aovN {EnvStats}

Compute Sample Size Necessary to Achieve Specified Power for One-Way Fixed-Effects Analysis of Variance

Description

Compute 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.

Usage

  aovN(mu.vec, sigma = 1, alpha = 0.05, power = 0.95, 
    round.up = TRUE, n.max = 5000, tol = 1e-07, maxiter = 1000)

Arguments

mu.vec

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.

sigma

optional numeric scalar specifying the population standard deviation (\sigma) for each group. The default value is sigma=1.

alpha

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.

power

optional numeric scalar between 0 and 1 indicating the power associated with the hypothesis test. The default value is power=0.95.

round.up

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.

n.max

positive integer greater then 1 indicating the maximum sample size per group. The default value is n.max=5000.

tol

optional numeric scalar indicating the tolerance to use in the uniroot search algorithm. The default value is tol=1e-7.

maxiter

optional positive integer indicating the maximum number of iterations to use in the uniroot search algorithm. The default value is maxiter=1000.

Details

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.

Value

numeric scalar indicating the required sample size for each group. (The number of groups is equal to the length of the argument mu.vec.)

Note

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.

Author(s)

Steven P. Millard ([e-mail])

References

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.

See Also

aovPower, plotAovDesign, Normal, aov.

Examples

  # 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 

  aovN(mu.vec = c(10, 12, 15), sigma = 5, power = 0.9) 
  #[1] 27 

  aovN(mu.vec = c(10, 12, 15), sigma = 5, power = 0.95) 
  #[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 

  aovN(mu.vec = c(10, 10, 15), sigma = 5) 
  #[1] 25 

  aovN(mu.vec = c(10, 13, 15), sigma = 5) 
  #[1] 33 

  aovN(mu.vec = c(10, 15, 20), sigma = 5) 
  #[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 

  aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.01) 
  #[1] 67 

  aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.05) 
  #[1] 50 

  aovN(mu.vec = c(10, 12, 14), sigma = 5, alpha = 0.1) 
  #[1] 42