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Non-parametric equivalents to the t-test

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# Non-parametric equivalents to the t-test - PowerPoint PPT Presentation

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

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 = 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
• 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
• 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 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
• 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…
• 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’
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.

-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
• 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
• 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