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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

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

• 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

### 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: 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

### 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

• 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