Minimal or Maximal Detectable Ratio of Means for One- or Two-Sample t-Test, Assuming Lognormal Data
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.
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)n.or.n1 |
numeric vector of sample sizes. When |
n2 |
numeric vector of sample sizes for group 2. The default value is the value of
|
cv |
numeric vector of positive value(s) specifying the coefficient of
variation. When |
alpha |
numeric vector of numbers between 0 and 1 indicating the Type I error level
associated with the hypothesis test. The default value is |
power |
numeric vector of numbers between 0 and 1 indicating the power
associated with the hypothesis test. The default value is |
sample.type |
character string indicating whether to compute power based on a one-sample or
two-sample hypothesis test. When |
alternative |
character string indicating the kind of alternative hypothesis. The possible values
are |
two.sided.direction |
character string indicating the direction (greater than 1 or less than 1) for the
detectable ratio of means when |
approx |
logical scalar indicating whether to compute the power based on an approximation to
the non-central t-distribution. The default value is |
tol |
numeric scalar indicating the toloerance to use in the
|
maxiter |
positive integer indicating the maximum number of iterations
argument to pass to the |
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.
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.
See tTestLnormAltPower.
Steven P. Millard (EnvStats@ProbStatInfo.com)
See tTestLnormAltPower.
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.