and diet
Control: standard drug treatment
Proportion of BP under control is assumed for both groups
Experiment therapy is considered non-inferior if the proportion of BP control is no more than 10% less than that of control group

30. Example

31. Example

32. Example

33. Two-sided equivalence
For two parallel groups

34. Two-sided equivalence
General Formulas for sample size per group

35. Two-sided equivalence

36. Account for Patient dropout due to loss-to-follow-up
Missing at random?
If not, the missing data introduce bias
If MAR, effective size is reduced
d is the dropout rate
n* = n/(1-d)

37. Account for Patient dropout due to nonaderence to treatment
Drop-out rate in group E

38. Sample Size Re-estimation
Requirement for sample size
Clinically meaningful difference
variability
Estimate of variability may not be adequate at planning stage of the trial
New class of drug
History of disease …
Adjustment of sample size during the trial

39. Sample Size Re-estimation

40. Sample Size Estimation
All formulas for sample size estimation provide the minimal number of patients required for trials.
Sample size increases in square as the expected difference decreases or the standard deviation increases.
Sample size increase four times as the expected difference decreases 50% or standard deviation doubles.
Sample size increases if the significance level decreases or power increases.

41. Sample Size Estimation
Website Resource
MD Anderson Cancer Center
Very comprehensive
SAS PROC POWER

42. Interim analyses and Data Monitoring
Limited and Finite Resource
Ethical
Scientific
Cost
Patients
$$$
Time
Optimization of available resource.

43. Information Sample mean of N observations
The variance of sample mean =2/N.
The information about the population mean provided by the sample mean is N/2.
If 2 =1, the information about the population mean provided by the sample of N observations is simply the sample size N.

44. Information
This is the definition of statistical information which can be also interpreted as the clinical information
The planned sample size is the minimum clinical and statistical information required to achieve the desired power for detecting a minimum clinically meaningful difference at a pre-determined risk of type I error.

45. Group Sequential Procedures
Trial
Two parallel groups, double-blinded randomized, FIXED SAMPLE.
Number of patients is sufficient large
Z statistics can be computed
Reject the null hypothesis and conclude there is a statistically significant difference between two groups at 5% nominal level if
absolute value of Z > 1.96
i.e., either Z < or Z > 1.96

46. Interim Analyses Group Sequential Procedures Two extreme Cases
Single-stage (Fixed sample)
Classical Sequential (Sequential likelihood ratio test)
Intermediate Case
Group sequential or multi-stage trials with some adjustments in type I error probabilities at successive stages.
WARNING !!!
Failure to do so
Actual probability of typeⅠerror >>> Nominal probability of typeⅠerror

47. Two parallel groups A total of N patients is planned to recruit
The number of interim analyses is pre-determined in advance in the protocol.
K interim analyses (stages) including the final analysis
n/2 patients per group at each stage, n = N/k
Stage
Test Drug
Placebo
Information Fraction
Z-statistic 1
n/2
1/k(n) Z1 2
2n/2
2/k(2n) Z2 3
3n/2
3/k(3n) Z3 K
kn/2
1(kn) Zk
At each of the K stages compute the Z-statistics.

48. Repeated Significance Test on Accumulated Data
The Number of Repeated Test at the 5% level
Overall Significance level 1
0.05 2
0.08 3
0.11 4
0.13 5
0.14 10
0.19 20
0.25 50
0.32
100
0.37
1000
0.53 ∞
1.00
Repeated testing increases probability of typeⅠerror
Must make adjustment for nominal significant level to be conservative

49. Group Sequential Procedures
Idea
Compute summary (Z) statistic at each interim analysis, based on additional groups of new patients
Compare statistics to a conservative critical value for an overall 0.05 level of type Ⅰ error probability

50. Haybittle-Peto method Haybittle (1971), Peto, et al. (1976)
Two parallel groups
A total of N patients is planned to recruit
K interim analyses (stages) including the final analysis
N/2 patients per group, at each stage, n = N/k
Simple, Ad Hoc
For interim analyses, use conservative critical values e.g. ± 3.0
For final analysis, no adjustment is required, i.e., use ± 1.96
At each stage compute Z-statistic
If Z-statistic crosses ±3.0, then reject the null hypothesis and recommend the possibility of terminating the trial.
Final use ± 1.96
Otherwise, continue the trial to the next stage

51. Pocock’s method Pocock (1977)
Two parallel groups
A total of N patients is planned to recruit
K interim analyses (stages) including the final analysis
N/2 patients per group, at each stage, n = N/k
For interim analyses, use a conservative critical values e.g. ± Ck
The critical values depend upon the number of planned interim analyses
At each stage compute Z-statistic
If Z-statistic crosses ± Ck, then reject the null hypothesis and recommend the possibility of terminating the trial.
Otherwise, continue the trial to the next stage

52. Boundaries for  = 0.025 and K = 5
Time
2*(t)
Pocock
0.2
2.44
2.41
0.4
2.43
0.6
0.8
2.40
1.0
2.39

53. O’Brien-Fleming’s Method O’Brien and Fleming (1979)
Two parallel groups
A total of N patients is planned to recruit
K interim analyses (stages) including the final analysis
N/2 patients per group, at each stage, n = N/k
For interim analyses, use a conservative critical values e.g. ± Cik
The critical values depend upon the number of planned interim analyses and stage of interim analysis
At each stage compute Z-statistic
If Z-statistic crosses ± Cik, then reject the null hypothesis and recommend the possibility of terminating the trial.
Otherwise, continue the trial to the next stage

54. Boundaries for  = 0.025 and K = 5
Time
1*(t)
O'Brien and Fleming
0.2
4.90
4.56
0.4
3.35
3.23
0.6
2.68
2.63
0.8
2.29
2.28
1.0
2.03
2.04

56. Group Sequential Procedures
Limitations
Number of planned interim analyses specified in advance
Require equal increment of patients for each stage
Limit Data Monitoring Committee
Ethical concerns
Scientific concerns

57. Spending Functions DeMets and Lan (1994)
Extend previous group sequential procedures to gain more flexibility
Define a function to spend the overall nominal significance level
The spending functionα*(t) is a increasing function of information fraction t
Specify the spending function in advance
Can not change the spending function during the trial

58. Spending Functions DeMets and Lan (1994)

59. Spending Functions DeMets and Lan (1994)
Compute the nominal significance level you need to spend at an interim analysis based on information fraction,
i.e., α*(t2) - α*(t1), t1<t2
Compute the corresponding critical values and perform the interim analyses as other group sequential methods
No need to specify the number of interim analyses
No need to specify the time to perform interim analyses

60. Different Forms of Alpha Spending Functions
Approximation
α1(s)=2{1-φ[z(α/2)/√s]}
O’Brien-Fleming
α2(s)=αln[1+(e-1)s]
Pocock
α3(s)=αsθ, θ>0
Lan-DeMets-Kim
α4(s)=α[(1-e-ζs)/(1-e-ζ)], ζ≠0
Hwang-Shih

61. Examples of Boundaries by Alpha Spending Function
Interim Analysis (s)
O’Brien-Fleming
α1(s)
Pocock
α2(s)
α3(s)[θ-1]
1(0.2)
4.56
4.90
2.41
2.44
2.58
2(0.4)
3.23
3.35
2.43
2.49
3(0.6)
2.63
2.68
4(0.8)
2.28
2.29
2.40
2.34
5(1.0)
2.04
2.03
2.39
Note: Number of interim Analyses =5.
Two-sided: 0.05, One-sided: 0.025

63. Cumulative Probability of TypeⅠError
Group Sequential Boundaries
Interim analysis(s)
Pocock
O’Brien-Fleming
Value
α(s)
Increment
1(0.2)
2.41
0.0079
4.56
0.0000
2.6×10-6
2(0.4)
0.0138
0.0059
3.23
0.0006
3(0.6)
0.0183
0.0045
2.63
0.0039
4(0.8)
0.0219
0.0036
2.28
0.0125
0.0080
5(1.0)
0.0250
0.0031
2.04

64. Computation of Spending Probability and Boundaries
P{Z(0.2) >4.56} = 2.6x10-6
P{Z(0.2) >4.56 or Z(0.4) > 3.23}
= P{Z(0.2) >4.56} + P{Z(0.2)  4.56 and Z(0.4) > 3.23}
= 2.6x =
P{Z(0.2) >4.56 or Z(0.4) > 3.23 or Z(0.6) > 2.63}
= P{Z(0.2) >4.56 or Z(0.4) > 3.23} + P{Z(0.2)  4.56 and Z(0.4)  3.23 and Z(0.6) > 2.63}
= =
P{Z(0.2) >4.56 or Z(0.4) > 3.23 or Z(0.6) > 2.63 or Z (0.8) > 2.28}
= P{Z(0.2) >4.56 or Z(0.4) > 3.23 or Z (0.6) > 2.63 } +
P{Z(0.2)  4.56 and Z(0.4)  3.23 and Z(0.6)  2.63 and Z(0.8) > 2.28}
= =
P{Z(0.2) >4.56 or Z(0.4) > 3.23 or Z(0.6) > 2.63 or Z (0.8) > 2.28 or Z(1.0) >2.04}
= P{Z(0.2) >4.56 or Z(0.4) > 3.23 or Z (0.6) > 2.63 or Z(0.8) > 2.28}
+ P{Z(0.2)  4.56 and Z(0.4)  3.23 and Z(0.6)  2.63 and Z(0.8)  2.28 and Z(1.0) > 2.04}
= =

65. Examples Beta-Blocker Heart Attack Trial (BHAT) Design Features
Informed consent
3,837 patients
Randomized
Male and females
Double-blinded
30-69 years of age
Placebo-controlled
5-21 days post M.I.
Extended follow-up
Propranolol or 240 mg/day
Preliminary report (1981) JAMA, 246:
Final report (1982) JAMA, 247:

66. BHAT Accumulating Survival Data
Date of DMC Meeting
Propranolol
Placebo
Z (log rank)
May 1979
22 / 860
34 / 848
1.68
October 1979
29 / 1080
48 / 1080
2.24
March 1980
50 / 1490
76 / 1486
2.37
October 1980
74 / 1846
103 / 1841
2.30
April 1981
106 / 1916
141 / 1921
2.34
October 1981
135 / 1916
183 / 1921
2.82*
* DMC terminated the BHAT

67. Women Health Initiatives (WHI)
A randomized, DB placebo-controlled primary prevention trial
Investigate the benefits and risks of hormone replacement therapy (HRT)
Estrogen + progestin vs. Placebo
16608 healthy postmenopausal women aged years with intact uterus recruited between 1993 and 1998 with expectation of final analysis in 2005 after an average of approximately 8.5 years of follow-up

68. Women Health Initiatives (WHI)
Primary efficacy endpoint
incidence of CHD
Primary safety endpoint
incidence of invasive breast cancer
Competing benefits and risks
colorectal cancer, hip fracture
stroke, pulmonary embolism, endometrial cancer, death due to other causes

69. Women Health Initiatives (WHI)
A weighted global index (Freedman et al, 1996)
W = w1d1 + … + w8d8
di: difference in proportions between the two groups for outcome i, i =1,…8
wi: weights for outcome i – expected proportion of diagnosed patients who will die of that disease within a specific years of diagnosis
Benefits and risks are not symmetric

70. Women Health Initiatives (WHI)
A mixed approach to early termination
1. O-B boundaries for each of eight outcomes and for global index
2. Asymmetric upper and lower boundaries:
one-sided  = for benefit
one-sided  = 0.05 for adverse effects
Adverse-effect boundaries were adjusted using Bonferroni method
3. Trial stops if the upper or lower boundaries were crossed and the result from global index was supportive at 0.20 level

71. Women Health Initiatives (WHI)
Semiannual DSMB meeting since the fall of 1997
The tenth interim analysis on May 31, 2002
1. The weighted log-rank test statistic z = crossed the lower boundary for adverse event z= -2.32
2. The global index is supportive (z = -1.62)
3. Additional evidence of risks on CHD, stroke, pulmonary embolism outweighed the evidence of benefit on hip fracture and colon cancer
4. DSMB recommended early termination of the estrogen plus progestin component of WHI
Writing group for WHI (2002) JAMA 288:

72. DMC in Pharmaceutical Industry
Pivotal phase III trials
Endpoints
Survival
Mortality
Irreversible morbidity
Myocardial infarction
stroke

73. Methods Used Totally in-house (not a good practice) External DMC
Internal analysis and data process (not a good practice)
Internal data process
External data analysis
External data process
Totally “hand off”
Representative from sponsor on DMC

74. External DMC Internal data process External data analysis
Sponsor at open session only
Sponsor at both open and closed session (not a good practice)
Limited representation and confidentiality

75. Decision Philosophy Ahead of time Session Format Positive trend
Negative trend
Session Format
Open (blinded) Progress
Recruiting, logistics
Closed (A vs. B) Clinical Endpoints
Safety, Efficacy
Executive Decisions
Who breaks the codes

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《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.449。本作品依據著作權法第 46、52、65 條合理使用。 12
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.455。本作品依據著作權法第 46、52、65 條合理使用。 19
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.485。本作品依據著作權法第 46、52、65 條合理使用。 39
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.518。本作品依據著作權法第 46、52、65 條合理使用。

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《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.427。本作品依據著作權法第 46、52、65 條合理使用。
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《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.428。本作品依據著作權法第 46、52、65 條合理使用。 55
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.429。本作品依據著作權法第 46、52、65 條合理使用。

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《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.430。本作品依據著作權法第 46、52、65 條合理使用。 58
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.431。本作品依據著作權法第 46、52、65 條合理使用。 59 60 61
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.432。本作品依據著作權法第 46、52、65 條合理使用。

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《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.433。本作品依據著作權法第 46、52、65 條合理使用。 63
《Design and analysis of clinical trials: concepts and methodologies》， 作者:Chow, SC, Liu, JP ，出版社: Wiley(third edition)，p.432。本作品依據著作權法第 46、52、65 條合理使用。