1. Input
Treatment Effect (\(\delta \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the offsite patients.
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a cross-sectional, fully decentralized clinical trial,
the randomization ratio between the experimental
arm and the control arm is 1:1. For offsite patients,
the standard deviation of the endpoint is 20.
The target treatment effect is an increase of 10 in the mean
of the endpoint with respect to the control. The goal is to determine the sample size
needed to detect this treatment effect with a power of 0.8 at a two-sided significance level of 0.05
Input:
\( \lambda=1, \sigma=20,
\delta = 10, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 128 participants
with 64 participants in the control arm and 64 participants in the experimental arm
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms, based on a weighted z-test.
1. Input
Treatment Effect (\(\delta \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the offsite patients.
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a cross-sectional, fully decentralized clinical trial,
the randomization ratio between the experimental
arm and the control arm is 1:1. For offsite patients,
the standard deviations of the endpoint for the control
and treatment arms are 20 and 25, respectively.
The target treatment effect is an increase of 10 in the mean
of the endpoint with respect to the control. The goal is to determine the sample size
needed to detect this treatment effect with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(\lambda=1, \sigma_{c}=20, \sigma_{e}=25,
\delta = 10, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 164 participants
with 82 participants in the control arm and 82 participants in the experimental arm
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms, based on a weighted z-test.
1. Input
Treatment Effect for Onsite Patients (\(\delta_1 \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the onsite patients.
Treatment Effect Relative Bias for Offsite Patients (\(\pi \)):
The relative bias of the treatment effect for the offsite patients,
suggesting that treatment effect (\(\delta_2\)) for the offsite patients is a biased value compared to
the treatment effect (\(\delta_1\)) for the onsite patients, i.e., \(\delta_2 = (1+\pi)\delta_1 \) .
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Allocation Ratio (\(r\)):
the ratio between the number
of offsite patients and the number of onsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a cross-sectional, partially decentralized clinical trial,
the ratio between the number of offsite patients and the number of
onsite patients is 3:1. The randomization ratio between the experimental
arm and the control arm is 1:1. The standard deviations of the endpoint for onsite
and offsite patients are 20 and 25, respectively. The target treatment effect (primary estimand) is an
increase of 10 in the mean of the endpoint with respect to the control for onsite patients.
The treatment effect for offsite patients will have a relative bias of -0.2. The goal is
to determine the sample size needed to detect the onsite treatment effect,
accounting for the corresponding biased estimate for the offsite patients,
with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(r=3, \lambda=1, \sigma_{1}=20, \sigma_{2}=25,
\delta_{1} = 10, \pi = -0.2, \delta_{2} = 8, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 228 participants
with 58 onsite participants (29 per arm)
and 170 offsite participants (85 per arm)
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms for the onsite patients, based on a weighted z-test, accounting for the
potential different data variability and relative bias for offsite patients.
1. Input
Treatment Effect (\(\delta_1 \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the onsite patients.
Treatment Effect Relative Bias for Offsite Patients (\(\pi \)):
The relative bias of the treatment effect for the offsite patients,
suggesting that treatment effect (\(\delta_2\)) for the offsite patients is a biased value compared to
the treatment effect (\(\delta_1\)) for the onsite patients, i.e., \(\delta_2 = (1+\pi)\delta_1 \) .
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Allocation Ratio (\(r\)):
the ratio between the number
of offsite patients and the number of onsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a cross-sectional, partially decentralized clinical trial,
the ratio between the number of offsite patients and the number of
onsite patients is 3:1. The randomization ratio between the experimental
arm and the control arm is 1:1. For onsite patients, the standard deviations
of the endpoint for the control and treatment arms are 20 and 25, respectively.
For offsite patients, the standard deviations of the endpoint for the control
and treatment arms are 25 and 30, respectively.The target treatment effect (primary estimand) is an
increase of 10 in the mean of the endpoint with respect to the control for onsite patients.
The treatment effect for offsite patients will have a relative bias of -0.2. The goal is
to determine the sample size needed to detect the onsite treatment effect,
accounting for the corresponding biased estimate for the offsite patients,
with a power of 0.8 at a two-sided significance level of 0.05
Input:
\( r=3, \lambda=1,\sigma_{1c}=20, \sigma_{1e}=25, \sigma_{2c}=25,
\sigma_{2e}=30, \delta_{1} = 10, \pi=-0.2, \delta_{2} = 8, 1-\beta=0.8, \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 284 participants
with 72 onsite participants (36 per arm)
and 212 offsite participants (106 per arm)
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms for the onsite patients, based on a weighted z-test, accounting for the
potential different data variability and relative bias for offsite patients.
1. Input
Treatment Effect (\(\delta \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the offsite patients.
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Intraclass correlation coefficient (\(\rho \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for offsite patients.
Number of repeated measures or cluster size (\(m \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered clinical trials for offsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a longitudinal, fully decentralized clinical trial,
the randomization ratio between the experimental
arm and the control arm is 1:1. For offsite patients,
the standard deviation of the endpoint is 20. The number of repeated measures per subject is 5,
and the anticipated intraclass correlation coefficient is 0.5 for offsite patients.
The target treatment effect is an increase of 10 in the mean
of the endpoint with respect to the control. The goal is to determine the sample size
needed to detect this treatment effect with a power of 0.8 at a two-sided significance level of 0.05
Input:
\( \lambda=1, \sigma=20,
\rho=0.5, m=5, \delta = 10, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 76 participants
with 38 participants in the control arm and 38 participants in the experimental arm
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms, based on a weighted z-test.
1. Input
Treatment Effect (\(\delta \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the offsite patients.
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Intraclass correlation coefficient (\(\rho \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for offsite patients.
Number of repeated measures or cluster size (\(m \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered clinical trials for offsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a longitudinal, fully decentralized clinical trial,
the randomization ratio between the experimental
arm and the control arm is 1:1. For offsite patients,
the standard deviations of the endpoint for the control
and treatment arms are 20 and 25, respectively. The number of repeated measures per subject is 5,
and the anticipated intraclass correlation coefficient is 0.5 for offsite patients.
The target treatment effect is an increase of 10 in the mean
of the endpoint with respect to the control. The goal is to determine the sample size
needed to detect this treatment effect with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(\lambda=1, \sigma_{c}=20, \sigma_{e}=25,
\rho=0.5, m=5, \delta = 10, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 98 participants
with 49 participants in the control arm and 49 participants in the experimental arm
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms, based on a weighted z-test.
1. Input
Treatment Effect (\(\delta_1 \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the onsite patients.
Treatment Effect Relative Bias for Offsite Patients (\(\pi \)):
The relative bias of the treatment effect for the offsite patients,
suggesting that treatment effect (\(\delta_2\)) for the offsite patients is a biased value compared to
the treatment effect (\(\delta_1\)) for the onsite patients, i.e., \(\delta_2 = (1+\pi)\delta_1 \) .
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Allocation Ratio (\(r\)):
the ratio between the number
of offsite patients and the number of onsite patients.
Intraclass correlation coefficient (\(\rho_1 \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for onsite patients.
Intraclass correlation coefficient (\(\rho_2 \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for offsite patients.
Number of repeated measures or cluster size (\(m_1 \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered trials for onsite patients.
Number of repeated measures or cluster size (\(m_2 \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered clinical trials for offsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a longitudinal, partially decentralized clinical trial,
the ratio between the number of offsite patients and the number of
onsite patients is 3:1. The randomization ratio between the experimental
arm and the control arm is 1:1. The standard deviations of the endpoint for onsite
and offsite patients are 20 and 25, respectively. The number of repeated measures
per subject is 5, and the anticipated intraclass correlation coefficient is 0.5 for both onsite
and offsite patients. The target treatment effect (primary estimand) is an
increase of 10 in the mean of the endpoint with respect to the control for onsite patients.
The treatment effect for offsite patients will have a relative bias of -0.2. The goal is
to determine the sample size needed to detect the onsite treatment effect,
accounting for the corresponding biased estimate for the offsite patients,
with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(r=3, \lambda=1, \sigma_{1}=20, \sigma_{2}=25,
\rho_1=\rho_2=0.5, m_1=m_2=5, \delta_{1} = 10, \pi=-0.2, \delta_{2} = 8, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 136 participants
with 34 onsite participants (17 per arm)
and 102 offsite participants (51 per arm)
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms for the onsite patients, based on a weighted z-test, accounting for the
potential different data variability and relative bias for offsite patients.
1. Input
Treatment Effect (\(\delta_1 \)):
The treatment effect as the true mean difference of the endpoint for
experimental and control arms for the onsite patients.
Treatment Effect Relative Bias for Offsite Patients (\(\pi \)):
The relative bias of the treatment effect for the offsite patients,
suggesting that treatment effect (\(\delta_2\)) for the offsite patients is a biased value compared to
the treatment effect (\(\delta_1\)) for the onsite patients, i.e., \(\delta_2 = (1+\pi)\delta_1 \) .
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Allocation Ratio (\(r\)):
the ratio between the number
of offsite patients and the number of onsite patients.
Intraclass correlation coefficient (\(\rho_1 \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for onsite patients.
Intraclass correlation coefficient (\(\rho_2 \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for offsite patients.
Number of repeated measures or cluster size (\(m_1 \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered trials for onsite patients.
Number of repeated measures or cluster size (\(m_2 \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered clinical trials for offsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a longitudinal, partially decentralized clinical trial,
the ratio between the number of offsite patients and the number of
onsite patients is 3:1. The randomization ratio between the experimental
arm and the control arm is 1:1. For onsite patients, the standard deviations
of the endpoint for the control and treatment arms are 20 and 25, respectively.
For offsite patients, the standard deviations of the endpoint for the control
and treatment arms are 25 and 30, respectively. The number of repeated measures per subject is 5,
and the anticipated intraclass correlation coefficient is 0.5 for both onsite
and offsite patients. The target treatment effect (primary estimand) is an
increase of 10 in the mean of the endpoint with respect to the control for onsite patients.
The treatment effect for offsite patients will have a relative bias of -0.2. The goal is
to determine the sample size needed to detect the onsite treatment effect,
accounting for the corresponding biased estimate for the offsite patients,
with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(r=3, \lambda=1, \sigma_{1c}=20, \sigma_{1e}=25, \sigma_{2c}=25, \sigma_{2e}=30,
\rho_1=\rho_2=0.5, m_1=m_2=5, \delta_{1} = 10, \pi=-0.2, \delta_{2} = 8,1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 172 participants
with 44 onsite participants (22 per arm)
and 128 offsite participants (64 per arm)
are needed to achieve 80% power to detect the treatment effect as the mean difference of 10 between
the experimental and the control arms for the onsite patients, based on a weighted z-test, accounting for the
potential different data variability and relative bias for offsite patients.
1. Input
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a cross-sectional, fully decentralized clinical trial,
the randomization ratio between the experimental
arm and the control arm is 1:1. For offsite patients,
the response rates of the binary endpoint for the control
and treatment arms are 0.2 and 0.4, respectively.
The target treatment effect is an increase of 0.2 in the mean
of the endpoint with respect to the control. The goal is to determine the sample size
needed to detect this treatment effect with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(\lambda=1, p_{c}=0.2, p_{e}=0.4,
1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 160 participants
with 80 participants in the control arm and 80 participants in the experimental arm
are needed to achieve 80% power to detect the treatment effect of 0.2 between
the experimental and the control arms, based on a weighted z-test.
1. Input
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Allocation Ratio (\(r\)):
the ratio between the number
of offsite patients and the number of onsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a cross-sectional, partially decentralized clinical trial,
the ratio between the number of offsite patients and the number of
onsite patients is 3:1. The randomization ratio between the experimental
arm and the control arm is 1:1. For onsite patients, the response rates
of the binary endpoint for the control and treatment arms are 0.2 and 0.4, respectively.
For offsite patients, the response rates of the binary endpoint for the control
and treatment arms are 0.25 and 0.4, respectively.The target treatment effect (primary estimand) is an
increase of 0.2 of the response rate with respect to the control for onsite patients.
The treatment effect for offsite patients will have a relative bias of (0.4-0.25)/(0.4-0.2)-1 = -0.25. The goal is
to determine the sample size needed to detect the onsite treatment effect,
accounting for the corresponding biased estimate for the offsite patients,
with a power of 0.8 at a two-sided significance level of 0.05
Input:
\( r=3, \lambda=1,p_{1c}=0.2, p_{1e}=0.4, p_{2c}=0.25,
p_{2e}=0.4, 1-\beta=0.8, \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 246 participants
with 62 onsite participants (31 per arm)
and 184 offsite participants (92 per arm)
are needed to achieve 80% power to detect the treatment effect of 0.2 between
the experimental and the control arms for the onsite patients, based on a weighted z-test, accounting for the
potential different data variability and relative bias for offsite patients.
1. Input
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Intraclass correlation coefficient (Analogue) (\(\rho \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for offsite patients.
Number of repeated measures or cluster size (\(m \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered clinical trials for offsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a longitudinal, fully decentralized clinical trial,
the randomization ratio between the experimental
arm and the control arm is 1:1. For offsite patients,
the response rates of the binary endpoint for the control
and treatment arms are 0.2 and 0.4, respectively. The number of repeated measures per subject is 5,
and the anticipated intraclass correlation coefficient is 0.5 for offsite patients.
The target treatment effect is an increase of 0.2 of the response rate with respect to the control. The goal is to determine the sample size
needed to detect this treatment effect with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(\lambda=1, p_{c}=0.2, p_{e}=0.4,
\rho=0.5, m=5, \delta = 10, 1-\beta=0.8 , \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 96 participants
with 48 participants in the control arm and 48 participants in the experimental arm
are needed to achieve 80% power to detect the treatment effect of 0.2 between
the experimental and the control arms, based on a weighted z-test.
1. Input
Randomization Ratio (\(\lambda\)):
the randomization ratio between the experimental arm and the control arm.
Allocation Ratio (\(r\)):
the ratio between the number
of offsite patients and the number of onsite patients.
Intraclass correlation coefficient (Analogue) (\(\rho_1 \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for onsite patients.
Intraclass correlation coefficient (Analogue) (\(\rho_2 \)):
the correlation coefficient among repeated measures from a subject (for longitudinal trials) or
among subjects in a cluster (for clustered trials) for offsite patients.
Number of repeated measures or cluster size (\(m_1 \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered trials for onsite patients.
Number of repeated measures or cluster size (\(m_2 \)):
the number of repeated measures for longitudinal trials or the cluster size
for clustered clinical trials for offsite patients.
Type I Error Rate (\(\alpha\)):
false positive rate.
Power \((1-\beta)\):
\(\beta\) is the type II error rate
(i.e., false negative rate).
2. Example
For a longitudinal, partially decentralized clinical trial,
the ratio between the number of offsite patients and the number of
onsite patients is 3:1. The randomization ratio between the experimental
arm and the control arm is 1:1. For onsite patients, the response rates
of the binary endpoint for the control and treatment arms are 0.2 and 0.4, respectively.
For offsite patients, the response rates of the binary endpoint for the control
and treatment arms are 0.25 and 0.4, respectively. The number of repeated measures per subject is 5,
and the anticipated intraclass correlation coefficient is 0.5 for both onsite
and offsite patients. The target treatment effect (primary estimand) is an
increase of 0.2 of the response rate with respect to the control for onsite patients.
The treatment effect for offsite patients will have a relative bias of (0.4-0.25)/(0.4-0.2)-1 = -0.25. The goal is
to determine the sample size needed to detect the onsite treatment effect,
accounting for the corresponding biased estimate for the offsite patients,
with a power of 0.8 at a two-sided significance level of 0.05
Input:
\(r=3, \ \lambda=1, \ p_{1c}=0.2, \ p_{1e}=0.4, \ p_{2c}=0.25, \ p_{2e}=0.4, \
\rho_1=\rho_2=0.5, \ m_1=m_2=5, \ 1-\beta=0.8 , \ \alpha=0.05 \), and assume a two-sided test.
Output:
At the two-sided significance level of 0.05, a total of 148 participants
with 38 onsite participants (19 per arm)
and 110 offsite participants (55 per arm)
are needed to achieve 80% power to detect the treatment effect of 0.2 between
the experimental and the control arms for the onsite patients, based on a weighted z-test, accounting for the
potential different data variability and relative bias for offsite patients.