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Categorical and discrete data. Non-parametric tests. Non-parametric tests: estimate sample differences when the known distribution shapes cannot help, or even confuse. Metrics of arbitrary distibutions. Median: the value that "splits the sample in half"

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slide2
Non-parametric tests:

estimate sample differences when the known distribution shapes cannot help, or even confuse

metrics of arbitrary distibutions
Metrics of arbitrary distibutions
  • Median: the value that "splits the sample in half"
  • Mode: the value that occurs with the greatest frequency.
  • n-th procentile: the value between n% of the sorted sample and the rest of it. Hence quantiles, quartiles, deciles etc.
non parametric correlations
Non-parametric correlations

Spearman R:

The closest analog of Pearson r with the difference that, instead of each raw value, its rank in the sample is used

X – Y

X – Z

2.4 -> 5

3.2 -> 1

2.7 -> 4

2.3 -> 6

2.2 -> 7

3.1 -> 2

2.0 -> 8

2.9 -> 3

5 – 6

1 – 1

4 – 3

6 – 5

7 – 8

2 – 2

8 – 7

3 – 4

5 – 8

1 – 6

4 – 3

6 – 2

7 – 4

2 – 5

8 – 1

3 – 7

non parametric correlations5
Non-parametric correlations

Spearman R:

A variability ratio

(X and Y vary synchronously) / (X and Y vary in total)

Kendall Tau:

A probability ratio

P(X and Y are related) / P(X and Y are NOT related)

comparing two independent samples
Comparing two independent samples

The Mann-Whitney U test:

The closest analog of the t-test with the difference that, instead of each raw value, its rank in the sample is used

Wald-Wolfowitz runs test:

Estimates if two samples differ in BOTH means and distribution shapes

– – + – – – – – – – + – + + + – + + – + + + + – + +

– – + – + – + + – – + + – + – + + + – + + – + + ––

comparing two dependent samples
Comparing two dependent samples

Sign test:

Wilcoxon matched pairs test:

The difference between two tied samples is significant, if a sum of pairwise differences (either positive or negative ones) is TOO BIG

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2 x 2 tables
2 x 2 tables

Chi-square test, Fisher’s exact test:

Χ2 =Σ(Observed - Expected)2 / Expected

slide9
Nonparametric methods are most appropriate when the sample sizes are small. When the data set is large (e.g., n > 100) it often makes little sense to use nonparametric statistics at all:

When the samples become very large, then the sample means will follow the normal distribution even if the respective variable is not normally distributed in the population