Title stata.com
power — Power and sample-size analysis for hypothesis tests
Description Menu Syntax Options
Remarks and examples Stored results Methods and formulas References
Also see
Description
The power command is useful for planning studies. It performs power and sample-size analysis for
studies that use hypothesis testing to form inferences about population parameters. You can compute
sample size given power and effect size, power given sample size and effect size, or the minimum
detectable effect size and the corresponding target parameter given power and sample size. You can
display results in a table ([PSS-2] power, table) and on a graph ([PSS-2] power, graph).
For precision and sample-size analysis for CIs, see [PSS-3] ciwidth.
Menu
Statistics > Power, precision, and sample size
Syntax
Compute sample size
power method . . .
, power(numlist) power options . . .
Compute power
power method . . . , n(numlist)
power options . . .
Compute effect size and target parameter
power method . . . , n(numlist) power(numlist)
power options . . .
1
2 power — Power and sample-size analysis for hypothesis tests
method Description
One sample
onemean One-sample mean test (one-sample t test)
oneproportion One-sample proportion test
onecorrelation One-sample correlation test
onevariance One-sample variance test
Two independent samples
twomeans Two-sample means test (two-sample t test)
twoproportions Two-sample proportions test
twocorrelations Two-sample correlations test
twovariances Two-sample variances test
Two paired samples
pairedmeans Paired-means test (paired t test)
pairedproportions Paired-proportions test (McNemar’s test)
Analysis of variance
oneway One-way ANOVA
twoway Two-way ANOVA
repeated Repeated-measures ANOVA
Linear regression
oneslope Slope test in a simple linear regression
rsquared R
2
test in a multiple linear regression
pcorr Partial-correlation test in a multiple linear regression
Contingency tables
cmh Cochran–Mantel–Haenszel test (stratified 2 × 2 tables)
mcc Matched case–control studies
trend Cochran–Armitage trend test (linear trend in J × 2 table)
Survival analysis
cox Cox proportional hazards model
exponential Two-sample exponential test
logrank Log-rank test
Cluster randomized design (CRD)
onemean, cluster One-sample mean test in a CRD
oneproportion, cluster One-sample proportion test in a CRD
twomeans, cluster Two-sample means test in a CRD
twoproportions, cluster Two-sample proportions test in a CRD
logrank, cluster Log-rank test in a CRD
User-defined methods
usermethod Add your own method to power
power — Power and sample-size analysis for hypothesis tests 3
power options Description
Main
∗
alpha(numlist) significance level; default is alpha(0.05)
∗
power(numlist) power; default is power(0.8)
∗
beta(numlist) probability of type II error; default is beta(0.2)
∗
n(numlist) total sample size; required to compute power or effect size
∗
n1(numlist) sample size of the control group
∗
n2(numlist) sample size of the experimental group
∗
nratio(numlist) ratio of sample sizes, N2/N1; default is nratio(1), meaning
equal group sizes
compute(N1 | N2) solve for N1 given N2 or for N2 given N1
nfractional allow fractional sample sizes
direction(upper|lower) direction of the effect for effect-size determination; default is
direction(upper), which means that the postulated value
of the parameter is larger than the hypothesized value
onesided one-sided test; default is two sided
parallel treat number lists in starred options or in command arguments as
parallel when multiple values per option or argument are
specified (do not enumerate all possible combinations of values)
Table
no
table
(tablespec)
suppress table or display results as a table;
see [PSS-2] power, table
saving(filename
, replace
) save the table data to filename; use replace to overwrite
existing filename
Graph
graph
(graphopts)
graph results; see [PSS-2] power, graph
Iteration
init(#) initial value of the estimated parameter; default is
method specific
iterate(#) maximum number of iterations; default is iterate(500)
tolerance(#) parameter tolerance; default is tolerance(1e-12)
ftolerance(#) function tolerance; default is ftolerance(1e-12)
no
log suppress or display iteration log
no
dots suppress or display iterations as dots
notitle suppress the title
∗
Specifying a list of values in at least two starred options, or at least two command arguments, or at least one
starred option and one argument results in computations for all possible combinations of the values; see
[U] 11.1.8 numlist. Also see the parallel option.
Options n1(), n2(), nratio(), and compute() are available only for two-independent-samples methods.
Iteration options are available only with computations requiring iteration.
collect is allowed; see [U] 11.1.10 Prefix commands.
notitle does not appear in the dialog box.
4 power — Power and sample-size analysis for hypothesis tests
Options
Main
alpha(numlist) sets the significance level of the test. The default is alpha(0.05).
power(numlist) sets the power of the test. The default is power(0.8). If beta() is specified, this
value is set to be 1 − beta(). Only one of power() or beta() may be specified.
beta(numlist) sets the probability of a type II error of the test. The default is beta(0.2). If power()
is specified, this value is set to be 1 −power(). Only one of beta() or power() may be specified.
n(numlist) specifies the total number of subjects in the study to be used for power or effect-size
determination. If n() is specified, the power is computed. If n() and power() or beta() are
specified, the minimum effect size that is likely to be detected in a study is computed.
n1(numlist) specifies the number of subjects in the control group to be used for power or effect-size
determination.
n2(numlist) specifies the number of subjects in the experimental group to be used for power or
effect-size determination.
nratio(numlist) specifies the sample-size ratio of the experimental group relative to the control
group, N2/N1, for two-sample tests. The default is nratio(1), meaning equal allocation between
the two groups.
compute(N1 | N2) requests that the power command compute one of the group sample sizes given
the other one, instead of the total sample size, for two-sample tests. To compute the control-group
sample size, you must specify compute(N1) and the experimental-group sample size in n2().
Alternatively, to compute the experimental-group sample size, you must specify compute(N2)
and the control-group sample size in n1().
nfractional specifies that fractional sample sizes be allowed. When this option is specified, fractional
sample sizes are used in the intermediate computations and are also displayed in the output.
Also see the description and the use of options n(), n1(), n2(), nratio(), and compute() for
two-sample tests in [PSS-4] Unbalanced designs.
direction(upper | lower) specifies the direction of the effect for effect-size determination. For most
methods, the default is direction(upper), which means that the postulated value of the parameter
is larger than the hypothesized value. For survival methods, the default is direction(lower),
which means that the postulated value is smaller than the hypothesized value.
onesided indicates a one-sided test. The default is two sided.
parallel requests that computations be performed in parallel over the lists of numbers specified for
at least two study parameters as command arguments, starred options allowing numlist, or both.
That is, when parallel is specified, the first computation uses the first value from each list of
numbers, the second computation uses the second value, and so on. If the specified number lists
are of different sizes, the last value in each of the shorter lists will be used in the remaining
computations. By default, results are computed over all combinations of the number lists.
For example, let a
1
and a
2
be the list of values for one study parameter, and let b
1
and b
2
be the list of values for another study parameter. By default, power will compute results for all
possible combinations of the two values in the two study parameters: (a
1
, b
1
), (a
1
, b
2
), (a
2
, b
1
),
and (a
2
, b
2
). If parallel is specified, power will compute results for only two combinations:
(a
1
, b
1
) and (a
2
, b
2
).
power — Power and sample-size analysis for hypothesis tests 5
Table
notable, table, and table() control whether or not results are displayed in a tabular format.
table is implied if any number list contains more than one element. notable is implied with
graphical output—when either the graph or the graph() option is specified. table() is used to
produce custom tables. See [PSS-2] power, table for details.
saving(filename
, replace
) creates a Stata data file (.dta file) containing the table values
with variable names corresponding to the displayed columns. replace specifies that filename be
overwritten if it exists. saving() is only appropriate with tabular output.
Graph
graph and graph() produce graphical output; see [PSS-2] power, graph for details.
The following options control an iteration procedure used by the power command for solving nonlinear
equations.
Iteration
init(#) specifies an initial value for the estimated parameter. Each power method sets its own
default value. See the documentation entry of the method for details.
iterate(#) specifies the maximum number of iterations for the Newton method. The default is
iterate(500).
tolerance(#) specifies the tolerance used to determine whether successive parameter estimates have
converged. The default is tolerance(1e-12). See Convergence criteria in [M-5] solvenl( ) for
details.
ftolerance(#) specifies the tolerance used to determine whether the proposed solution of a
nonlinear equation is sufficiently close to 0 based on the squared Euclidean distance. The default
is ftolerance(1e-12). See Convergence criteria in [M-5] solvenl( ) for details.
log and nolog specify whether an iteration log is to be displayed. The iteration log is suppressed
by default. Only one of log, nolog, dots, or nodots may be specified.
dots and nodots specify whether a dot is to be displayed for each iteration. The iteration dots are
suppressed by default. Only one of dots, nodots, log, or nolog may be specified.
The following option is available with power but is not shown in the dialog box:
notitle prevents the command title from displaying.
6 power — Power and sample-size analysis for hypothesis tests
Remarks and examples stata.com
Remarks are presented under the following headings:
Using the power command
Specifying multiple values of study parameters
One-sample tests
Two-sample tests
Paired-sample tests
Analysis of variance models
Linear regression
Contingency tables
Survival analysis
Cluster randomized designs
Tables of results
Power curves
Add your own methods to power
This section describes how to perform power and sample-size analysis using the power command.
For a software-free introduction to power and sample-size analysis, see [PSS-2] Intro (power).
Using the power command
The power command computes sample size, power, or minimum detectable effect size and the
corresponding target parameter for various hypothesis tests. You can also add your own methods to
the power command as described in [PSS-2] power usermethod.
All computations are performed for a two-sided hypothesis test where, by default, the significance
level is set to 0.05. You may change the significance level by specifying the alpha() option. You
can specify the onesided option to request a one-sided test.
By default, the power command computes sample size for the default power of 0.8. You may
change the value of power by specifying the power() option. Instead of power, you can specify the
probability of a type II error in the beta() option.
To compute power, you must specify the sample size in the n() option.
To compute power or sample size, you must also specify a magnitude of the effect desired to
be detected by a hypothesis test. power’s methods provide several ways in which an effect can be
specified. For example, for a one-sample mean test, you can specify either the target mean or the
difference between the target mean and a reference mean; see [PSS-2] power onemean.
You can also compute the smallest magnitude of the effect or the minimum detectable effect size
(MDES) and the corresponding target parameter that can be detected by a hypothesis test given power
and sample size. To compute MDES, you must specify both the desired power in the power() option
or the probability of a type II error in the beta() option and the sample size in the n() option.
In addition to the effect size, power also reports the estimated value of the parameter of interest,
such as the mean under the alternative hypothesis for a one-sample test or the experimental-group
proportion for a two-sample test of independent proportions. By default, when the postulated value
is larger than the hypothesized value, the power command assumes an effect in the upper direction,
the direction(upper) option. You may request an estimate of the effect in the opposite, lower,
direction by specifying the direction(lower) option.
For hypothesis tests comparing two independent samples, you can compute one of the group sizes
given the other one instead of the total sample size. In this case, you must specify the label of the
group size you want to compute in the compute() option and the value of the other group size in
the respective n#() option. For example, if we wanted to find the size of the second group given the
size of the first group, we would specify the combination of options compute(N2) and n1(#).
power — Power and sample-size analysis for hypothesis tests 7
A balanced design is assumed by default for two-independent-samples hypothesis tests, but you
can request an unbalanced design. For example, you can specify the allocation ratio n
2
/n
1
between
the two groups in the nratio() option or the individual group sizes in the n1() and n2() options.
See [PSS-4] Unbalanced designs for more details about various ways of specifying an unbalanced
design.
For sample-size determination, the reported integer sample sizes may not correspond exactly to
the specified power because of rounding. To obtain conservative results, the power command rounds
up the sample size to the nearest integer so that the corresponding power is at least as large as
the requested one. You can specify the nfractional option to obtain the corresponding fractional
sample size.
Some of power’s computations require iteration. The defaults chosen for the iteration procedure
should be sufficient for most situations. In a rare situation when you may want to modify the defaults,
the power command provides options to control the iteration procedure. The most commonly used
is the init() option for supplying an initial value of the estimated parameter. This option can be
useful in situations where the computations are sensitive to the initial values. If you are performing
computations for many combinations of various study parameters, you may consider reducing the
default maximum number of iterations of 500 in the iterate() option so that the command is
not spending time on calculations in difficult-to-compute regions of the parameter space. By default,
power suppresses the iteration log. If desired, you can specify the log option to display the iteration
log or the dots option to display iterations as dots to monitor the progress of the iteration procedure.
The power command can produce results for one study scenario or for multiple study scenarios when
multiple values of the parameters are specified; see Specifying multiple values of study parameters
below for details.
For a single result, power displays results as text. For multiple results or if the table option
is specified, power displays results in a table. You can also display multiple results on a graph by
specifying the graph option. Graphical output suppresses the table of the results; use the table option
to also see the tabular output. You can customize the default tables and graphs by specifying suboptions
within the respective options table() and graph(); see [PSS-2] power, table and [PSS-2] power,
graph for details.
You can also save the tabular output to a Stata dataset by using the saving() option.
Specifying multiple values of study parameters
The power command can produce results for one study scenario or for multiple study scenarios
when multiple values of the parameters are supplied to the supported options. The options that support
multiple values specified as a numlist are marked with a star in the syntax diagram.
For example, the n() option supports multiple values. You can specify multiple sample sizes as
individual values, n(100 150 200), or as a range of values, n(100(50)200); see [U] 11.1.8 numlist
for other specifications.
In addition to options, you may specify multiple values of command arguments, values specified
after the command name. For example, let #
1
and #
2
be the first and the second command arguments
in
. power twoproportions #
1
#
2
, . . .
If we want to specify multiple values for the command arguments, we must enclose these values
in parentheses. For example,
. power twoproportions (0.1 0.2) (0.1 0.2 0.3 0.4), . . .
8 power — Power and sample-size analysis for hypothesis tests
or, more generally,
. power twoproportions (numlist) (numlist), . . .
When multiple values are specified in multiple options or for multiple command arguments, the
power command computes results for all possible combinations formed by the values from every
option and command argument. In some cases, you may want to compute results in parallel for
specific sets of values of the specified parameters. To request this, you can specify the parallel
option. If the specified number lists are of varying sizes, numlist with the maximum size determines
the number of final results produced by power. The last value from numlist of smaller sizes will be
used in the subsequent computations.
For example,
. power twoproportions (0.1 0.2) 0.4, n(100 200)
is equivalent to
. power twoproportions 0.1 0.4, n(100)
. power twoproportions 0.2 0.4, n(100)
. power twoproportions 0.1 0.4, n(200)
. power twoproportions 0.2 0.4, n(200)
When the parallel option is specified,
. power twoproportions (0.1 0.2) 0.4, n(100 200) parallel
is equivalent to
. power twoproportions 0.1 0.4, n(100)
. power twoproportions 0.2 0.4, n(200)
When the parallel option is specified and numlist is of different sizes, the last value of the
shorter numlist is used in the subsequent computations. For example,
. power twoproportions (0.1 0.2 0.3) 0.4, n(100 200) parallel
is equivalent to
. power twoproportions 0.1 0.4, n(100)
. power twoproportions 0.2 0.4, n(200)
. power twoproportions 0.3 0.4, n(200)
One-sample tests
The power command provides PSS computations for four one-sample tests. power onemean
performs PSS analysis for a one-sample mean test; power oneproportion performs PSS analysis
for a one-sample proportion test; power onecorrelation performs PSS analysis for a one-sample
correlation test; and power onevariance performs PSS analysis for a one-sample variance test.
power onemean provides PSS computations for a one-sample t test assuming known or unknown
population standard deviation. It also provides a way to adjust computations for a finite population
sample. See [PSS-2] power onemean.
power — Power and sample-size analysis for hypothesis tests 9
power oneproportion provides PSS computations for a test that compares one proportion with a
reference value. By default, the computations are based on a large-sample z test that uses the normal
approximation of the distribution of the test statistic. You may choose between two large-sample
tests: the score test or Wald test. You may also compute power for the small-sample binomial test
by specifying the test(binomial) option. See [PSS-2] power oneproportion.
power onecorrelation provides PSS computations for a test that compares one correlation with a
reference value. The computations are based on a Fisher’s z transformation of a correlation coefficient.
See [PSS-2] power onecorrelation.
power onevariance provides PSS computations for a test that compares one variance with a
reference value. The computations are based on a χ
2
test of the ratio of the variance to its reference
value. You can perform computations in the variance or standard deviation metric. See [PSS-2] power
onevariance.
All one-sample methods compute sample size given power and target parameter, power given
sample size and target parameter, or MDES and the corresponding target parameter given power and
sample size.
For PSS determination, an effect may be supplied by specifying the null and alternative values of
the target parameter as command arguments #
0
and #
a
:
. power onesample #
0
#
a
, . . .
Instead of the alternative value #
a
, you can specify the ratio of the alternative value to the null
value in the ratio() option and the null value as #
0
for power onevariance,
. power onevariance #
0
, ratio(#) . . .
or you can specify the difference between the alternative value and the null value in the diff()
option and the null value as #
0
for other methods,
. power onesample #
0
, diff(#) . . .
For sample-size determination, the reported sample size is rounded up to the nearest integer. This
ensures that the corresponding actual power is at least as large as the specified power. You can specify
the nfractional option to obtain the corresponding fractional sample size, or you can recompute
the actual power using the reported rounded value; see Fractional sample sizes in [PSS-4] Unbalanced
designs for details.
Below we show a quick example of PSS analysis for a one-sample mean test. See entries of the
one-sample methods for more examples.
Example 1: PSS analysis for a one-sample mean test
A group of pediatricians would like to study the exposure of infants to television. The group
wants to investigate whether the average number of hours watched per day by infants between 3 and
12 months of age is greater than 2 hours. Before conducting a study, pediatricians would like to
determine how many infants they need to enroll in the study. The analysis will use the one-sample
t test to compare the mean of the obtained sample with the reference value. An earlier pilot study
reported an average of 2.5 hours watched per day with a standard deviation of 0.8. Pediatricians
would like to compute the sample size required to detect a mean of 2.5 using a two-sided test with
5% significance level and 80% power. Although pediatricians suspect that the effect is in the upper
direction—more than two hours watched on average—they prefer to obtain the required sample size
for a two-sided test instead of a one-sided test.
10 power — Power and sample-size analysis for hypothesis tests
We use power onemean to compute the required sample size. We specify the reference or null
value of 2 and the comparison or alternative value of 2.5 as command arguments. We also specify the
standard deviation of 0.8 in the sd() option. We omit the alpha(0.05) and power(0.8) options
because the desired values are the defaults for these options. The default test is two sided, so we do
not need to supply any additional information to the command.
. power onemean 2 2.5, sd(0.8)
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
Study parameters:
alpha = 0.0500
power = 0.8000
delta = 0.6250
m0 = 2.0000
ma = 2.5000
sd = 0.8000
Estimated sample size:
N = 23
All power commands have a similar output format. Information about the test and tested hypothesis is
displayed first. The input and implied values of the study parameters are displayed next under Study
parameters. The estimated parameters, such as the sample size in this example, are displayed last.
Pediatricians need to enroll 23 infants in the study to detect a standardized difference of 0.625
between the alternative mean of 2.5 and the null mean of 2 given a standard deviation of 0.8 using
a 5%-level two-sided one-sample t test with 80% power.
The pediatricians believe that they have resources to enroll more infants. They wish to compute
the power that corresponds to the sample size of 50. To compute the corresponding power, we specify
a sample size of 50 in the n() option:
. power onemean 2 2.5, sd(0.8) n(50)
Estimated power for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
Study parameters:
alpha = 0.0500
N = 50
delta = 0.6250
m0 = 2.0000
ma = 2.5000
sd = 0.8000
Estimated power:
power = 0.9911
The power increases to 99% for a larger sample of 50 infants.
The pediatricians also want to find out what is the smallest mean difference they can detect with
the larger sample of 50 infants while keeping the power at 80%. They assume the effect to be in
the upper direction for this computation. To compute the minimum detectable difference, we specify
both the sample size in the n() option and the power in the power() option.
power — Power and sample-size analysis for hypothesis tests 11
. power onemean 2, sd(0.8) n(50) power(0.8)
Performing iteration ...
Estimated target mean for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0; ma > m0
Study parameters:
alpha = 0.0500
power = 0.8000
N = 50
m0 = 2.0000
sd = 0.8000
Estimated effect size and target mean:
delta = 0.4042
ma = 2.3233
The smallest standardized difference that can be detected given the study parameters is about 0.4,
with a corresponding mean of 2.32.
Two-sample tests
The power command provides PSS computations for four two-sample tests. power twomeans
performs PSS analysis for a two-sample means test; power twoproportions performs PSS analysis
for a two-sample proportions test; power twocorrelations performs PSS analysis for a two-sample
correlations test; and power twovariances performs PSS analysis for a two-sample variances test.
power twomeans provides PSS computations for a two-sample means test that compares the means
of two independent populations. The computations provided assume known or unknown and equal or
unequal population standard deviations of the two groups. See [PSS-2] power twomeans.
power twoproportions provides PSS computations for a two-sample proportions test that compares
the proportions in two independent populations with binary outcomes. Three tests are supported: the
large-sample Pearson’s χ
2
test, the large-sample likelihood-ratio test, and the small-sample Fisher’s
exact test. Several effect specifications are available. For example, you can specify the effect of
interest as a risk difference, or a relative risk, or an odds ratio. See [PSS-2] power twoproportions.
power twocorrelations provides PSS computations for a two-sample correlations test that
compares the correlation coefficients of two independent populations. The computations are based on
a Fisher’s z transformation of a correlation coefficient. See [PSS-2] power twocorrelations.
power twovariances provides PSS computations for a two-sample variances test that compares
the variances of two independent populations. The computations are based on an F test of the
ratio of variances. You can perform computations in the variance or standard deviation metric. See
[PSS-2] power twovariances.
Also see Survival analysis for power and sample-size analysis for a two-sample comparison of
survivor functions using the power logrank and power exponential commands.
All two-sample methods compute sample size given power and the control-group and experimental-
group values of the target parameter, power given sample size and the control-group and experimental-
group values of the target parameter, or MDES and the corresponding target value of the parameter
in the experimental group given power, sample size, and the control-group parameter value.
To compute sample size or power, you can specify the magnitude of the effect of interest in
two ways: by directly specifying the alternative values of the target parameter in two groups or by
12 power — Power and sample-size analysis for hypothesis tests
specifying the control-group alternative value and the corresponding relation of the experimental-group
value to the control-group alternative value.
The two alternative values are specified as command arguments: the alternative value of the target
parameter in the control or reference group, #
a1
, and the alternative value of the target parameter in
the experimental or comparison group, #
a2
:
. power twosample #
a1
#
a2
, . . .
The experimental-group alternative value, #
a2
, may be omitted if an option containing the relationship
between the two alternative values is specified. For example, for power twomeans and power
twocorrelations, such an option is diff(), and it specifies the difference between the experimental-
group and control-group alternative values:
. power twomeans #
a1
, diff(#) . . .
For power twovariances, such an option is ratio(), and it contains the ratio of the experimental-
group alternative value to the control-group value:
. power twovariances #
a1
, ratio(#) . . .
power twoproportions provides several alternative specifications in which a difference between
the two populations may be expressed. For example, you can express the “difference” as an odds
ratio of the experimental group to the control group,
. power twoproportions #
a1
, oratio(#) . . .
or as a relative risk,
. power twoproportions #
a1
, rrisk() . . .
In addition to the total sample size, two-sample methods provide a way to solve for one of the
group sizes when the other group size is fixed. This can be achieved by specifying the compute()
option. To compute the size of the first group, you must specify the compute(N1) option and the
size of the second group in the n2() option. To compute the size of the second group, you must
specify the compute(N2) option and the size of the first group in the n1() option.
To compute power, you can specify a total sample size in the n() option, group sample sizes in
the n1() and n2() options, or one of the group sample sizes and its ratio, n
2
/n
1
, in the nratio()
option; see [PSS-4] Unbalanced designs for more specifications.
Below we show a quick example of PSS analysis for a two-sample means test. See entries of the
two-sample methods for more examples.
Example 2: PSS analysis for a two-sample mean test
A pharmaceutical company would like to conduct a study to compare a new weight-loss drug with
an older drug. Investigators are planning to use a two-sample t test to compare the average weight loss
for the two drugs. The average weight loss of people taking the old drug for 3 months is 12 pounds,
with a standard deviation of 5.5 pounds. The new drug is expected to produce an average weight loss
of 16 pounds, with a standard deviation of 5 pounds for the same period of time. Investigators want
to find out how many subjects they need to recruit into the study to detect the specified difference
using a 5% level two-sided test with 90% power.
We use power twomeans to perform PSS analyses. We specify the control-group mean 12 and
the experimental-group mean 16 as command arguments after the command name. We specify the
respective standard deviations in the sd1() and sd2() options. The default power is set to 0.8, so
we specify power(0.9) to request 90% power.
power — Power and sample-size analysis for hypothesis tests 13
. power twomeans 12 16, sd1(5.5) sd2(5) power(0.9)
Performing iteration ...
Estimated sample sizes for a two-sample means test
Satterthwaite’s t test assuming unequal variances
H0: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.9000
delta = 4.0000
m1 = 12.0000
m2 = 16.0000
sd1 = 5.5000
sd2 = 5.0000
Estimated sample sizes:
N = 76
N per group = 38
We need a sample of 76 subjects, 38 per group, to detect a difference of 4 between the control-group
mean of 12 and the experimental-group mean of 16 given the respective standard deviations of 5.5
and 5 with 90% power using a 5%-level two-sided two-sample means t test.
The default test is two sided. You may specify the onesided option to request a one-sided test.
The default design is also balanced; see [PSS-4] Unbalanced designs for examples of unbalanced
designs.
The investigators hope to keep the sample size under 60 and would like to compute the power
corresponding to this sample size. To compute the corresponding power, we specify the n(60) option
instead of the power() option:
. power twomeans 12 16, sd1(5.5) sd2(5) n(60)
Estimated power for a two-sample means test
Satterthwaite’s t test assuming unequal variances
H0: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
N = 60
N per group = 30
delta = 4.0000
m1 = 12.0000
m2 = 16.0000
sd1 = 5.5000
sd2 = 5.0000
Estimated power:
power = 0.8259
The power decreases to 83% for the smaller sample of 60 subjects.
To keep the power at 90%, the investigators want to compute the smallest difference between the
experimental-group mean and the control-group mean (in the upper direction) given the sample of 60
subjects. For this computation, we specify both options n(60) and power(0.9):
14 power — Power and sample-size analysis for hypothesis tests
. power twomeans 12, sd1(5.5) sd2(5) n(60) power(0.9)
Performing iteration ...
Estimated experimental-group mean for a two-sample means test
Satterthwaite’s t test assuming unequal variances
H0: m2 = m1 versus Ha: m2 != m1; m2 > m1
Study parameters:
alpha = 0.0500
power = 0.9000
N = 60
N per group = 30
m1 = 12.0000
sd1 = 5.5000
sd2 = 5.0000
Estimated effect size and experimental-group mean:
delta = 4.4744
m2 = 16.4744
The smallest detectable mean difference is 4.47, with a corresponding value of the experimental-group
mean of 16.47.
Paired-sample tests
The power command provides PSS computations for two tests of paired samples. power paired-
means performs PSS analysis for a two-sample paired-means test, and power pairedproportions
performs PSS analysis for a two-sample paired-proportions test.
power pairedmeans provides PSS computations for a two-sample paired t test assuming known
or unknown population standard deviation of the differences between paired observations. You can
specify standard deviations of each group and a correlation between paired observations, or you can
directly specify the standard deviation of the differences between observations. You can obtain results
for a nonzero null hypothesis of a difference between the two paired means. The command also
provides a way to adjust computations for a finite population sample. See [PSS-2] power pairedmeans.
power pairedproportions provides PSS computations for a two-sample paired-proportions
test that compares proportions in two paired (correlated) samples. The computations are based on
McNemar’s test of marginal homogeneity. You can specify either the discordant proportions or the
marginal proportions. A number of effect specifications are available. For example, you can specify
the effect of interest as a relative risk or an odds ratio. See [PSS-2] power pairedproportions.
Both paired methods compute sample size given power and target parameter, power given sample
size and target parameter, or MDES and the corresponding target parameter given power and sample
size.
For power and sample-size determination of power pairedmeans, an effect may be supplied
by specifying the alternative values of the two means, pretreatment and posttreatment, as command
arguments m
a1
and m
a2
:
power pairedmeans m
a1
m
a2
, . . .
power — Power and sample-size analysis for hypothesis tests 15
Instead of the alternative value m
a2
, you can specify the difference between the two alternative
values in the altdiff() option and the alternative pretreatment mean value m
a1
:
power pairedmeans m
a1
, altdiff() . . .
You may omit both alternative values and specify only the difference between them in the
altdiff() option:
power pairedmeans, altdiff() . . .
By default, the null value of the difference between the pretreatment and posttreatment means is
zero, but you may change it by specifying the nulldiff() option.
For PSS determination of power pairedproportions, there are a number of ways of specifying
an effect of interest; see Alternative ways of specifying effect in [PSS-2] power pairedproportions.
Two main specifications include the specification of discordant proportions and the specification of
marginal probabilities. Specifically, you can supply the information about the effect of interest as
discordant proportions p
12
and p
21
,
power pairedproportions p
12
p
21
, . . .
or as marginal proportions p
1+
and p
+1
:
power pairedproportions p
1+
p
+1
, corr(numlist) . . .
When you specify marginal proportions, you must also specify the correlation between paired
observations in the corr() option.
For sample-size determination, the reported sample size is rounded up to the nearest integer. This
ensures that the corresponding actual power is at least as large as the specified power. You can specify
the nfractional option to obtain the corresponding fractional sample size or you can recompute the
actual power using the reported rounded value; see Fractional sample sizes in [PSS-4] Unbalanced
designs for details.
Below we show a quick example of PSS analyses for a two-sample paired-means test. See
[PSS-2] power pairedmeans and [PSS-2] power pairedproportions for more examples.
Example 3: PSS analysis for a two-sample paired-means test
A forester would like to study the effects of a fertilizer treatment on heights of Virginia pine trees.
The trees are planted in pairs with only one of them receiving the fertilizer treatment. The average
height of untreated trees is 27.5 feet, with a standard deviation of 4.5 feet. The fertilizer treatment is
expected to increase the average height to 30 feet, with the same standard deviation of 4.5 feet. The
correlation between the paired tree heights is expected to be 0.4. The forester would like to know
how many pairs of pine trees need to be planted so that a 5%-level two-sided paired-means t test
detects the anticipated difference with 80% power.
We use power pairedmeans for power and sample-size analysis. We supply the alternative
pretreatment and posttreatment means of 27.5 and 30, respectively, as command arguments after the
command name. The standard deviations of the two groups are the same, so we specify their common
value in the sd() option. We specify the correlation of 0.4 in the corr() option. The default value
for power is 0.8 and for significance level is 0.05, so we omit the corresponding options power(0.8)
and alpha(0.05).
16 power — Power and sample-size analysis for hypothesis tests
. power pairedmeans 27.5 30, sd(4.5) corr(0.4)
Performing iteration ...
Estimated sample size for a two-sample paired-means test
Paired t test assuming sd1 = sd2 = sd
H0: d = d0 versus Ha: d != d0
Study parameters:
alpha = 0.0500 ma1 = 27.5000
power = 0.8000 ma2 = 30.0000
delta = 0.5072 sd = 4.5000
d0 = 0.0000 corr = 0.4000
da = 2.5000
sd_d = 4.9295
Estimated sample size:
N = 33
The forester needs 33 pairs of pine trees to run the experiment.
The forester has resources to plant more trees and would like to compute the power corresponding
to the larger sample. To compute power given sample size, we specify sample size in the n() option:
. power pairedmeans 27.5 30, sd(4.5) corr(0.4) n(50)
Estimated power for a two-sample paired-means test
Paired t test assuming sd1 = sd2 = sd
H0: d = d0 versus Ha: d != d0
Study parameters:
alpha = 0.0500 ma1 = 27.5000
N = 50 ma2 = 30.0000
delta = 0.5072 sd = 4.5000
d0 = 0.0000 corr = 0.4000
da = 2.5000
sd_d = 4.9295
Estimated power:
power = 0.9400
The power increases to 0.94.
The forester may also wish to know the smallest detectable difference between average tree heights
of the fertilized group and of the control group that can be detected with 80% power and sample size
of 50. To compute this value, we specify both options n(50) and power(0.8):
. power pairedmeans 27.5, sd(4.5) corr(0.4) n(50) power(0.8)
Performing iteration ...
Estimated target parameters for a two-sample paired-means test
Paired t test assuming sd1 = sd2 = sd
H0: d = d0 versus Ha: d != d0; da > d0
Study parameters:
alpha = 0.0500 ma1 = 27.5000
power = 0.8000 sd = 4.5000
N = 50 corr = 0.4000
d0 = 0.0000
sd_d = 4.9295
Estimated effect size and target parameters:
delta = 0.4042
da = 1.9924
ma2 = 29.4924
power — Power and sample-size analysis for hypothesis tests 17
The smallest detectable difference is 1.99, with a corresponding value of the average tree height for
the fertilized trees of 29.5.
Analysis of variance models
The power command provides PSS computations for three types of analyses of variance (ANOVA)
designs: one way, two way, and repeated measures. power oneway performs PSS analysis for a
one-way ANOVA. power twoway performs PSS analysis for a two-way ANOVA. power repeated
performs PSS analysis for a repeated-measures ANOVA.
power oneway provides PSS computations for a one-way ANOVA model. You can choose between
the overall F test of the equality of group means and a test of a mean contrast. You can either specify
group means or specify their variability in the computations. See [PSS-2] power oneway.
power twoway provides PSS computations for a two-way fixed-effects ANOVA model. You can
choose the overall F test of the main effect of a row factor, a column factor, or a row-by-column
interaction. You can either specify cell means or specify the variance explained by the tested effect.
See [PSS-2] power twoway.
power repeated provides PSS computations for one-way and two-way fixed-effects repeated-
measures ANOVA models. You can choose the overall F test of the main effect of a between-subjects
factor, a within-subject factor, or a between–within factor interaction. You can either specify cell
means or specify the variance explained by the tested effect. See [PSS-2] power repeated.
All methods compute sample size given power and effect size, power given sample size and effect
size, or effect size given power and sample size.
For power and sample-size determination of power oneway, an effect may be supplied by specifying
the alternative values of group means as command arguments m
a1
, m
a2
, m
a3
, and so on:
power oneway m
a1
m
a2
m
a3
. . .
, . . .
Instead of the alternative group means, you can specify the variance of the group means in the
varmeans() option and the number of groups in the ngroups() option:
power oneway, ngroups() varmeans() . . .
For power and sample-size determination of power twoway and power repeated, an effect may
be supplied by specifying the alternative values of cell means as command arguments m
a1,1
, m
a1,2
,
and so on, in a matrix form:
power twoway m
a1,1
m
a1,2
. . .
\ m
a2,1
m
a2,2
. . .
, . . .
power repeated m
a1,1
m
a1,2
. . .
\ m
a2,1
m
a2,2
. . .
, . . .
Instead of the alternative cell means, you can specify the variance of the tested effect in the
vareffect() option and the dimensions of the cell-means matrix: number of rows and columns for
power twoway and number of groups and repeated measures for power repeated:
power twoway, nrows() ncols() factor() vareffect() . . .
power repeated, ngroups() nrepeated() factor() vareffect() . . .
The means can also be supplied as a matrix at the command line. For example, suppose that we
have three groups.
power oneway m
a1
m
a2
m
a3
, . . .
18 power — Power and sample-size analysis for hypothesis tests
The above command would be equivalent to
matrix means = (m
a1
,m
a2
,m
a3
)
power oneway means, . . .
There are also other alternative specifications of an effect with these commands. See the specific
entry of each command.
For sample-size determination, the reported sample size is rounded up to the nearest integer. This
ensures that the corresponding actual power is at least as large as the specified power. You can specify
the nfractional option to obtain the corresponding fractional sample size, or you can recompute
the actual power using the reported rounded value; see Fractional sample sizes in [PSS-4] Unbalanced
designs for details.
Below we show a quick example of PSS analysis for a one-way ANOVA model. See [PSS-2] power
oneway, [PSS-2] power twoway, and [PSS-2] power repeated for more examples.
Example 4: PSS analysis for a one-way ANOVA model
Researchers would like to compare the effects of four drugs on systolic blood pressure. They
would like to use a one-way ANOVA model to test the equality of mean blood-pressure measurements
across four drugs. To conduct a study, the researchers need an estimate for the number of subjects to
be enrolled in a study. From a previous pilot study, the variance between group means was estimated
to be 57, and the error variance was estimated to be 115. The researchers would like to compute
the required sample size to detect the effect size of 0.7040 =
p
57/115 with 80% power using a
5%-level F test of the equality of means assuming a balanced design.
We use power oneway to compute the sample size. We specify the number of groups and the
estimates of variances in the corresponding options. The default value for power is 0.8 and for
significance level is 0.05, so we omit the corresponding options power(0.8) and alpha(0.05).
. power oneway, ngroups(4) varmeans(57) varerror(115)
Estimated sample size for one-way ANOVA
F test for group effect
H0: delta = 0 versus Ha: delta != 0
Study parameters:
alpha = 0.0500
power = 0.8000
delta = 0.7040
N_g = 4
Var_m = 57.0000
Var_e = 115.0000
Estimated sample sizes:
N = 28
N per group = 7
The researchers need to recruit 28 subjects, 7 subjects per group, for this study.
Unfortunately, the researchers can afford to recruit only 20 subjects. They wish to compute the
power corresponding to this smaller sample size. To compute power, we additionally specify sample
size in the n() option:
power — Power and sample-size analysis for hypothesis tests 19
. power oneway, n(20) ngroups(4) varmeans(57) varerror(115)
Estimated power for one-way ANOVA
F test for group effect
H0: delta = 0 versus Ha: delta != 0
Study parameters:
alpha = 0.0500
N = 20
N per group = 5
delta = 0.7040
N_g = 4
Var_m = 57.0000
Var_e = 115.0000
Estimated power:
power = 0.6400
The power decreases to 0.64.
The researchers are not satisfied with such a low power. They now would like to compute the
smallest effect size and the corresponding variance of means that can be detected with the power of
80% and the sample size of 20. To compute effect size, we specify both power and sample size in
respective options:
. power oneway, n(20) power(0.8) ngroups(4) varerror(115)
Performing iteration ...
Estimated between-group variance for one-way ANOVA
F test for group effect
H0: delta = 0 versus Ha: delta != 0
Study parameters:
alpha = 0.0500
power = 0.8000
N = 20
N per group = 5
N_g = 4
Var_e = 115.0000
Estimated effect size and between-group variance:
delta = 0.8353
Var_m = 80.2329
The smallest detectable effect size is 0.8353, with a corresponding value of the between-group variance
of 80.2329.
Linear regression
The power command provides PSS computations for a linear regression model. power oneslope
provides PSS computations for a slope test in a simple linear regression. power rsquared provides PSS
computations for an R
2
test in a multiple linear regression. power pcorr provides PSS computations
for a partial-correlation test in a multiple linear regression.
power oneslope provides estimates of sample size, power, or target slope in a simple linear
regression. It supports multiple ways of specifying the effect size, which is defined as the difference
between the alternative and null values of the slope multiplied by the ratio of standard deviations of
the covariate to the error term. Instead of specifying the standard deviation of the error term using the
sderror() option, users can specify the standard deviation of the dependent variable in sdy() or the
correlation between the dependent variable and the covariate of interest in corr(). See [PSS-2] power
oneslope.
20 power — Power and sample-size analysis for hypothesis tests
power rsquared reports estimates of sample size, power, or target R
2
in a multiple linear
regression using an R
2
test. An R
2
test is an F test of the coefficient of determination, R
2
, which
is used to test the significance of coefficients in a multiple linear regression. When the ncontrol()
option is not specified, the computation is based on a test of all coefficients in the model. When the
ncontrol() option is specified, the computation is based on a test of a subset of coefficients in the
full model against the reduced model. See [PSS-2] power rsquared.
power pcorr provides estimates of sample size, power, or target squared partial correlation for
a partial-correlation test in a multiple linear regression. power pcorr is an alternative to power
rsquared, ncontrol() for testing the significance of a subset of coefficients using a partial-
correlation test. See [PSS-2] power pcorr.
Below we show two examples of PSS analysis for a linear regression model.
Example 5: Sample size for the test of the slope in a simple linear regression model
Consider a hypothetical study for which the goal is to investigate the effect of average time spent
per day exercising on BMI, measured in kg/m
2
. The parameter of interest is the slope coefficient b,
which measures the effect of exercising on BMI. Our null hypothesis is H
0
: b = 0 versus a two-sided
alternative H
a
: b 6= 0.
We wish to compute the sample size required to detect a drop in BMI of 0.1 kg/m
2
per minute
of exercise, with 80% power using a 5%-level two-sided test. We assume a standard deviation of 10
minutes for time spent exercising in sdx() and 4.0 kg/m
2
for BMI in sdy().
. power oneslope 0 -0.1, sdx(10) sdy(4)
Performing iteration ...
Estimated sample size for a linear regression slope test
t test
H0: b = b0 versus Ha: b != b0
Study parameters:
alpha = 0.0500
power = 0.8000
delta = -0.2582
b0 = 0.0000
ba = -0.1000
sdx = 10.0000
sderror = 3.8730
sdy = 4.0000
Estimated sample size:
N = 120
The required sample size is 120. See [PSS-2] power oneslope for details.
Example 6: Power of an R
2
test in a multiple linear regression model
Consider a hypothetical study for which the goal is to investigate the effect of verbal aptitude and
extraversion on sales, controlling for age, education, and prior experience.
Suppose that all five variables—verbal aptitude, extraversion, age, education, and prior experience—
explain about 10% of the variance of the sales and that the three control variables—age, education,
and prior experience—explain about 6% of the variance of the sales. We want to compute the power of
detecting a 4% change in the R
2
after adding the two tested variables, verbal aptitude and extraversion,
to the model, with 100 subjects at a 5% significance level:
power — Power and sample-size analysis for hypothesis tests 21
. power rsquared .06 .1, ntested(2) ncontrol(3) n(100)
Estimated power for multiple linear regression
F test for R2 testing subset of coefficients
H0: R2_F = R2_R versus Ha: R2_F != R2_R
Study parameters:
alpha = 0.0500
N = 100
delta = 0.0444
R2_R = 0.0600
R2_F = 0.1000
R2_diff = 0.0400
ncontrol = 3
ntested = 2
Estimated power:
power = 0.4431
The achieved power is about 44%. See [PSS-2] power rsquared for details.
Contingency tables
The power command provides PSS computations for three types of analyses of contingency tables.
power cmh performs PSS analysis for a Cochran–Mantel–Haenszel (CMH) test of association in
stratified 2 × 2 tables. The command accommodates unbalanced stratum sizes and unbalanced group
sizes within each stratum. See [PSS-2] power cmh.
power mcc performs PSS analysis for a test of association between a risk factor and a disease in
1:M matched case–control studies. See [PSS-2] power mcc.
power trend performs PSS analysis for a test of a linear trend in a probability of response
in J × 2 tables, also known as a Cochran–Armitage test. It accommodates unbalanced designs and
unequally spaced exposure levels (doses). With equally spaced exposure levels, a continuity correction
is available. See [PSS-2] power trend.
All methods compute sample size given power and effect size and power given sample size and
effect size. power cmh and power mcc also compute effect size given power and sample size.
For sample-size determination, the reported sample sizes are rounded up to the nearest integer.
This ensures that the corresponding actual power is at least as large as the specified power. You
can specify the nfractional option to obtain the corresponding fractional sample sizes, or you
can recompute the actual power using the reported rounded values; see Fractional sample sizes in
[PSS-4] Unbalanced designs for details.
Below we show a quick example of PSS analysis for a Cochran–Armitage test by using power
trend; see [PSS-2] power trend for more examples.
Example 7: Sample size for a Cochran–Armitage trend test
Consider a study investigating the effectiveness of a new topical antibiotic for the treatment of
skin infections.
Suppose that in previous studies of the treatment, we observed the following proportions of
successfully treated cases at different doses. We may hypothesize that these represent the probability
of a successful treatment for each dose.
22 power — Power and sample-size analysis for hypothesis tests
Doses/day Proportion successes
1 0.80
2 0.85
3 0.90
We wish to determine the minimum sample size required for a clinical trial designed to detect a
dose–response trend with 80% power using a two-sided 5%-level Cochran–Armitage test.
To compute the required sample size, we specify the values 0.80, 0.85, and 0.90 as the alternative
success probabilities for each of the three doses after the command name. We omit the alpha(0.05)
and power(0.8) options because the specified values are their defaults.
. power trend .80 .85 .90
note: exposure levels are assumed to be equally spaced.
Performing iteration ...
Estimated sample size for a trend test
Cochran--Armitage trend test
H0: b = 0 versus Ha: b != 0; logit(p) = a + b*x
Study parameters:
alpha = 0.0500
power = 0.8000
N_g = 3
p1 = 0.8000
p2 = 0.8500
p3 = 0.9000
Estimated sample sizes:
N = 597
N per group = 199
A total sample of 597 individuals, 199 individuals per group, must be obtained to detect a linear trend
in probability of a successful treatment with 80% power using a two-sided 5%-level Cochran–Armitage
test.
Suppose that we can recruit only 300 subjects. We can check how such a reduction in sample size
affects the power. To compute power, we specify the alternative group probabilities, as before, and
the total sample size in the n() option.
. power trend .80 .85 .90, n(300)
note: exposure levels are assumed to be equally spaced.
Estimated power for a trend test
Cochran--Armitage trend test
H0: b = 0 versus Ha: b != 0; logit(p) = a + b*x
Study parameters:
alpha = 0.0500
N = 300
N per group = 100
N_g = 3
p1 = 0.8000
p2 = 0.8500
p3 = 0.9000
Estimated power:
power = 0.5082
With a sample of 300 subjects in this study, the power to detect a linear trend in probabilities decreases
dramatically from 0.8 to 0.5, which is unacceptably low for practical purposes.
power — Power and sample-size analysis for hypothesis tests 23
Survival analysis
The power command provides PSS computations for survival analysis comparing two survivor
functions using the log-rank test or the exponential test, as well as for more general survival analysis
investigating the effect of a single covariate in a Cox proportional hazards regression model, possibly
in the presence of other covariates. It provides the estimate of the number of events required to be
observed (or the expected number of events) in a study. The minimal effect size (minimal detectable
difference, expressed as the hazard ratio or the log hazard-ratio) may also be obtained for the log-rank
test and for the Wald test on a single coefficient from the Cox model.
power cox provides estimates of sample size, power, or the minimal detectable value of the
coefficient when an effect of a single covariate on subject survival is to be explored using Cox
proportional hazards regression. It is assumed that the effect is to be tested using the partial likelihood
from the Cox model (for example, score or Wald test) on the coefficient of the covariate of interest.
See [PSS-2] power cox.
power exponential reports estimates of sample size or power when the disparity in the two
exponential survivor functions is to be tested using the exponential test, the parametric test comparing
the two exponential hazard rates. In particular, we refer to the (exponential) hazard-difference test as
the exponential test for the difference between hazards and the (exponential) log hazard-ratio test as
the exponential test for the log of the hazard ratio or, equivalently, for the difference between log
hazards. See [PSS-2] power exponential.
power logrank reports estimates of sample size, power, or minimal detectable value of the hazard
ratio (or log hazard-ratio) in the case when the two survivor functions are to be compared using the
log-rank test. The only requirement about the distribution of the survivor functions is that the two
survivor functions must satisfy the proportional-hazards assumption. See [PSS-2] power logrank.
For sample-size and power computations, the default effect size corresponds to a value of the
hazard ratio of 0.5 and may be changed by specifying the hratio() option. The hazard ratio is
defined as a ratio of hazards of the experimental group to the control group (or the less favorable of
the two groups). Other ways of specifying the effect size are available, and these are particular to
each subcommand.
By default, all subcommands assume a type I study, that is, perform computations for uncensored
survival data. The censoring information may be taken into account by specifying the appropriate
arguments or options. See [PSS-2] power cox, [PSS-2] power logrank, and [PSS-2] power exponential
for details.
Example 8: Sample size for the test of the effect of a covariate in the Cox model
Consider a hypothetical study for which the goal is to investigate the effect of the expression of one
gene on subject survival with the Cox proportional hazards regression model. Suppose that the Wald
test is to be used to test the coefficient on the gene after fitting the Cox model. Gene expression values
measure the level of activity of the gene. Consider the scenario described in Simon, Radmacher, and
Dobbin (2002) in which the hazard ratio of 3 associated with a one-unit change in the log
2
intensity
of a gene (or, respectively, with a twofold change in gene expression level) is desired to be detected
with 95% power using a two-sided, 0.001-level test. The estimate of the standard deviation of the
log
2
-intensity level of the gene over the entire set of samples is assumed to be 0.75.
24 power — Power and sample-size analysis for hypothesis tests
. power cox, hratio(3) sd(0.75) power(0.95) alpha(0.001)
Estimated sample size for Cox PH regression
Wald test
H0: beta1 = 0 versus Ha: beta1 != 0
Study parameters:
alpha = 0.0010
power = 0.9500
delta = 1.0986 (coefficient)
hratio = 3.0000
sd = 0.7500
Censoring:
Pr_E = 1.0000
Estimated number of events and sample size:
E = 36
N = 36
Provided that all subjects experience an event in this study, a total of 36 events is required to be
observed in the study to ensure the specified power.
See [PSS-2] power cox for more details.
Example 9: Sample size for two-sample test of exponential survivor functions
Consider an example from Lachin (2011, 490) of a study comparing two therapies, the combination
of a new therapy with the standard one versus the standard alone, in the treatment of lupus nephritis
patients. From previous studies, the survivor function of the control group treated with the standard
therapy was log linear with a constant yearly hazard rate of 0.3. The number of events (failures)
required to ensure 90% power to detect a 50% risk reduction, ∆ = 0.5, (or, respectively, the log
hazard-ratio of ln(0.5) = −0.6931) with a one-sided test at a 0.05 significance level was obtained
to be 72 under equal-group allocation. In the absence of censoring, Lachin (2011) determined that
a total of 72 subjects (36 per group) would have to be recruited to the study. To obtain this same
estimate with power exponential, we supply the control hazard rate 0.3 as an argument and
specify the power(0.9), onesided, and loghazard options to request 90% power, a one-sided
test, and sample-size determination for the exponential log hazard-ratio test (or test for the log-hazard
difference), respectively.
. power exponential 0.3, power(0.9) onesided loghazard
note: input parameters are hazard rates.
Estimated sample sizes for two-sample comparison of survivor functions
Exponential test, log hazard-ratio, conditional
H0: ln(HR) = 0 versus Ha: ln(HR) < 0
Study parameters:
alpha = 0.0500
power = 0.9000
delta = -0.6931 (log hazard-ratio)
Survival information:
h1 = 0.3000
h2 = 0.1500
hratio = 0.5000
Estimated sample sizes:
N = 72
N per group = 36
power — Power and sample-size analysis for hypothesis tests 25
Further, the study was planned to continue for 6 years with a recruitment period of 4 years.
Subjects who did not experience an event by the end of 6 years were censored. For this fixed-duration
study with uniform entry (recruitment), the estimate of the sample size increases from 72 to 128.
We specify the length of the accrual and the follow-up periods in the aperiod() and fperiod()
options, respectively. We also request to display the expected number of events by using the show
option.
. power exponential 0.3, power(0.9) onesided loghazard aperiod(4) fperiod(2) show
note: input parameters are hazard rates.
Estimated sample sizes for two-sample comparison of survivor functions
Exponential test, log hazard-ratio, conditional
H0: ln(HR) = 0 versus Ha: ln(HR) < 0
Study parameters:
alpha = 0.0500
power = 0.9000
delta = -0.6931 (log hazard-ratio)
Accrual and follow-up information:
duration = 6.0000
follow-up = 2.0000
accrual = 4.0000 (uniform)
Survival information:
h1 = 0.3000
h2 = 0.1500
hratio = 0.5000
Estimated expected number of events:
E|Ha = 72 E|H0 = 74
E1|Ha = 44 E1|H0 = 37
E2|Ha = 28 E2|H0 = 37
Estimated sample sizes:
N = 128
N per group = 64
Under the alternative hypothesis of H
a
: ln(∆) = −0.6931, where ln(∆) denotes the log hazard-
ratio of the experimental group to the control group, we expect to observe 44 events in the control
group and 28 events in the experimental group. A total of 128 subjects (64 per group) is required to
be enrolled into the study to observe an expected total of 72 events under the alternative.
See [PSS-2] power exponential for more examples.
Example 10: Sample size for the log-rank test
Consider an example from Machin and Campbell (2005) of a study comparing two forms of surgical
resection for patients with gastric cancer. From a prestudy survey, the baseline 5-year survival rate was
expected to be 20% and an anticipated increase in survival in the experimental group expressed as a
hazard ratio of 0.6667 (corresponding to a 5-year survival rate of approximately 34%) was desired to
be detected with 90% power using a two-sided, 0.05 level, log-rank test under 1:1 randomization. To
obtain the estimate of the sample size for this study, we use power logrank with survival proportion
in the control group 0.2 supplied as an argument, the hratio(0.6667) option to request a hazard
ratio of 0.6667, and the power(0.9) option to request 90% power.
26 power — Power and sample-size analysis for hypothesis tests
. power logrank 0.2, hratio(0.6667) power(0.9)
Estimated sample sizes for two-sample comparison of survivor functions
Log-rank test, Freedman method
H0: HR = 1 versus Ha: HR != 1
Study parameters:
alpha = 0.0500
power = 0.9000
delta = 0.6667 (hazard ratio)
hratio = 0.6667
Censoring:
s1 = 0.2000
s2 = 0.3420
Pr_E = 0.7290
Estimated number of events and sample sizes:
E = 263
N = 362
N per group = 181
From the output, 263 events (failures) are required to be observed in this study to ensure 90%
power to detect a hazard ratio of 0.6667 by using the log-rank test. The respective estimate of the
total number of subjects required to observe 263 events in a 5-year study is 362 with 181 subjects
per surgical group. Our estimate, 181, of each group’s sample size is close to the manually computed
estimate of 180 from Machin and Campbell (2005). This is a fixed-duration study in which 20% of
subjects were expected to survive (be censored) by the end of the study.
See [PSS-2] power logrank for more detailed examples and other available methods of sample-size
computation for this type of analysis.
Cluster randomized designs
So far, all power analyses have assumed simple randomization of the subjects in the study. We
could instead have a cluster randomized design (CRD). In a CRD, groups of subjects or clusters are
randomized instead of individual subjects, so the sample size is determined by the number of clusters
and the cluster size. The sample-size determination thus consists of the determination of the number
of clusters given cluster size or the determination of cluster size given the number of clusters.
power supports CRDs with methods onemean, oneproportion, twomeans, twoproportions,
and logrank. To request computations for a CRD, you specify the cluster option, include the
number of clusters k() with one-sample methods and k1() or k2() with two-sample methods, or
include the cluster size m(), m1(), or m2(). In addition to power and effect size, all methods compute
the numbers of clusters given the cluster sizes or the cluster sizes given the numbers of clusters.
Two-sample methods can also compute the number of clusters or cluster size of one group given that
of the other group.
A CRD requires more subjects to obtain the same statistical power as the corresponding individual-
level design because the subjects within a cluster are correlated. Power and sample-size computations
in a CRD account for this intraclass correlation. All power, cluster methods use the default intraclass
correlation of 0.5, but you may change this by using the rho() option.
By default, all methods assume equal cluster sizes or equal numbers of subjects in each cluster.
In practice, cluster sizes often vary, in which case you may provide the coefficient of variation of the
cluster sizes in the cvcluster() option to account for varying cluster sizes.
power — Power and sample-size analysis for hypothesis tests 27
Below we show a short example of PSS analysis for power onemean for the one-sample case and
power twoproportions for the two-sample case. See [PSS-2] power onemean, cluster, [PSS-2] power
oneproportion, cluster, [PSS-2] power twomeans, cluster, [PSS-2] power twoproportions, cluster,
and [PSS-2] power logrank, cluster for more examples.
Example 11: Number of clusters for a one-sample mean test in a CRD, specifying cluster
size
Consider an example that studies the effectiveness of coaching programs in improving the verbal
part of SAT scores. Previous studies found that students retaking the SAT exams without any coaching
program improve their scores by 15 points on average with a standard deviation of about 40 points.
The population standard deviation is assumed to be 40. We assume that students are sampled from a
set of classes and that the scores of students from the same class are correlated. We plan on sampling
10 students from each class and assume that the intraclass correlation is 0.3.
A new coaching program claims to improve average SAT scores by 40 points. The changes in scores
are assumed to be approximately normally distributed. The parameter of interest in this example is
the mean change in the test scores. To test the claim, investigators wish to conduct another study and
compute the number of classes that is required to detect a mean change in scores of 40 points with
80% power using a 5%-level two-sided test:
. power onemean 15 40, m(10) sd(40) rho(0.3)
Performing iteration ...
Estimated number of clusters for a one-sample mean test
Cluster randomized design, z test
H0: m = m0 versus Ha: m != m0
Study parameters:
alpha = 0.0500
power = 0.8000
delta = 0.3249
m0 = 15.0000
ma = 40.0000
sd = 40.0000
Cluster design:
M = 10
rho = 0.3000
Estimated number of clusters and sample size:
K = 8
N = 80
We find that 8 classes with 10 students per class, a total of 80 students, are required to detect a shift
of 40 points in average SAT scores given the standard deviation of 40 points with 80% power using
a 5%-level two-sided test. See [PSS-2] power onemean, cluster for more information.
Example 12: Numbers of clusters for a two-sample proportions test in a CRD, specifying
cluster sizes
Consider a study investigating the effectiveness of a program to promote after-school activities in
increasing the rate of students participating in the after-school club. Schools that are involved in the
study will be randomly assigned either to the experimental group that participates in the program or
to the control group that does not. A researcher plans to recruit 50 students from each school and
assumes an intraclass correlation of 0.2. The researcher wants to be able to detect an increase of 0.2
in the anticipated control-group rate of 0.4, which corresponds to the experimental-group rate of 0.6.
28 power — Power and sample-size analysis for hypothesis tests
To compute the number of schools in each group required to detect the desired rate with 80%
power using a 5%-level two-sided test, we type
. power twoproportions 0.4 0.6, m1(50) m2(50) rho(0.2)
Performing iteration ...
Estimated numbers of clusters for a two-sample proportions test
Cluster randomized design, Pearson’s chi-squared test
H0: p2 = p1 versus Ha: p2 != p1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = 0.2000 (difference)
p1 = 0.4000
p2 = 0.6000
Cluster design:
M1 = 50
M2 = 50
rho = 0.2000
Estimated numbers of clusters and sample sizes:
K1 = 21
K2 = 21
N1 = 1,050
N2 = 1,050
We find that for 50 students, 21 schools per group, with a total of 1,050 students per group, are
required to detect a 0.2 difference in participation rates in the after-school club with 80% power using
a 5%-level two-sided test. See [PSS-2] power twoproportions, cluster for more information.
Tables of results
When power is used to perform computations for a single set of study parameters, the results can
be displayed either as text or in a table. The default is to display results as text:
. power onemean 0 0.2
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
Study parameters:
alpha = 0.0500
power = 0.8000
delta = 0.2000
m0 = 0.0000
ma = 0.2000
sd = 1.0000
Estimated sample size:
N = 199
power — Power and sample-size analysis for hypothesis tests 29
You can specify the table option to display results in a table:
. power onemean 0 0.2, table
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
alpha power N delta m0 ma sd
.05 .8 199 .2 0 .2 1
For multiple sets of study parameters, when command arguments or options contain number lists,
the results are automatically displayed in a table:
. power onemean 0 (0.2 0.5)
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
alpha power N delta m0 ma sd
.05 .8 199 .2 0 .2 1
.05 .8 34 .5 0 .5 1
In this example, we specified two values for the second argument.
Default tables can be modified by specifying the table() option. For example, we can change
the order in which the columns are displayed:
. power onemean 0 (0.2 0.5), table(alpha power N m0 ma sd delta)
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
alpha power N m0 ma sd delta
.05 .8 199 0 .2 1 .2
.05 .8 34 0 .5 1 .5
Or we can change column labels:
. power onemean 0 (0.2 0.5), table(, labels(N "Sample size"))
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
alpha power Sample size delta m0 ma sd
.05 .8 199 .2 0 .2 1
.05 .8 34 .5 0 .5 1
30 power — Power and sample-size analysis for hypothesis tests
Or we can select which columns we want to display:
. power onemean 0 (0.2 0.5), table(alpha beta N m0 ma sd)
Performing iteration ...
Estimated sample size for a one-sample mean test
t test
H0: m = m0 versus Ha: m != m0
alpha beta N m0 ma sd
.05 .2 199 0 .2 1
.05 .2 34 0 .5 1
For more examples, see [PSS-2] power, table.
Power curves
During the planning stage of a study, it is difficult to decide on a number of subjects to be enrolled
in a study on the basis of only one set of study parameters. It is common to investigate the effect of
various study scenarios on power. Power curves, or plots of estimated power versus one of the study
parameters, are commonly used for this purpose.
The power command provides the graph and graph() options to produce power curves.
More precisely, when graph is specified, the estimated parameter such as power or sample size
is plotted on the y axis, and the varying parameter is plotted on the x axis.
For example, we compute power and plot it as a function of sample size for a range of sample-size
values between 100 and 200 with a step size of 10:
. power onemean 0 0.2, n(100(10)200) graph
.5
.6
.7
.8
Power (1-β)
100 120 140 160 180 200
Sample size (N)
Parameters: α = .05, δ = .2, µ
0
= 0, µ
a
= .2, σ = 1
t test
H
0
: µ = µ
0
versus H
a
: µ ≠ µ
0
Estimated power for a one-sample mean test
power — Power and sample-size analysis for hypothesis tests 31
Or we can compute sample size and plot it as a function of the alternative mean when the mean
ranges between 0.2 and 1 with a step size of 0.1:
. power onemean 0 (0.2(0.1)1), graph
0
50
100
150
200
Sample size (N)
.2 .4 .6 .8 1
Alternative mean (µ
a
)
Parameters: α = .05, 1-β = .8, µ
0
= 0, σ = 1
t test
H
0
: µ = µ
0
versus H
a
: µ ≠ µ
0
Estimated sample size for a one-sample mean test
Or we can compute the alternative mean for a given power of 80% and a range of sample-size
values between 100 and 200 with a step size of 10, and plot it against the sample size:
. power onemean 0, n(100(10)200) power(0.8) graph
.2
.22
.24
.26
.28
Alternative mean (µ
a
)
100 120 140 160 180 200
Sample size (N)
Parameters: α = .05, 1-β = .8, µ
0
= 0, σ = 1
t test
H
0
: µ = µ
0
versus H
a
: µ ≠ µ
0
; µ
a
> µ
0
Estimated target mean for a one-sample mean test
The above graphs are the default graphs produced by power, graph. Similarly to tabular output,
you can customize graphical output by specifying the graph() option.
For example, we modify the look of the default graph by using the graph(nosimplelabels
legend(title(""))) option. nosimplelabels requests that the graph legend include the column
symbol and an equal sign; legend(title("")) requests that the legend not have a title.
32 power — Power and sample-size analysis for hypothesis tests
. power onemean 0 (0.2(0.1)1), sd(1 1.5) graph(nosimplelabels legend(title("")))
0
100
200
300
400
500
Sample size (N)
.2 .4 .6 .8 1
Alternative mean (µ
a
)
σ=1
σ=1.5
Parameters: α = .05, 1-β = .8, µ
0
= 0
t test
H
0
: µ = µ
0
versus H
a
: µ ≠ µ
0
Estimated sample size for a one-sample mean test
By default, when a graph is produced, the tabular output is suppressed. You can specify the table
option if you also want to see results in a table.
For more examples, see [PSS-2] power, graph.
Add your own methods to power
The power command provides many built-in methods, but sometimes, you may want to compute
sample size or power yourself. For example, you may need to do this by simulation, or you may
want to use a method that is not available in any software package. power makes it easy for you to
add your own method. All you need to do is to write a program that computes sample size, power,
or effect size, and the power command will do the rest for you. It will deal with the support of
multiple values in options and with automatic generation of graphs and tables of results.
Suppose you want to add the method called mymethod to the power command. Just follow these
three steps:
1. Create a program that computes sample size, power, or effect size and follows power’s
naming convention: power cmd mymethod.
2. Store results following power’s simple naming conventions for results. For example, store
the value of power in r(power), the value of sample size in r(N), and so on.
3. Place your program power cmd mymethod where Stata can find it.
To show how easy this all is, let’s write a program to compute power for a one-sample z test given
sample size, standardized difference, and significance level. For simplicity, we assume a two-sided
test.
power — Power and sample-size analysis for hypothesis tests 33
We will call our new method myztest.
program power_cmd_myztest, rclass
version 18.0 // (or version 18.5 for StataNow)
// parse options
syntax , n(integer) /// sample size
STDDiff(real) /// standardized diff.
Alpha(string) /// significance level
// compute power
tempname power
scalar ‘power’ = normal(‘stddiff’*sqrt(‘n’) - ///
invnormal(1-‘alpha’/2))
// return results
return scalar power = ‘power’
return scalar N = ‘n’
return scalar alpha = ‘alpha’
return scalar stddiff = ‘stddiff’
end
The computation in this program takes only one line, but it could be as complicated as we like.
It could even involve simulation to compute the power.
With our program in hand, we can type
. power myztest, n(20) stddiff(1) alpha(.05)
power will find our program, supply it with the n(20), stddiff(1), and alpha(.05) options,
and use its returned results to produce
. power myztest, n(20) stddiff(1) alpha(.05)
Estimated power
Two-sided test
alpha power N
.05 .994 20
That was not too impressive. Our program did all the work.
What if we supplied power with a list of sample sizes?
. power myztest, n(10 15 20 25) stddiff(1)
Estimated power
Two-sided test
alpha power N
.05 .8854 10
.05 .9721 15
.05 .994 20
.05 .9988 25
power has taken our list of sample sizes and computed powers for all of them—even though our
program could only compute a single power!
Moreover, we can use power’s standard table() option to control exactly how that table looks;
see Table of results for more examples of tables. power also has hooks that let our program determine
how the columns are labeled and how the table appears.
34 power — Power and sample-size analysis for hypothesis tests
We can supply both sample sizes and significance levels and request a graph instead of a table:
. power myztest, n(10(1)20) alpha(.05 .10 .25) stddiff(1) graph
.85
.9
.95
1
Power (1-β)
10 12 14 16 18 20
Sample size (N)
.05
.1
.25
Significance level (α)
Two-sided test
Estimated power
We can even request that the graph show α on the x axis with separate plots for each sample size.
. power myztest, n(10(2)20) alpha(.05 .10 .25) stddiff(1) graph(xdim(alpha))
.85
.9
.95
1
Power (1-β)
.05 .1 .25
Significance level (α)
10
12
14
16
18
20
Sample size (N)
Two-sided test
Estimated power
All this may make it worth writing more complicated programs to compute power for more
complicated tests and comparisons.
See [PSS-2] power usermethod for more examples.
power — Power and sample-size analysis for hypothesis tests 35
Stored results
power stores the following in r():
Scalars
r(alpha) significance level
r(power) power
r(beta) probability of a type II error
r(delta) effect size
r(N) total sample size
r(N a) actual sample size
r(N1) sample size of the control group
r(N2) sample size of the experimental group
r(nratio) ratio of sample sizes, N2/N1
r(nratio a) actual ratio of sample sizes
r(nfractional) 1 if nfractional is specified, 0 otherwise
r(onesided) 1 for a one-sided test, 0 otherwise
r(separator) number of lines between separator lines in the table
r(divider) 1 if divider is requested in the table, 0 otherwise
r(init) initial value of the estimated parameter
r(maxiter) maximum number of iterations
r(iter) number of iterations performed
r(tolerance) requested parameter tolerance
r(deltax) final parameter tolerance achieved
r(ftolerance) requested distance of the objective function from zero
r(function) final distance of the objective function from zero
r(converged) 1 if iteration algorithm converged, 0 otherwise
Macros
r(type) test
r(method) the name of the specified method
r(direction) upper or lower
r(columns) displayed table columns
r(labels) table column labels
r(widths) table column widths
r(formats) table column formats
Matrices
r(pss table) table of results
Also see Stored results in the method-specific manual entries for the full list of stored results.
Methods and formulas
By default, the power command rounds sample sizes to integers and uses integer values in the
computations. To ensure conservative results, the command rounds down the input sample sizes and
rounds up the output sample sizes. See Fractional sample sizes in [PSS-4] Unbalanced designs for
details.
Some of power’s methods require iteration. For example, the sample size for a two-sided test
is usually solved iteratively from the two-sided power equation. Most methods use Mata function
solvenl() and its Newton’s method described in Newton-type methods in [M-5] solvenl( ) to solve a
nonlinear power equation. Other methods use a bisection method to find a root of a nonlinear power
equation.
See Methods and formulas in the method-specific manual entries for details.
36 power — Power and sample-size analysis for hypothesis tests
References
Batistatou, E., C. Roberts, and S. Roberts. 2014. Sample size and power calculations for trials and quasi-experimental
studies with clustering. Stata Journal 14: 159–175.
Cattaneo, M. D., R. Titiunik, and G. Vazquez-Bare. 2019. Power calculations for regression-discontinuity designs.
Stata Journal 19: 210–245.
Earnest, A. 2017. Essentials of a Successful Biostatistical Collaboration. Boca Raton, FL: CRC Press.
Gallis, J. A., X. Wang, P. J. Rathouz, J. S. Preisser, F. Li, and E. L. Turner. 2022. power swgee: GEE-based power
calculations in stepped wedge cluster randomized trials. Stata Journal 22: 811–841.
Huber, C. 2019a. Calculating power using Monte Carlo simulations, part 1: The basics. The Stata Blog: Not Elsewhere
Classified. https://blog.stata.com/2019/01/10/calculating-power-using-monte-carlo-simulations-part-1-the-basics/.
. 2019b. Calculating power using Monte Carlo simulations, part 2: Running your simulation using power.
The Stata Blog: Not Elsewhere Classified. https://blog.stata.com/2019/01/29/calculating-power-using-monte-carlo-
simulations-part-2-running-your-simulation-using-power/.
Lachin, J. M. 2011. Biostatistical Methods: The Assessment of Relative Risks. 2nd ed. Hoboken, NJ: Wiley.
Ma, X., and Y. B. Cheung. 2022. crtrest: A command for ratio estimators of intervention effects on event rates in
cluster randomized trials. Stata Journal 22: 908–923.
Machin, D. 2004. On the evolution of statistical methods as applied to clinical trials. Journal of Internal Medicine
255: 521–528. https://doi.org/10.1111/j.1365-2796.2004.01319.x.
Machin, D., and M. J. Campbell. 2005. Design of Studies for Medical Research. Chichester, UK: Wiley.
Simon, R., R. D. Radmacher, and K. Dobbin. 2002. Design of studies using DNA microarrays. Genetic Epidemiology
23: 21–36. https://doi.org/10.1002/gepi.202.
Thompson, J., C. Davey, R. J. Hayes, J. Hargreaves, and K. Fielding. 2019. Permutation tests for stepped-wedge
cluster-randomized trials. Stata Journal 19: 803–819.
Wittes, J. 2002. Sample size calculations for randomized control trials. Epidemiologic Reviews 24: 39–53.
https://doi.org/10.1093/epirev/24.1.39.
Also see
[PSS-2] Intro (power) — Introduction to power and sample-size analysis for hypothesis tests
[PSS-5] Glossary
[ADAPT] GSD intro — Introduction to group sequential designs
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