Chen's Modified One-Sided t-test for Skewed Distributions
For a skewed distribution, estimate the mean, standard deviation, and skew; test the null hypothesis that the mean is equal to a user-specified value vs. a one-sided alternative; and create a one-sided confidence interval for the mean.
chenTTest(x, y = NULL, alternative = "greater", mu = 0, paired = !is.null(y),
conf.level = 0.95, ci.method = "z")x |
numeric vector of observations. Missing ( |
y |
optional numeric vector of observations that are paired with the observations in
|
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
|
paired |
character string indicating whether to perform a paired or one-sample t-test. The
possible values are |
conf.level |
numeric scalar between 0 and 1 indicating the confidence level associated with the
confidence interval for the population mean. The default value is |
ci.method |
character string indicating which critical value to use to construct the confidence
interval for the mean. The possible values are |
One-Sample Case (paired=FALSE)
Let \underline{x} = (x_1, x_2, …, x_n) be a vector of n independent
and identically distributed (i.i.d.) observations from some distribution with mean
μ and standard deviation σ.
Background: The Conventional Student's t-Test
Assume that the n observations come from a normal (Gaussian) distribution, and
consider the test of the null hypothesis:
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)
The test of the null hypothesis (1) versus any of the three alternatives (2)-(4) is usually based on the Student t-statistic:
t = \frac{\bar{x} - μ_0}{s/√{n}} \;\;\;\;\;\; (5)
where
\bar{x} = \frac{1}{n}∑_{i=1}^n x_i \;\;\;\;\;\; (6)
s^2 = \frac{1}{n-1} ∑_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (7)
(see the R help file for t.test). Under the null hypothesis (1),
the t-statistic in (5) follows a Student's t-distribution with
n-1 degrees of freedom (Zar, 2010, p.99; Johnson et al., 1995, pp.362-363).
The t-statistic is fairly robust to departures from normality in terms of
maintaining Type I error and power, provided that the sample size is sufficiently
large.
Chen's Modified t-Test for Skewed Distributions
In the case when the underlying distribution of the n observations is
positively skewed and the sample size is small, the sampling distribution of the
t-statistic under the null hypothesis (1) does not follow a Student's t-distribution,
but is instead negatively skewed. For the test against the upper alternative in (2)
above, this leads to a Type I error smaller than the one assumed and a loss of power
(Chen, 1995b, p.767).
Similarly, in the case when the underlying distribution of the n observations is negatively skewed and the sample size is small, the sampling distribution of the t-statistic is positively skewed. For the test against the lower alternative in (3) above, this also leads to a Type I error smaller than the one assumed and a loss of power.
In order to overcome these problems, Chen (1995b) proposed the following modified t-statistic that takes into account the skew of the underlying distribution:
t_2 = t + a(1 + 2t^2) + 4a^2(t + 2t^3) \;\;\;\;\;\; (8)
where
a = \frac{√{\hat{β}_1}}{6n} \;\;\;\;\;\; (9)
\hat{β}_1 = \frac{\hat{μ}_3}{\hat{σ}^3} \;\;\;\;\;\; (10)
\hat{μ}_3 = \frac{n}{(n-1)(n-2)} ∑_{i=1}^n (x_i - \bar{x})^3 \;\;\;\;\;\; (11)
\hat{σ}^3 = s^3 = [\frac{1}{n-1} ∑_{i=1}^n (x_i - \bar{x})^2]^{3/2} \;\;\;\;\;\; (12)
Note that the quantity √{\hat{β}_1} in (9) is an estimate of
the skew of the underlying distribution and is based on unbiased estimators of
central moments (see the help file for skewness).
For a positively-skewed distribution, Chen's modified t-test rejects the null hypothesis (1) in favor of the upper one-sided alternative (2) if the t-statistic in (8) is too large. For a negatively-skewed distribution, Chen's modified t-test rejects the null hypothesis (1) in favor of the lower one-sided alternative (3) if the t-statistic in (8) is too small.
Chen's modified t-test is not applicable to testing the two-sided alternative
(4). It should also not be used to test the upper one-sided alternative (2)
based on negatively-skewed data, nor should it be used to test the lower one-sided
alternative (3) based on positively-skewed data.
Determination of Critical Values and p-Values
Chen (1995b) performed a simulation study in which the modified t-statistic in (8)
was compared to a critical value based on the normal distribution (z-value),
a critical value based on Student's t-distribution (t-value), and the average of the
critical z-value and t-value. Based on the simulation study, Chen (1995b) suggests
using either the z-value or average of the z-value and t-value when n
(the sample size) is small (e.g., n ≤ 10) or α (the Type I error)
is small (e.g. α ≤ 0.01), and using either the t-value or the average
of the z-value and t-value when n ≥ 20 or α ≥ 0.05.
The function chenTTest returns three different p-values: one based on the
normal distribution, one based on Student's t-distribution, and one based on the
average of these two p-values. This last p-value should roughly correspond to a
p-value based on the distribution of the average of a normal and Student's t
random variable.
Computing Confidence Intervals
The function chenTTest computes a one-sided confidence interval for the true
mean μ based on finding all possible values of μ for which the null
hypothesis (1) will not be rejected, with the confidence level determined by the
argument conf.level. The argument ci.method determines which p-value
is used in the algorithm to determine the bounds on μ. When
ci.method="z", the p-value is based on the normal distribution, when
ci.method="t", the p-value is based on Student's t-distribution, and when
ci.method="Avg. of z and t" the p-value is based on the average of the
p-values based on the normal and Student's t-distribution.
Paired-Sample Case (paired=TRUE)
When the argument paired=TRUE, the arguments x and y are assumed
to have the same length, and the n differences
d_i = x_i - y_i, \;\; i = 1, 2, …, n
are assumed to be i.i.d. observations from some distribution with mean μ and standard deviation σ. Chen's modified t-test can then be applied to the differences.
a list of class "htest" containing the results of the hypothesis test. See
the help file for htest.object for details.
The presentation of Chen's (1995b) method in USEPA (2002d) and Singh et al. (2010b, p. 52) is incorrect for two reasons: it is based on an intermediate formula instead of the actual statistic that Chen proposes, and it uses the intermediate formula to compute an upper confidence limit for the mean when the sample data are positively skewed. As explained above, for the case of positively skewed data, Chen's method is appropriate to test the upper one-sided alternative hypothesis that the population mean is greater than some specified value, and a one-sided upper alternative corresponds to creating a one-sided lower confidence limit, not an upper confidence limit (see, for example, Millard and Neerchal, 2001, p. 371).
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 may 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.
The regulatory guidance document Soil Screening Guidance: Technical Background Document (USEPA, 1996c, Part 4) recommends using Chen's t-test as one possible method to compare chemical concentrations in soil samples to a soil screening level (SSL). The document notes that the distribution of chemical concentrations will almost always be positively-skewed, but not necessarily fit a lognormal distribution well (USEPA, 1996c, pp.107, 117-119). It also notes that using a confidence interval based on Land's (1971) method is extremely sensitive to the assumption of a lognormal distribution, while Chen's test is robust with respect to maintaining Type I and Type II errors for a variety of positively-skewed distributions (USEPA, 1996c, pp.99, 117-119, 123-125).
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)
Chen, L. (1995b). Testing the Mean of Skewed Distributions. Journal of the American Statistical Association 90(430), 767–772.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York, Chapters 28, 31.
Land, C.E. (1971). Confidence Intervals for Linear Functions of the Normal Mean and Variance. The Annals of Mathematical Statistics 42(4), 1187–1205.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL, pp.402–404.
Singh, A., N. Armbya, and A. Singh. (2010b). ProUCL Version 4.1.00 Technical Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.
USEPA. (1996c). Soil Screening Guidance: Technical Background Document. EPA/540/R-95/128, PB96963502. Office of Emergency and Remedial Response, U.S. Environmental Protection Agency, Washington, D.C., May, 1996.
USEPA. (2002d). Estimation of the Exposure Point Concentration Term Using a Gamma Distribution. EPA/600/R-02/084. October 2002. Technology Support Center for Monitoring and Site Characterization, Office of Research and Development, Office of Solid Waste and Emergency Response, U.S. Environmental Protection Agency, Washington, D.C.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.
# The guidance document "Calculating Upper Confidence Limits for # Exposure Point Concentrations at Hazardous Waste Sites" # (USEPA, 2002d, Exhibit 9, p. 16) contains an example of 60 observations # from an exposure unit. Here we will use Chen's modified t-test to test # the null hypothesis that the average concentration is less than 30 mg/L # versus the alternative that it is greater than 30 mg/L. # In EnvStats these data are stored in the vector EPA.02d.Ex.9.mg.per.L.vec. sort(EPA.02d.Ex.9.mg.per.L.vec) # [1] 16 17 17 17 18 18 20 20 20 21 21 21 21 21 21 22 #[17] 22 22 23 23 23 23 24 24 24 25 25 25 25 25 25 26 #[33] 26 26 26 27 27 28 28 28 28 29 29 30 30 31 32 32 #[49] 32 33 33 35 35 97 98 105 107 111 117 119 dev.new() hist(EPA.02d.Ex.9.mg.per.L.vec, col = "cyan", xlab = "Concentration (mg/L)") # The Shapiro-Wilk goodness-of-fit test rejects the null hypothesis of a # normal, lognormal, and gamma distribution: gofTest(EPA.02d.Ex.9.mg.per.L.vec)$p.value #[1] 2.496781e-12 gofTest(EPA.02d.Ex.9.mg.per.L.vec, dist = "lnorm")$p.value #[1] 3.349035e-09 gofTest(EPA.02d.Ex.9.mg.per.L.vec, dist = "gamma")$p.value #[1] 1.564341e-10 # Use Chen's modified t-test to test the null hypothesis that # the average concentration is less than 30 mg/L versus the # alternative that it is greater than 30 mg/L. chenTTest(EPA.02d.Ex.9.mg.per.L.vec, mu = 30) #Results of Hypothesis Test #-------------------------- # #Null Hypothesis: mean = 30 # #Alternative Hypothesis: True mean is greater than 30 # #Test Name: One-sample t-Test # Modified for # Positively-Skewed Distributions # (Chen, 1995) # #Estimated Parameter(s): mean = 34.566667 # sd = 27.330598 # skew = 2.365778 # #Data: EPA.02d.Ex.9.mg.per.L.vec # #Sample Size: 60 # #Test Statistic: t = 1.574075 # #Test Statistic Parameter: df = 59 # #P-values: z = 0.05773508 # t = 0.06040889 # Avg. of z and t = 0.05907199 # #Confidence Interval for: mean # #Confidence Interval Method: Based on z # #Confidence Interval Type: Lower # #Confidence Level: 95% # #Confidence Interval: LCL = 29.82 # UCL = Inf # The estimated mean, standard deviation, and skew are 35, 27, and 2.4, # respectively. The p-value is 0.06, and the lower 95% confidence interval # is [29.8, Inf). Depending on what you use for your Type I error rate, you # may or may not want to reject the null hypothesis.
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