Plots for a Sampling Design Based on a One- or Two-Sample t-Test

Create plots involving sample size, power, scaled difference, and significance level for a one- or two-sample t-test.

plotTTestDesign(x.var = "n", y.var = "power", range.x.var = NULL, 
    n.or.n1 = 25, n2 = n.or.n1, 
    delta.over.sigma = switch(alternative, greater = 0.5, less = -0.5, 
      two.sided = ifelse(two.sided.direction == "greater", 0.5, -0.5)), 
    alpha = 0.05, power = 0.95, 
    sample.type = ifelse(!missing(n2), "two.sample", "one.sample"), 
    alternative = "two.sided", two.sided.direction = "greater", approx = FALSE, 
    round.up = FALSE, n.max = 5000, tol = 1e-07, maxiter = 1000, plot.it = TRUE, 
    add = FALSE, n.points = 50, plot.col = "black", plot.lwd = 3 * par("cex"), 
    plot.lty = 1, digits = .Options$digits, ..., main = NULL, xlab = NULL, 
    ylab = NULL, type = "l")

Arguments

x.var

character string indicating what variable to use for the x-axis. Possible values are "n" (sample size; the default), "delta.over.sigma" (scaled minimal detectable difference), "power" (power of the test), and "alpha" (significance level of the test).

y.var

character string indicating what variable to use for the y-axis. Possible values are "power" (power of the test; the default), "delta.over.sigma" (scaled minimal detectable difference), and "n" (sample size).

range.x.var

numeric vector of length 2 indicating the range of the x-variable to use for the plot. The default value depends on the value of x.var. When x.var="n" the default value is c(2,50). When x.var="delta.over.sigma" and
alternative="greater" or alternative="two.sided" and
two.sided.direction="greater", the default value is c(0.5, 2). When x.var="delta.over.sigma" and alternative="less" or
alternative="two.sided" and two.sided.direction="less", the default value is -c(2, 0.5). When x.var="power" the default value is
c(alpha + .Machine$double.eps, 0.95). When x.var="alpha", the default value is c(0.01, 0.2).

n.or.n1

numeric scalar indicating the sample size. The default value is n.or.n1=25. When sample.type="one.sample", n.or.n1 denotes the number of observations in the single sample. When sample.type="two.sample", n.or.n1 denotes the number of observations from group 1. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed. This argument is ignored if either x.var="n" or y.var="n".

n2

numeric scalar indicating the sample size for group 2. The default value is the value of n.or.n1. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed. This argument is ignored when sample.type="one.sample".

delta.over.sigma

numeric scalar specifying the ratio of the true difference (\(\delta\)) to the population standard deviation (\(\sigma\)). This is also called the "scaled difference". When alternative="greater" or alternative="two.sided" and
two.sided.direction="greater", the default value is delta.over.sigma=0.5. When alternative="less" or alternative="two.sided" and
two.sided.direction="less", the default value is delta.over.sigma=-0.5. This argument is ignored when x.var="delta.over.sigma" or
y.var="delta.over.sigma".

alpha

numeric scalar between 0 and 1 indicating the Type I error level associated with the hypothesis test. The default value is alpha=0.05. This argument is ignored when x.var="alpha".

power

numeric scalar between 0 and 1 indicating the power associated with the hypothesis test. The default value is power=0.95. This argument is ignored when x.var="power" or y.var="power".

sample.type

character string indicating whether the design is based on a one-sample or two-sample t-test. When sample.type="one.sample", the computations for the plot are based on a one-sample t-test. When sample.type="two.sample", the computations for the plot are based on a two-sample t-test. The default value is sample.type="one.sample".

alternative

character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "less", and "greater".

two.sided.direction

character string indicating the direction (positive or negative) for the scaled minimal detectable difference when alternative="two.sided". When
two.sided.direction="greater" (the default), the scaled minimal detectable difference is positive. When two.sided.direction="less", the scaled minimal detectable difference is negative. This argument is ignored unless
alternative="two.sided" and either x.var="delta" or y.var="delta".

approx

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

round.up

logical scalar indicating whether to round up the values of the computed sample size(s) to the next smallest integer. The default value is round.up=FALSE. This argument is ignored unless y.var="n".

n.max

for the case when y.var="n", a positive integer greater than 1 indicating the maximum sample size when sample.type="one.sample" or the maximum sample size for group 1 when sample.type="two.sample". The default value is n.max=5000.

tol

numeric scalar relevant to the case when y.var="n" or y.var="delta.over.sigma". This argument is passed to the uniroot function and indicates the tolerance to use in the search algorithm. The default value is tol=1e-7.

maxiter

numeric scalar relevant to the case when y.var="n" and approx=FALSE (i.e., when the power is based on the exact test), or when y.var="delta.over.sigma". This argument is passed to the uniroot function and is a positive integer indicating the maximum number of iterations. The default value is maxiter=1000.

plot.it

a logical scalar indicating whether to create a new plot or add to the existing plot (see add) on the current graphics device. If plot.it=FALSE, no plot is produced, but a list of (x,y) values is returned (see VALUE). The default value is plot.it=TRUE.

add

a logical scalar indicating whether to add the design plot to the existing plot (add=TRUE), or to create a plot from scratch (add=FALSE). The default value is add=FALSE. This argument is ignored if plot.it=FALSE.

n.points

a numeric scalar specifying how many (x,y) pairs to use to produce the plot. There are n.points x-values evenly spaced between range.x.var[1] and
range.x.var[2]. The default value is n.points=50.

plot.col

a numeric scalar or character string determining the color of the plotted line or points. The default value is plot.col="black". See the entry for col in the help file for par for more information.

plot.lwd

a numeric scalar determining the width of the plotted line. The default value is 3*par("cex"). See the entry for lwd in the help file for par for more information.

plot.lty

a numeric scalar determining the line type of the plotted line. The default value is plot.lty=1. See the entry for lty in the help file for par for more information.

digits

a scalar indicating how many significant digits to print out on the plot. The default value is the current setting of options("digits").

main, xlab, ylab, type, ...

additional graphical parameters (see par).

Details

See the help files for tTestPower, tTestN, and tTestScaledMdd for information on how to compute the power, sample size, or scaled minimal detectable difference for a one- or two-sample t-test.

Value

plotTTestDesign invisibly returns a list with components x.var and y.var, giving coordinates of the points that have been or would have been plotted.

References

See the help files for tTestPower, tTestN, and tTestScaledMdd.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

See the help files for tTestPower, tTestN, and tTestScaledMdd.

See also

tTestPower, tTestN, tTestScaledMdd, t.test.

Examples

  # Look at the relationship between power and sample size for a two-sample t-test, 
  # assuming a scaled difference of 0.5 and a 5% significance level:

  dev.new()
  plotTTestDesign(sample.type = "two")

  #----------

  # For a two-sample t-test, plot sample size vs. the scaled minimal detectable 
  # difference for various levels of power, using a 5% significance level:

  dev.new()
  plotTTestDesign(x.var = "delta.over.sigma", y.var = "n", sample.type = "two", 
    ylim = c(0, 110), main="") 

  plotTTestDesign(x.var = "delta.over.sigma", y.var = "n", sample.type = "two", 
    power = 0.9, add = TRUE, plot.col = "red") 

  plotTTestDesign(x.var = "delta.over.sigma", y.var = "n", sample.type = "two", 
    power = 0.8, add = TRUE, plot.col = "blue") 

  legend("topright", c("95%", "90%", "80%"), lty = 1, 
    lwd = 3 * par("cex"), col = c("black", "red", "blue"), bty = "n")

  title(main = paste("Sample Size vs. Scaled Difference for", 
    "Two-Sample t-Test, with Alpha=0.05 and Various Powers", 
    sep="\n"))

  #==========

  # Modifying the example on pages 21-4 to 21-5 of USEPA (2009), look at 
  # power versus scaled minimal detectable difference for various sample 
  # sizes in the context of the problem of using a one-sample t-test to 
  # compare the mean for the well with the MCL of 7 ppb.  Use alpha = 0.01, 
  # assume an upper one-sided alternative (i.e., compliance well mean larger 
  # than 7 ppb).

  dev.new()
  plotTTestDesign(x.var = "delta.over.sigma", y.var = "power", 
    range.x.var = c(0.5, 2), n.or.n1 = 8, alpha = 0.01, 
    alternative = "greater", ylim = c(0, 1), main = "")

  plotTTestDesign(x.var = "delta.over.sigma", y.var = "power", 
    range.x.var = c(0.5, 2), n.or.n1 = 6, alpha = 0.01, 
    alternative = "greater", add = TRUE, plot.col = "red")

  plotTTestDesign(x.var = "delta.over.sigma", y.var = "power", 
    range.x.var = c(0.5, 2), n.or.n1 = 4, alpha = 0.01, 
    alternative = "greater", add = TRUE, plot.col = "blue")

  legend("topleft", paste("N =", c(8, 6, 4)), lty = 1, lwd = 3 * par("cex"), 
    col = c("black", "red", "blue"), bty = "n")

  title(main = paste("Power vs. Scaled Difference for One-Sample t-Test", 
    "with Alpha=0.01 and Various Sample Sizes", sep="\n"))

  #==========

  # Clean up
  #---------
  graphics.off()