WEEK 5: Non-Parametric ANOVA Flashcards
Lecture overview:
So, I’ve violated the assumptions of a 1 way ANOVA - what to do
• Non parametric ANOVAs
Parametric vs Non-Parametric Data
Parametric data – When data is normally distributed
Non-parametric data – When data is not normally distributed
Assumptions of parametric ANOVAs
- Continuous (Scale) dependent variable - can be almost any number on a scale
- Normal distribution
- No outliers (extreme scores)
- Equal variance (Sphericity)
–> Remember: ANOVA robust to violations if there are equal sample sizes
Learning Objectives:
- Understand and be able to explain the conditions under which a non-parametric ANOVA would be used
- Be able to report the results of a 1 way non-parametric ANOVA
- Understand and be able to explain how non-parametric ANOVAs rank scores
- Understand and be able to explain the difference between a within and between participants design from the perspective of the ranking used in each type of ANOVA
Introducing non-parametric ANOVAs
> Extension of non parametric tests of 2 conditions
- Between participants – Mann Witney
- Within participants – Wilcoxon
> Non-parametric ANOVA
- Between participants – Kruskall-Wallis
- Within participants – Freidman’s
> Non-parametric ANOVA
- Uses the rank of scores rather than the scores
- Tests whether there are differences between the ranks of scores across more than 2 groups (or Conditions)
If the outcome is significant follow up tests are required
If the outcome is not significant no action is required
Independent non-parametric ANOVA:
How it works - Ranking
Scores are ranked all together regardless of the groups
Scores are ranked according to their size, so the smallest at the top with subsequent items increasing in size.
If the scores are the same ranking go like this:
e.g. 4 scores that are the same, the value 13 has been recorded for 4 different participants and ranked 1-4. As they are all the same score we add up the ranks and take the mean value of the ranks.
So in this case we add up 1+2+3+4 divide it by 4 and end up with a rank of 2.5 for all of these values that share the same score.
Can do the same with the next set of scores that are the same, so here we’ve added up 6,7 & 8 to get an average rank of 7.
The ranks are then separated back into the groups that they should be in, and the mean rank for each group is calculated
*Kruskal-Wallis Output:
If the mean rank of one group is higher than the others this means that on average the scores in this group were higher than the scores in the other 2 groups.
The chi square tells us about the association between our levels and the significance of the p values tells us that the probability of this effect being observed if there is in fact no effect in the population
The non-parametric ANOVA makes use of the mean rank, but we also calculate and the median score for each group
(DON’T use the mean because it would be heavily skewed by non-parametric scores….)
Reporting:
X2(df)= (x2 value), p= (p value - use the exact sig)
Follow up tests
If the output is significant then follow up tests should be conducted - Mann Whitney U
If the output of the kruskall wallis is not significant then you should stop there
Significant non-parametric ANOVA
Why do we use box and whisker plots instead of error bar graphs?
> There are differences between the ranks of scores across more than 2 groups
Because the outcome is significant follow up tests are required WITH Bonferroni corrections, as with parametric data
If the outcome had not been significant no follow up tests would be required
The significant outcome tells us that we have a difference but not where ….
For parametric comparisons we looked at an error bar graph to get a feel for where the differences might lie. Error bar graphs are based on means and SEs which are appropriate measures for a parametric test but NOT for a non parametric test.
For this type of test we would look at the box and whisker plot to get a sense of where our medians lie and how the data is spread out
Box and Whisker plots - plot in which a rectangle is drawn to represent the second and third quartiles, usually with a vertical line inside to indicate the median value.
Inspecting the box and whisker plot
Look for the differences between the medians in each group. We are looking for one to be noticeably higher than the other conditions…
For your lab reports use a bar graph that represents the median values without error bars.
Post hoc tests for non-parametric ANOVAs
The non-parametric version of a between participants t-test is the Mann Whitney U test and this should be used for post hoc comparisons.
As you know from last year this test also ranks the scores for the two groups as 1 group and then works out which group has the highest rankings.
The test statistic is the Mann Whitney U and it comes with a value for W and Z but for the purposes of this course we will stick to the convention of reporting the U statistic in our write ups.
The associated p value should be bonferroni corrected for multiple comparisons just like you do for a t-test. This tells us the probability that we would observe this effect if there was no effect in the population.
Reporting the results of a between participants significant non-parametric ANOVA
As the data were not normally distributed a Kruskall-Wallis one-way ANOVA was performed on [Factor 1]. The scores for [Level 1] (median = __ ) were the highest followed by [Level 2] (median = __ ) and finally [level 3] (median = __ ).
The results showed that there was a significant difference between [3 levels] with a X2 of [X2 value] and an associated probability of p = [p value].
Or - X2(df) = [X2 value], p = [p value]
Bonferroni corrected post hoc Mann-Whitney U tests revealed that scores for [level 1] were significantly higher than scores for [level 2] (Mann-Whitney U = [U value], p = [p value]) and [level 3] (Mann-Whitney U = [U value], p = [p value]).
However, there was no difference between [level 2] and [level 3] (Mann-Whitney U = [U value], p = [p value]).
These results suggest that _____________________
Within participants non-parametric ANOVA
Ranking within ppt data
• Extension of non-parametric tests of 2 conditions
- Within participants test – WILCOXON
• Non-parametric ANOVA
- Within participants - FREIDMAN’S
So now the data is laid out for a within participants design, with ppts down one side and their condition performance in columns
***The way the data is ranked for the within participants design is quite different.
We work out the rank for the scores for each condition for a single participant. In this example for the first participant their classical music essay was scored lowest and that is therefore ranked as 1, their rock essay was the next highest value so that is ranked as 2, silence essay was highest so ranked 3.
We keep going until we have ranked all the scores for each participant.
Then we add up the ranks to calculate the sum of the ranks for each group and divide by the number of observations to calculate the mean ranks.
The mean ranks for each participant are higher for the condition where they completed the task in silence. This suggests that participants tended to score more highly in this condition compared to the other two conditions.
The exact significance value tells us that it is unlikely this effect has been observed due to chance.
It is unlikely that this effect would be observed if there is no effect in the population and we can therefore reject our null hypothesis and carry out follow up tests
The non-parametric version of a within participants t-test is the Wilcoxon test and this should be used for post hoc comparisons. As you know from last year the wilcoxon test takes the difference between the scores in each condition for a single participant and ranks the differences. Some Ps will end up with negative scores and some will have positive scores so the output gives you the mean ranks for the negative and the positive scores. In addition you get a z score and a p value that you report in your write up.
Reporting the results of a within participants non-parametric ANOVA
As the data were not normally distributed a Friedman’s one way ANOVA was performed on [Factor 1]. The scores for [Level 1] (median = __ ) were the highest followed by [Level 2] (median = __ ) and finally [level 3] (median = __ ).
The results showed that there was a significant difference between [3 levels] with a X2 of [X2 value] and an associated probability of p = [p value].
Or - X2(df) = [X2 value], p = [p value]
Bonferroni corrected post hoc Wilcoxon tests revealed that scores for [level 1] were significantly higher than scores for [level 2] (z = [z value], p = [p value]) and [level 3] (z = [z value], p = [p value]).
However, there was no difference between [level 2] and [level 3] (z = [z value], p = [p value]).
These results suggest that____________________
