tTestLnormAltRatioOfMeans: Minimal or Maximal Detectable Ratio of Means for One- or...

In EnvStats: Package for Environmental Statistics, Including US EPA Guidance

Minimal or Maximal Detectable Ratio of Means for One- or Two-Sample t-Test, Assuming Lognormal Data

Description

Compute the minimal or maximal detectable ratio of means associated with a one- or two-sample t-test, given the sample size, coefficient of variation, significance level, and power, assuming lognormal data.

Usage

  tTestLnormAltRatioOfMeans(n.or.n1, n2 = n.or.n1, cv = 1, alpha = 0.05, power = 0.95, 
    sample.type = ifelse(!missing(n2), "two.sample", "one.sample"), 
    alternative = "two.sided", two.sided.direction = "greater", approx = FALSE, 
    tol = 1e-07, maxiter = 1000)

Arguments

n.or.n1	numeric vector of sample sizes. When sample.type="one.sample", n.or.n1 denotes n, the number of observations in the single sample. When sample.type="two.sample", n.or.n1 denotes n_1, the number of observations from group 1. 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 the value of n.or.n1. This argument is ignored when sample.type="one.sample". Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
cv	numeric vector of positive value(s) specifying the coefficient of variation. When sample.type="one.sample", this is the population coefficient of variation. When sample.type="two.sample", this is the coefficient of variation for both the first and second population. The default value is cv=1.
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 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 mean. When sample.type="two.sample", the computed power is based on a hypothesis test for the difference between two means. The default value is sample.type="one.sample" unless the argument n2 is supplied.
alternative	character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "greater", and "less".
two.sided.direction	character string indicating the direction (greater than 1 or less than 1) for the detectable ratio of means when alternative="two.sided". When two.sided.direction="greater" (the default), the detectable ratio of means is greater than 1. When two.sided.direction="less", the detectable ratio of means is less than 1 (but greater than 0). This argument is ignored if alternative="less" or alternative="greater".
approx	logical scalar indicating whether to compute the power based on an approximation to the non-central t-distribution. The default value is FALSE.
tol	numeric scalar indicating the toloerance to use in the uniroot search algorithm. The default value is tol=1e-7.
maxiter	positive integer indicating the maximum number of iterations argument to pass to the uniroot function. The default value is maxiter=1000.

Details

If the arguments n.or.n1, n2, cv, alpha, and power are not all the same length, they are replicated to be the same length as the length of the longest argument.

Formulas for the power of the t-test for lognormal data for specified values of the sample size, ratio of means, and Type I error level are given in the help file for tTestLnormAltPower. The function tTestLnormAltRatioOfMeans uses the uniroot search algorithm to determine the required ratio of means for specified values of the power, sample size, and Type I error level.

Value

a numeric vector of computed minimal or maximal detectable ratios of means. When
alternative="less", or alternative="two.sided" and two.sided.direction="less", the computed ratios are less than 1 (but greater than 0). Otherwise, the ratios are greater than 1.

Note

See tTestLnormAltPower.

Author(s)

Steven P. Millard ([e-mail])

References

See tTestLnormAltPower.

See Also

tTestLnormAltPower, tTestLnormAltN, plotTTestLnormAltDesign, LognormalAlt, t.test, Hypothesis Tests.

Examples

  # Look at how the minimal detectable ratio of means for the one-sample t-test 
  # increases with increasing required power:

  seq(0.5, 0.9, by = 0.1) 
  #[1] 0.5 0.6 0.7 0.8 0.9 

  ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = 20, 
    power = seq(0.5, 0.9, by = 0.1)) 

  round(ratio.of.means, 2) 
  #[1] 1.47 1.54 1.63 1.73 1.89

  #----------

  # Repeat the last example, but compute the minimal detectable ratio of means 
  # based on the approximate power instead of the exact:

  ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = 20, 
    power = seq(0.5, 0.9, by = 0.1), approx = TRUE) 

  round(ratio.of.means, 2) 
  #[1] 1.48 1.55 1.63 1.73 1.89

  #==========

  # Look at how the minimal detectable ratio of means for the two-sample t-test 
  # decreases with increasing sample size:

  seq(10, 50, by = 10) 
  #[1] 10 20 30 40 50 

  ratio.of.means <- tTestLnormAltRatioOfMeans(seq(10, 50, by = 10), sample.type="two") 

  round(ratio.of.means, 2) 
  #[1] 4.14 2.65 2.20 1.97 1.83

  #----------

  # Look at how the minimal detectable ratio of means for the two-sample t-test 
  # decreases with increasing values of Type I error:

  ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = 20, 
    alpha = c(0.001, 0.01, 0.05, 0.1), sample.type = "two") 

  round(ratio.of.means, 2) 
  #[1] 4.06 3.20 2.65 2.42

  #==========

  # The guidance document Soil Screening Guidance: Technical Background Document 
  # (USEPA, 1996c, Part 4) discusses sampling design and sample size calculations 
  # for studies to determine whether the soil at a potentially contaminated site 
  # needs to be investigated for possible remedial action. Let 'theta' denote the 
  # average concentration of the chemical of concern.  The guidance document 
  # establishes the following goals for the decision rule (USEPA, 1996c, p.87):
  #
  #     Pr[Decide Don't Investigate | theta > 2 * SSL] = 0.05
  #
  #     Pr[Decide to Investigate | theta <= (SSL/2)] = 0.2
  #
  # where SSL denotes the pre-established soil screening level.
  #
  # These goals translate into a Type I error of 0.2 for the null hypothesis
  #
  #     H0: [theta / (SSL/2)] <= 1
  #
  # and a power of 95% for the specific alternative hypothesis
  #
  #     Ha: [theta / (SSL/2)] = 4
  #
  # Assuming a lognormal distribution, the above values for Type I and power, and a 
  # coefficient of variation of 2, determine the minimal detectable increase above 
  # the soil screening level associated with various sample sizes for the one-sample 
  # test.  Based on these calculations, you need to take at least 6 soil samples to 
  # satisfy the requirements for the Type I and Type II errors when the coefficient 
  # of variation is 2.

  N <- 2:8
  ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = N, cv = 2, alpha = 0.2, 
    alternative = "greater") 

  names(ratio.of.means) <- paste("N=", N, sep = "")
  round(ratio.of.means, 1) 
  # N=2  N=3  N=4  N=5  N=6  N=7  N=8 
  #19.9  7.7  5.4  4.4  3.8  3.4  3.1

  #----------

  # Repeat the last example, but use the approximate power calculation instead of 
  # the exact.  Using the approximate power calculation, you need 7 soil samples 
  # when the coefficient of variation is 2.  Note how poorly the approximation 
  # works in this case for small sample sizes!

  ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = N, cv = 2, alpha = 0.2, 
    alternative = "greater", approx = TRUE) 

  names(ratio.of.means) <- paste("N=", N, sep = "")
  round(ratio.of.means, 1) 
  #  N=2   N=3   N=4   N=5   N=6   N=7   N=8 
  #990.8  18.5   8.3   5.7   4.6   3.9   3.5

  #==========

  # Clean up
  #---------
  rm(ratio.of.means, N)

EnvStats documentation built on Sept. 11, 2024, 6:03 p.m.