propTestN.RdCompute the sample size necessary to achieve a specified power for a one- or two-sample proportion test, given the true proportion(s) and significance level.
propTestN(p.or.p1, p0.or.p2, alpha = 0.05, power = 0.95,
sample.type = "one.sample", alternative = "two.sided",
ratio = 1, approx = TRUE,
correct = sample.type == "two.sample",
round.up = TRUE, warn = TRUE, return.exact.list = TRUE,
n.min = 2, n.max = 10000, tol.alpha = 0.1 * alpha,
tol = 1e-7, maxiter = 1000)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 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.
numeric vector of numbers between 0 and 1 indicating the power associated with
the hypothesis test. The default value is power=0.95.
character string indicating whether to compute sample size based on a one-sample or
two-sample hypothesis test.
When sample.type="one.sample", the computed sample size is based
on a hypothesis test for a single proportion.
When sample.type="two.sample", the computed sample size 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".
numeric vector indicating the ratio of sample size in group 2 to sample size
in group 1 (\(n_2/n_1\)). The default value is ratio=1.
All values of ratio must be greater than or equal to 1.
This argument is ignored if sample.type="one.sample".
logical scalar indicating whether to compute the sample size 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 round up the values of the computed sample size(s)
to the next smallest integer. The default value is round.up=TRUE.
logical scalar indicating whether to issue a warning. The default value is warn=TRUE. When approx=TRUE (sample size based on the normal approximation)
and warn=T, a warning is issued for cases when the normal approximation to the
binomial distribution probably is not accurate.
When approx=FALSE (sample size 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, propTestN returns only a vector
containing the computed sample size(s) (see the VALUE section below). When return.exact.list=TRUE (the default) and approx=FALSE, propTestN returns a list with components indicating the required sample size,
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).
integer relevant to the case when approx=FALSE
(i.e., when the power is based on the exact test). This argument indicates the minimum
allowed value for n to use in the search algorithm. The default value is n.min=2.
integer relevant to the case when approx=FALSE
(i.e., when the power is based on the exact test). This argument indicates the maximum
allowed value for n to use in the search algorithm. The default value is n.max=10000.
numeric vector relevant to the case when approx=FALSE
(i.e., when the power is based on the exact test). This argument indicates the tolerance
on alpha to use in the search algorithm (i.e., how close the actual Type
I error level is to the value prescribed by alpha). The default value is tol.alpha=0.1*alpha.
numeric scalar relevant to the case when approx=FALSE
(i.e., when the power is based on the exact test). This argument is passed to the
uniroot function and indicates the tolerance to use in the search algorithm.
The default value is tol=1e-7.
integer relevant to the case when approx=FALSE
(i.e., when the power is based on the exact test). This argument is passed to the
uniroot function and indicates the maximum number of iterations to use
in the search algorithm. The default value is maxiter=1000.
If the arguments p.or.p1, p0.or.p2, alpha, power, ratio,
and tol.alpha are not all the same length, they are replicated to be the same length
as the length of the longest argument.
The computed sample size 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,
sample size 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 the associated power is presented in
standard statistics texts, including Zar (2010, pp. 534-537, 539-541).
These equations can be inverted to solve for the sample size, given a specified power,
significance level, hypothesized proportion, and true proportion.
approx=FALSE.When sample.type="one.sample" and approx=FALSE,
sample size 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 standard statistics texts,
including Zar (2010, pp. 532-534, 539) and
Millard and Neerchal (2001, pp. 385-386, 504-506). The formula for the power involves
five quantities: the hypothesized proportion (\(p_0\)), the true proportion (\(p\)),
the significance level (\(alpha\)), the power, and the sample size (\(n\)).
In this case the function propTestN uses a search algorithm to determine the
required sample size to attain a specified power, given the values of the
hypothesized and true proportions and the significance level.
Two-Sample Case (sample.type="two.sample").
When sample.type="two.sample", sample size 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).
These equations can be inverted to solve for the sample size, given a specified power,
significance level, true proportions, and ratio of sample size in group 2 to sample size in
group 1.
Approximate Test (approx=TRUE).
When sample.type="one.sample", or sample.type="two.sample"
and ratio=1 (i.e., equal sample sizes for each group), propTestN
returns a numeric vector of sample sizes. When sample.type="two.sample" and at least one element of ratio is
greater than 1, propTestN returns a list with two components called
n1 and n2, specifying the sample sizes for each group.
Exact Test (approx=FALSE).
If return.exact.list=FALSE, propTestN returns a numeric vector of sample sizes.
If return.exact.list=TRUE, propTestN returns a list with the following components:
numeric vector of sample sizes.
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 required sample size of the one-sample
# proportion test with a two-sided alternative and Type I error
# set to 5% increases with increasing power:
seq(0.5, 0.9, by = 0.1)
#> [1] 0.5 0.6 0.7 0.8 0.9
#[1] 0.5 0.6 0.7 0.8 0.9
propTestN(p.or.p1 = 0.7, p0.or.p2 = 0.5,
power = seq(0.5, 0.9, by = 0.1))
#> [1] 25 31 38 47 62
#[1] 25 31 38 47 62
#----------
# Repeat the last example, but compute the sample size based on
# the exact test instead of the approximation. Note that because
# we require the actual Type I error (alpha) to be within
# 10% of the supplied value of alpha (which is 0.05 by default),
# due to the discrete nature of the exact binomial test
# we end up with more power then we specified.
n.list <- propTestN(p.or.p1 = 0.7, p0.or.p2 = 0.5,
power = seq(0.5, 0.9, by = 0.1), approx = FALSE)
lapply(n.list, round, 3)
#> $n
#> [1] 37 37 44 51 65
#>
#> $power
#> [1] 0.698 0.698 0.778 0.836 0.910
#>
#> $alpha
#> [1] 0.047 0.047 0.049 0.049 0.046
#>
#> $q.critical.lower
#> [1] 12 12 15 18 24
#>
#> $q.critical.upper
#> [1] 24 24 28 32 40
#>
#$n
#[1] 37 37 44 51 65
#
#$power
#[1] 0.698 0.698 0.778 0.836 0.910
#
#$alpha
#[1] 0.047 0.047 0.049 0.049 0.046
#
#$q.critical.lower
#[1] 12 12 15 18 24
#
#$q.critical.upper
#[1] 24 24 28 32 40
#----------
# Using the example above, see how the sample size changes
# if we allow the Type I error to deviate by more than 10 percent
# of the value of alpha (i.e., by more than 0.005).
n.list <- propTestN(p.or.p1 = 0.7, p0.or.p2 = 0.5,
power = seq(0.5, 0.9, by = 0.1), approx = FALSE, tol.alpha = 0.01)
lapply(n.list, round, 3)
#> $n
#> [1] 25 35 42 49 65
#>
#> $power
#> [1] 0.512 0.652 0.743 0.810 0.910
#>
#> $alpha
#> [1] 0.043 0.041 0.044 0.044 0.046
#>
#> $q.critical.lower
#> [1] 7 11 14 17 24
#>
#> $q.critical.upper
#> [1] 17 23 27 31 40
#>
#$n
#[1] 25 35 42 49 65
#
#$power
#[1] 0.512 0.652 0.743 0.810 0.910
#
#$alpha
#[1] 0.043 0.041 0.044 0.044 0.046
#
#$q.critical.lower
#[1] 7 11 14 17 24
#
#$q.critical.upper
#[1] 17 23 27 31 40
#----------
# Clean up
#---------
rm(n.list)
#==========
# Look at how the required sample size for the two-sample
# proportion test decreases with increasing difference between
# the two population proportions:
seq(0.4, 0.1, by = -0.1)
#> [1] 0.4 0.3 0.2 0.1
#[1] 0.4 0.3 0.2 0.1
propTestN(p.or.p1 = seq(0.4, 0.1, by = -0.1),
p0.or.p2 = 0.5, sample.type = "two")
#> Warning: The computed 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:
#> 4
#>
#> [1] 661 163 70 36
#[1] 661 163 70 36
#Warning message:
#In propTestN(p.or.p1 = seq(0.4, 0.1, by = -0.1), p0.or.p2 = 0.5, :
# The computed 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:
# 4
#----------
# Look at how the required sample size for the two-sample
# proportion test decreases with increasing values of Type I error:
propTestN(p.or.p1 = 0.7, p0.or.p2 = 0.5,
sample.type = "two",
alpha = c(0.001, 0.01, 0.05, 0.1))
#> [1] 299 221 163 137
#[1] 299 221 163 137
#==========
# Modifying the example on pages 8-5 to 8-7 of USEPA (1989b),
# determine the required sample size to detect a difference in the
# proportion of detects of cadmium between the background and
# compliance wells. Set the complicance well to "group 1" and
# the backgound 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 and 0.5, use a 5%
# significance level and 95% power, 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 required sample size decreases from about
# 1160 at each well to about 200 at each well as the difference in
# proportions changes from (0.4 - 1/3) to (0.5 - 1/3), but both of
# these sample sizes are enormous compared to the number of samples
# usually collected in the field.
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
propTestN(p.or.p1 = c(0.4, 0.50), p0.or.p2 = p.hat.back,
alt="greater", sample.type="two")
#> [1] 1159 199
#[1] 1159 199
#----------
# Clean up
#---------
rm(p.hat.back, p.hat.comp, n.back, n.comp)