Inferential Statistics - Test 2 - Notes 1/2 - Grade: 7.4

Inhoudsopgave
Lecture 8: More about means – Non-parametric tests (alternatives) ............................................................. 2
Wilcoxon Signed Rank Test (W+) (= related / paired samples) ......................................................................... 3
Wilcoxon Rank Sum Test (= two independent samples) ...................................................................................... 7

Exercises: Chapter 20 and 21 – Practicing with Wilcoxon tests .................................................................... 10

Canvas 8.1: Exercises Non-parametric tests ............................................................................................... 16

Lecture 9: Regression: Using the model ..................................................................................................... 23

9.1: Investigating assumptions Regression: (Exercises) ............................................................................... 36

9.2: Using the regression model ................................................................................................................ 44

Chapter 23 ............................................................................................................................................... 49

Lecture 10: Introduction to Multiple Regression ........................................................................................ 55

Chapter 23 ............................................................................................................................................... 62




1

,Lecture 8: More about means – Non-parametric tests (alternatives)
YOU WILL GET A QUESTION ON THE EXAM WHERE YOU HAVE TO USE EITHER
THE WILCOXON SIGNED RANK TEST OR WILCOXON RANK SUM TEST!
- Alternatives when assumptions for tests are not fulfilled
o Normal population assumption
- When sample size N is relatively small (no CLM)

Main objectives
Knowing how to construct a test for the difference between two means if the assumptions for
a parametric test are not fulfilled

If assumptions t-procedure are not fulfilled, e.g. low measurement level. It can still be
numerical variables such as income, but questioned in categories that can be ranked
Then: ordinal measurement procedures → ranking of the data (!)

1. Nonparametric tests (NOT VIA SPSS!!!!!!)
o For ranked data with low measurement level: they don’t look to means of
original variables but to ranking. You can’t say ‘it’s two times as much’ but
you can say ‘it’s higher’
o For high measurement level if you don’t trust the expected value for that
situation. E.g. if you don’t trust the mean for a certain distribution, because
the variable is skewed and will probably differs a lot from sample to sample (if
N is small)

- Wilcoxon Signed Rank test (= related / paired samples)
- Wilcoxon Rank Sum test (= two independent samples)
----------------------------------------------------------------------------------------------------
- Kruskal Wallis test (more in lecture 11)




- We don’t work with the ‘sign test’ but we work with the ‘Wilcoxon Sign Rank test’


2

, - If you say that the measurement level is not good and it’s a skewed distribution, the
problem of equal variance assumption is not so relevant anymore → strictly taken
one type of data set and two independent samples → we use ‘Wilcoxon Rank Sum
test’

Non-parametric tests




- Both test statistics helps standardizing the values into z-scores
- Never asked to write down what I given between brackets (N(muw(+), sigmaw(+)) )
- If your sample size is big enough and you have at least 5 or more cases, then you can
already work with the Z-distribution and the answers won’t differ a lot from the more
precise complex test tables for the Wilcoxon Tests

Wilcoxon Signed Rank Test (W+) (= related / paired samples)




- For the ordering, it doesn’t matter if it’s positive or negative → therefore, we order it
in absolute values and we don’t look to the negative sign
- In terms of ranking, we talk about medians and NOT means



3

, - If we have a sample size of at least 5 → use normal approximation with z-scores
(standardized test statistic) INSTEAD of using W+ (blue part, unstandardized test
statistic)
- When doing step 3: give test statistic (z) plus distribution (= N(0,1)-distribution)

Example: Retelling two fairytales (n = 10) – Wilcoxon Signed
Rank Test (W+)
Pre-school children (n = 10):
- They are asked to retell two fairytales that were read aloud to them earlier in the
week

Each child is told two stories:
- Story 1: only read aloud
- Story 2: read aloud + illustrated with pictures

- Observations: The retelling of the children was recorded. An expert assigned a score
to each child for each story (a score for certain aspects of the retelling)

Do illustrations improve how children retell a story? (we assume that the stories are
comparable) →
- Because we expect that children know more from story 2, we calculate the difference
as: story 2 score – story 1 (so we get positive differences)




1. Rethink the problem
- Two means, is it paired or independent samples? → We see that we have 10 units of
analysis (children) and for each child there are 2 observations for which we calculate
difference (story 2 score – story 1 score) = paired samples!
- We want to use a non-parametric test about paired samples. So, Wilcoxon Signed
Rank Test (W+) because → problems with the distribution, given the sample size of N
= 10 (maybe also say that for one group there is a score which is far away from all the
other scores, such as 0, which can be indication of skewness and you don’t trust the
mean anymore and you want to work with a non-parametric test and ranking)

2. Formulate H0 and HA, and define sigma
- H0: median(story 1) = median(story 2)
- HA: median(story 2) > median(story 1)




4