Non parametric equivalents to the t test
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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

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Non parametric equivalents to the t test

Non-parametric equivalents to the t-test

Sam Cromie

Parametric assumptions
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
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
What type of scale?

  • Education level

  • County of Birth

  • Reaction time

  • IQ

Between groups design
Between groups design

  • Non-parametric equivalent = Mann-Whitney U-test

Mann whitney u test

Group A Scores

Group B Scores

7, 10, 10, 12

3, 4, 4, 9

Mann-Whitney U-test

  • Based on ordinal data

  • If differences exist scores in one group should be larger than in the other

Rank ordering the data
Rank ordering the data

  • Scores must be combined and rank ordered to carry out the analysis e.g.,

    Original scores: 3 4 47910 10 12

    Ordinal scores: 1 2 3456 7 8

    Final Ranks: 1 2.5 2.5456.5 6.5 8

  • 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

Null hypothesis
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


  • For each data point, need to identify how many data points in the other group have a largerrank 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
Calculating U scores

Determining significance
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 toor lower than to be considered sig

Mann whitney u table 2 groups of 4 two tailed
Mann-Whitney U table (2 groups of 4 two-tailed),

Note extremes
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
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
Formula for calculation

  • Previous process can be tedious and therefore using a formula is more ‘straight forward’

Repeated measures wilcoxon t
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

  • Calculate difference score

  • Assign rank independent of sign

  • Add ranks for each sign separately

  • T = lowest rank total











Interpreting results
Interpreting results

  • Look up the critical value of T

  • You result must be equal toor lowerthan 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
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