Non-parametric methods

1. Non-parametric methods t-test (et cetera) tests hypotheses about parameters of distribution (in t-test about μ as a parameter of normal distribution); there are other approaches too

2. What to do, if datahave not normal distribution?and disturbance of normality is so large, that I cannot rely on test robustness • There are transformations improving the normality and homoscedascity [we will go through it later] • If data have such a distribution, which can be approximated with selected types of distribution, then special methods can be used developed for them (generalized linear modes) • We use non-parametric tests

3. Non-parametric methods • Most often • Permutation [commonly randomized] tests • Rank-based tests

4. Permutation tests • Basic idea (for t-test): • Reached level of significance is probability, that so different samples I get just by chance, if from one population. So, I can try it – I put all the observations from both groups together, and then randomly assign their group membership (e.g. by tossing from a hat or by computer random number generator):

5. And so on, at least thousand times I look how often is |t| from randomly generated groups bigger than from data. So, I try to simulate it here. I don’t believe this P as I don’t know, if assumptions are fulfilled

6. Reached level of significance (P) is computed then Number of random permutations, where “it was better” than in data (so where |tpermut | > |tdata |

7. Attention • I test hypothesis, that both samples are from one (and same) population. If I want to interpret the test as location test, then I have to add an assumption that both populations have the same distribution shape. If they differ after that, they can differ in the location parameter.

8. Rank-based tests • Basic idea – We don’t know, what the distribution is, so we forgot real values and replace them with their rank • Many parametric methods have their non-parametric counterparts

9. Mann-Whitney testnon-parametric analogue of two-sample t-test • All values from both samples are arrayed (and so they get numbers from 1 to n, where n=n1+n2 • It doesn’t matter, if the arrangement is made from top or from bottom, but I must pay attention on it, if one-tailed tests are used.

10. compute it gives especially high value, if ranks in the first group are low or it gives especially high value, if ranks in the second group are low holds U + U' = n1n2,

11. Male and female students are the same high. Male and female students aren’t the same high. High of males High of females High of males rank High of females rank As we refuse H0 Mann-Whitney test for non-parametric testing if two-tailed hypothesis, that there is no difference between heights of male and female students.

12. Attention All sorts of values are tabulated, so pay attention, what is tabulated and how Statistica prints 2*1sided exact p (if I want one-tailed test, if deviation goes in the right direction, I divide by two)

13. Normal approximation – if there is great number of observations, holds Z = (U-U)/ Uhas near normal distribution. At it is easy job to find corresponding p to it – Statistica prints - Attention – if I have exact p, this value is never more of interest.

14. Similar to permutation test • even M-W has its presumptions: • It is either test of null hypothesis, that the samples are from the same population • If it is formulated as a location test, then there is an assumption that samples have the same distribution shape

15. It is thus absurd to write • As we had not homogeneity of variances, we had to use non-parametric test. • 1. to test, if it is the same population, when I have proved inhomogeneity of variancepreviously, doesn’t make any sense • 2. for location test, inhomogeneity of variance is the same problem for MW as for t-test.

16. Another presumption - data can be ranked Ties are averaged – deviation from original presumption can make problem, some tests use equalities correction “ties”

17. Median test • I compute median for all observations and how much observations is in each group above and how much below this median. I analyse it then with classic 2 x 2 table. So, it is test about overall median and it has not any further assumptions, but it is very weak.

18. Wilcoxon test • Analogue of pair t-test • Attention, more tests are called Wilcoxon, thus it is sometimes written as Wilcoxon for pair observations

19. Wilcoxon test • First, we count differencesamong observations, then we rank them according to the size of their absolute value from the smallest to the largest one. After that we total of positive differences ranks and number of negative differences ranks (marked asT+ andT-). (As the sum of series numerical from 1 tonisn(n+1)/2, we can easily computeT+={n(n+1)/2}-T-) Thus, test reflects number as well as quantity of positive and negative differences.

20. Length of foreleg and hind leg is the same in roe-deer. Length of foreleg and hind leg isn’t the same in roe-deer. Roe-deer Hind leg L. Foreleg L. Difference Rank Rank with mark As is rejected or Wilcoxon pair test applied upon data of roe-deer legs’ length

21. Approximation can be used again (for large samples) and from this compute Z. Attention, Statistica shows just normal approximation, does not print exact p – look for it in tables, if needed. tables can be found here: http://fsweb.berry.edu/academic/education/vbissonnette/tables/wilcox_t.pdf Test has assumption about symmetric distribution of differences.

22. Sign test Compares numbers of positive and negative differences Has no assumptions, but very weak

23. Non-parametric tests • If assumptions for parametric test are fulfilled, non-parametric tests are weaker than corresponding parametric test. • Common idea about no assumptions for nonparametric test is not true. • Generally – the more observations I have, the more robust parametric tests used to be to disturbances of their presumptions • The stronger assumptions are fulfilled, the more powerful test I can usually use