NON-PARAMETRIC TESTS

NON-PARAMETRIC TESTS

• Does not require assumptions of normality or homogeneity of variance to be met 

• Can be used with non-normal data or when groups have unequal variances 

• They are 'distribution-free' tests 

ADVANTAGES

• Can be used on data that does not meet the assumptions of parametric tests 

• Ideal for analysing data from small samples  

• Reduce effect of outliers 

• Can be used to analyse nominal and ordinal scales 

DISADVANTAGES

If used for normally distributed data, they have less power  

• Can increase type II error  

• Require larger N 

• Less likely to detect power 

PARAMETRIC TESTS EQUIVALENT

Independent Groups t-test → Wilcoxon’s Rank Sum Test

Repeated Measures t-test → Wilcoxon’s Matched Pairs Signed-Ranks Test

WILCOXON’S RANK SUM TEST

• Test compares the sum of ranks (R) between groups  

• Start by ranking scores as a whole, regardless of groups  

• If H0 is true, the rankings in each group will be unsystematically 'mixed' in with each other 

• If H0 is not true, the rankings for one group tend to be in the upper/lower positions relative to other groups 

DEALING WITH TIED SCORES (WILCOXON’S RANK SUM)

Assign the average of the ranks that the scores would have received of they hadn't been tied.

UNEQUAL GROUP SIZES (WILCOXON’S RANK SUM)

If unequal n’s:

• Ws is the rank sum of the smaller group

If equal n’s:

• Ws is the smaller rank sum

WILCOXON’S RANK SUM TEST DECISION

H0:

Expect roughly equal sums of ranks in both groups.

H1:

If group 1 scores < group 2 scores… expect low ranks to be assigned to group 1, and high ranks assigned to group 2

Wcrit:

If critical value (.025) is LESS THAN or EQUAL to Ws obtained, the result is significant.

NON-PARAMETRIC POWER

• Have less statistical power than their parametric equivalent

• Higher type 2 error rate

• require larger n for same power

• ONLY use if needed

CHECKING SCORES

The sum of all ranks should equal N(N+1)/2

WILCOXON’S RANK SUM TEST STEPS

1. State hypothesis

2. Rank all scores

3. Verify rankings

4. Calculate rank sum per group

5. Identify Ws

6. Find critical value

7. Decision

8. Interpretation

WILCOXON RANK SUM CRITICAL VALUE

In table, n1 = smaller group, n2 = bigger group. a = .025.

WHAT IF SMALLER GROUP HAS LARGER RANK SUM?

USE:

W’s = n1 (n1 + n2 + 1) - Ws

WILCOXON’S MATCHED PAIRS SIGNED RANK TEST

• Used when differences between pairs of scores are not normally distributed

WILCOXON’S MATCHED-PAIRS SIGNED RANK HYPOTHESIS

H0 = No average difference between matched pairs of scores

H1 = average difference between matched pairs of scores

WILCOXON MATCHED-PAIRS SIGNED RANKS STEPS

1. Calculate differences between scores

2. Rank differences from lowest to highest (disregard sign)

3. Restore signs to ranks

4. Find total T+ for positive ranks

5. Find total T- for negative ranks

6. The test statistic T is the smallest of T+ or T-

7. Find critical T

8. Decision and interpretation

DIFFERENCE OF SCORES = 0

If difference of scores = 0, the pair is deleted from analysis.

n = number of non-zero differences.

FINDING T CRIT

n = number of non-zero difference scores

alpha = .025

T CRIT DECISION

If Tobt is smaller than/ equal to Tcrit = reject H0