Summary of Discovering statistics using IBM SPSS statistics by Field - 5th edition
- 11956 keer gelezen
Statistics
Chapter 7
Non-parametric models
Sometimes you can’t correct problems in your data.
This is especially irksome if you have a small sample and can’t rely on the central limit theorem to get you out of trouble.
The four most common non-parametric procedures:
All four tests overcome distributional problems by ranking the data.
Ranking the data: finding the lowest score and giving it a rank 1, then finding the next highest score and giving it the rank 3, and so on.
This process results in high scores being represented by large ranks, and low scores being represented by small ranks.
The model is then fitted to the ranks and not to the raw scores.
There are two choices to compare the distributions in two conditions containing scores from different entities:
Both tests are equivalent.
There is also a second Wilcoxon test that does something different.
Theory
If you were to rank the data ignoring the group to which a person belonged from lowest to highest, if there’s no difference between the groups, ten you should find a similar number of high and low ranks in each group.
If you were to rank the data ignoring the group to which a person belonged from lowest to highest, if there’s a difference between the groups, ten you should not find a similar number of high and low ranks in each group.
The Mann-Whitney and Wilcoxon rank-sum test use the principles above.
Starting at the lowest score, we assign potential ranks starting with 1 and going up the number of scores we have.
One you’ve ranked the data, we add the ranks for the two groups.
Wilcoxon rank-sum test
The mean (Ẅs) and the standard error (SEẄs) of this test statistic can be calculated from the sample sizes of each group.
n1 is the sample size of group 1
n2 is the sample size of group 2
Ẅs = (n1 + n2 +1)/2
SEẄs = square root((n1n2(n1 + n2 +1))/12)
z = (X – Ẍ)/s = (Ws – Ẅs)/ SEẄx
The Mann-Whitney test
Basically the same, but uses a test statistic U, which has a direct relationship with the Wilcoxon tst statistic.
R1 is the sum of ranks for group 1
U = n1 n2 + (n1 (n1 +1)/2) – R1
Output from the Mann-Whitney test
The Mann-Whitney test works by looking at differences in the ranked positions of scores in different groups.
With all non-parametric tests, the output contains a summary table that you need to double-click to open the model viewer window.
The model viewer is divided into two panels:
The group that has the highest mean rank should have a greater number of high scores within it.
Underneath the graph a table shows the test statistics for the Mann-Whitney test, the Wilcoxon procedure and the corresponding z-score.
Calculating effect size
r: effect size estimate
r = z/ square root(N)
z: the z-score SPSS procedures
N: the size of the study (number of total observations)
Writing results
For the Mann-Whetney test, report only the test statistic (denoted by U) and its significance. Include the effect size and report exact values of p.
Summary
The Wilcoxon signed-rank test: used in situations where you want to compare two sets of scores that are related in some way (they come from the same entities).
Theory of the Wilcoxon signed-rank test
The Wilcoxon signed-rank test is based on ranking the differences between scores in the two conditions you’re comparing.
Doing the test
To calculate significance of the test statistic (T), we look at the mean (Ť), and the standard error (SET)
Ť = (n(n+1))/4
SET = square root (n(n+1)(2n+1))/ 24
z = (T – Ť) / SET
Output
If you have split the file, the first set of results obtained will be for the fist group.
If you double-click this table to enter the model viewer you will see a histogram of the distribution of differences.
Calculating the effect size
The effect size can be calculated in the same way as for the Mann-Whitney test.
Writing the results
For the Wilcoxon test, we report the test statistic (denoted by the letter T), its exact significance and an effect size
Summary
The Kruskal-Wallis test: compares groups or conditions containing independent scores.
Assesses the hypothesis that multiple independent groups come form different populations.
Theory of the Kruskal-Wallis test
The Kruskal-Wallis test is used with ranked data.
Once the ranks has been calculated within each group, the test statistic H is calculated.
H = 12/(N(N+1)) Σki=1 ((R2i/ni)-3(N+1))
N is the total sample size
ni is the sample size within each group
This test statistic has a distribution from the family of chi-square distributions.
Follow-up analysis
The Kruswal-Wallis test tells us that, overall, groups come from different populations.
It doesn’t tell us which groups differ.
Output from the Kruskal-Wallis test
Double-click on the summary table to open up the model viewer, which contains:
The right-hand pane of the model viewer shows the main output by default (the Independent Samples Test View).
The column labelled Adj.Sig contains the adjusted p-values.
Testing for trends: the Jockheere-Terpstra test
Jockheere-Terpstra test: tests for an ordered pattern to the medians of the groups you’re comparing.
It does the same thing as the Kruskal-Wallis test (test for a difference between the medians of the groups), but it incorporates information about whether the order of the group is meaningful.
Two options of the test:
The test determines whether the medians of the groups ascend or descend in the order specified by the coding variable.
The coding variable must code groups in the order that you expect the medians to change.
Calculating effect size
There isn’t an easy way to convert a Kruskal-Wallis test statistic that has more than 2 degree of freedom to an effect size, r.
You should us the significance value of the Kruskal-Wallis test statistic to find an associated value of z from a table of probability values for the normal distribution.
From this you could use the conversion to r.
r = z/square root (N)
Jockheere-Terpstra
rJ-T = z/square root (N)
Writing and interpreting the results
For the Kruskal-Wallis test, report the test statistic H, its degrees of freedom and its significance.
Report the follow-up statistics as well.
Summary
Friedman’s ANOVA: tests differences between three or more conditions when the scores across conditions are related (usually the same entities have provided scores in all conditions).
Theory of Friedman’s ANOVA
Friedman’s ANOVA works on ranked data.
Once the sum of ranks has been calculated for each group, the test statistic Fr is calculated:
Fr = [(12/Nk(k+1)) Σki=1R2i] – 3N(k+1)
Ri is the sum of ranks for each group
N is the total sample size
k is the number of conditions
When the number of people tested is greater than 10, this test statistic has a chi-square distribution with degrees of freedom that is (k-1).
Output from Friedman’s ANOVA
Double-click the summary table to display more details in the model viewer window.
We now see:
Following up Friedman’s ANOVA
We can follow up a Friedman’s ANOVA by comparing all groups, or using a step-down procedure.
Calculating an effect size
We can do a series of Wilcoxon tests form which we extract a z-score.
Then get an effect size r from the Wilcoxon signed-rank test for each comparison.
r = z/square root (N)
Writing and interpreting results
For Friedman’s ANOVA we report the test statistic, denoted by X2F, its degrees of freedom and its significance.
Summary
Join with a free account for more service, or become a member for full access to exclusives and extra support of WorldSupporter >>
This is a summary of the book "Discovering statistics using IBM SPSS statistics" by A. Field. In this summary, everything students at the second year of psychology at the Uva will need is present. The content needed in the thirst three blocks are already online, and the rest
...There are several ways to navigate the large amount of summaries, study notes en practice exams on JoHo WorldSupporter.
Do you want to share your summaries with JoHo WorldSupporter and its visitors?
Main summaries home pages:
Main study fields:
Business organization and economics, Communication & Marketing, Education & Pedagogic Sciences, International Relations and Politics, IT and Technology, Law & Administration, Medicine & Health Care, Nature & Environmental Sciences, Psychology and behavioral sciences, Science and academic Research, Society & Culture, Tourisme & Sports
Main study fields NL:
JoHo can really use your help! Check out the various student jobs here that match your studies, improve your competencies, strengthen your CV and contribute to a more tolerant world
5573 |
Add new contribution