propTestPower {EnvStats} R Documentation

## Compute the Power of a One- or Two-Sample Proportion Test

### Description

Compute the power of a one- or two-sample proportion test, given the sample size(s), true proportion(s), and significance level.

### Usage

  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)


### Arguments

 n.or.n1 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. 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. n2 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. 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. sample.type 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". alternative character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "less", and "greater". approx 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"). 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. warn 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. 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, 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).

### Details

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).

### Value

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 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.

propTestN, propTestMdd, plotPropTestDesign, prop.test, binom.test.

### Examples

  # Look at how the power of the one-sample proportion test
# increases with increasing sample size:

seq(20, 50, by=10)
# 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)
# 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)
#$power: # 0.42 0.59 0.70 0.78 # #$alpha:
# 0.04 0.04 0.04 0.03
#
#$q.critical.lower: # 5 9 13 17 # #$q.critical.upper:
# 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)
# 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)
# 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)
# 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.

#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
# 0.3333333

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

p.hat.comp
# 0.375

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

n.back
# 24

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

n.comp
# 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")
# 0.08953013 0.12421135

#----------

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


[Package EnvStats version 2.8.1 Index]