Power of a One- or Two-Sample t-Test Assuming Lognormal Data

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Description

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

Usage

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  tTestLnormAltPower(n.or.n1, n2 = n.or.n1, ratio.of.means = 1, cv = 1, alpha = 0.05, 
    sample.type = ifelse(!missing(n2), "two.sample", "one.sample"), 
    alternative = "two.sided", approx = FALSE)

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.

ratio.of.means

numeric vector specifying the ratio of the first mean to the second mean. When sample.type="one.sample", this is the ratio of the population mean to the hypothesized mean. When sample.type="two.sample", this is the ratio of the mean of the first population to the mean of the second population. The default value is ratio.of.means=1.

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.

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".

approx

logical scalar indicating whether to compute the power based on an approximation to the non-central t-distribution. The default value is FALSE.

Details

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

One-Sample Case (sample.type="one.sample")
Let \underline{x} = x_1, x_2, …, x_n denote a vector of n observations from a lognormal distribution with mean θ and coefficient of variation τ, and consider the null hypothesis:

H_0: θ = θ_0 \;\;\;\;\;\; (1)

The three possible alternative hypotheses are the upper one-sided alternative (alternative="greater"):

H_a: θ > θ_0 \;\;\;\;\;\; (2)

the lower one-sided alternative (alternative="less")

H_a: θ < θ_0 \;\;\;\;\;\; (3)

and the two-sided alternative (alternative="two.sided")

H_a: θ \ne θ_0 \;\;\;\;\;\; (4)

To test the null hypothesis (1) versus any of the three alternatives (2)-(4), one might be tempted to use Student's t-test based on the log-transformed observations. Unlike the two-sample case with equal coefficients of variation (see below), in the one-sample case Student's t-test applied to the log-transformed observations will not test the correct hypothesis, as now explained.

Let

y_i = log(x_i), \;\; i = 1, 2, …, n \;\;\;\;\;\; (5)

Then \underline{y} = y_1, y_2, …, y_n denote n observations from a normal distribution with mean μ and standard deviation σ, where

μ = log(\frac{θ}{√{τ^2 + 1}}) \;\;\;\;\;\; (6)

σ = [log(τ^2 + 1)]^{1/2} \;\;\;\;\;\; (7)

θ = exp[μ + (σ^2/2)] \;\;\;\;\;\; (8)

τ = [exp(σ^2) - 1]^{1/2} \;\;\;\;\;\; (9)

(see the help file for LognormalAlt). Hence, by Equations (6) and (8) above, the Student's t-test on the log-transformed data would involve a test of hypothesis on both the parameters θ and τ, not just on θ.

To test the null hypothesis (1) above versus any of the alternatives (2)-(4), you can use the function elnormAlt to compute a confidence interval for θ, and use the relationship between confidence intervals and hypothesis tests. To test the null hypothesis (1) above versus the upper one-sided alternative (2), you can also use Chen's modified t-test for skewed distributions.

Although you can't use Student's t-test based on the log-transformed observations to test a hypothesis about θ, you can use the t-distribution to estimate the power of a test about θ that is based on confidence intervals or Chen's modified t-test, if you are willing to assume the population coefficient of variation τ stays constant for all possible values of θ you are interested in, and you are willing to postulate possible values for τ.

First, let's re-write the hypotheses (1)-(4) as follows. The null hypothesis (1) is equivalent to:

H_0: \frac{θ}{θ_0} = 1 \;\;\;\;\;\; (10)

The three possible alternative hypotheses are the upper one-sided alternative (alternative="greater")

H_a: \frac{θ}{θ_0} > 1 \;\;\;\;\;\; (11)

the lower one-sided alternative (alternative="less")

H_a: \frac{θ}{θ_0} < 1 \;\;\;\;\;\; (12)

and the two-sided alternative (alternative="two.sided")

H_a: \frac{θ}{θ_0} \ne 1 \;\;\;\;\;\; (13)

For a constant coefficient of variation τ, the standard deviation of the log-transformed observations σ is also constant (see Equation (7) above). Hence, by Equation (8), the ratio of the true mean to the hypothesized mean can be written as:

R = \frac{θ}{θ_0} = \frac{exp[μ + (σ^2/2)]}{exp[μ_0 + (σ^2/2)]} = \frac{e^μ}{e^μ_0} = e^{μ - μ_0} \;\;\;\;\;\; (14)

which only involves the difference

μ - μ_0 \;\;\;\;\;\; (15)

Thus, for given values of R and τ, the power of the test of the null hypothesis (10) against any of the alternatives (11)-(13) can be computed based on the power of a one-sample t-test with

\frac{δ}{σ} = \frac{log(R)}{√{log(τ^2 + 1)}} \;\;\;\;\;\; (16)

(see the help file for tTestPower). Note that for the function tTestLnormAltPower, R corresponds to the argument ratio.of.means, and τ corresponds to the argument cv.

Two-Sample Case (sample.type="two.sample")
Let \underline{x}_1 = x_{11}, x_{12}, …, x_{1n_1} denote a vector of n_1 observations from a lognormal distribution with mean θ_1 and coefficient of variaiton τ, and let \underline{x}_2 = x_{21}, x_{22}, …, x_{2n_2} denote a vector of n_2 observations from a lognormal distribution with mean θ_2 and coefficient of variation τ, and consider the null hypothesis:

H_0: θ_1 = θ_2 \;\;\;\;\;\; (17)

The three possible alternative hypotheses are the upper one-sided alternative (alternative="greater"):

H_a: θ_1 > θ_2 \;\;\;\;\;\; (18)

the lower one-sided alternative (alternative="less")

H_a: θ_1 < θ_2 \;\;\;\;\;\; (19)

and the two-sided alternative (alternative="two.sided")

H_a: θ_1 \ne θ_2 \;\;\;\;\;\; (20)

Because we are assuming the coefficient of variation τ is the same for both populations, the test of the null hypothesis (17) versus any of the three alternatives (18)-(20) can be based on the Student t-statistic using the log-transformed observations.

To show this, first, let's re-write the hypotheses (17)-(20) as follows. The null hypothesis (17) is equivalent to:

H_0: \frac{θ_1}{θ_2} = 1 \;\;\;\;\;\; (21)

The three possible alternative hypotheses are the upper one-sided alternative (alternative="greater")

H_a: \frac{θ_1}{θ_2} > 1 \;\;\;\;\;\; (22)

the lower one-sided alternative (alternative="less")

H_a: \frac{θ_1}{θ_2} < 1 \;\;\;\;\;\; (23)

and the two-sided alternative (alternative="two.sided")

H_a: \frac{θ_1}{θ_2} \ne 1 \;\;\;\;\;\; (24)

If coefficient of variation τ is the same for both populations, then the standard deviation of the log-transformed observations σ is also the same for both populations (see Equation (7) above). Hence, by Equation (8), the ratio of the means can be written as:

R = \frac{θ_1}{θ_2} = \frac{exp[μ_1 + (σ^2/2)]}{exp[μ_2 + (σ^2/2)]} = \frac{e^μ_1}{e^μ_2} = e^{μ_1 - μ_2} \;\;\;\;\;\; (25)

which only involves the difference

μ_1 - μ_2 \;\;\;\;\;\; (26)

Thus, for given values of R and τ, the power of the test of the null hypothesis (21) against any of the alternatives (22)-(24) can be computed based on the power of a two-sample t-test with

\frac{δ}{σ} = \frac{log(R)}{√{log(τ^2 + 1)}} \;\;\;\;\;\; (27)

(see the help file for tTestPower). Note that for the function tTestLnormAltPower, R corresponds to the argument ratio.of.means, and τ corresponds to the argument cv.

Value

a numeric vector powers.

Note

The normal distribution and lognormal distribution are probably the two most frequently used distributions to model environmental data. Often, you need to determine whether a population mean is significantly different from a specified standard (e.g., an MCL or ACL, USEPA, 1989b, Section 6), or whether two different means are significantly different from each other (e.g., USEPA 2009, Chapter 16). When you have lognormally-distributed data, you have to be careful about making statements regarding inference for the mean. For the two-sample case with assumed equal coefficients of variation, you can perform the Student's t-test on the log-transformed observations. For the one-sample case, you can perform a hypothesis test by constructing a confidence interval for the mean using elnormAlt, or use Chen's t-test modified for skewed data.

In the course of designing a sampling program, an environmental scientist may wish to determine the relationship between sample size, significance level, power, and scaled difference if one of the objectives of the sampling program is to determine whether a mean differs from a specified level or two means differ from each other. The functions tTestLnormAltPower, tTestLnormAltN, tTestLnormAltRatioOfMeans, and plotTTestLnormAltDesign can be used to investigate these relationships for the case of lognormally-distributed observations.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

van Belle, G., and D.C. Martin. (1993). Sample Size as a Function of Coefficient of Variation and Ratio of Means. The American Statistician 47(3), 165–167.

Also see the list of references in the help file for tTestPower.

See Also

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

Examples

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  # Look at how the power of the one-sample test increases with increasing 
  # sample size:

  seq(5, 30, by = 5) 
  #[1]  5 10 15 20 25 30 

  power <- tTestLnormAltPower(n.or.n1 = seq(5, 30, by = 5), 
    ratio.of.means = 1.5, cv = 1) 

  round(power, 2) 
  #[1] 0.14 0.28 0.42 0.54 0.65 0.73

  #----------

  # Repeat the last example, but use the approximation to the power instead of the 
  # exact power.  Note how the approximation underestimates the true power for 
  # the smaller sample sizes:

  power <- tTestLnormAltPower(n.or.n1 = seq(5, 30, by = 5), 
    ratio.of.means = 1.5, cv = 1, approx = TRUE) 

  round(power, 2) 
  #[1] 0.09 0.25 0.40 0.53 0.64 0.73

  #==========

  # Look at how the power of the two-sample t-test increases with increasing 
  # ratio of means:

  power <- tTestLnormAltPower(n.or.n1 = 20, sample.type = "two", 
    ratio.of.means = c(1.1, 1.5, 2), cv = 1) 

  round(power, 2) 
  #[1] 0.06 0.32 0.73

  #----------

  # Look at how the power of the two-sample t-test increases with increasing 
  # values of Type I error:

  power <- tTestLnormAltPower(30, sample.type = "two", ratio.of.means = 1.5, 
    cv = 1, alpha = c(0.001, 0.01, 0.05, 0.1)) 

  round(power, 2) 
  #[1] 0.07 0.23 0.46 0.59

  #==========

  # 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 with a coefficient of variation of 2, 
  # determine the power associated with various sample sizes for this 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.

  power <- tTestLnormAltPower(n.or.n1 = 2:8, ratio.of.means = 4, cv = 2, 
    alpha = 0.2, alternative = "greater") 

  names(power) <- paste("N=", 2:8, sep = "")

  round(power, 2) 
  # N=2  N=3  N=4  N=5  N=6  N=7  N=8 
  #0.65 0.80 0.88 0.93 0.96 0.97 0.98

  #----------

  # Repeat the last example, but use the approximate power calculation instead of 
  # the exact one.  Using the approximate power calculation, you need at least 
  # 7 soil samples instead of 6 (because the approximation underestimates the power).

  power <- tTestLnormAltPower(n.or.n1 = 2:8, ratio.of.means = 4, cv = 2, 
    alpha = 0.2, alternative = "greater", approx = TRUE) 

  names(power) <- paste("N=", 2:8, sep = "")

  round(power, 2)
  # N=2  N=3  N=4  N=5  N=6  N=7  N=8 
  #0.55 0.75 0.84 0.90 0.93 0.95 0.97

  #==========

  # Clean up
  #---------
  rm(power)

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