Power of a t-Test for Linear Trend
Compute the power of a parametric test for linear trend, given the sample size or predictor variable values, scaled slope, and significance level.
linearTrendTestPower(n, x = lapply(n, seq), slope.over.sigma = 0, alpha = 0.05,
alternative = "two.sided", approx = FALSE)n |
numeric vector of sample sizes. All values of |
x |
numeric vector of predictor variable values, or a list in which each component is
a numeric vector of predictor variable values. Usually, the predictor variable is
time (e.g., days, months, quarters, etc.). The default value is
|
slope.over.sigma |
numeric vector specifying the ratio of the true slope to the standard deviation of
the error terms (σ). This is also called the "scaled slope". The
default value is |
alpha |
numeric vector of numbers between 0 and 1 indicating the Type I error level
associated with the hypothesis test. The default value is |
alternative |
character string indicating the kind of alternative hypothesis. The possible values
are |
approx |
logical scalar indicating whether to compute the power based on an approximation to
the non-central t-distribution. The default value is |
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 = β_0 + β_1 X + ε \;\;\;\;\;\; (1)
where X denotes the predictor variable (observed without error), β_0 denotes the intercept, β_1 denotes the slope, and the error term ε is assumed to be a random variable from a normal distribution with mean 0 and standard deviation σ. Let
(\underline{x}, \underline{y}) = (x_1, y_1), (x_2, y_2), …, (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 β_0 and β_1 with the Type II error of the
hypothesis test, which is denoted by β.
Estimation of Coefficients and Confidence Interval for Slope
The standard least-squares estimators of the slope and intercept are given by:
\hat{β}_1 = \frac{S_{xy}}{S_{xx}} \;\;\;\;\;\; (3)
\hat{β}_0 = \bar{y} - \hat{β}_1 \bar{x} \;\;\;\;\;\; (4)
where
S_{xy} = ∑_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) \;\;\;\;\;\; (5)
S_{xx} = ∑_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (6)
\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\;\;\; (7)
\bar{y} = \frac{1}{n} ∑_{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{β}_1) = σ_{\hat{β}_1}^2 = \frac{σ^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-α)100\% two-sided confidence interval for the slope is given by:
[ \hat{β}_1 - t_{n-2}(1-α/2) \hat{σ}_{\hat{β}_1}, \;\; \hat{β}_1 + t_{n-2}(1-α/2) \hat{σ}_{\hat{β}_1} ] \;\;\;\;\;\; (10)
where
\hat{σ}_{\hat{β}_1} = \frac{\hat{σ}}{√{S_{xx}}} \;\;\;\;\;\; (11)
\hat{σ}^2 = s^2 = \frac{1}{n-2} ∑_{i=1}^n (y_i - \hat{y}_i)^2 \;\;\;\;\;\; (12)
\hat{y}_i = \hat{β}_0 + \hat{β}_1 x_i \;\;\;\;\;\; (13)
and t_{ν}(p) denotes the p'th quantile of
Student's t-distribution with ν 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: β_1 = 0 \;\;\;\;\;\; (14)
The three possible alternative hypotheses are the upper one-sided alternative
(alternative="greater"):
H_a: β_1 > 0 \;\;\;\;\;\; (15)
the lower one-sided alternative (alternative="less")
H_a: β_1 < 0 \;\;\;\;\;\; (16)
and the two-sided alternative (alternative="two.sided")
H_a: β_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{β}_1}{\hat{σ}_{\hat{β}_1}} = \frac{\hat{β}_1}{s/√{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 β_1 or σ
(the population standard deviation of the error terms), only the ratio
β_1/σ. 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-α if
t ≥ t_{ν}(1 - α) \;\;\;\;\;\; (19)
where
ν = n - 2 \;\;\;\;\;\; (20)
and, as noted previously, t_{ν}(p) denotes the p'th quantile of Student's t-distribution with ν degrees of freedom. 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] \;\;\;\;\;\; (21)
where
Δ = √{S_{xx}} \frac{β_1}{σ} \;\;\;\;\;\; (22)
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). 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, …, n \;\;\;\;\;\; (23)
then the non-centrality parameter in Equation (22) becomes:
Δ = √{\frac{(n-1)n(n+1)}{12}} \frac{β_1}{σ} \;\;\;\;\;\; (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-α if
t ≤ t_{ν}(α) \;\;\;\;\;\; (25)
and the power of this test is given by:
1 - β = Pr[t_{ν, Δ} ≤ t_{ν}(α)] = G[t_{ν}(α), ν, Δ] \;\;\;\;\;\; (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-α if
|t| ≥ t_{ν}(1 - α/2) \;\;\;\;\;\; (27)
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] \;\;\;\;\;\; (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_{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} \;\;\;\;\;\; (29)
where \cong denotes “equal in distribution”. Thus, it follows that
[t_{ν}(1 - α/2)]^2 = F_{1, ν}(1 - α) \;\;\;\;\;\; (30)
so the formula for the power of the t-test given in Equation (28) 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] \;\;\;\;\;\; (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
√{S_{xx}} \frac{β_1}{s} \approx √{SS_{xx}} \frac{β_1}{σ} = Δ \;\;\;\;\;\; (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, ν, Δ) \approx G(x - Δ, ν, 0) = G(x - Δ, ν) \;\;\;\;\;\; (33)
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 (15) given in Equation (21) can be
approximated as:
1 - β = Pr[t ≥ t_{ν}(1 - α)]
= Pr[\frac{\hat{β}_1}{s/√{S_{xx}}} ≥ t_{ν}(1 - α) - √{S_{xx}}\frac{β_1}{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] \;\;\;\;\;\; (34)
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 (16) given in Equation (26) can be
approximated as:
1 - β = Pr[t ≤ t_{ν}(α)]
= Pr[\frac{\hat{β}_1}{s/√{S_{xx}}} ≤ t_{ν}(α) - √{S_{xx}}\frac{β_1}{s}]
\approx Pr[t_{ν} ≤ t_{ν}(α) - Δ]
= G[t_{ν}(α) - Δ, ν] \;\;\;\;\;\; (35)
Two-sided alternative (alternative="two.sided")
The power for the two-sided alternative (17) given in Equation (28) can be
approximated as:
1 - β = Pr[t ≤ t_{ν}(α/2)] + Pr[t ≥ t_{ν}(1 - α/2)]
= Pr[\frac{\hat{β}_1}{s/√{S_{xx}}} ≤ t_{ν}(α/2) - √{SS_{xx}}\frac{β_1}{s}] + Pr[\frac{\hat{β}_1}{s/√{S_{xx}}} ≥ t_{ν}(1 - α) - √{SS_{xx}}\frac{β_1}{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] \;\;\;\;\;\; (36)
a numeric vector powers.
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 represent measurements taken over time. The slope then represents the change in the average of the response variable per one unit of time.
You can use the parametric model (1) to model your data, then use the R function
lm to fit the regression coefficients and the summary.lm
function to perform a test for the significance of the slope coefficient. The
function linearTrendTestPower computes the power of this t-test, given a
fixed value of the sample size, scaled slope, and significance level.
You can also use Kendall's nonparametric test for trend
if you don't want to assume the error terms are normally distributed. When the
errors are truly normally distributed, the asymptotic relative efficiency of
Kendall's test for trend versus the parametric t-test for a zero slope is 0.98,
and Kendall's test can be more powerful than the parametric t-test when the errors
are not normally distributed. Thus the function linearTrendTestPower can
also be used to estimate the power of Kendall's test for trend.
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 slope if one of the objectives of the sampling program is to determine
whether a trend is occurring. The functions linearTrendTestPower,
linearTrendTestN, linearTrendTestScaledMds, and plotLinearTrendTestDesign can be used to investigate these
relationships.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Second Edition. Lewis Publishers, Boca Raton, FL.
Draper, N., and H. Smith. (1998). Applied Regression Analysis. Third Edition. John Wiley and Sons, New York, Chapter 1.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, Chapter 9.
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.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.
# Look at how the power of the t-test for zero slope increases with increasing
# sample size:
seq(5, 30, by = 5)
#[1] 5 10 15 20 25 30
power <- linearTrendTestPower(n = seq(5, 30, by = 5), slope.over.sigma = 0.1)
round(power, 2)
#[1] 0.06 0.13 0.34 0.68 0.93 1.00
#----------
# Repeat the last example, but compute the approximate power instead of the
# exact:
power <- linearTrendTestPower(n = seq(5, 30, by = 5), slope.over.sigma = 0.1,
approx = TRUE)
round(power, 2)
#[1] 0.05 0.11 0.32 0.68 0.93 0.99
#----------
# Look at how the power of the t-test for zero slope increases with increasing
# scaled slope:
seq(0.05, 0.2, by = 0.05)
#[1] 0.05 0.10 0.15 0.20
power <- linearTrendTestPower(15, slope.over.sigma = seq(0.05, 0.2, by = 0.05))
round(power, 2)
#[1] 0.12 0.34 0.64 0.87
#----------
# Look at how the power of the t-test for zero slope increases with increasing
# values of Type I error:
power <- linearTrendTestPower(20, slope.over.sigma = 0.1,
alpha = c(0.001, 0.01, 0.05, 0.1))
round(power, 2)
#[1] 0.14 0.41 0.68 0.80
#----------
# Show that for a simple regression model, you get a greater power of detecting
# a non-zero slope if you take all the observations at two endpoints, rather than
# spreading the observations evenly between two endpoints.
# (Note: This design usually cannot work with environmental monitoring data taken
# over time since usually observations taken close together in time are not
# independent.)
linearTrendTestPower(x = 1:10, slope.over.sigma = 0.1)
#[1] 0.1265976
linearTrendTestPower(x = c(rep(1, 5), rep(10, 5)), slope.over.sigma = 0.1)
#[1] 0.2413823
#==========
# Clean up
#---------
rm(power)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.