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 data have 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 dont believe this P as I dont 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 gt
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 dont 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 nn1n2
• It doesnt 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 arent 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 21sided 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)/ ?U has 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 variance
  previously, doesnt 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 differences among 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 as T and T-). (As the sum of series
  numerical from 1 to n is n(n1)/2, we can easily
  compute Tn(n1)/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 isnt 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/vbissonn
ette/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