Non-parametric equivalents
to the t-test
Sam Cromie
Parametric assumptions
• Normal distribution
– (Kolmogorov-Smirnov test)
• For between groups designs homogeneity of
variance
– (Levene’s test)
• Data must be of interval quality or above
Scales of measurement - NOIR
• Nominal
– Label that is attached to someone or something
– Can be arbitrary or have meaning e.g., number
on a football shirt as opposed to gender
– Has no numerical meaning
• Ordinal
– Organised in magnitude according to some
variable e.g., place in class, world ranking
– Tells us nothing about the distance between
adjacent scores
Scales of measurement - NOIR
• Interval
– adjacent data points are separated by equivalent
amounts e.g., going from an IQ of 100 to 110 is the
same increase as going from 110-120
• Ratio data
– adjacent data points are separated by the same
amount but the scales also has an absolute zero e.g.,
height or weight
– When we talk about attractiveness on a scale of 0-5,
0 does not mean that the person has zero
attractiveness it means we cannot measure it
– Psychological data is rarely of ratio quality
What type of scale?
•
•
•
•
Education level
County of Birth
Reaction time
IQ
Between groups design
• Non-parametric equivalent = MannWhitney U-test
Mann-Whitney U-test
• Based on ordinal data
• If differences exist scores in one group
should be larger than in the other
Group A
Scores
Group B
Scores
3, 4, 4, 9
7, 10, 10, 12
Rank ordering the data
• Scores must be combined and rank ordered to
carry out the analysis e.g.,
Original scores: 3
Ordinal scores: 1
Final Ranks: 1
4
2
2.5
4
3
2.5
7
4
4
9
5
5
10
6
6.5
10
7
6.5
• If there is a difference, scores for one
group should be concentrated at one
end (e.g., end which represents a high
score) while the scores for the second
group are concentrated at the other end
12
8
8
Null hypothesis
• H0: There is no tendency for ranks in one
treatment condition to be systematically
higher or lower than the ranks in the other
treatment condition.
• Could also be thought of as
– Mean rank for inds in the first treatment is the
same as the mean rank for the inds in the
second treatment
• Less accurate since average rank is not calculated
Calculation
• For each data point, need to identify how
many data points in the other group have
a larger rank order
• Sum these for each group - referred to as
U scores
• As difference between two Gs increases
so the difference between these two sum
scores (U values) increases
Calculating U scores
Rank
Score
No of data points in
alternative G with
larger rank
1
2.5
2.5
4
5
6.5
6.5
8
U score for
both Gs
3
4
4
7
9
10
10
12
UA
UB
4
4
4
1
3
0
0
0
15
1
Determining significance
• Mann-Whitney U value = the smaller of the
two U values calculated - here it is 1
• With the specified n for each group you
can look up a value of U which your result
should be equal to or lower than to be
considered sig
Mann-Whitney U table
(2 groups of 4 two-tailed),
Note extremes…
– At the extreme there should be no
overlap and therefore the Mann-Whitney
U value should be = 0
– As the two groups become more alike
then the ranks begin to intermix and U
becomes larger
Reporting the result
• Critical U = 0
• Critical value is dependent on n for each
group
• U=1 (n=4,4), p>.05, two tailed
Formula for calculation
• Previous process can be tedious and
therefore using a formula is more ‘straight
forward’
U A  nA nB 
Where
R
A
n A ( n A  1)
2

R
A
is the sum of ranks for Group A
Repeated measures - Wilcoxon T
• H0 = In the general population there is no
tendency for the signs of the difference
scores to be systematically positive or
negative. There is no difference between
the means.
• H1= the difference scores are
systematically positive or negative. There
is a difference between the means.
Table showing calculation
Trea tments
Subject
1
2
diff
1
18
43
+25
6
2
9
14
+5
2
3
21
20
-1
1
4
30
48
+18
5
5
14
21
+7
3
6
12
4
-8
4
R

16
R

5


rank
• Calculate
difference score
• Assign rank
independent of
sign
• Add ranks for
each sign
separately
• T = lowest rank
total
T=5
Interpreting results
• Look up the critical value of T
• You result must be equal to or lower than it
in order to be considered significant
• With n = 6 critical T is 0 and therefore the
result here is not significant.
• As either sum of ranks approaches 0 the presence of
that direction of change is limited
• If the sum of negative ranks is small there are
obviously very few decreases indicating that most
scores increased
Non-parametric Pros and Cons
• Advantages of non-parametric tests
– Shape of the underlying distribution is irrelevant - does
not have to be normal
– Large outliers have no effect
– Can be used with data of ordinal quality
• Disadvantages
– Less Power - less likely to reject H0
– Reduced analytical sophistication. With nonparametric
tests there are not as many options available for
analysing your data
– Inappropriate to use with lots of tied ranks