Formulae for power analyses for one-sample t and sign tests

Noether (1987) gives various formulae related to the classic formula for powering a one sample t-test but applied to various commonly used nonparametric tests. One of these (Sign test) together with the one-sample t-test is computed using this spreadsheet. Kendall's tau correlation is also powered but the input (numbers of concordant/discordant pairs) is not commonly available. For this reason it is not computed in the spreadsheet.

I have compared my spreadsheet computations from Noether (1987) with another method using Field (2005) for effect sizes and a conversion formula from DeCoster (2012). There is agreement with the Mann-Whitney test but varying degrees of agreement for the Wilcoxon one-sample signed ranks test.

Example 1a (One sample Wilcoxon signed rank test comparing differences of two paired groups)

Six differences comparing two groups each of of size 6: -2, -4, -6, -4, 1, 1

From Field r= 1.581/sqrt(12)= 0.456, this gives (DeCoster spreadsheet) a d=1.025 and yields 11 differences needed. (Following Field N=12= total number of observations in the two groups being compared)

From Noether using outputted W=3 (sum of ranks for positive differences) and one third of the 6 differences being positive we have p' = 3 - (6 x 0.333) / (0.5*6*(6-1)) = 0.066, 14 differences needed (N=6=total number of differences)

Example 1b (not so close agreement for Wilcoxon test)

Ten differences comparing two groups each of size 10: 2, -4, 7, 3, -1, 1, -1, 2, 5, 5 using Field r=1.688/sqrt(20) =0.377 and DeCoster gives a d= 0.814 with 15 differences required

Using the formula for p' with W=44 and 70% of differences being positive we have from Noether(1987) (44 - 10x0.7) /(0.5x10x(10-1)) = 0.822 which yields for his N(W) 26 differences needed

Example 2 (Mann-Whitney two sample test)

Group 1 = 1,5,6,4 Group 2 = 4,3,2,1,2

From a standard stats package: U=5 It follows from Noether (1987) that p double prime =5/(4*5) = 0.25 and prop(n1)=0.5 giving 42 in total (21 per group) required for 80% power, 5% two-sided type I error

This compares with From Field Using r=z/sqrt(N) = 1.24/sqrt(4+5) = 0.4133 giving a d=0.9078 and a total of 44 required for 80% power, two-sided type I error

References

DeCoster, J. (2012) Spreadsheet for converting effect size measures. Available from: http://www.stat-help.com/spreadsheets/Converting%20effect%20sizes%202012-06-19.xls (accessed 04.09.2014).

Field, A. (2005) Discovering Statistics using SPSS, Sage:London.

Noether, G.E. (1987) Sample size determination for some common nonparametric tests Journal of the American Statistical Association 82(398) 645-647.