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Make a list of formulae that are used in non-parametric tests.your list should contain name of the test, its purpose and the test statistic used.

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4   ]Make a list of formulae that are used in non-parametric tests. Your list should contain name of the test, its purpose and the test statistic used.
•    Non-parametric tests
•   These are sometimes referred to as “distribution free” tests, because they do not make assumptions about the normality or variance of the data
•   The Mann Whitney U test is appropriate for a 2 condition independent samples design
•   The Wilcoxon Signed Rank test is appropriate for a 2 condition related samples design
•   If you have decided to use a non-parametric test then the most appropriate measure of central tendency will probably be the median
•    Mann-Whitney U test
•   To avoid making the assumptions about the data that are made by parametric tests, the Mann-Whitney U test first converts the data to ranks.
•   If the data were originally measured on an interval or ratio scale then after converting to ranks the data will have an ordinal level of measurement
Mann-Whitney U test: ranking the data
Sample 1      Sample 2   
Score   Rank 1   Score   Rank 2
7   3   6   2
13   8   12   7
8   4   4   1
9   5.5   9   5.5

Mann-Whitney U test: ranking the data
Sample 1      Sample 2   
Score   Rank 1   Score   Rank 2
7   3   6   2
13   8   12   7
8   4   4   1
9   5.5   9   5.5
Scores are ranked irrespective of which experimental group they come from
Mann-Whitney U test: ranking the data
Sample 1      Sample 2   
Score   Rank 1   Score   Rank 2
7   3   6   2
13   8   12   7
8   4   4   1
9   5.5   9   5.5
Tied scores take the mean of the ranks they occupy. In this example, ranks 5 and 6 are shared in this way between 2 scores. (Then the next highest score is ranked 7)
•    Rationale of Mann-Whitney U
•   Imagine two samples of scores drawn at random from the same population
•   The two samples are combined into one larger group and then ranked from lowest to highest
•   In this case there should be a similar number of high and low ranked scores in each original group
–   if you sum the ranks in each group the totals should be about the same
–   this is the null hypothesis
•   If however, the two samples are from different populations with different medians then most of the scores from one sample will be lower in the ranked list than most of the scores from the other sample
–   the sum of ranks in each group will differ
Mann-Whitney U test: sum of ranks
Sample 1      Sample 2   
Score   Rank 1   Score   Rank 2
7   3   6   2
13   8   12   7
8   4   4   1
9   5.5   9   5.5
Sum of ranks   20.5      15.5

The next step in computing the Mann-Whitney U is to sum the ranks in the two groups
Mann Whitney U – SPSS
•   Mann Whitney U - reporting
•   “As the data was skewed, and the two sample sizes were unequal, the most appropriate statistical test was Mann-Whitney. Descriptive statistics showed that group 1 (median = ____ ) scored higher on the DV than group 2 (median = ____). However, the Mann-Whitney U was found to be 51 (Z = -1.21), p > 0.05, and so the null hypothesis that the difference between the medians arose through sampling effects cannot be rejected.”
•   For a significant result: “….. Mann-Whitney U was found to be 276.5 (Z = -2.56), p = 0.01 (one-tailed), and so the null hypothesis that the difference between the medians arose through sampling effects can be rejected in favour of the alternative hypothesis that the IV had an influence on the DV.”
•    Wilcoxon signed ranks test
•   This is appropriate for within participants designs
•   The t test lecture used a within participants example based upon testing reaction time in the morning and in the afternoon, using the same group of participants in both conditions
•   The Wilcoxon test is conceptually similar to the related samples t test
–   between subjects variation is minimised by calculation of difference scores
Wilcoxon test: ranking the data
Score cond 1   Score cond 2   Difference   Ranked dif ignoring + /-   
3   7   -4   3.5   
5   6   -1   1   
5   3   2   2   
4   8   -4   3.5   
First rank the difference scores, ignoring the sign of the difference. Differences of 0 receive no rank
•    Rationale of Wilcoxon test
•   Some difference scores will be large, others will be small
•   Some difference scores will be positive, others negative
•   If there is no difference between the two experimental conditions then there will be similar numbers of positive and negative difference scores
•   If there is no difference between the two experimental conditions then the numbers and sizes of positive and negative differences will be equal
–   this is the null hypothesis
•   If there is a differences between the two experimental conditions then there will either be more positive ranks than negative ones, or the other way around
–   Also, the larger ranks will tend to lie in one direction
Wilcoxon test: ranking the data
Score cond 1   Score cond 2   Difference   Ranked dif ignoring + /-   Ranked dif +/- reattached
3   7   -4   3.5   -3.5
5   6   -1   1   -1
5   3   2   2   2
4   8   -4   3.5   -3.5
Add the sign of the difference back into the ranks
Wilcoxon test: ranking the data
Score cond 1   Score cond 2   Difference   Ranked dif ignoring + /-   Ranked dif +/- reattached
3   7   -4   3.5   -3.5
5   6   -1   1   -1
5   3   2   2   2
4   8   -4   3.5   -4
Separately, sum the positive ranks and the negative ranks. In this example the positive sum is 2 and the negative sum is     -8.5. The Wilcoxon T is whichever is smaller (2 in this case)
Wilcoxon T – SPSS
The value of T is equal to whichever of the mean ranks is lower
T is converted to a Z score by SPSS, taking into account sample size, and the p value is derived from the standard normal distribution
•    Wilcoxon T - reporting
•   “As the difference scores were not normally distributed,  the most appropriate statistical test was the Wilcoxon signed-rank test. Descriptive statistics showed that measurement in condition 1 (median = ____ ) produced higher scores than in condition 2 (median = ____). The Wilcoxon test (T = 2.17) was converted into a Z score of -2.73, p = 0.006 (two tailed). It can therefore be concluded that the experimental and control treatments produced different scores.”
•   Limitations of non-parametric methods
•   Converting ratio level data to ordinal ranked data entails a loss of information
•   This reduces the sensitivity of the non-parametric test compared to the parametric alternative in most circumstances
–   sensitivity is the power to reject the null hypothesis, given that it is false in the population
–   lower sensitivity gives a higher type 2 error rate
•   Many parametric tests have no non-parametric equivalent
–   e.g. Two way ANOVA, where two IV’s and their interaction are considered simultaneously
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