Topic 15 Nonparametric/Distribution Free Methods

1. Topic 15Nonparametric/Distribution Free Methods • Does not depend on distributional assumptions such as normality • Distribution-free property achieved by using signs and/or ranks in place of the original measurements

2. Potencies (in mg) of 10 tablets

3. The mean potency should be 100 mg according to the manufacturer

4. Sign Test

5. Fairly strong evidence that the median potency is greater than 100 mg

6. Calculating p value • Use binomial table if n is small • Use normal approximation for moderate or large n

7. Wilcoxon two sample test • Nonparametric counterpart of the 2-sample t test • For use with 2 independent samples and should be distinguished from W+ which is calculated from a single sample or paired data • Does not require normality or even symmetry assumption • The null hypothesis is that the two populations being compared have the same distribution and hence also the same mean/median. • The alternative hypothesis is that the distribution of the population 2 has the same shape as that of population 1 but is shifted to the left or right so that the mean/median is shifted as well • By convention, sample 1 is the smaller sample

8. Normalized mental age scores (nMA) for 2 samples of children suffering from phenylketonuria (PKU)

9. Under H0 For large nS , nL (>10) is approximately standard normal Use Table A.7 for small samples

10. Example: W = 313 p = 2(0.093) = 0.186

11. We do not need to assume that the data follow a normal distribution Less sensitive to outliers because of the use of signs and ranks instead of the actual measurements Only slight loss of efficiency compared with t test when data are in fact normally distributed T tests fairly robust to non-normality Nonparametric tests are not without assumptions e.g., symmetry for W+ and same shape for W Argument for using nonparametric test Nonparametric versus T test Argument for using t test