RMS - Non-parametric

Central Limit Theorem makes it so that we can study all kinds of distributions because their sampling distributions converge into a normalising distribution

• → this means that we only need a limited set of statistical tools to draw inferences about our data.
  • Parametric statistics

It is not always though, that sample is normally distributed.

• Data may be skewed
• Test is not robust to violation of assumption

→ small samples

Non-parametric → unable to apply normal sampling distribution. We cannot assume that the sampling distribution is normal.

Non-parametric testing

• Converting data to ranks

• Eliminates outliers

• Transform the (quantitative) data to rank orders

• Sample space of rank orders can be easily determined: it ranges from 1 to n

• This gives the sampling distribution of the test statistic

^ The lowest score gets rank 1

^^^ when two score are tied:^^

• ^^Scores share 2 places now, 2nd and 3rd. Because they are tied they get rank 2.5; average value of the places they share.^^

Wilcoxon test

• Non-parametric, two-sample t-test
• Example used: A clinical trial of a rare disease, 5 patients: 3 receive treatment, 2 control

Small sample → hard to determine whether assumption of normality is met

We look across all groups when it comes to ranking!

Wilcoxon test steps

1. Assumptions:

1. Data are rank-ordered

2. Two independent samples

3. No assumption with respect to distribution

   2. Hypotheses

1. H0: The population distributions of the (quantitative) score are identical

       1. → average rank in the groups is thew same

2. HA: The populations distributions of the (quantitative) score differ in such a way that the expected average ranks also differ

       1. This implies that the average rank in the group differs

• Statement about the distributions of the two populations → equality between two groups

1. Test statistic

1. Just the difference in average rank

2 - 4,5 = -2.5.

• Is this an extreme outcome, given the sample space?

Sample space: set of all possible outcomes

Under H0: all events are equally likely

Some events have same outcomes

W = difference in average rank

4. P-value:

1. Test statistic: -2.5

2. P(diff=-2.5 or more extreme) = 0.1

3. Two-sided = 0.1 x 2 = 20

   5. P > 0.05 → can’t reject H0

• Limited Power?
  • After all, more extreme outcomes were not even possible in this example!

1 sided statement:

• which of these is stated, what am I interested in
• How is test statistic defined
• Distribution drawing

Kruskal-Wallis Test

• Non-parametric ANOVA
• Example: 18 students, have to memorise words while listening to 3 types of music and recalls as many words as possible next day
  • Condition 1: 6 Bach
  • 2: 6 Venom
  • 3: 6 silence
• Small sample → non-parametric
• Convert data into ranks
  • Lowest value: lowest rank (1)

We look across all groups when it comes to ranking!

Steps

1. Compute average overall rank R
2. Compute average rank for the individual groups
3. Fill in KW-formula
4. Chi-squared distribution

1. df = groups - 1

Sign Test

• Non-parametric paired t-test
• Wilcoxon: independent, sign test: independent

Steps

1. Hypotheses:

1. H0: The proportion of ‘‘+’’P