Compute the Power of a One- or Two-Sample Proportion Test
Compute the power of a one- or two-sample proportion test, given the sample size(s), true proportion(s), and significance level.
propTestPower(n.or.n1, p.or.p1 = 0.5, n2 = n.or.n1,
p0.or.p2 = 0.5, alpha = 0.05, sample.type = "one.sample",
alternative = "two.sided", approx = TRUE,
correct = sample.type == "two.sample", warn = TRUE,
return.exact.list = TRUE)n.or.n1 |
numeric vector of sample sizes. When |
p.or.p1 |
numeric vector of proportions. When |
n2 |
numeric vector of sample sizes for group 2. The default value is |
p0.or.p2 |
numeric vector of proportions. When |
alpha |
numeric vector of numbers between 0 and 1 indicating the Type I error level
associated with the hypothesis test. The default value is |
sample.type |
character string indicating whether to compute power based on a one-sample or
two-sample hypothesis test. When |
alternative |
character string indicating the kind of alternative hypothesis.
The possible values are |
approx |
logical scalar indicating whether to compute the power based on the normal
approximation to the binomial distribution. The default value is |
correct |
logical scalar indicating whether to use the continuity correction when |
warn |
logical scalar indicating whether to issue a warning. The default value is |
return.exact.list |
logical scalar relevant to the case when |
If the arguments n.or.n1, p.or.p1, n2, p0.or.p2, and
alpha are not all the same length, they are replicated to be the same length
as the length of the longest argument.
The power is based on the difference p.or.p1 - p0.or.p2.
One-Sample Case (sample.type="one.sample").
approx=TRUE When sample.type="one.sample" and approx=TRUE,
power is computed based on the test that uses the normal approximation to the
binomial distribution; see the help file for prop.test.
The formula for this test and its associated power is presented in most standard statistics
texts, including Zar (2010, pp. 534-537, 539-541).
approx=FALSE When sample.type="one.sample" and approx=FALSE,
power is computed based on the exact binomial test; see the help file for binom.test.
The formula for this test and its associated power is presented in most standard statistics
texts, including Zar (2010, pp. 532-534, 539) and
Millard and Neerchal (2001, pp. 385-386, 504-506).
Two-Sample Case (sample.type="two.sample").
When sample.type="two.sample", power is computed based on the test that uses the
normal approximation to the binomial distribution;
see the help file for prop.test.
The formula for this test and its associated power is presented in standard statistics texts,
including Zar (2010, pp. 549-550, 552-553) and
Millard and Neerchal (2001, pp. 443-445, 508-510).
By default, propTestPower returns a numeric vector of powers.
For the one-sample proportion test (sample.type="one.sample"),
when approx=FALSE and return.exact.list=TRUE, propTestPower
returns a list with the following components:
power |
numeric vector of powers. |
alpha |
numeric vector containing the true significance levels.
Because of the discrete nature of the binomial distribution, the true significance
levels usually do not equal the significance level supplied by the user in the
argument |
q.critical.lower |
numeric vector of lower critical values for rejecting the null
hypothesis. If the observed number of "successes" is less than or equal to these values,
the null hypothesis is rejected. (Not present if |
q.critical.upper |
numeric vector of upper critical values for rejecting the null
hypothesis. If the observed number of "successes" is greater than these values,
the null hypothesis is rejected. (Not present if |
The binomial distribution is used to model processes with binary (Yes-No, Success-Failure, Heads-Tails, etc.) outcomes. It is assumed that the outcome of any one trial is independent of any other trial, and that the probability of “success”, p, is the same on each trial. A binomial discrete random variable X is the number of "successes" in n independent trials. A special case of the binomial distribution occurs when n=1, in which case X is also called a Bernoulli random variable.
In the context of environmental statistics, the binomial distribution is sometimes used to model the proportion of times a chemical concentration exceeds a set standard in a given period of time (e.g., Gilbert, 1987, p.143), or to compare the proportion of detects in a compliance well vs. a background well (e.g., USEPA, 1989b, Chapter 8, p.3-7).
In the course of designing a sampling program, an environmental scientist may wish to determine the
relationship between sample size, power, significance level, and the difference between the
hypothesized and true proportions if one of the objectives of the sampling program is to
determine whether a proprtion differs from a specified level or two proportions differ from each other.
The functions propTestPower, propTestN, propTestMdd, and
plotPropTestDesign can be used to investigate these relationships for the case of
binomial proportions.
Studying the two-sample proportion test, Haseman (1978) found that the formulas used to estimate the power that do not incorporate the continuity correction tend to underestimate the power. Casagrande, Pike, and Smith (1978) found that the formulas that do incorporate the continuity correction provide an excellent approximation.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Berthouex, P.M., and L.C. Brown. (1994). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton, FL, Chapter 15.
Casagrande, J.T., M.C. Pike, and P.G. Smith. (1978). An Improved Approximation Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions. Biometrics 34, 483-486.
Fleiss, J. L. (1981). Statistical Methods for Rates and Proportions. Second Edition. John Wiley and Sons, New York, Chapters 1-2.
Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY.
Haseman, J.K. (1978). Exact Sample Sizes for Use with the Fisher-Irwin Test for 2x2 Tables. Biometrics 34, 106-109.
Millard, S.P., and N. Neerchal. (2001). Environmental Statistics with S-Plus. CRC Press, Boca Raton, FL.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.
# Look at how the power of the one-sample proportion test
# increases with increasing sample size:
seq(20, 50, by=10)
#[1] 20 30 40 50
power <- propTestPower(n.or.n1 = seq(20, 50, by=10),
p.or.p1 = 0.7, p0 = 0.5)
round(power, 2)
#[1] 0.43 0.60 0.73 0.83
#----------
# Repeat the last example, but compute the power based on
# the exact test instead of the approximation.
# Note that the significance level varies with sample size and
# never attains the requested level of 0.05.
prop.test.list <- propTestPower(n.or.n1 = seq(20, 50, by=10),
p.or.p1 = 0.7, p0 = 0.5, approx=FALSE)
lapply(prop.test.list, round, 2)
#$power:
#[1] 0.42 0.59 0.70 0.78
#
#$alpha:
#[1] 0.04 0.04 0.04 0.03
#
#$q.critical.lower:
#[1] 5 9 13 17
#
#$q.critical.upper:
#[1] 14 20 26 32
#==========
# Look at how the power of the two-sample proportion test
# increases with increasing difference between the two
# population proportions:
seq(0.5, 0.1, by=-0.1)
#[1] 0.5 0.4 0.3 0.2 0.1
power <- propTestPower(30, sample.type = "two",
p.or.p1 = seq(0.5, 0.1, by=-0.1))
#Warning message:
#In propTestPower(30, sample.type = "two", p.or.p1 = seq(0.5, 0.1, :
#The sample sizes 'n1' and 'n2' are too small, relative to the given
# values of 'p1' and 'p2', for the normal approximation to work well
# for the following element indices:
# 5
round(power, 2)
#[1] 0.05 0.08 0.26 0.59 0.90
#----------
# Look at how the power of the two-sample proportion test
# increases with increasing values of Type I error:
power <- propTestPower(30, sample.type = "two",
p.or.p1 = 0.7,
alpha = c(0.001, 0.01, 0.05, 0.1))
round(power, 2)
#[1] 0.02 0.10 0.26 0.37
#==========
# Clean up
#---------
rm(power, prop.test.list)
#==========
# Modifying the example on pages 8-5 to 8-7 of USEPA (1989b),
# determine how adding another 20 observations to the background
# well to increase the sample size from 24 to 44 will affect the
# power of detecting a difference in the proportion of detects of
# cadmium between the background and compliance wells. Set the
# compliance well to "group 1" and set the background well to
# "group 2". Assume the true probability of a "detect" at the
# background well is 1/3, set the probability of a "detect" at the
# compliance well to 0.4, use a 5% significance level, and use the
# upper one-sided alternative (probability of a "detect" at the
# compliance well is greater than the probability of a "detect" at
# the background well).
# (The original data are stored in EPA.89b.cadmium.df.)
#
# Note that the power does increase (from 9% to 12%), but is relatively
# very small.
EPA.89b.cadmium.df
# Cadmium.orig Cadmium Censored Well.type
#1 0.1 0.100 FALSE Background
#2 0.12 0.120 FALSE Background
#3 BDL 0.000 TRUE Background
# ..........................................
#86 BDL 0.000 TRUE Compliance
#87 BDL 0.000 TRUE Compliance
#88 BDL 0.000 TRUE Compliance
p.hat.back <- with(EPA.89b.cadmium.df,
mean(!Censored[Well.type=="Background"]))
p.hat.back
#[1] 0.3333333
p.hat.comp <- with(EPA.89b.cadmium.df,
mean(!Censored[Well.type=="Compliance"]))
p.hat.comp
#[1] 0.375
n.back <- with(EPA.89b.cadmium.df,
sum(Well.type == "Background"))
n.back
#[1] 24
n.comp <- with(EPA.89b.cadmium.df,
sum(Well.type == "Compliance"))
n.comp
#[1] 64
propTestPower(n.or.n1 = n.comp,
p.or.p1 = 0.4,
n2 = c(n.back, 44), p0.or.p2 = p.hat.back,
alt="greater", sample.type="two")
#[1] 0.08953013 0.12421135
#----------
# Clean up
#---------
rm(p.hat.back, p.hat.comp, n.back, n.comp)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.