# 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. The spreadsheet also computes sample sizes for the Wilcoxon one-sample signed ranks test and the Mann-Whitney two group test using the approaches in Noether (1987). Kendall's tau correlation is also powered in the Noether paper 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 power computations for the Wilcoxon and Mann-Whitney tests using Noether (1987) with another method using Field (2005) to compute nonparametric effect sizes and then using a conversion formula from DeCoster (2012) to yield a Cohen's d which one can then run through traditional power software. There is agreement between the two methods powering the Mann-Whitney test but varying degrees of agreement when powering 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 of 12 used in the calculation equals the 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 giving 14 differences needed (N=6 used in the calculation=total number of differences).

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

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) p' = (44 - 10x0.7) /(0.5x10x(10-1)) = 0.822. This yields Noether's 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.

Using Field we have 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.