Non-parametric tests

1
Non-parametric tests

• Minimal population assumptions
• Distribution-free tests

2
Non-parametric tests...

• The sign test
• The Mann-Whitney U test (Chapter 11)
• Chi-squared
• Wilcoxon matched-pairs signed-ranks test
• Kruskal-Wallis test
• In general, non-parametric tests are less
  powerful than parametric tests.

3
Chi-squared (c2)

• c2 requires frequencies in groups.
• Two uses of c2
• Test of goodness of fit
• Test of independence
• c2 compares an observed frequency distribution
  with a theoretical model expected frequency
  distribution.

4
Models for c2

• The theoretical model may be chance, which leads
  us to expect the same number of observations in
  each category or group.
• Thus, if we have 40 observations and four
  categories or groups, we expect 10 observations
  in each group.
• If we have 107 observations and three categories,
  we expect 107/3 35.667 observations in each
  category.

5
A non-chance model

• We may have a basis for a non-chance theoretical
  model, if prior research or theory tells us the
  proportions or percentages of observations to
  expect in each group.

6
Examples of theoretical models

• For example, a genetic model might predict
  proportions of 9331 for four kinds of pea
  hybrids. A sociological model of a neighborhood
  might predict 40 single-parent homes, 55
  dual-parent homes, and 5 three-parent homes.

7
The goodness of fit c2

• Form your data into columns, one for observed
  frequencies fo and one for expected frequencies
  fe.
• Ensure that the sum of fo the sum of fe.
• Form a third column, fo - fe.
• Form a fourth column, (fo - fe)2
• Form a fifth column, (fo - fe)2 / fe
• c2 is the sum of column 5.

8
An example

• c2 S(fo - fe)2 / fe
• Here is a chi-squared comparing the distribution
  of men and women in a class with a theoretical
  5050 model
• fo fe fo - fe (fo - fe)2
  fo - fe)2 / fe
• Men 12 15 -3 9 0.6
• Women 18 15 3 9 0.6
• c2 1.2
• df k-11. c2crit 3.84, so retain H0 .

9
c2 test of independence

• The contingency table
• Two grouping variables Are they related or
  independent?
• Republican Democrat SRows
• Men 10 5 15
• Women 20 30 50
• SColumns 30 35 65

10
Worked example

• Republican Democrat SRows
• Men 10 5 15
• Women 20 30 50
• SColumns 30 35 65
• Obtain fe from the marginal sums. The fe for a
  particular cell equals its row marginal times its
  column marginal divided by the grand total (30 x
  15)/65 for Republican men, for example

11
Example...

• Republican Democrat SRows
• Men 10 5 15
• Women 20 30 50
• SColumns 30 35 65
• Republican men (30 x 15)/65 450/65 6.923
• Republican women (30 x 50)/65 1500/65 23.077
• Democrat men (35 x 15)/65 525/65 8.077
• Democrat women (35 x 50)/65 1750/65 26.923

12
And the c2 is...

• Group fo fe (fo - fe) (fo - fe)2 (fo - fe)2/fe
• RM 10 6.923 3.077 9.468 1.368
• RF 20 23.077 3.077 9.468 0.410
• DM 5 8.077 -3.077 9.468 1.172
• DF 30 26.923 -3.077 9.468 0.352
• c2 3.302
• df (r-1)(c-1)1. c2crit 3.84, so retain H0 .

13
Interpreting c2

• In the goodness of fit test, H0 states that the
  observed frequencies are essentially the same
  (within chance variation) as the expected
  frequencies.
• If we reject H0, we believe that the observed
  frequency distribution is different from the
  model of expected frequencies.

14
Interpreting c2...

• In the test of independence, H0 states that the
  frequencies for each group on one grouping
  variable are unrelated to (independent of) the
  groups of the other variable Political party is
  unrelated to gender.
• If we reject H0, we believe that the frequncies
  for each group on one grouping variable depend on
  (are different for) the groups of the other
  variable Men and women differ in their political
  party.