aovN.RdCompute the sample sizes necessary to achieve a specified power for a one-way fixed-effects analysis of variance test, given the population means, population standard deviation, and significance level.
aovN(mu.vec, sigma = 1, alpha = 0.05, power = 0.95,
round.up = TRUE, n.max = 5000, tol = 1e-07, maxiter = 1000)required numeric vector of population means. The length of
mu.vec must be at least 2. Missing (NA),
undefined (NaN), and infinite (Inf, -Inf)
values are not allowed.
optional numeric scalar specifying the population standard
deviation (\(\sigma\)) for each group. The default value
is sigma=1.
optional numeric scalar between 0 and 1 indicating the Type I
error level associated with the hypothesis test. The default
value is alpha=0.05.
optional numeric scalar between 0 and 1 indicating the power
associated with the hypothesis test. The default value
is power=0.95.
optional logical scalar indicating whether to round up the value of the
computed sample size to the next smallest integer. The default
value is round.up=TRUE.
positive integer greater then 1 indicating the maximum sample size per group.
The default value is n.max=5000.
optional numeric scalar indicating the tolerance to use in the
uniroot search algorithm. The default value is
tol=1e-7.
optional positive integer indicating the maximum number of iterations to use in the
uniroot search algorithm. The default value is
maxiter=1000.
The F-statistic to test the equality of \(k\) population means
assuming each population has a normal distribution with the same
standard deviation \(\sigma\) is presented in most basic
statistics texts, including Zar (2010, Chapter 10),
Berthouex and Brown (2002, Chapter 24), and Helsel and Hirsh (1992, pp.164-169).
The formula for the power of this test is given in Scheffe
(1959, pp.38-39,62-65). The power of the one-way fixed-effects ANOVA depends
on the sample sizes for each of the \(k\) groups, the value of the
population means for each of the \(k\) groups, the population
standard deviation \(\sigma\), and the significance level
\(\alpha\). See the help file for aovPower.
The function aovN assumes equal sample
sizes for each of the \(k\) groups and uses a search
algorithm to determine the sample size \(n\) required to
attain a specified power, given the values of the population
means and the significance level.
numeric scalar indicating the required sample size for each
group. (The number of groups is equal to the length of the
argument mu.vec.)
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Second Edition. Lewis Publishers, Boca Raton, FL.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, Chapter 7.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York, Chapters 27, 29, 30.
Millard, S.P., and Neerchal, N.K. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, Florida.
Scheffe, H. (1959). The Analysis of Variance. John Wiley and Sons, New York, 477pp.
USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C. p.6-38.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ, Chapter 10.
The normal and lognormal distribution are probably the two most frequently used distributions to model environmental data. Sometimes it is necessary to compare several means to determine whether any are significantly different from each other (e.g., USEPA, 2009, p.6-38). In this case, assuming normally distributed data, you perform a one-way parametric analysis of variance.
In the course of designing a sampling program, an environmental
scientist may wish to determine the relationship between sample
size, Type I error level, power, and differences in means if
one of the objectives of the sampling program is to determine
whether a particular mean differs from a group of means. The
functions aovPower, aovN, and
plotAovDesign can be used to investigate these
relationships for the case of normally-distributed observations.
# Look at how the required sample size for a one-way ANOVA
# increases with increasing power:
aovN(mu.vec = c(10, 12, 15), sigma = 5, power = 0.8)
#> [1] 21
#[1] 21
aovN(mu.vec = c(10, 12, 15), sigma = 5, power = 0.9)
#> [1] 27
#[1] 27
aovN(mu.vec = c(10, 12, 15), sigma = 5, power = 0.95)
#> [1] 33
#[1] 33
#----------------------------------------------------------------
# Look at how the required sample size for a one-way ANOVA,
# given a fixed power, decreases with increasing variability
# in the population means:
aovN(mu.vec = c(10, 10, 11), sigma=5)
#> [1] 581
#[1] 581
aovN(mu.vec = c(10, 10, 15), sigma = 5)
#> [1] 25
#[1] 25
aovN(mu.vec = c(10, 13, 15), sigma = 5)
#> [1] 33
#[1] 33
aovN(mu.vec = c(10, 15, 20), sigma = 5)
#> [1] 10
#[1] 10
#----------------------------------------------------------------
# Look at how the required sample size for a one-way ANOVA,
# given a fixed power, decreases with increasing values of
# Type I error:
aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.001)
#> [1] 89
#[1] 89
aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.01)
#> [1] 67
#[1] 67
aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.05)
#> [1] 50
#[1] 50
aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.1)
#> [1] 42
#[1] 42