Type I Error Level for a One- or Two-Sample t-Test

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Description

Compute the Type I Error level necessary to achieve a specified power for a one- or two-sample t-test, given the sample size(s) and scaled difference.

Usage

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  tTestAlpha(n.or.n1, n2 = n.or.n1, delta.over.sigma = 0, power = 0.95, 
    sample.type = ifelse(!missing(n2) && !is.null(n2), "two.sample", "one.sample"), 
    alternative = "two.sided", approx = FALSE, tol = 1e-07, maxiter = 1000)

Arguments

n.or.n1

numeric vector of sample sizes. When sample.type="one.sample", n.or.n1 denotes n, the number of observations in the single sample. When
sample.type="two.sample", n.or.n1 denotes n_1, the number of observations from group 1. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

n2

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

delta.over.sigma

numeric vector specifying the ratio of the true difference (δ) to the population standard deviation (σ). This is also called the “scaled difference”.

power

numeric vector of numbers 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".

approx

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

tol

numeric scalar indicating the tolerance argument to pass to the uniroot function. 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.

Details

Formulas for the power of the t-test for specified values of the sample size, scaled difference, and Type I error level are given in the help file for tTestPower. The function tTestAlpha uses the uniroot search algorithm to determine the required Type I error level for specified values of the sample size, power, and scaled difference.

Value

numeric vector of Type I error levels.

Note

See tTestPower.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

See tTestPower.

See Also

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

Examples

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  # Look at how the required Type I error level for the one-sample t-test 
  # decreases with increasing sample size.  Set the power to 80% and 
  # the scaled difference to 0.5.

  seq(5, 30, by = 5) 
  #[1] 5 10 15 20 25 30 

  alpha <- tTestAlpha(n.or.n1 = seq(5, 30, by = 5), 
    power = 0.8, delta.over.sigma = 0.5) 

  round(alpha, 2) 
  #[1] 0.65 0.45 0.29 0.18 0.11 0.07


  #----------


  # Repeat the last example, but use the approximation.
  # Note how the approximation underestimates the power 
  # for the smaller sample sizes.
  #----------------------------------------------------

  alpha <- tTestAlpha(n.or.n1 = seq(5, 30, by = 5), 
    power = 0.8, delta.over.sigma = 0.5, approx = TRUE) 

  round(alpha, 2)
  #[1] 0.63 0.46 0.30 0.18 0.11 0.07


  #----------


  # Look at how the required Type I error level for the two-sample 
  # t-test decreases with increasing scaled difference.  Use 
  # a power of 90% and a sample size of 10 in each group.

  seq(0.5, 2, by = 0.5) 
  #[1] 0.5 1.0 1.5 2.0 

  alpha <- tTestAlpha(10, sample.type = "two.sample", 
    power = 0.9, delta.over.sigma = seq(0.5, 2, by = 0.5)) 

  round(alpha, 2) 
  #[1] 0.82 0.35 0.06 0.01


  #----------


  # Look at how the required Type I error level for the two-sample 
  # t-test increases with increasing values of required power.  Use 
  # a sample size of 20 for each group and a scaled difference of 
  # 1.

  alpha <- tTestAlpha(20, sample.type = "two.sample", delta.over.sigma = 1, 
    power = c(0.8, 0.9, 0.95)) 

  round(alpha, 2)
  #[1] 0.03 0.07 0.14
 

  #----------


  # Clean up
  #---------
  rm(alpha)

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