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 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

Based on ordinal data If differences exist scores in one group should be larger than in the other Group A Scores 3, 4, 4, 9 Group B Scores 7, 10, 10, 12 Mann-Whitney U-test

Scores must be combined and rank ordered to carry out the analysis e.g., Original scores: Ordinal scores: Final Ranks: 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 Rank ordering the data

Null hypothesis H 0 : 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 larger rank 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

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 to or lower than to be considered sig

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

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’

H 0 = 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. H 1 = the difference scores are systematically positive or negative. There is a difference between the means. Repeated measures - Wilcoxon T

Table showing calculation T=5 Calculate difference score Assign rank independent of sign Add ranks for each sign separately T = lowest rank total

Interpreting results Look up the critical value of T You result must be equal to or lower than 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 H 0 –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