Description

Compute the power of a one- or two-sample t-test, given the sample size, scaled difference, and significance level.

Usage

Arguments

Details

If the arguments n.or.n1, n2, delta.over.sigma, 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 normal distribution with mean μ and standard deviation σ, and consider 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 (alternative="two.sided")

H_a: μ \ne μ_0 \;\;\;\;\;\; (4)

The test of the null hypothesis (1) versus any of the three alternatives (2)-(4) is 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)

Under the null hypothesis (1), the t-statistic in (5) follows a Student's t-distribution with n-1 degrees of freedom (Zar, 2010, Chapter 7; Johnson et al., 1995, pp.362-363).

The formula for the power of the test depends on which alternative is being tested. The two subsections below describe exact and approximate formulas for the power of the one-sample t-test. Note that none of the equations for the power of the t-test requires knowledge of the values δ (Equation (12) below) or σ (the population standard deviation), only the ratio δ/σ. The argument delta.over.sigma is this ratio, and it is referred to as the “scaled difference”.

Exact Power Calculations (approx=FALSE)
This subsection describes the exact formulas for the power of the one-sample t-test.

Upper one-sided alternative (alternative="greater")
The standard Student's t-test rejects the null hypothesis (1) in favor of the upper alternative hypothesis (2) at level-α if

t ≥ t_{ν}(1 - α) \;\;\;\;\;\; (8)

where

ν = n - 1 \;\;\;\;\;\; (9)

and t_{ν}(p) denotes the p'th quantile of Student's t-distribution with ν degrees of freedom (Zar, 2010; Berthouex and Brown, 2002). The power of this test, denoted by 1-β, where β denotes the probability of a Type II error, is given by:

1 - β = Pr[t_{ν, Δ} ≥ t_{ν}(1 - α)] = 1 - G[t_{ν}(1 - α), ν, Δ] \;\;\;\;\;\; (10)

where

Δ = √{n} (\frac{δ}{σ}) \;\;\;\;\;\; (11)

δ = μ - μ_0 \;\;\;\;\;\ (12)

and t_{ν, Δ} denotes a non-central Student's t-random variable with ν degrees of freedom and non-centrality parameter Δ, and G(x, ν, Δ) denotes the cumulative distribution function of this random variable evaluated at x (Johnson et al., 1995, pp.508-510).

Lower one-sided alternative (alternative="less")
The standard Student's t-test rejects the null hypothesis (1) in favor of the lower alternative hypothesis (3) at level-α if

t ≤ t_{ν}(α) \;\;\;\;\;\; (13)

and the power of this test is given by:

1 - β = Pr[t_{ν, Δ} ≤ t_{ν}(α)] = G[t_{ν}(α), ν, Δ] \;\;\;\;\;\; (14)

Two-sided alternative (alternative="two.sided")
The standard Student's t-test rejects the null hypothesis (1) in favor of the two-sided alternative hypothesis (4) at level-α if

|t| ≥ t_{ν}(1 - α/2) \;\;\;\;\;\; (15)

and the power of this test is given by:

1 - β = Pr[t_{ν, Δ} ≤ t_{ν}(α/2)] + Pr[t_{ν, Δ} ≥ t_{ν}(1 - α/2)]

= G[t_{ν}(α/2), ν, Δ] + 1 - G[t_{ν}(1 - α/2), ν, Δ] \;\;\;\;\;\; (16)

The power of the t-test given in Equation (16) can also be expressed in terms of the cumulative distribution function of the non-central F-distribution as follows. Let F_{ν_1, ν_2, Δ} denote a non-central F random variable with ν_1 and ν_2 degrees of freedom and non-centrality parameter Δ, and let H(x, ν_1, ν_2, Δ) denote the cumulative distribution function of this random variable evaluated at x. Also, let F_{ν_1, ν_2}(p) denote the p'th quantile of the central F-distribution with ν_1 and ν_2 degrees of freedom. It can be shown that

(t_{ν, Δ})^2 \cong F_{1, ν, Δ^2} \;\;\;\;\;\; (17)

where \cong denotes “equal in distribution”. Thus, it follows that

[t_{ν}(1 - α/2)]^2 = F_{1, ν}(1 - α) \;\;\;\;\;\; (18)

so the formula for the power of the t-test given in Equation (16) can also be written as:

1 - β = Pr\{(t_{ν, Δ})^2 ≥ [t_{ν}(1 - α/2)]^2\}

= Pr[F_{1, ν, Δ^2} ≥ F_{1, ν}(1 - α)] = 1 - H[F_{1, ν}(1-α), 1, ν, Δ^2] \;\;\;\;\;\; (19)

Approximate Power Calculations (approx=TRUE)
Zar (2010, pp.115–118) presents an approximation to the power for the t-test given in Equations (10), (14), and (16) above. His approximation to the power can be derived by using the approximation

√{n} (\frac{δ}{s}) \approx √{n} (\frac{δ}{σ}) = Δ \;\;\;\;\;\; (20)

where \approx denotes “approximately equal to”. Zar's approximation can be summarized in terms of the cumulative distribution function of the non-central t-distribution as follows:

G(x, ν, Δ) \approx G(x - Δ, ν, 0) = G(x - Δ, ν) \;\;\;\;\;\; (21)

where G(x, ν) denotes the cumulative distribution function of the central Student's t-distribution with ν degrees of freedom evaluated at x.

The following three subsections explicitly derive the approximation to the power of the t-test for each of the three alternative hypotheses.

Upper one-sided alternative (alternative="greater")
The power for the upper one-sided alternative (2) given in Equation (10) can be approximated as:

1 - β = Pr[t ≥ t_{ν}(1 - α)]

= Pr[\frac{\bar{x} - μ}{s/√{n}} ≥ t_{ν}(1 - α) - √{n}\frac{δ}{s}]

\approx Pr[t_{ν} ≥ t_{ν}(1 - α) - Δ]

= 1 - Pr[t_{ν} ≤ t_{ν}(1 - α) - Δ]

= 1 - G[t_{ν}(1-α) - Δ, ν] \;\;\;\;\;\; (22)

where t_{ν} denotes a central Student's t-random variable with ν degrees of freedom.

Lower one-sided alternative (alternative="less")
The power for the lower one-sided alternative (3) given in Equation (14) can be approximated as:

1 - β = Pr[t ≤ t_{ν}(α)]

= Pr[\frac{\bar{x} - μ}{s/√{n}} ≤ t_{ν}(α) - √{n}\frac{δ}{s}]

\approx Pr[t_{ν} ≤ t_{ν}(α) - Δ]

= G[t_{ν}(α) - Δ, ν] \;\;\;\;\;\; (23)

Two-sided alternative (alternative="two.sided")
The power for the two-sided alternative (4) given in Equation (16) can be approximated as:

1 - β = Pr[t ≤ t_{ν}(α/2)] + Pr[t ≥ t_{ν}(1 - α/2)]

= Pr[\frac{\bar{x} - μ}{s/√{n}} ≤ t_{ν}(α/2) - √{n}\frac{δ}{s}] + Pr[\frac{\bar{x} - μ}{s/√{n}} ≥ t_{ν}(1 - α) - √{n}\frac{δ}{s}]

\approx Pr[t_{ν} ≤ t_{ν}(α/2) - Δ] + Pr[t_{ν} ≥ t_{ν}(1 - α/2) - Δ]

= G[t_{ν}(α/2) - Δ, ν] + 1 - G[t_{ν}(1-α/2) - Δ, ν] \;\;\;\;\;\; (24)

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 normal distribution with mean μ_1 and standard deviation σ, and let \underline{x}_2 = x_{21}, x_{22}, …, x_{2n_2} denote a vector of n_2 observations from a normal distribution with mean μ_2 and standard deviation σ, and consider the null hypothesis:

H_0: μ_1 = μ_2 \;\;\;\;\;\; (25)

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

H_a: μ_1 > μ_2 \;\;\;\;\;\; (26)

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

H_a: μ_1 < μ_2 \;\;\;\;\;\; (27)

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

H_a: μ_1 \ne μ_2 \;\;\;\;\;\; (28)

The test of the null hypothesis (25) versus any of the three alternatives (26)-(28) is based on the Student t-statistic:

t = \frac{\bar{x}_1 - \bar{x}_2}{s_p√{\frac{1}{n_1} + \frac{1}{n_2}}} \;\;\;\;\;\; (29)

where

\bar{x}_1 = \frac{1}{n_1} ∑_{i=1}^{n_1} x_{1i} \;\;\;\;\;\; (30)

\bar{x}_2 = \frac{1}{n_2} ∑_{i=1}^{n_2} x_{2i} \;\;\;\;\;\; (31)

s_p^2 = \frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2} \;\;\;\;\;\; (32)

s_1^2 = \frac{1}{n_1 - 1} ∑_{i=1}^{n_1} (x_{1i} - \bar{x}_1)^2 \;\;\;\;\;\; (33)

s_2^2 = \frac{1}{n_2 - 1} ∑_{i=1}^{n_2} (x_{2i} - \bar{x}_2)^2 \;\;\;\;\;\; (34)

Under the null hypothesis (25), the t-statistic in (29) follows a Student's t-distribution with n_1 + n_2 - 2 degrees of freedom (Zar, 2010, Chapter 8; Johnson et al., 1995, pp.508–510, Helsel and Hirsch, 1992, pp.124–128).

The formulas for the power of the two-sample t-test are precisely the same as those for the one-sample case, with the following modifications:

ν = n_1 + n_2 - 2 \;\;\;\;\;\; (35)

Δ = √{\frac{n_1 n_2}{n_1 + n_2}} (\frac{δ}{σ}) \;\;\;\;\;\; (36)

δ = μ_1 - μ_2 \;\;\;\;\;\; (37)

Note that none of the equations for the power of the t-test requires knowledge of the values δ or σ (the population standard deviation for both populations), only the ratio δ/σ. The argument delta.over.sigma is this ratio, and it is referred to as the “scaled difference”.

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). In this case, assuming normally distributed data, you can perform the Student's t-test.

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 tTestPower, tTestN, tTestScaledMdd, and plotTTestDesign can be used to investigate these relationships for the case of normally-distributed observations.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Second Edition. Lewis Publishers, Boca Raton, FL.

Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, Chapter 7.

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

Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL.

USEPA. (1989b). Statistical Analysis of Ground-Water Monitoring Data at RCRA Facilities, Interim Final Guidance. EPA/530-SW-89-026. Office of Solid Waste, U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.

Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.

See Also

tTestN, tTestScaledMdd, tTestAlpha, plotTTestDesign, Normal, t.test, Hypothesis Tests.

Examples