# Look at how the required sample size for the one-sample 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
tTestLnormAltN(ratio.of.means = 1.5, power = seq(0.5, 0.9, by = 0.1))
# [1] 19 23 28 36 47
#----------
# Repeat the last example, but compute the sample size based on the approximate
# power instead of the exact power:
tTestLnormAltN(ratio.of.means = 1.5, power = seq(0.5, 0.9, by = 0.1), approx = TRUE)
# [1] 19 23 29 36 47
#==========
# Look at how the required sample size for the two-sample t-test decreases with
# increasing ratio of means:
seq(1.5, 2, by = 0.1)
#[1] 1.5 1.6 1.7 1.8 1.9 2.0
tTestLnormAltN(ratio.of.means = seq(1.5, 2, by = 0.1), sample.type = "two")
#[1] 111 83 65 54 45 39
#----------
# Look at how the required sample size for the two-sample t-test decreases with
# increasing values of Type I error:
tTestLnormAltN(ratio.of.means = 1.5, alpha = c(0.001, 0.01, 0.05, 0.1),
sample.type = "two")
#[1] 209 152 111 92
#----------
# For the two-sample t-test, compare the total sample size required to detect a
# ratio of means of 2 for equal sample sizes versus the case when the sample size
# for the second group is constrained to be 30. Assume a coefficient of variation
# of 1, a 5% significance level, and 95% power. Note that for the case of equal
# sample sizes, a total of 78 samples (39+39) are required, whereas when n2 is
# constrained to be 30, a total of 84 samples (54 + 30) are required.
tTestLnormAltN(ratio.of.means = 2, sample.type = "two")
#[1] 39
tTestLnormAltN(ratio.of.means = 2, n2 = 30)
#$n1:
#[1] 54
#
#$n2:
#[1] 30
#==========
# 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 and the above values for Type I error and
# power, determine the required samples sizes associated with various values of
# the coefficient of variation 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.
cv <- c(0.5, 1, 2)
N <- tTestLnormAltN(ratio.of.means = 4, cv = cv, alpha = 0.2,
alternative = "greater")
names(N) <- paste("CV=", cv, sep = "")
N
#CV=0.5 CV=1 CV=2
# 2 3 6
#----------
# 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 (because the approximation underestimates the
# true power).
N <- tTestLnormAltN(ratio.of.means = 4, cv = cv, alpha = 0.2,
alternative = "greater", approx = TRUE)
names(N) <- paste("CV=", cv, sep = "")
N
#CV=0.5 CV=1 CV=2
# 3 5 7
#----------
# Repeat the last example, but use a Type I error of 0.05.
N <- tTestLnormAltN(ratio.of.means = 4, cv = cv, alternative = "greater",
approx = TRUE)
names(N) <- paste("CV=", cv, sep = "")
N
#CV=0.5 CV=1 CV=2
# 4 6 12
#==========
# Reproduce the second column of Table 2 in van Belle and Martin (1993, p.167).
tTestLnormAltN(ratio.of.means = 1.10, cv = seq(0.1, 0.8, by = 0.1),
power = 0.8, sample.type = "two.sample", approx = TRUE)
#[1] 19 69 150 258 387 533 691 856
#==========
# Clean up
#---------
rm(cv, N)Run the code above in your browser using DataLab