Scaled Minimal Detectable Difference for One- or Two-Sample t-Test

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

tTestScaledMdd(n.or.n1, n2 = n.or.n1, alpha = 0.05, power = 0.95, 
    sample.type = ifelse(!missing(n2) && !is.null(n2), "two.sample", "one.sample"), 
    alternative = "two.sided", two.sided.direction = "greater", 
    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.

alpha

numeric vector of numbers between 0 and 1 indicating the Type I error level associated with the hypothesis test. The default value is alpha=0.05.

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). \(H_a: \mu \ne \mu_0\) for the one-sample case and \(H_a: \mu_1 \ne \mu_2\) for the two-sample case.

  • "greater". \(H_a: \mu > \mu_0\) for the one-sample case and \(H_a: \mu_1 > \mu_2\) for the two-sample case.

  • "less". \(H_a: \mu < \mu_0\) for the one-sample case and \(H_a: \mu_1 < \mu_2\) for the two-sample case.

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 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 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 tTestScaledMdd uses the uniroot search algorithm to determine the required scaled minimal detectable difference for specified values of the sample size, power, and Type I error level.

Value

numeric vector of scaled minimal detectable differences.

References

See tTestPower.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

See tTestPower.

Examples

  # Look at how the scaled minimal detectable difference for the 
  # one-sample t-test increases with increasing required power:

  seq(0.5, 0.9, by = 0.1) 
#> [1] 0.5 0.6 0.7 0.8 0.9
  #[1] 0.5 0.6 0.7 0.8 0.9 

  scaled.mdd <- tTestScaledMdd(n.or.n1 = 20, power = seq(0.5,0.9,by=0.1)) 

  round(scaled.mdd, 2) 
#> [1] 0.46 0.52 0.59 0.66 0.76
  #[1] 0.46 0.52 0.59 0.66 0.76

  #----------

  # Repeat the last example, but compute the scaled minimal detectable 
  # differences based on the approximation to the power instead of the 
  # exact formula:

  scaled.mdd <- tTestScaledMdd(n.or.n1 = 20, power = seq(0.5, 0.9, by = 0.1), 
    approx = TRUE) 

  round(scaled.mdd, 2) 
#> [1] 0.47 0.53 0.59 0.66 0.76
  #[1] 0.47 0.53 0.59 0.66 0.76

  #==========

  # Look at how the scaled minimal detectable difference for the two-sample 
  # t-test decreases with increasing sample size:

  seq(10,50,by=10) 
#> [1] 10 20 30 40 50
  #[1] 10 20 30 40 50 

  scaled.mdd <- tTestScaledMdd(seq(10, 50, by = 10), sample.type = "two") 

  round(scaled.mdd, 2) 
#> [1] 1.71 1.17 0.95 0.82 0.73
  #[1] 1.71 1.17 0.95 0.82 0.73

  #----------

  # Look at how the scaled minimal detectable difference for the two-sample 
  # t-test decreases with increasing values of Type I error:

  scaled.mdd <- tTestScaledMdd(20, alpha = c(0.001, 0.01, 0.05, 0.1), 
    sample.type="two") 

  round(scaled.mdd, 2) 
#> [1] 1.68 1.40 1.17 1.06
  #[1] 1.68 1.40 1.17 1.06

  #==========

  # Modifying the example on pages 21-4 to 21-5 of USEPA (2009), 
  # determine the minimal mean level of aldicarb at the third compliance 
  # well necessary to detect a mean level of aldicarb greater than the 
  # MCL of 7 ppb, assuming 90%, 95%, and 99% power.  Use a 99% significance 
  # level and assume an upper one-sided alternative (third compliance well 
  # mean larger than 7).  Use the estimated standard deviation from the 
  # first four months of data to estimate the true population standard 
  # deviation in order to determine the minimal detectable difference based 
  # on the computed scaled minimal detectable difference, then use this 
  # minimal detectable difference to determine the mean level of aldicarb 
  # necessary to detect a difference.  (The data are stored in 
  # EPA.09.Ex.21.1.aldicarb.df.) 
  #
  # Note that the scaled minimal detectable difference changes from 3.4 to 
  # 3.9 to 4.7 as the power changes from 90% to 95% to 99%.  Thus, the 
  # minimal detectable difference changes from 7.2 to 8.1 to 9.8, and the 
  # minimal mean level of aldicarb changes from 14.2 to 15.1 to 16.8.

  EPA.09.Ex.21.1.aldicarb.df
#>    Month   Well Aldicarb.ppb
#> 1      1 Well.1         19.9
#> 2      2 Well.1         29.6
#> 3      3 Well.1         18.7
#> 4      4 Well.1         24.2
#> 5      1 Well.2         23.7
#> 6      2 Well.2         21.9
#> 7      3 Well.2         26.9
#> 8      4 Well.2         26.1
#> 9      1 Well.3          5.6
#> 10     2 Well.3          3.3
#> 11     3 Well.3          2.3
#> 12     4 Well.3          6.9
  #   Month   Well Aldicarb.ppb
  #1      1 Well.1         19.9
  #2      2 Well.1         29.6
  #3      3 Well.1         18.7
  #4      4 Well.1         24.2
  #5      1 Well.2         23.7
  #6      2 Well.2         21.9
  #7      3 Well.2         26.9
  #8      4 Well.2         26.1
  #9      1 Well.3          5.6
  #10     2 Well.3          3.3
  #11     3 Well.3          2.3
  #12     4 Well.3          6.9

  sigma <- with(EPA.09.Ex.21.1.aldicarb.df, 
    sd(Aldicarb.ppb[Well == "Well.3"]))

  sigma
#> [1] 2.101388
  #[1] 2.101388

  scaled.mdd <- tTestScaledMdd(n.or.n1 = 4, alpha = 0.01, 
    power = c(0.90, 0.95, 0.99), sample.type="one", alternative="greater") 

  scaled.mdd 
#> [1] 3.431501 3.853682 4.668749
  #[1] 3.431501 3.853682 4.668749

  mdd <- scaled.mdd * sigma 
  mdd 
#> [1] 7.210917 8.098083 9.810856
  #[1] 7.210917 8.098083 9.810856

  minimal.mean <- mdd + 7 

  minimal.mean 
#> [1] 14.21092 15.09808 16.81086
  #[1] 14.21092 15.09808 16.81086


  #==========

  # Clean up
  #---------
  rm(scaled.mdd, sigma, mdd, minimal.mean)