propTestN {EnvStats} R Documentation

## Compute Sample Size Necessary to Achieve a Specified Power for a One- or Two-Sample Proportion Test

### Description

Compute 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.

### Usage

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)

### Arguments

 p.or.p1 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. p0.or.p2 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. alpha 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. power numeric vector of numbers between 0 and 1 indicating the power associated with the hypothesis test. The default value is power=0.95. sample.type 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". alternative character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "less", and "greater". ratio 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". approx 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"). correct 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. round.up 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. warn 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. return.exact.list 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). n.min 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. n.max 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. tol.alpha 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. tol 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. maxiter 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.

### Details

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.

### Value

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:

 n numeric vector of sample sizes. 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 alpha. 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 alternative="greater".) 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 alternative="less".)

### Note

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.

### Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

### References

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.

### Examples

# 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

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

#----------

# 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
#
#$q.critical.upper #[1] 24 24 28 32 40 #---------- # Using the example above, see how the sample size changes # if we allow the Type I error to deviate by more than 10 percent # of the value of alpha (i.e., by more than 0.005). n.list <- propTestN(p.or.p1 = 0.7, p0.or.p2 = 0.5, power = seq(0.5, 0.9, by = 0.1), approx = FALSE, tol.alpha = 0.01) lapply(n.list, round, 3) #$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

#----------

# 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

propTestN(p.or.p1 = seq(0.4, 0.1, by = -0.1),
p0.or.p2 = 0.5, sample.type = "two")
#[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

#==========

# 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.

#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

mean(!Censored[Well.type=="Background"]))

p.hat.back
#[1] 0.3333333

mean(!Censored[Well.type=="Compliance"]))

p.hat.comp
#[1] 0.375

sum(Well.type == "Background"))

n.back
#[1] 24

sum(Well.type == "Compliance"))

n.comp
#[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

#----------

# Clean up
#---------
rm(p.hat.back, p.hat.comp, n.back, n.comp)

[Package EnvStats version 2.8.1 Index]