1. Input
Median Survival Time for Historical Control \((m_c)\):
median survival time for historical control.
Median Survival Time for Treatment (\(m_t)\):
median survival time for treatment.
Length of Accrual Period \(A\):
the
time from starting patient enrollment to completing patient enrollment.
Maximum Follow-up Time \(L\):
\(L=A+F\) where \(A\) is length of accrual period, and \(F\) the additional
follow-up time after accrual completes.
The following figure shows the three notations in an intuitive way.
Type I Error (\(\alpha\)):
false positive rate.
Power (\(1-\beta\)):
where \(\beta\) is false negative rate.
2. Example
(a one-sample design)
Suppose an investigator is interested in determining whether a new treatment is better than
a historical control in terms of survival time. The median survival time for the historical control is 3 months,
and the investigator hypothesizes that the median survival time for the new treatment is 7 months.
The trial will have an accrual period of 3 months, and patients are followed for an additional 6 months. At the significance level
of 0.05, how many subjects are needed to have a power of 0.8 to test his hypothesis with a one-sided test?
Input:
\( m_c=3, m_t=7, A=3, L=3+6=9,\alpha=0.05, 1-\beta=0.8\).
Output:
Given an accrual period of 3 months, a maximum follow-up time of 9 months, at the significance level of 0.05,
to achieve the power of 0.8, the number of events required is
9
. Equivalently, a sample size of
17
is needed. The result is based on a one-sided test with exponential assumption for survival time.
Control (\(m_c)\):
median survival time for control;
Treatment (\(m_t)\):
median survival time for treatment.
Control (\(\lambda_c)\):
hazard rate for control;
Treatment (\(\lambda_t)\):
hazard rate for treatment.
Subject Allocation Ratio of Treatment vs. Control (\(k=n_t/n_c\)):
the ratio of number of subjects assigned to
treatment to the number subjects in control
where \(n_t, n_c\) are sample size for treatment and control, respectively.
Length of Accrual Period \(A\):
the
time from starting patient enrollment to completing patient enrollment.
Maximum Follow-up Time \(L\):
\(L=A+F\) where \(A\) is length of accrual period, and \(F\) the additional
follow-up time after accrual completes.
The following figure shows the three notations in an intuitive way.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power (\(1-\beta\)):
where \(\beta\) is false negative rate.
2. Example
(a two-sample design)
Suppose an investigator is interested in determining if a treatment is
better than a control. The trial will have an accrual
period of 12 months, and patients are followed for an additional 6 months.
The investigator hypothesizes that the median survival time for control is 10 months and for treatment is 20 months.
At the significance level of 0.05,
how many subjects are needed for each group, in a one-sided test, to achive the power 0.8 using equal allocation?
Input:
\(m_c=10, m_t=20, k=n_t/n_c=1, A=12, L=12+6=18, \alpha=0.05, 1-\beta=0.8\).
Output:
Given an accrual period of 12 months, a maximum follow-up time of 18 months, at the significance level of 0.05,
to achieve the power of 0.8, the number of events required is
52
. Or equivalently,
a total number of
118
subjects are needed. With equal allocation, each treatment needs to enroll
59
subjects.
The result is based on a one-sided test with exponential assumption for survival time.
Control (\(m_c)\):
median survival time for control;
Treatment (\(m_t)\):
median survival time for treatment.
Control (\(\lambda_c)\):
hazard rate for control;
Treatment (\(\lambda_t)\):
hazard rate for treatment.
Equivalence Limit \((\delta>0)\):
length of margin, which is called (1) equivalence limit in equivalence test,
(2) noninferiority margin in nonferiority test, (3) supriority margin when supriority test is of interest. The difference of the three types of tests can be shown intuitively
in the following Figure. When \(log(m_t/m_c)\)=0, the treatment is the same as the control, as \(log(m_t/m_c)\) increase, treatment is better than the control
(more details can be found under Document).
Subject Allocation Ratio of Treatment vs. Control (\(k=n_t/n_c\)):
the ratio of number of subjects assigned to
treatment to the number subjects in control
where \(n_t, n_c\) are sample size for treatment and control, respectively.
Length of Accrual Period \(A\):
the
time from starting patient enrollment to completing patient enrollment.
Maximum Follow-up Time \(L\):
\(L=A+F\) where \(A\) is length of accrual period, and \(F\) the additional
follow-up time after accrual completes.
The following figure shows the three notations in an intuitive way.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power (\(1-\beta\)):
where \(\beta\) is false negative rate.
2. Example
(Equivalence test for two-sample design)
Suppose an investigator is interested in demonstrating that a treatment is as good as the current control. The trial will have an accrual
period of 12 months, and patients are followed for an additional 6 months.
The investigator hypothesizes that the median survival time for control is 9 months and for treatment is 10 months.
At the significance level of 0.05, given an equivalence limit of 0.2,
how many subjects are needed for each group to achive the power 0.8 using equal allocation?
Input:
\( \delta=0.4, m_c=9, m_t=10, k=n_t/n_c=1, A=12, L=12+6=18, \alpha=0.05, 1-\beta=0.8\).
Output:
Given an accrual period of 12 months, a maximum follow-up time of 18 months, at the significance level of 0.05,
to achieve the power of 0.8, the number of events required is
395
. Or equivalently,
a total number of
694
subjects are needed. With equal allocation, each treatment needs to enroll
347
subjects.
The result is based on the assumption that the median survival time for treatment is 9.5 months, and for control is 10 months.
Control (\(m_c)\):
median survival time for control;
Treatment (\(m_t)\):
median survival time for treatment.
Control (\(\lambda_c)\):
hazard rate for control;
Treatment (\(\lambda_t)\):
hazard rate for treatment.
Noninferiority Margin \((\delta>0)\):
length of margin, which is called (1) equivalence limit in equivalence test,
(2) noninferiority margin in nonferiority test, (3) supriority margin when supriority test is of interest. The difference of the three types of tests can be shown intuitively
in the following Figure. When \(log(m_t/m_c)\)=0, the treatment is the same as the control, as \(log(m_t/m_c)\) increase, treatment is better than the control
(more details can be found under Document).
Subject Allocation Ratio of Treatment vs. Control (\(k=n_t/n_c\)):
the ratio of number of subjects assigned to
treatment to the number subjects in control
where \(n_t, n_c\) are sample size for treatment and control, respectively.
Length of Accrual Period \(A\):
the
time from starting patient enrollment to completing patient enrollment.
Maximum Follow-up Time \(L\):
\(L=A+F\) where \(A\) is length of accrual period, and \(F\) the additional
follow-up time after accrual completes.
The following figure shows the three notations in an intuitive way.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power (\(1-\beta\)):
where \(\beta\) is false negative rate.
2. Example
( Noninferiority test for two-sample design)
Suppose an investigator is interested in determining if a new treatment is not appreciably worse than
a control. The trial will have an accrual
period of 12 months, and patients are followed for an additional 6 months.
The investigator hypothesizes that the median survival time for control is 10 months and for treatment is 9 months.
At the significance level of 0.05,given a noninferiority margin of 0.2,
how many subjects are needed for each group to achive the power 0.8 using equal allocation?
Input:
\( \delta=0.5, m_c=10, m_t=9.5, k=n_t/n_c=1, A=12, L=12+6=18, \alpha=0.05, 1-\beta=0.8\).
Output:
Given an accrual period of 12 months, a maximum follow-up time of 18 months, at the significance level of 0.05,
to achieve the power of 0.8, the number of events required is
123
. Or equivalently,
a total number of
220
subjects are needed. With equal allocation, each treatment needs to enroll
110
subjects.
Control (\(m_c)\):
median survival time for control;
Treatment (\(m_t)\):
median survival time for treatment.
Control (\(\lambda_c)\):
hazard rate for control;
Treatment (\(\lambda_t)\):
hazard rate for treatment.
Superiority Margin \((\delta>0)\):
length of margin, which is called (1) equivalence limit in equivalence test,
(2) noninferiority margin in nonferiority test, (3) supriority margin when supriority test is of interest. The difference of the three types of tests can be shown intuitively
in the following Figure. When \(log(m_t/m_c)\)=0, the treatment is the same as the control, as \(log(m_t/m_c)\) increase, treatment is better than the control
(more details can be found under Document).
Subject Allocation Ratio of Treatment vs. Control (\(k=n_t/n_c\)):
the ratio of number of subjects assigned to
treatment to the number subjects in control
where \(n_t, n_c\) are sample size for treatment and control, respectively.
Length of Accrual Period \(A\):
the
time from starting patient enrollment to completing patient enrollment.
Maximum Follow-up Time \(L\):
\(L=A+F\) where \(A\) is length of accrual period, and \(F\) the additional
follow-up time after accrual completes.
The following figure shows the three notations in an intuitive way.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power (\(1-\beta\)):
where \(\beta\) is false negative rate.
2. Example
( Superiority test for two-sample design)
Suppose an investigator is interested in demonstrating that a new treatment is superior to the current control.
The trial will have an accrual
period of 12 months, and patients are followed for an additional 6 months.
The investigator hypothesizes that the median survival time for control is 10 months and for treatment is 18 months.
At the significance level of 0.05, given a superiority margin of 0.2,
how many subjects are needed for each group to achive the power 0.8 using equal allocation?
Input:
\( \delta=0.2, m_c=10, m_t=18, k=n_t/n_c=1, A=12, L=12+6=18, \alpha=0.05, 1-\beta=0.8\).
Output:
Given an accrual period of 12 months, a maximum follow-up time of 18 months, at the significance level of 0.05,
to achieve the power of 0.8, the number of events required is
165
. Or equivalently,
a total number of
362
subjects are needed. With equal allocation, each treatment needs to enroll
181
subjects.