Title stata.com
power twomeans — Power analysis for a two-sample means test
Description Quick start Menu Syntax
Options Remarks and examples Stored results Methods and formulas
References Also see
Description
power twomeans computes sample size, power, or the experimental-group mean for a two-sample
means test. By default, it computes sample size for the given power and the values of the control-group
and experimental-group means. Alternatively, it can compute power for given sample size and values
of the control-group and experimental-group means or the experimental-group mean for given sample
size, power, and the control-group mean. For power and sample-size analysis in a cluster randomized
design, see [PSS] power twomeans, cluster. Also see [PSS] power for a general introduction to the
power command using hypothesis tests.
Quick start
Sample size for a test of H
0
: µ
1
= µ
2
versus H
a
: µ
1
6= µ
2
given alternative control-group mean
m
1
= 8 and alternative experimental-group mean m
2
= 12 with shared standard deviation of 9
using default power of 0.8 and significance level α = 0.05
power twomeans 8 12, sd(9)
As above, but for m
2
equal to 10, 11, 12, 13, and 14
power twomeans 8 (10(1)14), sd(9)
As above, but display results in a graph of sample size versus m
2
power twomeans 8 (10(1)14), sd(9) graph
As above, but specify different standard deviations s
1
= 7 and s
2
= 10
power twomeans 8 (10(1)14), sd1(7) sd2(10) graph
Sample size for one-sided test with power of 0.9
power twomeans 8 12, sd(9) power(.9) onesided
Same as above, specified as µ
1
and difference between means m
2
− m
1
= 4
power twomeans 8, sd(9) power(.9) onesided diff(4)
Power for a total sample size of 74 with balanced group sizes
power twomeans 8 12, sd(9) n(74)
As above, but for sample sizes of 45 and 30 in groups 1 and 2, respectively
power twomeans 8 12, sd(9) n1(45) n2(30)
Effect size and target mean difference for a sample size of 200 with power of 0.8
power twomeans 8, sd(9) power(.8) n(200)
1
2 power twomeans — Power analysis for a two-sample means test
Menu
Statistics > Power and sample size
Syntax
Compute sample size
power twomeans m
1
m
2
, power(numlist) options
Compute power
power twomeans m
1
m
2
, n(numlist)
options
Compute effect size and experimental-group mean
power twomeans m
1
, n(numlist) power(numlist)
options
where m
1
is the mean in the control (reference) group and m
2
is the mean in the experimental
(comparison) group. m
1
and m
2
may each be specified either as one number or as a list of values
in parentheses (see [U] 11.1.8 numlist).
power twomeans — Power analysis for a two-sample means test 3
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
∗
diff(numlist) difference between the experimental-group mean and the
control-group mean, m
2
− m
1
; specify instead of the
experimental-group mean m
2
∗
sd(numlist) common standard deviation of the control and the
experimental groups assuming equal standard deviations in
both groups; default is sd(1)
∗
sd1(numlist) standard deviation of the control group; requires sd2()
∗
sd2(numlist) standard deviation of the experimental group; requires sd1()
knownsds request computation assuming known standard deviations for
both groups; default is to assume unknown standard
deviations
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] power, table
saving(filename
, replace
) save the table data to filename; use replace to overwrite
existing filename
Graph
graph
(graphopts)
graph results; see [PSS] power, graph
4 power twomeans — Power analysis for a two-sample means test
Iteration
init(#) initial value for sample sizes or experimental-group mean
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
cluster perform computations for a CRD;
see [PSS] power twomeans, cluster
noti
tle 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.
cluster and notitle do not appear in the dialog box.
where tablespec is
column
:label
column
:label
. . .
, tableopts
column is one of the columns defined below, and label is a column label (may contain quotes and
compound quotes).
column Description Symbol
alpha significance level α
power power 1 − β
beta type II error probability β
N total number of subjects N
N1 number of subjects in the control group N
1
N2 number of subjects in the experimental group N
2
nratio ratio of sample sizes, experimental to control N
2
/N
1
delta effect size δ
m1 control-group mean µ
1
m2 experimental-group mean µ
2
diff difference between the experimental-group mean and µ
2
− µ
1
the control-group mean
sd common standard deviation σ
sd1 control-group standard deviation σ
1
sd2 experimental-group standard deviation σ
2
target target parameter; synonym for m2
all display all supported columns
Column beta is shown in the default table in place of column power if specified.
Columns nratio, diff, sd, sd1, and sd2 are shown in the default table if specified.
power twomeans — Power analysis for a two-sample means test 5
Options
Main
alpha(), power(), beta(), n(), n1(), n2(), nratio(), compute(), nfractional; see
[PSS] power.
diff(numlist) specifies the difference between the experimental-group mean and the control-group
mean, m
2
− m
1
. You can specify either the experimental-group mean m
2
as a command argument
or the difference between the two means in diff(). If you specify diff(#), the experimental-
group mean is computed as m
2
= m
1
+ #. This option is not allowed with the effect-size
determination.
sd(numlist) specifies the common standard deviation of the control and the experimental groups
assuming equal standard deviations in both groups. The default is sd(1).
sd1(numlist) specifies the standard deviation of the control group. If you specify sd1(), you must
also specify sd2().
sd2(numlist) specifies the standard deviation of the experimental group. If you specify sd2(), you
must also specify sd1().
knownsds requests that standard deviations of each group be treated as known in the computations.
By default, standard deviations are treated as unknown, and the computations are based on a
two-sample t test, which uses a Student’s t distribution as a sampling distribution of the test
statistic. If knownsds is specified, the computation is based on a two-sample z test, which uses
a normal distribution as the sampling distribution of the test statistic.
direction(), onesided, parallel; see [PSS] power.
Table
table, table(), notable; see [PSS] power, table.
saving(); see [PSS] power.
Graph
graph, graph(); see [PSS] power, graph. Also see the column table for a list of symbols used by
the graphs.
Iteration
init(#) specifies the initial value for the estimated parameter. For sample-size determination, the
estimated parameter is either the control-group size n
1
or, if compute(N2) is specified, the
experimental-group size n
2
. For the effect-size determination, the estimated parameter is the
experimental-group mean m
2
. The default initial values for a two-sided test are obtained as a
closed-form solution for the corresponding one-sided test with the significance level α/2. The
default initial values for the t test computations are based on the corresponding large-sample
normal approximation.
iterate(), tolerance(), ftolerance(), log, nolog, dots, nodots; see [PSS] power.
The following options are available with power twomeans but are not shown in the dialog box:
cluster; see [PSS] power twomeans, cluster.
notitle; see [PSS] power.
6 power twomeans — Power analysis for a two-sample means test
Remarks and examples stata.com
Remarks are presented under the following headings:
Introduction
Using power twomeans
Computing sample size
Computing power
Computing effect size and experimental-group mean
Testing a hypothesis about two independent means
This entry describes the power twomeans command and the methodology for power and sample-
size analysis for a two-sample means test. See [PSS] intro for a general introduction to power
and sample-size analysis and [PSS] power for a general introduction to the power command using
hypothesis tests. Also see [PSS] power twomeans, cluster for power and sample-size analysis in a
cluster randomized design.
Introduction
The analysis of means is one of the most commonly used approaches in a wide variety of statistical
studies. Many applications lead to the study of two independent means, such as studies comparing
the average mileage of foreign and domestic cars, the average SAT scores obtained from two different
coaching classes, the average yields of a crop due to a certain fertilizer, and so on. The two populations
of interest are assumed to be independent.
This entry describes power and sample-size analysis for the inference about two population means
performed using hypothesis testing. Specifically, we consider the null hypothesis H
0
: µ
2
= µ
1
versus
the two-sided alternative hypothesis H
a
: µ
2
6= µ
1
, the upper one-sided alternative H
a
: µ
2
> µ
1
, or
the lower one-sided alternative H
a
: µ
2
< µ
1
.
The considered two-sample tests rely on the assumption that the two random samples are normally
distributed or that the sample size is large. Suppose that the two samples are normally distributed. If
variances of the considered populations are known a priori, the test statistic has a standard normal
distribution under the null hypothesis, and the corresponding test is referred to as a two-sample z test.
If variances of the two populations are not known, then the null sampling distribution of the test
statistic depends on whether the two variances are assumed to be equal. If the two variances are
assumed to be equal, the test statistic has an exact Student’s t distribution under the null hypothesis.
The corresponding test is referred to as a two-sample t test. If the two variances are not equal, then
the distribution can only be approximated by a Student’s t distribution; the degrees of freedom is
approximated using Satterthwaite’s method. We refer to this test as Satterthwaite’s t test. For a large
sample, the distribution of the test statistic is approximately normal, and the corresponding test is a
large-sample z test.
The power twomeans command provides power and sample-size analysis for the above tests.
Using power twomeans
power twomeans computes sample size, power, or experimental-group mean for a two-sample
means test. 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, all computations
assume a balanced- or equal-allocation design; see [PSS] unbalanced designs for a description of
how to specify an unbalanced design.
power twomeans — Power analysis for a two-sample means test 7
By default, all computations are for a two-sample t test, which assumes equal and unknown
standard deviations. By default, the common standard deviation is set to one but may be changed by
specifying the sd() option. To specify different standard deviations, use the respective sd1() and
sd2() options. These options must be specified together and may not be used in combination with
sd(). When sd1() and sd2() are specified, the computations are based on Satterthwaite’s t test,
which assumes unequal and unknown standard deviations. If standard deviations are known, use the
knownsds option to request that computations be based on a two-sample z test.
To compute the total sample size, you must specify the control-group mean m
1
, the experimental-
group mean m
2
, and, optionally, the power of the test in the power() option. The default power is
set to 0.8.
Instead of the total sample size, you can compute one of the group sizes given the other one. To
compute the control-group sample size, you must specify the compute(N1) option and the sample
size of the experimental group in the n2() option. Likewise, to compute the experimental-group
sample size, you must specify the compute(N2) option and the sample size of the control group in
the n1() option.
To compute power, you must specify the total sample size in the n() option, the control-group
mean m
1
, and the experimental-group mean m
2
.
Instead of the experimental-group mean m
2
, you may specify the difference m
2
− m
1
between the
experimental-group mean and the control-group mean in the diff() option when computing sample
size or power.
To compute effect size, the difference between the experimental-group mean and the null mean,
and the experimental-group mean, you must specify the total sample size in the n() option, the power
in the power() option, the control-group mean m
1
, and, optionally, the direction of the effect. The
direction is upper by default, direction(upper), which means that the experimental-group mean
is assumed to be larger than the specified control-group value. You can change the direction to be
lower, which means that the experimental-group mean is assumed to be smaller than the specified
control-group value, by specifying the direction(lower) option.
Instead of the total sample size n(), you can specify individual group sizes in n1() and n2(), or
specify one of the group sizes and nratio() when computing power or effect size. Also see Two
samples in [PSS] unbalanced designs for more details.
In the following sections, we describe the use of power twomeans accompanied by examples for
computing sample size, power, and experimental-group mean.
Computing sample size
To compute sample size, you must specify the control-group mean m
1
, the experimental-group
mean m
2
, and, optionally, the power of the test in the power() option. A default power of 0.8 is
assumed if power() is not specified.
Example 1: Sample size for a two-sample means test
Consider a study investigating the effects of smoking on lung function of males. The response
variable is forced expiratory volume (FEV), measured in liters (L), where better lung function implies
higher values of FEV. We wish to test the null hypothesis H
0
: µ
1
= µ
2
versus a two-sided alternative
hypothesis H
a
: µ
1
6= µ
2
, where µ
1
and µ
2
are the mean FEV for nonsmokers and smokers, respectively.
Suppose that the mean FEV from previous studies was reported to be 3 L for nonsmokers and
2.7 L for smokers. We are designing a new study and wish to find out how many subjects we need
8 power twomeans — Power analysis for a two-sample means test
to enroll so that the power of a 5%-level two-sided test to detect the specified difference between
means is at least 80%. We assume equal numbers of subjects in each group and a common standard
deviation of 1.
. power twomeans 3 2.7
Performing iteration ...
Estimated sample sizes for a two-sample means test
t test assuming sd1 = sd2 = sd
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = -0.3000
m1 = 3.0000
m2 = 2.7000
sd = 1.0000
Estimated sample sizes:
N = 352
N per group = 176
We need a total sample of 352 subjects, 176 per group, to detect the specified mean difference between
the smoking and nonsmoking groups with 80% power using a two-sided 5%-level test.
The default computation is for the case of equal and unknown standard deviations, as indicated
by the output. You can specify the knownsds option to request the computation assuming known
standard deviations.
Example 2: Sample size assuming unequal standard deviations
Instead of assuming equal standard deviations as in example 1, we use the estimates of the standard
deviations from previous studies as our hypothetical values. The standard deviation of FEV for the
nonsmoking group was reported to be 0.8 L and that for the smoking group was reported to be 0.7 L.
We specify standard deviations in the sd1() and sd2() options.
. power twomeans 3 2.7, sd1(0.8) sd2(0.7)
Performing iteration ...
Estimated sample sizes for a two-sample means test
Satterthwaite’s t test assuming unequal variances
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = -0.3000
m1 = 3.0000
m2 = 2.7000
sd1 = 0.8000
sd2 = 0.7000
Estimated sample sizes:
N = 200
N per group = 100
The specified standard deviations are smaller than one, so we obtain a smaller required total sample
size of 200 compared with example 1.
power twomeans — Power analysis for a two-sample means test 9
Example 3: Specifying difference between means
Instead of the mean FEV of 2.7 for the smoking group as in example 2, we can specify the
difference between the two means of 2.7 − 3 = −0.3 in the diff() option.
. power twomeans 3, sd1(0.8) sd2(0.7) diff(-0.3)
Performing iteration ...
Estimated sample sizes for a two-sample means test
Satterthwaite’s t test assuming unequal variances
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = -0.3000
m1 = 3.0000
m2 = 2.7000
diff = -0.3000
sd1 = 0.8000
sd2 = 0.7000
Estimated sample sizes:
N = 200
N per group = 100
We obtain the same results as in example 2. The difference between means is now also reported in
the output following the individual means.
Example 4: Computing one of the group sizes
Suppose we anticipate a sample of 120 nonsmoking subjects. We wish to compute the required
number of subjects in the smoking group, keeping all other study parameters as in example 2.
We specify the number of subjects in the nonsmoking group in the n1() option and specify the
compute(N2) option.
. power twomeans 3 2.7, sd1(0.8) sd2(0.7) n1(120) compute(N2)
Performing iteration ...
Estimated sample sizes for a two-sample means test
Satterthwaite’s t test assuming unequal variances
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = -0.3000
m1 = 3.0000
m2 = 2.7000
sd1 = 0.8000
sd2 = 0.7000
N1 = 120
Estimated sample sizes:
N = 202
N2 = 82
We need a sample of 82 smoking subjects given a sample of 120 nonsmoking subjects.
10 power twomeans — Power analysis for a two-sample means test
Example 5: Unbalanced design
By default, power twomeans computes sample size for a balanced- or equal-allocation design. If
we know the allocation ratio of subjects between the groups, we can compute the required sample
size for an unbalanced design by specifying the nratio() option.
Continuing with example 2, we will suppose that we anticipate to recruit twice as many smokers
than nonsmokers; that is, n
2
/n
1
= 2. We specify the nratio(2) option to compute the required
sample size for the specified unbalanced design.
. power twomeans 3 2.7, sd1(0.8) sd2(0.7) nratio(2)
Performing iteration ...
Estimated sample sizes for a two-sample means test
Satterthwaite’s t test assuming unequal variances
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = -0.3000
m1 = 3.0000
m2 = 2.7000
sd1 = 0.8000
sd2 = 0.7000
N2/N1 = 2.0000
Estimated sample sizes:
N = 237
N1 = 79
N2 = 158
We need a total sample size of 237 subjects, which is larger than the required total sample size for
the corresponding balanced design from example 2.
Also see Two samples in [PSS] unbalanced designs for more examples of unbalanced designs for
two-sample tests.
Computing power
To compute power, you must specify the total sample size in the n() option, the control-group
mean m
1
, and the experimental-group mean m
2
.
Example 6: Power of a two-sample means test
Continuing with example 1, we will suppose that we have resources to enroll a total of only 250
subjects, assuming equal-sized groups. To compute the power corresponding to this sample size given
the study parameters from example 1, we specify the total sample size in n():
power twomeans — Power analysis for a two-sample means test 11
. power twomeans 3 2.7, n(250)
Estimated power for a two-sample means test
t test assuming sd1 = sd2 = sd
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
N = 250
N per group = 125
delta = -0.3000
m1 = 3.0000
m2 = 2.7000
sd = 1.0000
Estimated power:
power = 0.6564
With a total sample of 250 subjects, we obtain a power of only 65.64%.
Example 7: Multiple values of study parameters
In this example, we assess the effect of varying the common standard deviation (assuming equal
standard deviations in both groups) of FEV on the power of our study.
Continuing with example 6, we compute powers for a range of common standard deviations
between 0.5 and 1.5 with the step size of 0.1. We specify the corresponding numlist in the sd()
option.
. power twomeans 3 2.7, sd(0.5(0.1)1.5) n(250)
Estimated power for a two-sample means test
t test assuming sd1 = sd2 = sd
Ho: m2 = m1 versus Ha: m2 != m1
alpha power N N1 N2 delta m1 m2 sd
.05 .9972 250 125 125 -.3 3 2.7 .5
.05 .976 250 125 125 -.3 3 2.7 .6
.05 .9215 250 125 125 -.3 3 2.7 .7
.05 .8397 250 125 125 -.3 3 2.7 .8
.05 .747 250 125 125 -.3 3 2.7 .9
.05 .6564 250 125 125 -.3 3 2.7 1
.05 .5745 250 125 125 -.3 3 2.7 1.1
.05 .5036 250 125 125 -.3 3 2.7 1.2
.05 .4434 250 125 125 -.3 3 2.7 1.3
.05 .3928 250 125 125 -.3 3 2.7 1.4
.05 .3503 250 125 125 -.3 3 2.7 1.5
The power decreases from 99.7% to 35.0% as the common standard deviation increases from 0.5 to
1.5 L.
For multiple values of parameters, the results are automatically displayed in a table, as we see
above. For more examples of tables, see [PSS] power, table. If you wish to produce a power plot,
see [PSS] power, graph.
12 power twomeans — Power analysis for a two-sample means test
Computing effect size and experimental-group mean
Effect size δ for a two-sample means test is defined as the difference between the experimental-group
mean and the control-group mean δ = µ
2
− µ
1
.
Sometimes, we may be interested in determining the smallest effect and the corresponding
experimental-group mean that yield a statistically significant result for prespecified sample size and
power. In this case, power, sample size, and control-group mean must be specified. In addition,
you must also decide on the direction of the effect: upper, meaning m
2
> m
1
, or lower, meaning
m
2
< m
1
. The direction may be specified in the direction() option; direction(upper) is the
default.
Example 8: Minimum detectable change in the experimental-group mean
Continuing with example 6, we compute the smallest change in the mean of the smoking group
that can be detected given a total sample of 250 subjects and 80% power, assuming equal-group
allocation. To solve for the mean FEV of the smoking group, after the command name, we specify
the nonsmoking-group mean of 3, total sample size n(250), and power power(0.8).
Because our initial study was based on the hypothesis that FEV for the smoking group is lower
than that of the nonsmoking group, we specify the direction(lower) option to compute the
smoking-group mean that is lower than the specified nonsmoking-group mean.
. power twomeans 3, n(250) power(0.8) direction(lower)
Performing iteration ...
Estimated experimental-group mean for a two-sample means test
t test assuming sd1 = sd2 = sd
Ho: m2 = m1 versus Ha: m2 != m1; m2 < m1
Study parameters:
alpha = 0.0500
power = 0.8000
N = 250
N per group = 125
m1 = 3.0000
sd = 1.0000
Estimated effect size and experimental-group mean:
delta = -0.3558
m2 = 2.6442
We find that the minimum detectable value of the effect size is −0.36, which corresponds to the
mean FEV of 2.64 for the smoking group.
Testing a hypothesis about two independent means
After data are collected, we can use the ttest command to test the equality of two independent
means using a t test; see [R] ttest for details. In this section, we demonstrate the use of ttesti,
the immediate form of the test command, which can be used to test a hypothesis using summary
statistics instead of the actual data values.
power twomeans — Power analysis for a two-sample means test 13
Example 9: Two-sample t test
Consider an example from van Belle et al. (2004, 129), where newborn infants were divided into
two groups: a treatment group, where infants received daily “walking stimulus” for eight weeks, and
a control group, where no stimulus was provided. The goal of this study was to test whether receiving
the walking stimulus during stages of infancy induces the walking ability to develop sooner.
The average number of months before the infants started walking was recorded for both groups.
The authors provide estimates of the average of 10.125 months for the treatment group with estimated
standard deviation of 1.447 months and 12.35 months for the control group with estimated standard
deviation of 0.9618 months. The sample sizes for treatment and control groups were 6 and 5,
respectively. We supply these estimates to the ttesti command and use the unequal option to
perform a t test assuming unequal variances.
. ttesti 6 10.125 1.447 5 12.35 0.9618, unequal
Two-sample t test with unequal variances
Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
x 6 10.125 .5907353 1.447 8.606467 11.64353
y 5 12.35 .43013 .9618 11.15577 13.54423
combined 11 11.13636 .501552 1.66346 10.01884 12.25389
diff -2.225 .7307394 -3.887894 -.562106
diff = mean(x) - mean(y) t = -3.0449
Ho: diff = 0 Satterthwaite’s degrees of freedom = 8.66326
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Pr(T < t) = 0.0073 Pr(|T| > |t|) = 0.0145 Pr(T > t) = 0.9927
We reject the null hypothesis of H
0
: µ
C
= µ
T
against the two-sided alternative H
a
: µ
C
6= µ
T
at
the 5% significance level; the p-value = 0.0145.
We use the estimates of this study to perform a sample-size analysis we would have conducted
before a new study. In our analysis, we assume equal-group allocation.
. power twomeans 10.125 12.35, power(0.8) sd1(1.447) sd2(0.9618)
Performing iteration ...
Estimated sample sizes for a two-sample means test
Satterthwaite’s t test assuming unequal variances
Ho: m2 = m1 versus Ha: m2 != m1
Study parameters:
alpha = 0.0500
power = 0.8000
delta = 2.2250
m1 = 10.1250
m2 = 12.3500
sd1 = 1.4470
sd2 = 0.9618
Estimated sample sizes:
N = 14
N per group = 7
We find that the sample size required to detect a difference of 2.225 (12.35 − 10.125 = 2.225) given
the control-group standard deviation of 1.447 and the experimental-group standard deviation of 0.9618
using a 5%-level two-sided test is 7 in each group.
14 power twomeans — Power analysis for a two-sample means test
Stored results
power twomeans 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(m1) control-group mean
r(m2) experimental-group mean
r(diff) difference between the experimental- and control-group means
r(sd) common standard deviation of the control and experimental groups
r(sd1) standard deviation of the control group
r(sd2) standard deviation of the experimental group
r(knownsds) 1 if option knownsds is specified; 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 for sample sizes or experimental-group mean
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) twomeans
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
Methods and formulas
Consider two independent samples with n
1
subjects in the control group and n
2
subjects in the
experimental group. Let x
11
, . . . , x
1n
1
be a random sample of size n
1
from a normal population with
mean µ
1
and variance σ
2
1
. Let x
21
, . . . , x
2n
2
be a random sample of size n
2
from a normal population
with mean µ
2
and variance σ
2
2
. Let effect size δ be the difference between the experimental-group
mean and the control-group mean, δ = µ
2
− µ
1
. The sample means and variances for the two
independent samples are
x
1
=
1
n
1
n
1
X
i=1
x
1i
and s
2
1
=
1
n
1
− 1
n
1
X
i=1
(x
1i
− x
1
)
2
power twomeans — Power analysis for a two-sample means test 15
x
2
=
1
n
2
n
2
X
i=1
x
2i
and s
2
2
=
1
n
2
− 1
n
2
X
i=1
(x
2i
− x
2
)
2
where x
j
and s
2
j
are the respective sample means and sample variances of the two samples.
A two-sample means test involves testing the null hypothesis H
0
: µ
2
= µ
1
versus the two-sided
alternative hypothesis H
a
: µ
2
6= µ
1
, the upper one-sided alternative H
a
: µ
2
> µ
1
, or the lower
one-sided alternative H
a
: µ
2
< µ
1
.
The two-sample means test can be performed under four different assumptions: 1) population
variances are known and not equal; 2) population variances are known and equal; 3) population
variances are unknown and not equal; and 4) population variances are unknown and equal.
Let σ
D
denote the standard deviation of the difference between the two sample means. The test
statistic of the form
TS =
(x
2
− x
1
) − (µ
2
− µ
1
)
σ
D
(1)
is used in each of the four cases described above. Each case, however, determines the functional form
of σ
D
and the sampling distribution of the test statistic (1) under the null hypothesis.
Let R = n
2
/n
1
denote the allocation ratio. Then n
2
= R × n
1
and power can be viewed as
a function of n
1
. Therefore, for sample-size determination, the control-group sample size n
1
is
computed first. The experimental-group size n
2
is then computed as R × n
1
, and the total sample size
is computed as n = n
1
+ n
2
. By default, sample sizes are rounded to integer values; see Fractional
sample sizes in [PSS] unbalanced designs for details.
The following formulas are based on Armitage, Berry, and Matthews (2002); Chow, Shao, and
Wang (2008); and Dixon and Massey (1983).
Methods and formulas are presented under the following headings:
Known standard deviations
Unknown standard deviations
Unequal standard deviations
Equal standard deviations
Known standard deviations
Below we present formulas for the computations that assume unequal standard deviations. When
standard deviations are equal, the corresponding formulas are special cases of the formulas below
with σ
1
= σ
2
= σ.
When the standard deviations of the control and the experimental groups are known, the test
statistic in (1) is a z test statistic
z =
(x
2
− x
1
) − (µ
2
− µ
1
)
p
σ
2
1
/n
1
+ σ
2
2
/n
2
with σ
D
=
p
σ
2
1
/n
1
+ σ
2
2
/n
2
. The sampling distribution of this test statistic under the null hypothesis
is standard normal. The corresponding test is referred to as a z test.
Let α be the significance level, β be the probability of a type II error, and z
1−α
and z
β
be the
(1 − α)th and the βth quantiles of a standard normal distribution.
16 power twomeans — Power analysis for a two-sample means test
The power π = 1 − β is computed using
π =
Φ
δ
σ
D
− z
1−α
for an upper one-sided test
Φ
−
δ
σ
D
− z
1−α
for a lower one-sided test
Φ
δ
σ
D
− z
1−α/2
+ Φ
−
δ
σ
D
− z
1−α/2
for a two-sided test
(2)
where Φ(·) is the cdf of a standard normal distribution.
For a one-sided test, the control-group sample size n
1
is computed as follows:
n
1
=
z
1−α
− z
β
µ
2
− µ
1
2
σ
2
1
+
σ
2
2
R
(3)
For a one-sided test, if one of the group sizes is known, the other one is computed using the
following formula. For example, to compute n
1
given n
2
, we use the following formula:
n
1
=
σ
2
1
µ
2
−µ
1
z
1−α
−z
β
2
−
σ
2
2
n
2
(4)
For a two-sided test, sample sizes are computed by iteratively solving the two-sided power equation
in (2). The default initial values for the iterative procedure are calculated from the respective equations
(3) and (4), with α replaced with α/2.
The absolute value of the effect size for a one-sided test is obtained by inverting the corresponding
one-sided power equation in (2):
|δ| = σ
D
(z
1−α
− z
β
)
Note that the magnitude of the effect size is the same regardless of the direction of the test.
The experimental-group mean for a one-sided test is then computed as
µ
2
=
µ
1
+ (z
1−α
− z
β
)
p
σ
2
1
/n
1
+ σ
2
2
/n
2
when µ
2
> µ
1
µ
1
− (z
1−α
− z
β
)
p
σ
2
1
/n
1
+ σ
2
2
/n
2
when µ
2
< µ
1
For a two-sided test, the experimental-group mean is computed by iteratively solving the two-sided
power equation in (2) for µ
2
. The default initial value is obtained from the corresponding one-sided
computation with α/2.
Unknown standard deviations
When the standard deviations of the control group and the experimental group are unknown, the
test statistic in (1) is a t test statistic
t =
(x
2
− x
1
) − (µ
2
− µ
1
)
s
D
power twomeans — Power analysis for a two-sample means test 17
where s
D
is the estimated standard deviation of the sample mean difference. The sampling distribution
of this test statistic under the null hypothesis is (approximately) a Student’s t distribution with ν
degrees of freedom. Parameters ν and s
D
are defined below, separately for the case of equal and
unequal standard deviations.
Let t
ν,α
denote the αth quantile of a Student’s t distribution with ν degrees of freedom. Under the
alternative hypothesis, the test statistic follows a noncentral Student’s t distribution with ν degrees
of freedom and noncentrality parameter λ.
The power is computed from the following equations:
π =
1 − T
ν,λ
(t
ν,1−α
) for an upper one-sided test
T
ν,λ
(−t
ν,1−α
) for a lower one-sided test
1 − T
ν,λ
t
ν,1−α/2
+ T
ν,λ
−t
ν,1−α/2
for a two-sided test
(5)
In the equations above, λ = |µ
2
− µ
1
|/s
D
.
Sample sizes and the experimental-group mean are obtained by iteratively solving the nonlinear
equation (5) for n
1
, n
2
, and µ
2
, respectively. For sample-size and effect-size computations, the default
initial values for the iterative procedure are calculated using the corresponding formulas assuming
known standard deviations from the previous subsection.
Unequal standard deviations
In the case of unequal standard deviations,
s
D
=
q
s
2
1
/n
1
+ s
2
2
/n
2
and the degrees of freedom ν of the test statistic is obtained by Satterthwaite’s formula:
ν =
s
2
1
n
1
+
s
2
2
n
2
2
(s
2
1
/n
1
)
2
n
1
−1
+
(s
2
2
/n
2
)
2
n
2
−1
The sampling distribution of the test statistic under the null hypothesis is an approximate Student’s
t distribution. We refer to the corresponding test as Satterthwaite’s t test.
Equal standard deviations
In the case of equal standard deviations,
s
D
= s
p
p
1/n
1
+ 1/n
2
where s
p
=
P
n
1
i=1
(x
1i
− x
1
)
2
+
P
n
2
i=1
(x
2i
− x
2
)
2
/(n
1
+ n
2
− 2) is the pooled-sample standard
deviation.
The degrees of freedom ν is
ν = n
1
+ n
2
− 2
18 power twomeans — Power analysis for a two-sample means test
The sampling distribution of the test statistic under the null hypothesis is exactly a Student’s t
distribution. We refer to the corresponding test as a two-sample t test.
References
Armitage, P., G. Berry, and J. N. S. Matthews. 2002. Statistical Methods in Medical Research. 4th ed. Oxford:
Blackwell.
Chow, S.-C., J. Shao, and H. Wang. 2008. Sample Size Calculations in Clinical Research. 2nd ed. New York: Dekker.
Dixon, W. J., and F. J. Massey, Jr. 1983. Introduction to Statistical Analysis. 4th ed. New York: McGraw–Hill.
van Belle, G., L. D. Fisher, P. J. Heagerty, and T. S. Lumley. 2004. Biostatistics: A Methodology for the Health
Sciences. 2nd ed. New York: Wiley.
Also see
[PSS] power twomeans, cluster — Power analysis for a two-sample means test, CRD
[PSS] power — Power and sample-size analysis for hypothesis tests
[PSS] power oneway — Power analysis for one-way analysis of variance
[PSS] power twoway — Power analysis for two-way analysis of variance
[PSS] power, graph — Graph results from the power command
[PSS] power, table — Produce table of results from the power command
[PSS] Glossary
[R] ttest — t tests (mean-comparison tests)
