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Non-parametric equivalents to the t-test

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Non-parametric equivalents to the t-test

Sam Cromie

- 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

- 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

- Education level
- County of Birth
- Reaction time
- IQ

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

Group A Scores

Group B Scores

7, 10, 10, 12

3, 4, 4, 9

- Based on ordinal data
- If differences exist scores in one group should be larger than in the other

- Scores must be combined and rank ordered to carry out the analysis e.g.,
Original scores: 34479101012

Ordinal scores: 12345678

Final Ranks: 12.52.5456.56.58

- 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

- 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

- Mean rank for inds in the first treatment is the same as the mean rank for the inds in the second treatment

- 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

- 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

- 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

- Critical U = 0
- Critical value is dependent on n for each group
- U=1 (n=4,4), p>.05, two tailed

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

- 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.

-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

- 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

- 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