# Minimal or Maximal Detectable Ratio of Means for One- or Two-Sample t-Test, Assuming Lognormal Data Compute the minimal or maximal detectable ratio of means associated with a one- or two-sample t-test, given the sample size, coefficient of variation, significance level, and power, assuming lognormal data. ```r tTestLnormAltRatioOfMeans(n.or.n1, n2 = n.or.n1, 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, 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. - `cv`: numeric vector of positive value(s) 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 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), `"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 `FALSE`. - `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`. ## Details If the arguments `n.or.n1`, `n2`, `cv`, `alpha`, and `power` are not all the same length, they are replicated to be the same length as the length of the longest argument. Formulas for the power of the t-test for lognormal data for specified values of the sample size, ratio of means, and Type I error level are given in the help file for `tTestLnormAltPower`. The function `tTestLnormAltRatioOfMeans` uses the `uniroot` search algorithm to determine the required ratio of means for specified values of the power, sample size, and Type I error level. ## Returns a numeric vector of computed minimal or maximal detectable ratios of means. When `alternative="less"`, or `alternative="two.sided"` and `two.sided.direction="less"`, the computed ratios are less than 1 (but greater than 0). Otherwise, the ratios are greater than 1. ## References See `tTestLnormAltPower`. ## Author(s) Steven P. Millard (EnvStats@ProbStatInfo.com ) ## Note See `tTestLnormAltPower`. ## See Also `tTestLnormAltPower`, `tTestLnormAltN`, `plotTTestLnormAltDesign`, LognormalAlt , `t.test`, Hypothesis Tests . ## Examples ```r # Look at how the minimal detectable ratio of means 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 ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = 20, power = seq(0.5, 0.9, by = 0.1)) round(ratio.of.means, 2) #[1] 1.47 1.54 1.63 1.73 1.89 #---------- # Repeat the last example, but compute the minimal detectable ratio of means # based on the approximate power instead of the exact: ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = 20, power = seq(0.5, 0.9, by = 0.1), approx = TRUE) round(ratio.of.means, 2) #[1] 1.48 1.55 1.63 1.73 1.89 #========== # Look at how the minimal detectable ratio of means for the two-sample t-test # decreases with increasing sample size: seq(10, 50, by = 10) #[1] 10 20 30 40 50 ratio.of.means <- tTestLnormAltRatioOfMeans(seq(10, 50, by = 10), sample.type="two") round(ratio.of.means, 2) #[1] 4.14 2.65 2.20 1.97 1.83 #---------- # Look at how the minimal detectable ratio of means for the two-sample t-test # decreases with increasing values of Type I error: ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = 20, alpha = c(0.001, 0.01, 0.05, 0.1), sample.type = "two") round(ratio.of.means, 2) #[1] 4.06 3.20 2.65 2.42 #========== # 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, the above values for Type I and power, and a # coefficient of variation of 2, determine the minimal detectable increase above # the soil screening level associated with various sample sizes for the one-sample # test. Based on these calculations, you need to take at least 6 soil samples to # satisfy the requirements for the Type I and Type II errors when the coefficient # of variation is 2. N <- 2:8 ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = N, cv = 2, alpha = 0.2, alternative = "greater") names(ratio.of.means) <- paste("N=", N, sep = "") round(ratio.of.means, 1) # N=2 N=3 N=4 N=5 N=6 N=7 N=8 #19.9 7.7 5.4 4.4 3.8 3.4 3.1 #---------- # Repeat the last example, but use the approximate power calculation instead of # the exact. Using the approximate power calculation, you need 7 soil samples # when the coefficient of variation is 2. Note how poorly the approximation # works in this case for small sample sizes! ratio.of.means <- tTestLnormAltRatioOfMeans(n.or.n1 = N, cv = 2, alpha = 0.2, alternative = "greater", approx = TRUE) names(ratio.of.means) <- paste("N=", N, sep = "") round(ratio.of.means, 1) # N=2 N=3 N=4 N=5 N=6 N=7 N=8 #990.8 18.5 8.3 5.7 4.6 3.9 3.5 #========== # Clean up #--------- rm(ratio.of.means, N) ```

tTestLnormAltRatioOfMeans function