Introduction to Sample size determination
In an
experiment, experimenter is interested in the effect of certain process,
intervention or change (treatment) on targeted objects (experimental units).
Sample size determination
is to decide an appropriate sample size to achieve a desired probability that
the clinical trial give statistically significant result,
which is known as the power of test.
Go to:
(Terms in hypothesis test) (2 types of Experimental design) (3 types of test
hypothesis)
Procedures to identify the appropriate program to use
1. Identify the aspects you want
There are 6 aspects:
Means, Proportions, Survival Analysis, Phase II
Clinical Trial, Confidential Interval and Others
The test for correlation coefficient and standard
normal calculator is provided in the “Others” category.
2. For Means, Proportions,
i.
Select the
design you want: One sample design, Two samples
parallel design or Two samples
crossover design.
ii.
Select the test
you want: Equality, NonInferiority
/ Superiority, or
Equivalence.
iii.
Please kindly
follow the detailed procedures on the calculator to work out the sample size.
Examples are also provided as reference.
iv.
For two
proportions, besides choosing from designs,
You can also choose from Confidence Interval – Bristol
and Compare
Two Proportions – Casagrande, Pike & Smith
Note:
Please kindly find the detailed guidance and explanation of terms via
the links.
3. For Survival Analysis,
i.
Select the
comparison you want: One survival curve or two survival curves
ii.
There are four
choices:
1.
Comparison
of Survival Curves Using Historical Controls
2.
Comparison
of Two Survival Curves Allowing for Stratification
3. Comparison of Two Survival Curves –
Rubinstein
4. Comparison of Two Survival Curves –
Lachin
iii.
Please kindly
follow the detailed procedures on the page to work out the sample size.
Examples are also provided as reference.
The detailed calculation and
formulae are shown in the Formula section on the calculator page.
Note:
For some terms of survival analysis appear on the calculator, please
kindly find detailed explanation via the links.
To know more about survival analysis and the common terms, please
click: Survival
Analysis
4. For Phase II Clinical Trials,
i.
Select the
technique you want:
1. Fleming's
Phase II Procedure
3. Simon’s
Randomized Phase II Design
ii.
Please kindly follow
the detailed procedures on the page to work out the sample size. Examples are
also provided as reference.
Detailed explanation to the principle, choice of sample size, hypotheses,
decision boundaries and
stopping criteria of the Phase II trial is shown in the ‘theory’ section on the
calculator page.
Note:
For some terms of Phase II Clinical Trials on the calculator, please
kindly find detailed explanation via the links.
To know more about Phase II Clinical Trials, please click: Phase II
Clinical Trials
For the difference between Fleming’s method and Bayesian method, please
click: Difference
between Fleming’s and Bayesian method
5. For Confidential
Interval
i.
Select the
technique you want:
3.
Correlation
5. Relative Risk and Attributable Risk
6. Odds Ratios, ARR, RRR, NNT, PEER
ii Please
kindly follow the detailed procedures on the page to work out the sample size.
Examples are also provided as reference.
Formulae and the
definition of terms are shown on the page also.
6. For
Others,
i.
Select the
technique you want:
1. Correlation Coefficient
using ztransformation
2. Standard Normal
Calculator
ii.
Please kindly
follow the detailed procedures on the page to work out the sample size.
Examples are also provided as reference.
A statistical hypothesis test is a method of making
decisions using data from a scientific study.
We have an original belief and
now we have some evidence to suspect that the original belief is wrong and need
to be updated.
Then we carry out hypothesis
test to see whether the evidence is significant or not to disprove the
original belief.
In statistics, a result is called statistically
significant if it is unlikely to occur by chance alone
according to a predetermined threshold probability called the significance level.
Definition of terms for hypothesis test and scientific experiment ^{[4] }Top
1. Sample size (N)
The
number of patients or experimental units required for the trial
2. Treatment
The
effect of certain process, intervention or change on objects
3. Null hypothesis (H_{0})
A
general or default position, i.e. no relationship between two treatments
or a proposed medical treatment has no effect.
Experiment
aims at rejecting the null hypothesis in a scientifically and statistically
significant sense,
i.e.
to prove the original belief to be false and update our conception. We reject
the null hypothesis if it is not likely to occur.
4. Alternative hypothesis (H_{1}/H_{a})
It
is the alternative to the null hypothesis, suggesting there is relationship
between two treatments in an unknown direction (twosided)
or
a specific direction (positive or negative, one side).
It
is the position that our new evidence is suggested or is the position that we
want to update our original belief.
An
example of null hypothesis and twosided alternative hypothesis is testing
equality of means in one sample design:
5. Test statistic
It
is a numerical summary or function of the observation, e.g. the mean of
sample
In
hypothesis test, we consider whether the value of observed test statistic is
extreme by its distribution under the null hypothesis.
6. Statistically significance
Statistical significance
is a statistical assessment of whether observations reflect a meaningful
pattern
rather than a pattern by chance.
In
statistics, a test statistic is believed to be statistically significant if it
is more extreme than critical value, i.e. in the rejection region.
Test
Statistic
7. Significance level (α)
It is a desired parameter of a cutoff
probability in experimental design to determine whether an observed test
statistic is extreme or not.
α is usually set to be 0.05, 0.025 or 0.01.
We reject the null hypothesis if the
probability of the observed test statistic to appear is smaller than α.
8. Critical value
It is the marginal value corresponding
to a given significance level α.
This
cutoff value determines the boundary that leads to the decision of rejecting or not the null
hypothesis.
9. pvalue
It
is the probability to obtain a new test statistic which is equal or more
extreme than the original observed test statistic.
A
small pvalue indicates that it is unlikely to get the value of
the observed test statistic.
We
reject the null hypothesis if pvalue is smaller than α.
10. Type I error
It is rejecting
the null hypothesis when it is true, i.e. false positive.
α is the probability of type I error. It
equals to the significance level in a simple null hypothesis.
11. Type II error
It
is not rejecting the null hypothesis when it is false, i.e. false
negative.
Or say, not accepting the alternative
hypothesis when it is true.
β is the probability of type II error.
The power of test equals to 1β
12. Power of test
The probability that a clinical
trial will have a significant result, i.e. have a pvalue
less than the specified significance level α.
This probability is
computed under the assumption that the treatment difference or strength of
association equals the minimal detectable difference.
The above figure shows the
distribution of a test statistic X under null and alternative hypothesis.
As one increases sample size,
the spread of the
distributions in the above figure decreases, i.e.βdecreases (power
increases).
Thus if the statistical test fails to reach significance, the power of the test becomes a critical factor in reaching
an inference.
It is not widely appreciated that the failure to achieve
statistical significance may often be related more to the low power
of the trial than to an actual lack of
difference between the competing therapies. Clinical trials with inadequate sample size
are thus doomed to failure before they begin.
Thereforeone should take steps
to ensure that the power of the clinical trial is sufficient to
justify the effort involved.^{[21]}
13. Minimal detectable
difference
The smallest
difference between treatments
you desire to be able to detect.
It is the smallest
difference to be
clinically important and biologically plausible in
clinical trial.
14. Onesided test
It
is a test for particular direction, stated in the alternative
hypothesis.
For
example, it can be,
choosing one of the directions in alternative hypothesis.
15. Twosided test
It
is a test for both directions, stated in the alternative hypothesis. For
example,
E.g.
Onesided test and twosided test with same significance level = 0.05:
Two types of experimental design Top
It is a design for a clinical trial in which a patient is assigned to receive only one of the study treatments.
It compares
the results of a treatment on two separate groups of patients.
The
experimental units (patients) are put into 2 groups randomly and each group
receives one and only one treatment.
Then the
results of treatment in two groups are compared.
Conducted properly, it provides assurance that any difference
between treatments is in fact due to treatment effects (or random
chance),
rather than some systematic
differences between the groups of subjects.
For example, let and be the mean of the response
of the study endpoint of interest.
Also let and be the intersubject variance
and intrasubject variance, respectively.
Assuming the equivalence limit is
,
, where and (by Chow and Wang,2001)
Crossover design ^{[10]}
It is a design for a clinical trial
in which a patient is assigned to receive more than one of the study treatments.
It is a repeated measurements design such that each patient receives different treatments during
the different time periods.
It compares the results of a
set of
treatments
on the same group of experimental units (patients).
So in the design
each
patient serves as his/her own matched control.
The sequence of treatment
received in each experimental unit is random.
For example, subject 1 first receives
treatment A, then treatment B, then treatment C.
Subject 2 might receive
treatment B, then treatment A, then treatment C.
It has the advantage of eliminating individual subject differences
from the overall treatment effect, thus enhancing statistical power.
On
the other hand, it is important in a crossover study that the underlying
condition not changes over time,
and that the effects of one treatment disappear before the next is
applied.
Therefore,
it is usually use to study chronic disease and there is a washout period between each treatment to prevent carryover
effect.
For
example, define and assume that the equivalence limit is , then
, where (by Chow and Wang,2001)
Various types of test hypothesis ^{[2]} ^{[11] }Top
E.g.
Recall that the null hypothesis is a
general or default position and is the position that we want to disprove or
reject.
Alternative
hypothesis is the position opposite to the null hypothesis and is the position
that the new belief is suggested.
Hypothesis
test check whether the evidence is significant or not to reject the null
hypothesis (the original belief)
and
establish a new belief (the alternative hypothesis).
It tests for the equality of a sample
value with a targeted constant value or tests for the equality between treatment
and active control/placebo.
Assume larger value indicates better performance. Null
hypothesis states that the sample value equals the targeted value.
Alternative hypothesis is the
sample value is not equal to the targeted value in either direction.
In two samples cases, testing equality is testing whether the values
from 2 samples equal or not, i.e.
2. Testing noninferiority/ superiority ^{[14]}
Noninferiority:
Superiority:
Where δ>0 is the noninferiority margin, or called superiority margin
Here _{ }represents the standard approved
treatment/product; T represents the new treatment/product.
Or say,
Test 
Null hypothesis 
Alternative hypothesis 
Noninferiority 


Superiority 


Equivalence 


Where
T:
Treatment
C:
Control
Assuming that the values to
the right of zero correspond to a better response with the new drug
so that the values to the
left indicate that the control is better,
Noninferiority means a treatment at
least not appreciably worse than an active control/placebo by the noninferiority
margin δ.
That
means the new treatment does not perform poorer than the active control/placebo
appreciably.
This
corresponds to the inequality suggested in the alternative hypothesis.
Conversely,
inferiority means that a
treatment is poorer than an
active control/placebo by the noninferiority margin δ.
Superiority means a
treatment is more effective than the active control by the superiority margin δ, stated
in the alternative hypothesis.
Conversely,
nonsuperiority means that a
treatment is not better than an
active control/placebo by the superiority margin δ.
There
are two types of superiority hypotheses, the above hypotheses are known as
hypothesis for testing clinical
superiority.
When
δ=0,
the above hypotheses are referred to as hypotheses for testing statistical superiority.
Q&A:
It may be
confusing if you see this
title the first time as when something, say “A”, is not inferior to “B”,
it
means that “A” is not too worse than “B”, but not necessarily to be superior to
(better than) “B” and vice versa.
Then how
come we are testing for noninferiority/ superiority.
Actually Testing
noninferiority/ superiority are two separate
tests using the same setting of H_{0
}and H_{a}, but with different signs in margin.
Assume that larger value of T represents better
performance,
if the margin is δ,
then H_{0} means that test drug
is inferior to the control. H_{1} is the
noninferiority of the test drug.
If the margin is δ, then H_{0 }means test drug is not superior to
control. H_{1}
is the superiority of the test drug.
There
is also confusion with the above Testing equality.
For
Testing Equality, the equation corresponds to equality is stated in null
hypothesis and it is what we want to reject.
Actually
we expect to have difference between two treatments, as stated in the alternative
hypothesis.
But
by convention, we call this as testing equality.
Compared
to Testing equality, in testing noninferiority/ superiority, noninferiority
/superiority is stated in the alternative hypothesis.
The
opposite, which is inferiority/non superiority of treatment and control,
is stated in the null hypothesis.
That
is we expect to the test drug to be superior/ not inferior to the control.
We put what we expect in the alternative hypothesis.
E.g.
In a test of superiority, to examine the effect of a test drug,
H_{0
}is the response of test drug is less than that of placebo by δ.
H_{a
}is the response of test drug is greater than that of placebo.
The
test helps us to see whether the test drug is superior to the placebo by an
amount of δ.
In
two sample cases, testing noninferiority/ superiority compares the values from
two samples, i.e.
Noninferiority:
Superiority:
Sample Size Determination
^{[11]}
For a superiority trial (S), the
necessary sample size (N) depends on δs, the clinically important difference.
For a noninferiority
trial (NI), the
necessary sample size depends on δ_{NI}, the upper bound for noninferiority.
When δ_{NI} =δ_{S}, the necessary sample size for the noninferiority trial is the same as superiority trial under the assumption of TT_{0} = 0
On the other hand, δ_{S} is
typically larger than δ_{NI}, which causes the sample size for a noninferiority trial often to be much larger
than that of a superiority trial.
E.g.
where δ > 0 is the margin of
clinically accepted difference, called equivalence margin.
Here
equality and equivalence are two different concepts.
Equality
only focuses on whether the values are equal or not.
Equivalence means the difference of treatment and active control is within specific
amount (δ)
in either direction (positive or negative)
Note
that the statement of equivalence is stated in the alternative hypothesis.
The
inequality in the null hypothesis means that the treatment and
control are not equivalence.
That
means this test aims at proving the treatment and control are equivalence,
therefore this new belief is put in the alternative hypothesis.
Null hypothesis states that the difference is at least δ.
Alternative
hypothesis states that the difference is less than δ, i.e.
equivalence.
In
two sample cases, testing equivalence compares the values from two samples,
i.e.
Proportions Top
It is used to determine the required sample size
for a desired power of test and control the length of confidence
intervals
of the
difference of proportions not exceeding certain value,
compared with two samples parallel design test that only tests for the
difference of proportions with a desired power.
The value of length is a bound to the length of
confidential interval.
It is chosen relative to the expected length of the
confidence interval, which is calculated by the formula on the webpage.
E.g. the expected length calculated is 0.141. Then
you can find the sample size with the bound of length to be, say 0.2.
For
binomial success probabilities, let π_{1} andπ_{2 }denote the success
probabilities of interest and let
Here are the largesample
normal approximations, as the exact results are very complicated
and the approximate results usually suffice for sample size determination.
For reference: ^{[17]}
First consider the
problem of testing H_{0}:Δ=0 against H_{1} :Δ0. Based on a sample
of size n
from each distribution, let p_{1} and p_{2} denote the observed
proportions of successes,
, With the hypothesis testing problem,
Fleiss uses approximations based on the asymptotic normality
of the estimates to construct a confidence interval for A.
The approximate (1 α) 100 percent confidence interval forΔis
(1)
and thus the associated hypothesis testing procedure is:
Rule: Reject Ho in favour of H1, if 0 is not an element of I, where I is the interval given in (1).
The length of the
confidence interval given in (1) is
Of course, this is
a random variable and thus cannot be controlled. If the variance of the two normal distributions
described in the previous section had been unknown, then the length of the resulting
confidence interval
for the difference of the means is based on the Student’s tdistribution and is also a random variable.
One approach to this problem is the determination of the
expected length. The exact result is
difficult to obtain and is unnecessary for the problem of sample
size determination.
An approximation to
this expected length is:
Let n_{L} denote the sample
size required to have L*=L_{0}, a specified
positive value. It is straightforward to show that
where .
Note that, for fixedπ_{1, }n_{L} is maximized at, and symmetric about, π_{2} =0.5. Further, n_{L}_{ }is symmetric in π_{1} and π_{2}
and is maximized at
Casagrande, Pike & Smith method ^{[18]}
It is a simple but accurate sample size approximation
method for comparing two binomial probabilities.
It is shown that over fairly wide ranges of
parameter values and ratios of sample sizes,
the percentage
error which results from using the approximation is no greater than 1%.
You can
choose onesided or twosided test in this method.
To find
the minimum n to achieve a power of 100βpercent an iterative procedure is
required.
This
involves very extensive calculations and numerous approximations have thus been suggested.
The two most commonly employed are:
1. The
"arcsin formula" as given, for example, in
Cochran and Cox (1957)
2. The
"uncorrected x2 formula" as given, for example, in Fleiss (1973)
Casagrande, Pike & Smith method is a Derivation of Corrected χ^{2}^{ }
and is
tested to be of good approximation over fairly wide ranges
of values.
For details in calculation, please read “Casagrande, Pike and Smith (1978) Biometrics 34: 483486”
Survival Analysis^{ [5] }Top
Survival analysis is
the study of time between the entry into observation and the substantial
events.
That means we observe the time needed until an event occur.
The
events include death, relapse from
remission, onset of a new disease and recovery.
It usually involves following the patients for a long time.
Some common terms in survival analysis: ^{[12]}
1.
Survival function [S(t)]
where T is the time of failure or death
It is the chance that
the subject survives longer than some specified time t.
2.
Hazard rate /Hazard function [λ(t)]
It is the
instantaneous risk of
occurrence of an event
at time t, given the subject survives until time t or
later.
Or it is
the probability of failure in an infinitesimally small time period (t, t+Δt) conditional on survival
until time t or later (that is, T ≥ t)
It is a
risk measure. The higher the hazard function, the higher the chance of failure
in the particular period.
Hazard
function is a nonnegative function. It can be increasing or decreasing.
3.
Hazard ratio (δ)
It is
the ratio of the hazard rate in control group and hazard rate in
experimental group.
It
gives an instantaneous comparison between risk of failure in
experimental group and control group.
4.
Censoring
This refers to the value of an observation is only
partly known. In survival analysis,
that is we have some information about the survival time of a
subject, but do not know exactly when it fails.
It happen when a person does not encounter the
event before the study ends, which is called “administratively censored”
or a person
is lost to followup during the study.
Prospective
study
is a study that follows over time a group of similar individuals who
differ in certain
factors under study
so as
to determine the effect of these
factors on the rate of outcome.
For survival analysis, the usual
observation strategy is prospective study.
You start to observe in certain
welldefined time point and the follow the patients for some substantial period
and finding out the time needed for an event to occur.
Note that the 4 methods below are not exact.
They describe the power function of the
logrank test only under the assumptions of a population model, but not distributionfree.
Thus,
these methods have been proposed as approximations under the restrictive
assumptions of proportional constant hazards,
i.e., under the exponential model.
In reality, the hazards will not be constant, nor
exactly proportional, over time.
The logrank test will still be
applicable but may not be maximally efficient.
Thus,
it is our view that these methods should be cautiously applied using
"worstcase" assumptions,
such as using
the lowest plausible hazard rate for the control group, the smallest clinically
relevant reduction in mortality,
and
appropriate adjustments for departures
from the usual assumptions of uniform patient recruitment, complete followup,
full
compliance, and homogeneity over prognostic strata. But each model has certain generalization to release the assumption. ^{[19]}
Reference: Lachin and Foulkes (1986) Biometrics
42: 508
Prospective study and retrospective study ^{[1] }Top
Prospective
A prospective study watches for outcomes, such as the development of a disease,
during the study period
and relates this to other factors such as suspected risk or protection
factor(s).
The study usually involves taking a cohort of subjects and watching
them over a long period.
The outcome of interest should be common; otherwise, the number of outcomes
observed will be too small to be
statistically meaningful (indistinguishable
from those that may have arisen by chance).
All efforts should be made to avoid sources of bias such as the loss of
individuals to follow up during the study.
Prospective studies usually have fewer potential sources of bias and
confounding than retrospective studies.
Retrospective
A retrospective study looks backwards and examines exposures to
suspected risk or protection factors
in relation to an outcome that is
established at the start of the study.
Most sources of error due to confounding and
bias are more common in retrospective studies than in prospective studies.
For this reason, retrospective investigation is often criticized although it tends to be cheaper and faster than prospective study.
In addition,
retrospective cohort studies are limited to outcomes and prognostic factors
that have already been collected,
and
may not be the factors that are important to answer the clinical question.
If the outcome of interest is uncommon and the required size of prospective study to estimate
relative risk is often too large to be feasible,
the odds ratio in retrospective study provides an estimate of relative risk.
You should take special care to avoid sources of bias and confounding in retrospective studies.
Comparison of Survival Curves Using Historical Controls Top
It determines the number of patients needed in a prospective comparison of survival curves,
when the control group patients
have already been followed for some period.
Explanation to variables:
α is the significance level for the test, usually 0.05
δ is the minimum hazard ratio, which is
calculated by dividing the estimated hazard rate of control group by that of
experimental group
M_{S}_{ }is the median survival time in month in the control group, which
can be estimated from existing control data.
r is the accrual rate. It is the rate of
arrival of patients per month. It is estimated for future accrual.
n_{C} and
y_{C} are the number
of deaths observed and the number of patients still at risk in the historical control respectively.
Both are obtained from existing control data.
τ is the length of the planned continuation period for the study in months
T is the length of accrual period for the new
study. It is the time needed to recruit patients into the trial.
Base on the accrual rate and the required number of
accrual target and power of test,
you can find an appropriate accrual period to achieve desired power
of test and required sample size.
So you adjust the variable T (accrual period) until desire power is obtained (e.g. 80%).
Several assumptions are made in this model.
Firstly, It assumes time
to survival is exponential distributed with hazard rate λ.
Secondly, It assumes prospective
studies are used. It also assumes no withdrawal or losses to followup throughout the study.
The detailed calculation and formulae are shown in
the Formula section on the calculator page.
Large randomized trials require longer time and higher cost, therefore the pilot
investigations should be
carefully designed and analyzed. There are also diseases in which outcome is very predictable based on
known prognostic features and historically controlled studies
may be viewed as an alternative to randomized
clinical trials.
For some rare diseases, historically controlled study is suitable.^{ }
^{ }
The accrual requirement declines as (i) the accrual rate declines,(ii) δ increases,
(iii) median survival in the
controls decreases, (iv) the number of historical controls
increases,
and (v) the number of failures already observed in the control group increases.^{ [20]}
Reference: Dixon & Simon (1988) J Clin Epidemiol 41:12091213
Comparison of Two Survival Curves
Allowing for Stratification Top
Stratification means patients are divided into homogeneous subgroups
called strata by a prognostic factor such as severity of
disease.
Other properties can also be used such as Age > 50 or not, or male
and female. In the calculator 2 strata can be set.
Explanation to variables:
α is the significance level for the test
β is the probability of type II error, or (1power)
of the test
K is the weight assigned to each stratum,
identifying which one is more significant to the result.
It is usually proportional to sample size in each
stratum.
δ is the minimum hazard ratio.
It is calculated by dividing the estimated hazard
rate of control group by that of experimental group
M_{S}_{ }is the median survival time in month in the control group, which
can be estimated from existing control data.
The sample fractions of control group (Q_{C})_{ }and
experimental group (Q_{E}) can be difference in each stratum and across two
strata:
T_{0} is the accrual period in month. It is the length of time to recruit patients for study in
each stratum.
TT_{0} is the followup
period
in month. It is the
continuation
period of all recruited patients to the
end of study T in each stratum.
For detailed formula and theory, please check the Formula
section on the calculator page.
Comparison of Two Survival Curves – Rubinstein Top
It is the determination of the
number of patients needed in a prospective comparison of survival curves with losses to followup,
when the control group
patients have already been followed for some period.
The explanation
to variables is the same as above “Allowing for Stratification” one.
Unlike the other model that
only assume time to survival is exponential distributed with hazard rate
λ, several
assumptions are made under this model.
First, the arrival of patients is modeled by
a Poisson process with rate n per year.
Then the patient is randomly assigned to the
experimental group or control group, with equal probability each.
Second, the survival time for a patient is
assumed to follow exponential distribution and is independent to each
other.
Third, the time until loss to followup is
assumed to follow exponential distribution and is also independent to
each other.
The
explanation to variables is the same as above “Allowing for
Stratification” one.
For detailed formula and theory, please check the Theory section on the calculator page.
Comparison of Two
Survival Curves – Lachin Top
It determines the number of patients needed in a prospective comparison of survival curves,
when the control group patients have already been followed for some period.
It only
assumes the
survival time is
exponential distributed with hazard rate λ.
In determination of sample size, it specifies the
minimal relevant difference
The
explanation to variables is the same as above “Allowing for
Stratification” one.
For detailed formula and theory, please check the Formula section on the calculator page.
Phase II clinical trials ^{[3]} Top
Phase II clinical trial typically investigates preliminary evidence of efficacy and continues to monitor safety.
There are three
main objectives in treating patients in Phase II clinical trials.
The primary
objective is to test whether the therapeutic intervention benefits the patient.
The second
objective is to screen the experimental treatment for the response activity in a
given type of cancer.
The final
objective is to extend our knowledge of the toxicology and pharmacology of the
treatment.
It
involves usually fewer than 50 patients. Patients accrue in several
stages in
a multiple testing procedure,
testing being performed at each
stage after appropriate patient accrual has been completed.
The number of patients accumulates.
This feature is particularly
appealing in a clinical setting where there are compelling ethical
reasons to terminate a Phase II trial early
if the initial proportion
of patients experiencing a tumor regression is too low or too high.
Phase
II trials decide whether the new treatment is promising and warrants further investigation in a
largescale randomized Phase III clinical trial.
Phase
II clinical trials are generally singlearm studies, but may take the form of multiplearm
trials.
Multiplearm
trials can be randomized or nonrandomized
with or without control arms.
It aims at estimating the activity of a new
treatment.
These
“pilot” studies are commonly applied to anticancer drugs to assess the
therapeutic efficacy and toxicity of new treatment regimens.
Phase
II clinical trials are only able to detect
a large treatment improvement, e.g. greater than 10%.
To
detect a small difference in treatment, e.g. less than 5%, one would require a
much larger sample size,
which
is not possible in Phase II studies due to the limited number of subjects
eligible for the study
and
the large number of treatments awaiting study.
Phase
II studies are prominent in cancer therapeutics as new treatments
frequently arise
from
combinations of existing therapies or by varying dose or radiation schedules.
An important
characteristic of some Phase II trial designs is the use of early stopping rules.
If there is sufficient
evidence that one of the treatments under study has a positive treatment
effect,
then patient accrual is terminated and
this treatment is declared promising.
Also, if a
treatment is sufficiently shown not to have a desirable effect,
then patient accrual is terminated and this
treatment is declared not promising.
Difference between Fleming’s procedure and Bayesian design of Phase II
clinical trials^{ }Top
This section describes both the hypotheses and design for
Fleming’s and the Bayesian approach to singlearm Phase II clinical trials.
Both designs are used for Phase II clinical trials with
binary outcomes and continuous monitoring.
The fundamental difference between the two designs is the
frequentist
basis for Fleming’s
procedure
only depends on the observed results whereas the Bayesian approach uses prior
information
(Information from previous studies).
The testing procedure for Fleming’s procedure is
based on the normal approximation to the
binomial distribution of the observed number of treatment responses.
The resulting decision boundaries, r_{g}
and a_{g}, are
solved analytically.
The Bayesian
design incorporates prior information about the treatment being
investigated with
the observed results to yield revised beliefs about the treatment.
The testing procedure is based on the posterior
probability of the experimental treatment given the observed data.
The posterior probability is a conditional probability
computed from a beta distribution which results in the upper and lower decision
boundaries,
U_{n} and L_{n}, which are evaluated using
numerical integration, namely “Simpson’s Composite Algorithm”.
Another difference between the two designs is that Fleming’s
procedure has only two outcomes at the final recruitment stage, i.e. reject or
accept H_{0},
while the Bayesian
design traditionally allows for an inconclusive
trial at the final stage (After attaining the maximum sample size set).
Simon’s Randomized Phase II Design ^{[8]} Top
In phase II clinical trial,
randomized design is proposed to establish the sample size for the study
to obtain the treatment
with greatest response rate for further / phase III clinical
trial.
There are some advantages for randomized design:
1.
Randomization
helps ensure that patients are centrally registered before treatment starts.
Establishment of a reliable mechanism to ensure patient registration
prior to treatment is of fundamental importance for all clinical trials.
2.
Comparing to
independent phase II studies, the differences in results obtained for the two
agents will more likely represent
real differences in
toxicity or antitumor effects rather than differences in patient selection,
response evaluation, or other factors.
3.
In randomized phase II clinical trials, one is merely making
a rational choice of one arm
and is
free of any burden to prove statistically that the selected arm is superior.
Although it is desirable to select
the best treatment,
selecting an arm
that is equivalent to another or even slightly worse is not considered too
grave a mistake.
Hence, the error rate to control is the probability of erroneously
selecting an arm whose response rate is lower than
that of the
best arm by an amount with medical importance (for example, 10%).
Similarly, the relevant power is the probability
of correctly selecting an arm whose response rate is larger than
that of the secondbest arm by an amount with medical importance (for example, 15%).^{[13] }
The
formulae for the probability are shown on the calculator page.
Explanation to variables:
p is the Lowest
response rate among all k treatments
k is the number of treatment
arms
D is the difference
in true response rates of the best and the next best
treatment
Confidence Interval Top
Confidence interval (C.I.) is a
range providing an interval estimate to true but unknown population parameter
and is
used to indicate the reliability of an estimate.
It is an observed interval
calculated from the particular sample, in
principle different from sample to sample.
The confidence level (1α) is the proportion of
confidence intervals that cover the true parameter,
i.e. a 95% C.I.
is the interval that you are 95%
certain contains the unknown
population true value.
Its relation with hypothesis test is that the 100(1α)% confidence interval
of the test statistic is the acceptance region of a 2sided hypothesis test.
If the test statistic is more
extreme than the upper or lower bound of the confidence interval, the null
hypothesis is rejected.
The significance level of the
test is the complement of the confidence level.
One
sample proportion Top
Proportion is the number
of success divided by the sample size. The calculator gives a confidential
interval for the estimate.
Two sample proportions Top
It
compares two proportions from independent samples and provides a confidential
interval.
Confidence intervals of difference not
containing 0 imply that there is a statistically significant difference between
the population proportions.
Correlation Top
Correlation
indicates whether two variables are associated.
It is a value from 1 to 1
with 1 representing perfectly negative correlation and 1
representing perfectly positive correlation.
The two variables should come
from random samples and have a Normal distribution (or after transformation).
The confidence interval is
a range which contains the true correlation with 100(1α)%
confidence.
Single
incidence rate Top
Incidence
rate is the rate at which new clinical events
occur in a population.
It is the
number of new events divided by the population at risk of an event in a
specific time period, sometimes it is the persontime
at risk.
Incidence is different from prevalence, which measures the total number of cases of disease in a population.
Thus, incidence carries information about the risk of having the disease, while
prevalence indicates how widespread the disease is.
Relative
Risk and Attributable Risk Top
Disease 
No disease 
Totals 

Exposed 
a 
b 
n_{1}=a+b 
Nonexposed 
c 
d 
n_{2}=c+d 
Totals 
m_{1}=a+c 
m_{2}=b+d 
N=n_{1+}n_{2} 
Relative Risk
is the ratio of incidence of disease in Exposed group to that in Nonexposed
group from a cohort/prospective study.
If Relative Risk is
larger than 1, it is a positive association. If it is smaller than 1,
it is a negative association.
Attributable Risk is
the amount of disease incidence which can be attributed to an exposure in a
prospective study.
Population Attributable
Risk is the reduction in incidence if the whole population
were unexposed, comparing with actual exposure pattern.
Relative Risk
compares the risk of having a disease for not receiving a medical treatment
against people receiving a treatment.
It can also compare the
risk of having side effect in drug treatment against the people not receiving
the treatment.
Attributable Risk
and Population Attributable Risk tell the amount of risk prevented if we
do not have certain exposure.
Exposed group is the group of patients exposed to certain
factors of interest such as a new treatment, age 45 or above or smoking for 10
years or above.
Odds
Ratios, ARR, RRR, NNT, PEER ^{[6] }Top
Outcome Positive 
Outcome Negative 
Totals 

Feature Present 
a 
b 
n_{1}=a+b 
Feature Absent 
c 
d 
n_{2}=c+d 
Totals 
m_{1}=a+c 
m_{2}=b+d 
N=n_{1}+n_{2} 
Odds Ratio (OR)
refers to the ratio of the odds of the outcome in two groups in a retrospective
study.
It
is an estimate for the relative risk in a prospective study.
Absolute Risk Reduction (ARR) is the change in risk in the 2 groups and its inverse is the Number
Needed to Treat (NNT).
Patient expected event rate (PEER) is the expected rate of events in a patient received no treatment or
conventional treatment.
The Ztest for Odds
Ratio shows whether the exposure affects
the odds of outcome.
OR=1 means
exposure has no effect on the odds of outcome.
OR>1
means exposure leads to higher odds of outcome and vice versa.
The Ztest for 2
Proportions shows whether there is difference between the proportions of
events in 2 groups.
The Chisquare test for
Association tests the association between the groups of feature and test
result.
Diagnostic
Statistics ^{[7]} Top
Disease 
No disease 
Totals 

Test Outcome Positive 
a (True Positive) 
b (False Positive) 
n_{1}=a+b 
Test Outcome Negative 
c (False Negative) 
d (True Negative) 
n_{2}=c+d 
Totals 
m_{1}=a+c 
m_{2}=b+d 
N=n_{1}+n_{2} 
Sensitivity is the ability of the test to pick up
what it is testing for
and Specificity is ability
to reject what it is not
testing for.
Likelihood ratios determine how the
test result changes the probability of certain outcomes and events.
Pretest and Posttest probabilities are the subjective
probabilities of the presence of a clinical event or status before and after
the diagnostic test.
For positive test, we find
the positive posttest probability and for negative test, we find the negative
posttest probability.
McNemar’s Test Top
Test 2 Positive 
Test 2 Negative 
Totals 

Test 1 Positive 
a 
b 
n_{1}=a+b 
Test 1 Negative 
c 
d 
n_{2}=c+d 
Totals 
m_{1}=a+c 
m_{2}=b+d 
N=n_{1}+n_{2} 
McNemar’s
Test is a test on a 2x2 contingency table. It checks the marginal
homogeneity of two dichotomous variables.
It is used for data of the two groups coming from the same participants, i.e. paired data
For
example, it is used to analyze tests performed before and after
treatment in a population.
1. http://www.statsdirect.com/help/basics/prospective.htm
2. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2701110/
3. https://onlinecourses.science.psu.edu/stat509/node/22
4. http://hedwig.mgh.harvard.edu/sample_size/quan_measur/defs.html
5. http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf
6. http://www.cebm.net/index.aspx?o=1044
7. http://ceaccp.oxfordjournals.org/content/8/6/221.full
8. http://www.nihtraining.com/cc/ippcr/current/downloads/SV.pdf
9. http://www.statistics.com/index.php?page=glossary&term_id=439
10. http://www.statistics.com/index.php?page=glossary&term_id=424
11. http://www.nyuhjdbulletin.org/mod/bulletin/v66n2/docs/v66n2_16.pdf
12. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3227332/
13. http://onlinelibrary.wiley.com/doi/10.1002/sim.5829/full
14. http://www.scielo.br/scielo.php?pid=S167754492010000300009&script=sci_arttext&tlng=en
15. http://annals.org/article.aspx?articleid=736284
16. https://onlinecourses.science.psu.edu/stat504/node/19\
17. Bristol
(1989) Statistics in Med, 8:803811
18. Casagrande, Pike
and Smith (1978) Biometrics 34:
483486
19. Lachin and Foulkes
(1986) Biometrics 42: 507519
20. Dixon & Simon
(1988) J Clin Epidemiol 41:12091213
21. Lachin (1981) Controlled Clinical Trials 2: 94