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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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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
slide3

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
calculation
Calculation
  • 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
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
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.
slide18

-1

+18

+7

-8

T=5

Table showing calculation

  • Calculate difference score
  • Assign rank independent of sign
  • Add ranks for each sign separately
  • T = lowest rank total

+25

6

+5

2

1

5

3

4

16

5

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