Statistical power and study design

This vignette surveys the functions available for planning studies, then derives power calculations for t-tests and tests of a linear trend. The overview includes sample-size and precision tools for confidence, prediction, and tolerance intervals.

Related vignettes:

• Prediction intervals for environmental monitoring
• Trend tests and serial correlation
• Nonparametric estimation and two-sample tests

For monitoring-specific power calculations, see normal prediction interval power and nonparametric prediction interval power.

FcnsByCatPower

Help topic: FcnsByCatPower.

Confidence Intervals

Function Name	Description
ciTableProp	Confidence intervals for binomial proportion, or
	difference between two proportions, following Bacchetti (2010)
ciBinomHalfWidth	Compute the half-width of a confidence interval for a
	Binomial proportion or the difference between two proportions.
ciBinomN	Compute the sample size necessary to achieve a specified
	half-width of a confidence interval for a Binomial proportion or
	the difference between two proportions.
plotCiBinomDesign	Create plots for a sampling design based on a confidence interval
	for a Binomial proportion or the difference between two proportions.
ciTableMean	Confidence intervals for mean of normal distribution, or
	difference between two means, following Bacchetti (2010)
ciNormHalfWidth	Compute the half-width of a confidence interval for the mean of a
	Normal distribution or the difference between two means.
ciNormN	Compute the sample size necessary to achieve a specified half-width
	of a confidence interval for the mean of a Normal distribution or
	the difference between two means.
plotCiNormDesign	Create plots for a sampling design based on a confidence interval
	for the mean of a Normal distribution or the difference between
	two means.
ciNparConfLevel	Compute the confidence level associated with a nonparametric
	confidence interval for a percentile.
ciNparN	Compute the sample size necessary to achieve a specified
	confidence level for a nonparametric confidence interval for
	a percentile.
plotCiNparDesign	Create plots for a sampling design based on a nonparametric
	confidence interval for a percentile.

Hypothesis Tests

Function Name	Description
aovN	Compute the sample sizes necessary to achieve a
	specified power for a one-way fixed-effects analysis
	of variance test.
aovPower	Compute the power of a one-way fixed-effects analysis of
	variance test.
plotAovDesign	Create plots for a sampling design based on a one-way
	analysis of variance.
propTestN	Compute the sample size necessary to achieve a specified
	power for a one- or two-sample proportion test.
propTestPower	Compute the power of a one- or two-sample proportion test.
propTestMdd	Compute the minimal detectable difference associated with
	a one- or two-sample proportion test.
plotPropTestDesign	Create plots involving sample size, power, difference, and
	significance level for a one- or two-sample proportion test.
tTestAlpha	Compute the Type I Error associated with specified values for
	for power, sample size(s), and scaled MDD for a one- or
	two-sample t-test.
tTestN	Compute the sample size necessary to achieve a specified
	power for a one- or two-sample t-test.
tTestPower	Compute the power of a one- or two-sample t-test.
tTestScaledMdd	Compute the scaled minimal detectable difference
	associated with a one- or two-sample t-test.
plotTTestDesign	Create plots for a sampling design based on a one- or
	two-sample t-test.
tTestLnormAltN	Compute the sample size necessary to achieve a specified
	power for a one- or two-sample t-test, assuming lognormal
	data.
tTestLnormAltPower	Compute the power of a one- or two-sample t-test, assuming
	lognormal data.
tTestLnormAltRatioOfMeans	Compute the minimal or maximal detectable ratio of means
	associated with a one- or two-sample t-test, assuming
	lognormal data.
plotTTestLnormAltDesign	Create plots for a sampling design based on a one- or
	two-sample t-test, assuming lognormal data.
linearTrendTestN	Compute the sample size necessary to achieve a specified
	power for a t-test for linear trend.
linearTrendTestPower	Compute the power of a t-test for linear trend.
linearTrendTestScaledMds	Compute the scaled minimal detectable slope for a t-test
	for linear trend.
plotLinearTrendTestDesign	Create plots for a sampling design based on a t-test for
	linear trend.

Prediction Intervals

Normal Distribution Prediction Intervals

Function Name	Description
predIntNormHalfWidth	Compute the half-width of a prediction
	interval for a normal distribution.
predIntNormK	Compute the required value of $K$ for
	a prediction interval for a Normal
	distribution.
predIntNormN	Compute the sample size necessary to
	achieve a specified half-width for a
	prediction interval for a Normal
	distribution.
plotPredIntNormDesign	Create plots for a sampling design
	based on the width of a prediction
	interval for a Normal distribution.
predIntNormTestPower	Compute the probability that at least
	one future observation (or mean)
	falls outside a prediction interval
	for a Normal distribution.
plotPredIntNormTestPowerCurve	Create plots for a sampling
	design based on a prediction interval
	for a Normal distribution.
predIntNormSimultaneousTestPower	Compute the probability that at
	least one set of future observations
	(or means) violates the given rule
	based on a simultaneous prediction
	interval for a Normal distribution.
plotPredIntNormSimultaneousTestPowerCurve	Create plots for a sampling design
	based on a simultaneous prediction
	interval for a Normal distribution.

Lognormal Distribution Prediction Intervals

Function Name	Description
predIntLnormAltTestPower	Compute the probability that at least
	one future observation (or geometric
	mean) falls outside a prediction
	interval for a lognormal distribution.
plotPredIntLnormAltTestPowerCurve	Create plots for a sampling design
	based on a prediction interval for a
	lognormal distribution.
predIntLnormAltSimultaneousTestPower	Compute the probability that at least
	one set of future observations (or
	geometric means) violates the given
	rule based on a simultaneous
	prediction interval for a lognormal
	distribution.
plotPredIntLnormAltSimultaneousTestPowerCurve	Create plots for a sampling design
	based on a simultaneous prediction
	interval for a lognormal distribution.

Nonparametric Prediction Intervals

Function Name	Description
predIntNparConfLevel	Compute the confidence level associated with
	a nonparametric prediction interval.
predIntNparN	Compute the required sample size to achieve
	a specified confidence level for a
	nonparametric prediction interval.
plotPredIntNparDesign	Create plots for a sampling design based on
	the confidence level and sample size of a
	nonparametric prediction interval.
predIntNparSimultaneousConfLevel	Compute the confidence level associated with
	a simultaneous nonparametric prediction
	interval.
predIntNparSimultaneousN	Compute the required sample size for a
	simultaneous nonparametric prediction
	interval.
plotPredIntNparSimultaneousDesign	Create plots for a sampling design based on
	a simultaneous nonparametric prediction
	interval.
predIntNparSimultaneousTestPower	Compute the probability that at least one
	set of future observations violates the
	given rule based on a nonparametric
	simultaneous prediction interval.
plotPredIntNparSimultaneousTestPowerCurve	Create plots for a sampling design based on
	a simultaneous nonparametric prediction
	interval.

Tolerance Intervals

Function Name	Description
tolIntNormHalfWidth	Compute the half-width of a tolerance
	interval for a normal distribution.
tolIntNormK	Compute the required value of $K$ for a
	tolerance interval for a Normal distribution.
tolIntNormN	Compute the sample size necessary to achieve a
	specified half-width for a tolerance interval
	for a Normal distribution.
plotTolIntNormDesign	Create plots for a sampling design based on a
	tolerance interval for a Normal distribution.
tolIntNparConfLevel	Compute the confidence level associated with a
	nonparametric tolerance interval for a specified
	sample size and coverage.
tolIntNparCoverage	Compute the coverage associated with a
	nonparametric tolerance interval for a specified
	sample size and confidence level.
tolIntNparN	Compute the sample size required for a nonparametric
	tolerance interval with a specified coverage and
	confidence level.
plotTolIntNparDesign	Create plots for a sampling design based on a
	nonparametric tolerance interval.

tTestPower

Help topic: tTestPower.

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 $\underline{x} = x_1, x_2, \ldots, x_n$ denote a vector of $n$ observations from a normal distribution with mean $\mu$ and standard deviation $\sigma$, and consider the null hypothesis:

$$H_0: \mu = \mu_0 \;\;\;\;\;\; (1)$$

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

$$H_a: \mu > \mu_0 \;\;\;\;\;\; (2)$$

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

$$H_a: \mu < \mu_0 \;\;\;\;\;\; (3)$$

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

$$H_a: \mu \ne \mu_0 \;\;\;\;\;\; (4)$$

The test of the null hypothesis (1) versus any of the three alternatives (2)-(4) is based on the Student t-statistic:

$$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \;\;\;\;\;\; (5)$$

where

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \;\;\;\;\;\; (6)$$

$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (7)$$

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 $\delta$ (Equation (12) below) or $\sigma$ (the population standard deviation), only the ratio $\delta/\sigma$. 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-$\alpha$ if

$$t \ge t_{\nu}(1 - \alpha) \;\;\;\;\;\; (8)$$

where

$$\nu = n - 1 \;\;\;\;\;\; (9)$$

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

$$1 - \beta = Pr[t_{\nu, \Delta} \ge t_{\nu}(1 - \alpha)] = 1 - G[t_{\nu}(1 - \alpha), \nu, \Delta] \;\;\;\;\;\; (10)$$

where

$$\Delta = \sqrt{n} (\frac{\delta}{\sigma}) \;\;\;\;\;\; (11)$$

$$\delta = \mu - \mu_0 \;\;\;\;\;\ (12)$$

and $t_{\nu, \Delta}$ denotes a non-central Student’s t-random variable with $\nu$ degrees of freedom and non-centrality parameter $\Delta$, and $G(x, \nu, \Delta)$ 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-$\alpha$ if

$$t \le t_{\nu}(\alpha) \;\;\;\;\;\; (13)$$

and the power of this test is given by:

$$1 - \beta = Pr[t_{\nu, \Delta} \le t_{\nu}(\alpha)] = G[t_{\nu}(\alpha), \nu, \Delta] \;\;\;\;\;\; (14)$$

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-$\alpha$ if

$$|t| \ge t_{\nu}(1 - \alpha/2) \;\;\;\;\;\; (15)$$

and the power of this test is given by:

$$1 - \beta = Pr[t_{\nu, \Delta} \le t_{\nu}(\alpha/2)] + Pr[t_{\nu, \Delta} \ge t_{\nu}(1 - \alpha/2)]$$

$$= G[t_{\nu}(\alpha/2), \nu, \Delta] + 1 - G[t_{\nu}(1 - \alpha/2), \nu, \Delta] \;\;\;\;\;\; (16)$$

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 $F_{\nu_1, \nu_2, \Delta}$ denote a non-central F random variable with $\nu_1$ and $\nu_2$ degrees of freedom and non-centrality parameter $\Delta$, and let $H(x, \nu_1, \nu_2, \Delta)$ denote the cumulative distribution function of this random variable evaluated at $x$. Also, let $F_{\nu_1, \nu_2}(p)$ denote the $p$’th quantile of the central F-distribution with $\nu_1$ and $\nu_2$ degrees of freedom. It can be shown that

$$(t_{\nu, \Delta})^2 \cong F_{1, \nu, \Delta^2} \;\;\;\;\;\; (17)$$

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

$$[t_{\nu}(1 - \alpha/2)]^2 = F_{1, \nu}(1 - \alpha) \;\;\;\;\;\; (18)$$

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

$$1 - \beta = Pr\{(t_{\nu, \Delta})^2 \ge [t_{\nu}(1 - \alpha/2)]^2\}$$

$$= Pr[F_{1, \nu, \Delta^2} \ge F_{1, \nu}(1 - \alpha)] = 1 - H[F_{1, \nu}(1-\alpha), 1, \nu, \Delta^2] \;\;\;\;\;\; (19)$$

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

$$\sqrt{n} (\frac{\delta}{s}) \approx \sqrt{n} (\frac{\delta}{\sigma}) = \Delta \;\;\;\;\;\; (20)$$

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:

$$G(x, \nu, \Delta) \approx G(x - \Delta, \nu, 0) = G(x - \Delta, \nu) \;\;\;\;\;\; (21)$$

where $G(x, \nu)$ denotes the cumulative distribution function of the central Student’s t-distribution with $\nu$ 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:

$$1 - \beta = Pr[t \ge t_{\nu}(1 - \alpha)]$$

$$= Pr[\frac{\bar{x} - \mu}{s/\sqrt{n}} \ge t_{\nu}(1 - \alpha) - \sqrt{n}\frac{\delta}{s}]$$

$$\approx Pr[t_{\nu} \ge t_{\nu}(1 - \alpha) - \Delta]$$

$$= 1 - Pr[t_{\nu} \le t_{\nu}(1 - \alpha) - \Delta]$$

$$= 1 - G[t_{\nu}(1-\alpha) - \Delta, \nu] \;\;\;\;\;\; (22)$$

where $t_{\nu}$ denotes a central Student’s t-random variable with $\nu$ 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:

$$1 - \beta = Pr[t \le t_{\nu}(\alpha)]$$

$$= Pr[\frac{\bar{x} - \mu}{s/\sqrt{n}} \le t_{\nu}(\alpha) - \sqrt{n}\frac{\delta}{s}]$$

$$\approx Pr[t_{\nu} \le t_{\nu}(\alpha) - \Delta]$$

$$= G[t_{\nu}(\alpha) - \Delta, \nu] \;\;\;\;\;\; (23)$$

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

$$1 - \beta = Pr[t \le t_{\nu}(\alpha/2)] + Pr[t \ge t_{\nu}(1 - \alpha/2)]$$

$$= Pr[\frac{\bar{x} - \mu}{s/\sqrt{n}} \le t_{\nu}(\alpha/2) - \sqrt{n}\frac{\delta}{s}] + Pr[\frac{\bar{x} - \mu}{s/\sqrt{n}} \ge t_{\nu}(1 - \alpha) - \sqrt{n}\frac{\delta}{s}]$$

$$\approx Pr[t_{\nu} \le t_{\nu}(\alpha/2) - \Delta] + Pr[t_{\nu} \ge t_{\nu}(1 - \alpha/2) - \Delta]$$

$$= G[t_{\nu}(\alpha/2) - \Delta, \nu] + 1 - G[t_{\nu}(1-\alpha/2) - \Delta, \nu] \;\;\;\;\;\; (24)$$

Two-Sample Case (sample.type="two.sample")
Let $\underline{x}_1 = x_{11}, x_{12}, \ldots, x_{1n_1}$ denote a vector of $n_1$ observations from a normal distribution with mean $\mu_1$ and standard deviation $\sigma$, and let $\underline{x}_2 = x_{21}, x_{22}, \ldots, x_{2n_2}$ denote a vector of $n_2$ observations from a normal distribution with mean $\mu_2$ and standard deviation $\sigma$, and consider the null hypothesis:

$$H_0: \mu_1 = \mu_2 \;\;\;\;\;\; (25)$$

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

$$H_a: \mu_1 > \mu_2 \;\;\;\;\;\; (26)$$

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

$$H_a: \mu_1 < \mu_2 \;\;\;\;\;\; (27)$$

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

$$H_a: \mu_1 \ne \mu_2 \;\;\;\;\;\; (28)$$

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

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

where

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

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

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

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

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

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:

$$\nu = n_1 + n_2 - 2 \;\;\;\;\;\; (35)$$

$$\Delta = \sqrt{\frac{n_1 n_2}{n_1 + n_2}} (\frac{\delta}{\sigma}) \;\;\;\;\;\; (36)$$

$$\delta = \mu_1 - \mu_2 \;\;\;\;\;\; (37)$$

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

linearTrendTestPower

Help topic: linearTrendTestPower.

If the argument x is a vector, it is converted into a list with one component. If the arguments n, x, slope.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.

Basic Model
Consider the simple linear regression model

$$Y = \beta_0 + \beta_1 X + \epsilon \;\;\;\;\;\; (1)$$

where $X$ denotes the predictor variable (observed without error), $\beta_0$ denotes the intercept, $\beta_1$ denotes the slope, and the error term $\epsilon$ is assumed to be a random variable from a normal distribution with mean 0 and standard deviation $\sigma$. Let

$$(\underline{x}, \underline{y}) = (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) \;\;\;\;\;\; (2)$$

denote $n$ independent observed $(X,Y)$ pairs from the model (1).

Often in environmental data analysis, we are interested in determining whether there is a trend in some indicator variable over time. In this case, the predictor variable $X$ is time (e.g., day, month, quarter, year, etc.), and the $n$ values of the response variable $Y$ represent measurements taken over time. The slope then represents the change in the average of the response variable per one unit of time.

When the argument x is a numeric vector, it represents the $n$ values of the predictor variable. When the argument x is a list, each component of x is a numeric vector that represents a set values of the predictor variable (and the number of elements may vary by component). By default, the argument x is a list for which the i’th component is simply the integers from 1 to the value of the i’th element of the argument n, representing, for example, Day 1, Day2, …, Day n[i].

In the discussion that follows, be sure not to confuse the intercept and slope coefficients $\beta_0$ and $\beta_1$ with the Type II error of the hypothesis test, which is denoted by $\beta$.

Estimation of Coefficients and Confidence Interval for Slope
The standard least-squares estimators of the slope and intercept are given by:

$$\hat{\beta}_1 = \frac{S_{xy}}{S_{xx}} \;\;\;\;\;\; (3)$$

$$\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \;\;\;\;\;\; (4)$$

where

$$S_{xy} = \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) \;\;\;\;\;\; (5)$$

$$S_{xx} = \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (6)$$

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \;\;\;\;\;\; (7)$$

$$\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i \;\;\;\;\;\; (8)$$

(Draper and Smith, 1998, p.25; Zar, 2010, p.332-334; Berthoux and Brown, 2002, p.297; Helsel and Hirsch, p.226). The estimator of slope in Equation (3) has a normal distribution with mean equal to the true slope, and variance given by:

$$Var(\hat{\beta}_1) = \sigma_{\hat{\beta}_1}^2 = \frac{\sigma^2}{S_{xx}} \;\;\;\;\;\; (9)$$

(Draper and Smith, 1998, p.35; Zar, 2010, p.341; Berthoux and Brown, 2002, p.299; Helsel and Hirsch, 1992, p.227). Thus, a $(1-\alpha)100\%$ two-sided confidence interval for the slope is given by:

$$[ \hat{\beta}_1 - t_{n-2}(1-\alpha/2) \hat{\sigma}_{\hat{\beta}_1}, \;\; \hat{\beta}_1 + t_{n-2}(1-\alpha/2) \hat{\sigma}_{\hat{\beta}_1} ] \;\;\;\;\;\; (10)$$

where

$$\hat{\sigma}_{\hat{\beta}_1} = \frac{\hat{\sigma}}{\sqrt{S_{xx}}} \;\;\;\;\;\; (11)$$

$$\hat{\sigma}^2 = s^2 = \frac{1}{n-2} \sum_{i=1}^n (y_i - \hat{y}_i)^2 \;\;\;\;\;\; (12)$$

$$\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i \;\;\;\;\;\; (13)$$

and $t_{\nu}(p)$ denotes the $p$’th quantile of Student’s t-distribution with $\nu$ degrees of freedom (Draper and Smith, 1998, p.36; Zar, 2010, p.343; Berthoux and Brown, 2002, p.300; Helsel and Hirsch, 1992, p.240).

Testing for a Non-Zero Slope
Consider the null hypothesis of a zero slope coefficient:

$$H_0: \beta_1 = 0 \;\;\;\;\;\; (14)$$

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

$$H_a: \beta_1 > 0 \;\;\;\;\;\; (15)$$

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

$$H_a: \beta_1 < 0 \;\;\;\;\;\; (16)$$

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

$$H_a: \beta_1 \ne 0 \;\;\;\;\;\; (17)$$

The test of the null hypothesis (14) versus any of the three alternatives (15)-(17) is based on the Student t-statistic:

$$t = \frac{\hat{\beta}_1}{\hat{\sigma}_{\hat{\beta}_1}} = \frac{\hat{\beta}_1}{s/\sqrt{S_{xx}}} \;\;\;\;\;\; (18)$$

Under the null hypothesis (14), the t-statistic in (18) follows a Student’s t-distribution with $n-2$ degrees of freedom (Draper and Smith, 1998, p.36; Zar, 2010, p.341; Helsel and Hirsch, 1992, pp.238-239).

The formula for the power of the test of a zero slope depends on which alternative is being tested. The two subsections below describe exact and approximate formulas for the power of the test. Note that none of the equations for the power of the t-test requires knowledge of the values $\beta_1$ or $\sigma$ (the population standard deviation of the error terms), only the ratio $\beta_1/\sigma$. The argument slope.over.sigma is this ratio, and it is referred to as the “scaled slope”.

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

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-$\alpha$ if

$$t \ge t_{\nu}(1 - \alpha) \;\;\;\;\;\; (19)$$

where

$$\nu = n - 2 \;\;\;\;\;\; (20)$$

and, as noted previously, $t_{\nu}(p)$ denotes the $p$’th quantile of Student’s t-distribution with $\nu$ degrees of freedom. The power of this test, denoted by $1-\beta$, where $\beta$ denotes the probability of a Type II error, is given by:

$$1 - \beta = Pr[t_{\nu, \Delta} \ge t_{\nu}(1 - \alpha)] = 1 - G[t_{\nu}(1 - \alpha), \nu, \Delta] \;\;\;\;\;\; (21)$$

where

$$\Delta = \sqrt{S_{xx}} \frac{\beta_1}{\sigma} \;\;\;\;\;\; (22)$$

and $t_{\nu, \Delta}$ denotes a non-central Student’s t-random variable with $\nu$ degrees of freedom and non-centrality parameter $\Delta$, and $G(x, \nu, \Delta)$ denotes the cumulative distribution function of this random variable evaluated at $x$ (Johnson et al., 1995, pp.508-510). Note that when the predictor variable $X$ represents equally-spaced measures of time (e.g., days, months, quarters, etc.) and

$$x_i = i, \;\; i = 1, 2, \ldots, n \;\;\;\;\;\; (23)$$

then the non-centrality parameter in Equation (22) becomes:

$$\Delta = \sqrt{\frac{(n-1)n(n+1)}{12}} \frac{\beta_1}{\sigma} \;\;\;\;\;\; (24)$$

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-$\alpha$ if

$$t \le t_{\nu}(\alpha) \;\;\;\;\;\; (25)$$

and the power of this test is given by:

$$1 - \beta = Pr[t_{\nu, \Delta} \le t_{\nu}(\alpha)] = G[t_{\nu}(\alpha), \nu, \Delta] \;\;\;\;\;\; (26)$$

Two-sided alternative (alternative="two.sided")
The standard Student’s t-test rejects the null hypothesis (14) in favor of the two-sided alternative hypothesis (17) at level-$\alpha$ if

$$|t| \ge t_{\nu}(1 - \alpha/2) \;\;\;\;\;\; (27)$$

and the power of this test is given by:

$$1 - \beta = Pr[t_{\nu, \Delta} \le t_{\nu}(\alpha/2)] + Pr[t_{\nu, \Delta} \ge t_{\nu}(1 - \alpha/2)]$$

$$= G[t_{\nu}(\alpha/2), \nu, \Delta] + 1 - G[t_{\nu}(1 - \alpha/2), \nu, \Delta] \;\;\;\;\;\; (28)$$

The power of the t-test given in Equation (28) can also be expressed in terms of the cumulative distribution function of the non-central F-distribution as follows. Let $F_{\nu_1, \nu_2, \Delta}$ denote a non-central F random variable with $\nu_1$ and $\nu_2$ degrees of freedom and non-centrality parameter $\Delta$, and let $H(x, \nu_1, \nu_2, \Delta)$ denote the cumulative distribution function of this random variable evaluated at $x$. Also, let $F_{\nu_1, \nu_2}(p)$ denote the $p$’th quantile of the central F-distribution with $\nu_1$ and $\nu_2$ degrees of freedom. It can be shown that

$$(t_{\nu, \Delta})^2 \cong F_{1, \nu, \Delta^2} \;\;\;\;\;\; (29)$$

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

$$[t_{\nu}(1 - \alpha/2)]^2 = F_{1, \nu}(1 - \alpha) \;\;\;\;\;\; (30)$$

so the formula for the power of the t-test given in Equation (28) can also be written as:

$$1 - \beta = Pr\{(t_{\nu, \Delta})^2 \ge [t_{\nu}(1 - \alpha/2)]^2\}$$

$$= Pr[F_{1, \nu, \Delta^2} \ge F_{1, \nu}(1 - \alpha)] = 1 - H[F_{1, \nu}(1-\alpha), 1, \nu, \Delta^2] \;\;\;\;\;\; (31)$$

Approximate Power Calculations (approx=TRUE)
Zar (2010, pp.115–118) presents an approximation to the power for the t-test given in Equations (21), (26), and (28) above. His approximation to the power can be derived by using the approximation

$$\sqrt{S_{xx}} \frac{\beta_1}{s} \approx \sqrt{SS_{xx}} \frac{\beta_1}{\sigma} = \Delta \;\;\;\;\;\; (32)$$

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:

$$G(x, \nu, \Delta) \approx G(x - \Delta, \nu, 0) = G(x - \Delta, \nu) \;\;\;\;\;\; (33)$$

where $G(x, \nu)$ denotes the cumulative distribution function of the central Student’s t-distribution with $\nu$ 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 (15) given in Equation (21) can be approximated as:

$$1 - \beta = Pr[t \ge t_{\nu}(1 - \alpha)]$$

$$= Pr[\frac{\hat{\beta}_1}{s/\sqrt{S_{xx}}} \ge t_{\nu}(1 - \alpha) - \sqrt{S_{xx}}\frac{\beta_1}{s}]$$

$$\approx Pr[t_{\nu} \ge t_{\nu}(1 - \alpha) - \Delta]$$

$$= 1 - Pr[t_{\nu} \le t_{\nu}(1 - \alpha) - \Delta]$$

$$= 1 - G[t_{\nu}(1-\alpha) - \Delta, \nu] \;\;\;\;\;\; (34)$$

where $t_{\nu}$ denotes a central Student’s t-random variable with $\nu$ degrees of freedom.

Lower one-sided alternative (alternative="less")
The power for the lower one-sided alternative (16) given in Equation (26) can be approximated as:

$$1 - \beta = Pr[t \le t_{\nu}(\alpha)]$$

$$= Pr[\frac{\hat{\beta}_1}{s/\sqrt{S_{xx}}} \le t_{\nu}(\alpha) - \sqrt{S_{xx}}\frac{\beta_1}{s}]$$

$$\approx Pr[t_{\nu} \le t_{\nu}(\alpha) - \Delta]$$

$$= G[t_{\nu}(\alpha) - \Delta, \nu] \;\;\;\;\;\; (35)$$

Two-sided alternative (alternative="two.sided")
The power for the two-sided alternative (17) given in Equation (28) can be approximated as:

$$1 - \beta = Pr[t \le t_{\nu}(\alpha/2)] + Pr[t \ge t_{\nu}(1 - \alpha/2)]$$

$$= Pr[\frac{\hat{\beta}_1}{s/\sqrt{S_{xx}}} \le t_{\nu}(\alpha/2) - \sqrt{SS_{xx}}\frac{\beta_1}{s}] + Pr[\frac{\hat{\beta}_1}{s/\sqrt{S_{xx}}} \ge t_{\nu}(1 - \alpha) - \sqrt{SS_{xx}}\frac{\beta_1}{s}]$$

$$\approx Pr[t_{\nu} \le t_{\nu}(\alpha/2) - \Delta] + Pr[t_{\nu} \ge t_{\nu}(1 - \alpha/2) - \Delta]$$

$$= G[t_{\nu}(\alpha/2) - \Delta, \nu] + 1 - G[t_{\nu}(1-\alpha/2) - \Delta, \nu] \;\;\;\;\;\; (36)$$