propTestPower.RdCompute 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)numeric vector of sample sizes. When sample.type="one.sample",
this argument denotes \(n\), the number of observations in the single sample.
When sample.type="two.sample", this argument denotes \(n_1\),
the number of observations from group 1.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector of proportions. When sample.type="one.sample",
this argument denotes the true value of \(p\), the probability of “success”. When sample.type="two.sample", this argument denotes the value of \(p_1\),
the probability of “success” in group 1. The default value is p.or.p1=0.5.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector of sample sizes for group 2. The default value is n2=n.or.n1.
This argument is ignored when sample.type="one.sample".
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector of proportions. When sample.type="one.sample",
this argument denotes the hypothesized value of \(p\), the probability of “success”.
When sample.type="two.sample", this argument denotes the value of \(p_2\),
the probability of “success” in group 2. The default value is p0.or.p2=0.5. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector of numbers between 0 and 1 indicating the Type I error level
associated with the hypothesis test. The default value is alpha=0.05.
character string indicating whether to compute power based on a one-sample or
two-sample hypothesis test. When sample.type="one.sample",
the computed power is based on a hypothesis test for a single proportion.
When sample.type="two.sample", the computed power is based on a
hypothesis test for the difference between two proportions.
The default value is sample.type="one.sample".
character string indicating the kind of alternative hypothesis.
The possible values are "two.sided" (the default), "less", and
"greater".
logical scalar indicating whether to compute the power based on the normal
approximation to the binomial distribution. The default value is approx=TRUE.
Currently, the exact method (approx=FALSE) is only available for the
one-sample case (i.e., sample.type="one.sample").
logical scalar indicating whether to use the continuity correction when approx=TRUE. The default value is approx=TRUE when sample.type="two.sample" and approx=FALSE when sample.type="one.sample". This argument is ignored when approx=FALSE.
logical scalar indicating whether to issue a warning. The default value is warn=TRUE. When approx=TRUE (power based on the normal approximation) and
warn=TRUE, a warning is issued for cases when the normal approximation to the
binomial distribution probably is not accurate. When approx=FALSE (power based on the exact test) and warn=TRUE,
a warning is issued when the user-supplied sample size is too small to
yield a significance level less than or equal to the user-supplied value of
alpha.
logical scalar relevant to the case when approx=FALSE
(i.e., when the power is based on the exact test). This argument indicates whether
to return a list containing extra information about the exact test in addition to
the power of the exact test. By default, propTestPower returns only a vector
containing the computed power(s) (see the VALUE section below). When return.exact.list=TRUE (the default) and approx=FALSE, propTestPower returns a list with components indicating the power of the
exact test, the true significance level associated with the exact test, and the
critical values associated with the exact test (see the DETAILS section for more information).
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=TRUEWhen 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=FALSEWhen 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:
numeric vector of powers.
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 alpha.
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 alternative="greater".)
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 alternative="less".)
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.
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.
# 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
#[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
#[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
#>
#> $q.critical.upper
#> [1] 14 20 26 32
#>
#$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
#[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: 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
#>
#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
#[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
#[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
#> 4 0.26 0.260 FALSE Background
#> 5 BDL 0.000 TRUE Background
#> 6 0.1 0.100 FALSE Background
#> 7 BDL 0.000 TRUE Background
#> 8 0.014 0.014 FALSE Background
#> 9 BDL 0.000 TRUE Background
#> 10 BDL 0.000 TRUE Background
#> 11 BDL 0.000 TRUE Background
#> 12 BDL 0.000 TRUE Background
#> 13 BDL 0.000 TRUE Background
#> 14 0.12 0.120 FALSE Background
#> 15 BDL 0.000 TRUE Background
#> 16 0.21 0.210 FALSE Background
#> 17 BDL 0.000 TRUE Background
#> 18 0.12 0.120 FALSE Background
#> 19 BDL 0.000 TRUE Background
#> 20 BDL 0.000 TRUE Background
#> 21 BDL 0.000 TRUE Background
#> 22 BDL 0.000 TRUE Background
#> 23 BDL 0.000 TRUE Background
#> 24 BDL 0.000 TRUE Background
#> 25 0.12 0.120 FALSE Compliance
#> 26 0.08 0.080 FALSE Compliance
#> 27 BDL 0.000 TRUE Compliance
#> 28 0.2 0.200 FALSE Compliance
#> 29 BDL 0.000 TRUE Compliance
#> 30 0.1 0.100 FALSE Compliance
#> 31 BDL 0.000 TRUE Compliance
#> 32 0.012 0.012 FALSE Compliance
#> 33 BDL 0.000 TRUE Compliance
#> 34 BDL 0.000 TRUE Compliance
#> 35 BDL 0.000 TRUE Compliance
#> 36 BDL 0.000 TRUE Compliance
#> 37 BDL 0.000 TRUE Compliance
#> 38 0.12 0.120 FALSE Compliance
#> 39 0.07 0.070 FALSE Compliance
#> 40 BDL 0.000 TRUE Compliance
#> 41 0.19 0.190 FALSE Compliance
#> 42 BDL 0.000 TRUE Compliance
#> 43 0.1 0.100 FALSE Compliance
#> 44 BDL 0.000 TRUE Compliance
#> 45 0.01 0.010 FALSE Compliance
#> 46 BDL 0.000 TRUE Compliance
#> 47 BDL 0.000 TRUE Compliance
#> 48 BDL 0.000 TRUE Compliance
#> 49 BDL 0.000 TRUE Compliance
#> 50 BDL 0.000 TRUE Compliance
#> 51 0.11 0.110 FALSE Compliance
#> 52 0.06 0.060 FALSE Compliance
#> 53 BDL 0.000 TRUE Compliance
#> 54 0.23 0.230 FALSE Compliance
#> 55 BDL 0.000 TRUE Compliance
#> 56 0.11 0.110 FALSE Compliance
#> 57 BDL 0.000 TRUE Compliance
#> 58 0.031 0.031 FALSE Compliance
#> 59 BDL 0.000 TRUE Compliance
#> 60 BDL 0.000 TRUE Compliance
#> 61 BDL 0.000 TRUE Compliance
#> 62 BDL 0.000 TRUE Compliance
#> 63 BDL 0.000 TRUE Compliance
#> 64 0.12 0.120 FALSE Compliance
#> 65 0.08 0.080 FALSE Compliance
#> 66 BDL 0.000 TRUE Compliance
#> 67 0.26 0.260 FALSE Compliance
#> 68 BDL 0.000 TRUE Compliance
#> 69 0.02 0.020 FALSE Compliance
#> 70 BDL 0.000 TRUE Compliance
#> 71 0.024 0.024 FALSE Compliance
#> 72 BDL 0.000 TRUE Compliance
#> 73 BDL 0.000 TRUE Compliance
#> 74 BDL 0.000 TRUE Compliance
#> 75 BDL 0.000 TRUE Compliance
#> 76 BDL 0.000 TRUE Compliance
#> 77 0.1 0.100 FALSE Compliance
#> 78 0.04 0.040 FALSE Compliance
#> 79 BDL 0.000 TRUE Compliance
#> 80 BDL 0.000 TRUE Compliance
#> 81 0.1 0.100 FALSE Compliance
#> 82 BDL 0.000 TRUE Compliance
#> 83 0.01 0.010 FALSE Compliance
#> 84 BDL 0.000 TRUE Compliance
#> 85 BDL 0.000 TRUE Compliance
#> 86 BDL 0.000 TRUE Compliance
#> 87 BDL 0.000 TRUE Compliance
#> 88 BDL 0.000 TRUE Compliance
# 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
#[1] 0.3333333
p.hat.comp <- with(EPA.89b.cadmium.df,
mean(!Censored[Well.type=="Compliance"]))
p.hat.comp
#> [1] 0.375
#[1] 0.375
n.back <- with(EPA.89b.cadmium.df,
sum(Well.type == "Background"))
n.back
#> [1] 24
#[1] 24
n.comp <- with(EPA.89b.cadmium.df,
sum(Well.type == "Compliance"))
n.comp
#> [1] 64
#[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
#[1] 0.08953013 0.12421135
#----------
# Clean up
#---------
rm(p.hat.back, p.hat.comp, n.back, n.comp)