Fisher's One-Sample Randomization (Permutation) Test for Location
Perform Fisher's one-sample randomization (permutation) test for location.
oneSamplePermutationTest(x, alternative = "two.sided", mu = 0, exact = FALSE,
n.permutations = 5000, seed = NULL, ...)x |
numeric vector of observations.
Missing ( |
alternative |
character string indicating the kind of alternative hypothesis. The possible values
are |
mu |
numeric scalar indicating the hypothesized value of the mean.
The default value is |
exact |
logical scalar indicating whether to perform the exact permutation test
(i.e., enumerate all possible permutations) or simply sample from the permutation
distribution. The default value is |
n.permutations |
integer indicating how many times to sample from the permutation distribution when
|
seed |
positive integer to pass to the R function |
... |
arguments that can be supplied to the |
Randomization Tests
In 1935, R.A. Fisher introduced the idea of a randomization test
(Manly, 2007, p. 107; Efron and Tibshirani, 1993, Chapter 15), which is based on
trying to answer the question: “Did the observed pattern happen by chance,
or does the pattern indicate the null hypothesis is not true?” A randomization
test works by simply enumerating all of the possible outcomes under the null
hypothesis, then seeing where the observed outcome fits in. A randomization test
is also called a permutation test, because it involves permuting the
observations during the enumeration procedure (Manly, 2007, p. 3).
In the past, randomization tests have not been used as extensively as they are now
because of the “large” computing resources needed to enumerate all of the
possible outcomes, especially for large sample sizes. The advent of more powerful
personal computers and software has allowed randomization tests to become much
easier to perform. Depending on the sample size, however, it may still be too
time consuming to enumerate all possible outcomes. In this case, the randomization
test can still be performed by sampling from the randomization distribution, and
comparing the observed outcome to this sampled permutation distribution.
Fisher's One-Sample Randomization Test for Location
Let \underline{x} = x_1, x_2, …, x_n be a vector of n independent
and identically distributed (i.i.d.) observations from some symmetric distribution
with mean μ. Consider the test of the null hypothesis that the mean
μ is equal to some specified value μ_0:
H_0: μ = μ_0 \;\;\;\;\;\; (1)
The three possible alternative hypotheses are the upper one-sided alternative
(alternative="greater")
H_a: μ > μ_0 \;\;\;\;\;\; (2)
the lower one-sided alternative (alternative="less")
H_a: μ < μ_0 \;\;\;\;\;\; (3)
and the two-sided alternative
H_a: μ \ne μ_0 \;\;\;\;\;\; (4)
To perform the test of the null hypothesis (1) versus any of the three alternatives (2)-(4), Fisher proposed using the test statistic
T = ∑_{i=1}^n y_i \;\;\;\;\;\; (5)
where
y_i = x_i - μ_0 \;\;\;\;\;\; (6)
(Manly, 2007, p. 112). The test assumes all of the observations come from the same distribution that is symmetric about the true population mean (hence the mean is the same as the median for this distribution). Under the null hypothesis, the y_i's are equally likely to be positive or negative. Therefore, the permutation distribution of the test statistic T consists of enumerating all possible ways of permuting the signs of the y_i's and computing the resulting sums. For n observations, there are 2^n possible permutations of the signs, because each observation can either be positive or negative.
For a one-sided upper alternative hypothesis (Equation (2)), the p-value is computed
as the proportion of sums in the permutation distribution that are greater than or
equal to the observed sum T. For a one-sided lower alternative hypothesis
(Equation (3)), the p-value is computed as the proportion of sums in the permutation
distribution that are less than or equal to the observed sum T. For a
two-sided alternative hypothesis (Equation (4)), the p-value is computed by using
the permutation distribution of the absolute value of T (i.e., |T|)
and computing the proportion of values in this permutation distribution that are
greater than or equal to the observed value of |T|.
Confidence Intervals Based on Permutation Tests
Based on the relationship between hypothesis tests and confidence intervals, it is
possible to construct a two-sided or one-sided (1-α)100\% confidence
interval for the mean μ based on the one-sample permutation test by finding
the values of μ_0 that correspond to obtaining a p-value of α
(Manly, 2007, pp. 18–20, 113). A confidence interval based on the bootstrap
however, will yield a similar type of confidence interval
(Efron and Tibshirani, 1993, p. 214); see the help file for
boot in the R package boot.
A list of class "permutationTest" containing the results of the hypothesis
test. See the help file for permutationTest.object for details.
A frequent question in environmental statistics is “Is the concentration of chemical X greater than Y units?”. For example, in groundwater assessment (compliance) monitoring at hazardous and solid waste sites, the concentration of a chemical in the groundwater at a downgradient well must be compared to a groundwater protection standard (GWPS). If the concentration is “above” the GWPS, then the site enters corrective action monitoring. As another example, soil screening at a Superfund site involves comparing the concentration of a chemical in the soil with a pre-determined soil screening level (SSL). If the concentration is “above” the SSL, then further investigation and possible remedial action is required. Determining what it means for the chemical concentration to be “above” a GWPS or an SSL is a policy decision: the average of the distribution of the chemical concentration must be above the GWPS or SSL, or the median must be above the GWPS or SSL, or the 95'th percentile must be above the GWPS or SSL, or something else. Often, the first interpretation is used.
Hypothesis tests you can use to perform tests of location include: Student's t-test, Fisher's randomization test, the Wilcoxon signed rank test, Chen's modified t-test, the sign test, and a test based on a bootstrap confidence interval. For a discussion comparing the performance of these tests, see Millard and Neerchal (2001, pp.408-409).
Steven P. Millard (EnvStats@ProbStatInfo.com)
Efron, B., and R.J. Tibshirani. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York, pp.224–227.
Manly, B.F.J. (2007). Randomization, Bootstrap and Monte Carlo Methods in Biology. Third Edition. Chapman & Hall, New York, pp.112-113.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL, pp.404–406.
# Generate 10 observations from a logistic distribution with parameters
# location=7 and scale=2, and test the null hypothesis that the true mean
# is equal to 5 against the alternative that the true mean is greater than 5.
# Use the exact permutation distribution.
# (Note: the call to set.seed() allows you to reproduce this example).
set.seed(23)
dat <- rlogis(10, location = 7, scale = 2)
test.list <- oneSamplePermutationTest(dat, mu = 5,
alternative = "greater", exact = TRUE)
# Print the results of the test
#------------------------------
test.list
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: Mean (Median) = 5
#
#Alternative Hypothesis: True Mean (Median) is greater than 5
#
#Test Name: One-Sample Permutation Test
# (Exact)
#
#Estimated Parameter(s): Mean = 9.977294
#
#Data: dat
#
#Sample Size: 10
#
#Test Statistic: Sum(x - 5) = 49.77294
#
#P-value: 0.001953125
# Plot the results of the test
#-----------------------------
dev.new()
plot(test.list)
#==========
# The guidance document "Supplemental Guidance to RAGS: Calculating the
# Concentration Term" (USEPA, 1992d) contains an example of 15 observations
# of chromium concentrations (mg/kg) which are assumed to come from a
# lognormal distribution. These data are stored in the vector
# EPA.92d.chromium.vec. Here, we will use the permutation test to test
# the null hypothesis that the mean (median) of the log-transformed chromium
# concentrations is less than or equal to log(100 mg/kg) vs. the alternative
# that it is greater than log(100 mg/kg). Note that we *cannot* use the
# permutation test to test a hypothesis about the mean on the original scale
# because the data are not assumed to be symmetric about some mean, they are
# assumed to come from a lognormal distribution.
#
# We will sample from the permutation distribution.
# (Note: setting the argument seed=542 allows you to reproduce this example).
test.list <- oneSamplePermutationTest(log(EPA.92d.chromium.vec),
mu = log(100), alternative = "greater", seed = 542)
test.list
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: Mean (Median) = 4.60517
#
#Alternative Hypothesis: True Mean (Median) is greater than 4.60517
#
#Test Name: One-Sample Permutation Test
# (Based on Sampling
# Permutation Distribution
# 5000 Times)
#
#Estimated Parameter(s): Mean = 4.378636
#
#Data: log(EPA.92d.chromium.vec)
#
#Sample Size: 15
#
#Test Statistic: Sum(x - 4.60517) = -3.398017
#
#P-value: 0.7598
# Plot the results of the test
#-----------------------------
dev.new()
plot(test.list)
#----------
# Clean up
#---------
rm(test.list)
graphics.off()Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.