Plots for a Sampling Design Based on a One- or Two-Sample t-Test, Assuming Lognormal Data

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Description

Create plots involving sample size, power, ratio of means, coefficient of variation, and significance level for a one- or two-sample t-test, assuming lognormal data.

Usage

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  plotTTestLnormAltDesign(x.var = "n", y.var = "power", range.x.var = NULL, 
    n.or.n1 = 25, n2 = n.or.n1, 
    ratio.of.means = switch(alternative, greater = 2, less = 0.5, 
      two.sided = ifelse(two.sided.direction == "greater", 2, 0.5)), 
    cv = 1, 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, cex.main = par("cex"), ..., 
    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), "ratio.of.means" (minimal or maximal detectable ratio of means), "cv" (coefficient of variaiton), "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), "ratio.of.means" (minimal or maximal detectable ratio of means), 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="ratio.of.means" and alternative="greater" or
    alternative="two.sided" and two.sided.direction="greater", the default value is c(1, 2).

  • When x.var="delta" and alternative="less" or
    alternative="two.sided" and two.sided.direction="less", the default value is c(0.5, 1).

  • When x.var="cv" the default value is c(0.5, 2).

  • 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".

ratio.of.means

numeric scalar 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.

When alternative="greater" or alternative="two.sided" and
two.sided.direction="greater", the default value is ratio.of.means=2. When alternative="less" or
alternative="two.sided" and two.sided.direction="less", the default value is ratio.of.means=0.5. This argument is ignored when
x.var="ratio.of.means" or y.var="ratio.of.means".

cv

numeric scalar: a positive value 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 scalar between 0 and 1 indicating the Type I error level associated with the hypothesis test. The default value is alpha=0.05.

power

numeric scalar between 0 and 1 indicating the power associated with the hypothesis test. The default value is power=0.95.

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".

two.sided.direction

character string indicating the direction (greater than 1 or less than 1) for the detectable ratio of means when alternative="two.sided". When
two.sided.direction="greater" (the default), the detectable ratio of means is greater than 1. When
two.sided.direction="less", the detectable ratio of means is less than 1 (but greater than 0). This argument is ignored if alternative="less" or
alternative="greater".

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 TRUE.

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 indicating the toloerance to use in the uniroot search algorithm. The default value is tol=1e-7.

maxiter

positive integer indicating the maximum number of iterations argument to pass to the uniroot function. 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=100.

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").

cex.main, main, xlab, ylab, type, ...

additional graphical parameters (see par).

Details

See the help files for tTestLnormAltPower, tTestLnormAltN, and tTestLnormAltRatioOfMeans for information on how to compute the power, sample size, or ratio of means for a one- or two-sample t-test assuming lognormal data.

Value

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

Note

See the help files for tTestLnormAltPower, tTestLnormAltN, and tTestLnormAltRatioOfMeans.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

See the help files for tTestLnormAltPower, tTestLnormAltN, and tTestLnormAltRatioOfMeans.

See Also

tTestLnormAltPower, tTestLnormAltN, tTestLnormAltRatioOfMeans, t.test.

Examples

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  # Look at the relationship between power and sample size for a two-sample t-test, 
  # assuming lognormal data, a ratio of means of 2, a coefficient of variation 
  # of 1, and a 5% significance level:

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

  #----------

  # For a two-sample t-test based on lognormal data, plot sample size vs. the 
  # minimal detectable ratio for various levels of power, assuming a coefficient 
  # of variation of 1 and using a 5% significance level:

  dev.new()
  plotTTestLnormAltDesign(x.var = "ratio.of.means", y.var = "n", 
    range.x.var = c(1.5, 2), sample.type = "two", ylim = c(20, 120), main="") 

  plotTTestLnormAltDesign(x.var = "ratio.of.means", y.var = "n", 
    range.x.var = c(1.5, 2), sample.type="two", power = 0.9, 
    add = TRUE, plot.col = "red") 

  plotTTestLnormAltDesign(x.var = "ratio.of.means", y.var = "n", 
    range.x.var = c(1.5, 2), 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. Ratio of Lognormal Means for", 
    "Two-Sample t-Test, with CV=1, Alpha=0.05 and Various Powers", 
    sep="\n")) 

  #==========

  # 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, a coefficient of variation of 2, and the above 
  # values for Type I error and power, create a performance goal diagram 
  # (USEPA, 1996c, p.89) showing the power of a one-sample test versus the minimal 
  # detectable ratio of theta/(SSL/2) when the sample size is 6 and the exact power 
  # calculations are used.

  dev.new()
  plotTTestLnormAltDesign(x.var = "ratio.of.means", y.var = "power", 
    range.x.var = c(1, 5), n.or.n1 = 6, cv = 2, alpha = 0.2, 
    alternative = "greater", approx = FALSE, ylim = c(0.2, 1), 
    xlab = "theta / (SSL/2)") 

  #==========

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

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