# 6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test

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1 6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test Having finally arrived at the bottom of our decision tree, we are now going to learn how to look for differences between two groups. The four tests covered in this course are very commonly used in many publications, however, as always, other statistical tests are available and these might be correct to use, too (it is your own responsibility as a scientist to understand the underlying assumption criteria if you decide to use a different test). The Null Hypothesis question/statement for any tests looking at differences between groups is: There are no differences between the groups.? p < 0.05 = there is a significant difference between the groups! Note that in contrast to all tests we have covered so far, we are now looking for a p value <0.05 to be able to state that we have a significant difference between groups. Experimental data (for Exercise ): We have collected serum from male and female subjects at two different timepoints and we have measured the concentration of a protein (in pg/ml) in this serum in the laboratory. Student s T-test Exercise 6.1: 1. In the Excel spreadsheet Excercises_2, select the tab Student s T-test I. 2. Copy and paste the raw data into the SPSS Data View tab. 3. Label the data appropriately in the Variable View tab. 4. In the menu bar, go to Analyse-> Compare Means->Independent samples T-test (see screen shot). 1

2 5. As soon as you select Independent samples T-test, a new small window called Independent samples T-test opens up, with several selection options. 6. Using the top blue arrow, transfer Serum Protein over into the box called Test Variable (s) (see screen shot). 7. Using the bottom blue arrow, transfer Gender over into the box called Grouping Variable (see screen shot). 2

3 8. Select the blue button Define Groups underneath the Grouping Variable box. 9. As soon as you click the button, a new small window called Define Groups opens up. 10. Type the number 1 in the box for Group 1 and type the number 2 in the box for Group 2 (see screen shot). These are the numbers that we have defined in Gender. 3

4 11. Select Continue and Ok. 12. The SPSS Output Window contains 2 tables. The top table shows Group Statistics and tells you which groups you have compared (e.g. we have compared the serum protein concentrations between male and female) (see screen shot). It also tells you useful information such as the sample size (N), mean and Standard Deviation. This is similar to the Summary Statistics we have learned at the beginning of the course. 13. The second table shows the results of the Student s t-test (Independent Samples Test). The p-value can be found under the heading Sig. (2-tailed) (see screen shot). The p-value is <0.05, which means that we have a significant difference between the serum protein concentrations of the two genders. 14. In the table, you can also see a section called the Levene s Test for Equality of Variances. This is the exact same test that we have learned previously. Since this test is so important, SPSS automatically performs it for you and you don t have to do it yourself every time. However, you can also see from the table, that SPSS returns two rows and even two p values for the t-test. The first row is Equal variances assumed and in this row you find the result of the Levene test (see screen shot). 4

5 15. The second row is Equal variances not assumed. This test is a different test than the Student s t-test and you should ignore all results in this row (especially the p-value of the T- test, even if it is significant!). In case that the Levene test shows a p-value <0.05 in the top row, you should also ignore the results of the top row and follow our decision tree to the Mann-Whitney U test. Exercise 6.2: 1. In the Excel spreadsheet Excercises_2, select the tab Student s T-test II. 2. Repeat Exercise 6.1 with this new dataset. Solution Exercise 6.2: The p-value for the Levene test is highly significant (p<0.001), meaning there is no homogeneity of variance between the two genders. Therefore, we cannot take the results of the T-test as valid and have to continue using the Mann-Whitney U test. Mann-Whitney U test Exercise 6.3: 1. In the Excel spreadsheet Excercises_2, select the tab Mann-Whitney-U I. 2. Copy and paste the raw data into the SPSS Data View tab. 3. Label the data appropriately in the Variable View tab. 4. In the menu bar, go to Analyse-> Nonparametric Tests-> Legacy dialogs->2 Independent Samples (see screen shot). 5

6 5. As soon as you select 2 Independent Samples, a new small window called Two Independent Samples opens up, with several selection options. 6. Using the top blue arrow, transfer Serum Protein over into the box called Test Variable List. 7. Using the bottom blue arrow, transfer Gender over into the box called Grouping Variable. 8. Select the blue button Define Groups underneath the Grouping Variable box. 9. Define the groups (1 and 2) as described in exercise Select Continue and Ok. 11. The SPSS Output Window again contains 2 tables. The first table (Ranks) gives an overview of the group comparison (it does not contain the summary statistics that the T-test does, see screen shot). 12. The second table contains the Test Statistics. You can find the relevant p-value for the Mann-Whitney U test in the row called Asymp. Sig. (2-tailed) (Asymptotic Significance), see screen shot. The p-value is <0.05, meaning that there are differences in the serum protein concentrations between the two genders. 6

7 Exercise 6.4: 1. In the Excel spreadsheet Excercises_2, select the tab Mann-Whitney U II. 2. Repeat Exercise 6.3 with this new dataset. Solution Exercise 6.4: After completing this dataset, in the Test Statistics table you find, in addition to Asymp. Sig., another p-value called Exact Sig. (Exact Significance). Whenever this p-value appears, you should be using this instead of the Asymp. Sig. The Asymptotic p-value is to be used with a larger sample size, the Exact p-value with small sample sizes. We will discuss later what a small and what a large sample size is. The p-value for the Mann-Whitney U test is not significant (p>0.05), meaning there is no difference in serum protein concentrations between the two genders. 7

8 Paired samples T-test Exercise 6.5: 1. In the Excel spreadsheet Excercises_2, select the tab Paired T-test I. 2. Copy and paste the raw data into the SPSS Data View tab. 3. Label the data appropriately in the Variable View tab. 4. In the menu bar, go to Analyse-> Compare Means->Paired-Samples T Test (see screen shot). 5. As soon as you select Paired-Samples T Test, a new small window called Paired- Samples T test opens up, with several selection options. 6. Using the blue arrow, transfer Serum Protein Month 0 and Serum Protein Month 3 over into the box called Paired Variables (see screen shot). 8

9 7. Note that the variables still stay in the left hand box. This allows for doing several comparisons with multiple pairs in one go. 8. Select Ok. 9. The SPSS Output Window contains 3 tables. The first table (Paired Samples Statistics) is again a summary statistics (see screen shot). 10. The 3 rd table contains the result of our Paired samples T-test (we can ignore the 2 nd table). The p-value can be found at the very end of the table under the heading Sig. (2- tailed), see screen shot. It is highly significant (p<0.001), meaning that there is a significant difference in serum protein concentrations between the timepoint Month 0 and the timepoint Month 3. 9

10 Exercise 6.6: 1. In the Excel spreadsheet Excercises_2, select the tab Paired T-test II. 2. Repeat Exercise 6.5 with this new dataset. Solution Exercise 6.6: The p-value is highly significant (p<0.001), meaning that there is a significant difference in serum protein concentrations between the timepoint Month 0 and the timepoint Month 3. Wilcoxon test Exercise 6.7: 1. In the Excel spreadsheet Excercises_2, select the tab Wilcoxon test I. 2. Copy and paste the raw data into the SPSS Data View tab. 3. Label the data appropriately in the Variable View tab. 4. In the menu bar, go to Analyse-> Nonparametric Tests-> Legacy dialogs->2 Related Samples (see screen shot). 10

11 5. As soon as you select 2 Related Samples, a new small window called Two Related Samples Tests opens up, with several selection options. 6. Using the blue arrow, transfer Serum Protein Month 0 and Serum Protein Month 3 over into the box called Test Pairs (see screen shot). 7. Note that similar to the paired samples t-test, the variables still stay in the left hand box. This allows for doing several comparisons with multiple pairs in one go. 8. Select Ok. 9. The SPSS Output Window looks similar to the Output of the Mann-Whitney U test. The p- value of the Wilcoxon test can be found at the bottom row of the second table (see screen shot). It is not significant (p>0.05), meaning that there is no difference in serum protein concentrations between the timepoints Month 0 and Month 3. 11

12 Exercise 6.8: 1. In the Excel spreadsheet Excercises_2, select the tab Wilcoxon test II. 2. Repeat Exercise 6.7 with this new dataset. Solution Exercise 6.8: The p-value is highly significant (p<0.001), meaning that there is a significant difference in serum protein concentrations between the timepoint Month 0 and the timepoint Month 3. We have now completed our decision tree for comparisons between two groups. 12

13 By following the decision tree, you have successfully learned how to make the right decision (depending on the type of your dataset) when to use which test to compare two groups for differences between them. However, many experiments have more than two groups and I will now introduce a more complex decision tree to compare differences for more than two groups (in theory, as many groups/conditions as you like ). 13

14 The beginning and end of this decision tree is the same as the decision tree for comparisons of 2 groups. We have first decisions about the type of dataset we have, followed by decisions on which assumption criteria our dataset meets. At the end of the decision tree we have our standard 2-group comparison tests, the Student s T-test, Mann-Whitney U- Test, Paired Samples T-test and Wilcoxon Test. You can ignore the bit about Bonferroni correction for now, we will cover this later. Above the 2 group tests we have four new tests, which are used to detect differences between more than 2 groups. On the left-hand side (e.g. the independent samples side) of the tree, we have the Oneway ANOVA and the Kruskal-Wallis test. On the right-hand side (e.g. the dependent samples side) of the tree, we have the Repeated Measurement ANOVA and the Friedman test. These four tests can be used to simultaneously test if there are differences between more than 2 groups. The tests can be very powerful and sometimes detect differences when a normal 2 group test doesn t. However, as you will see later on, these tests only will give you an answer to the question if there are differences somewhere between any of the tested groups. They will not tell you where these differences are and you need to do a 2 group test to find out exactly this. This is so-called post-hoc testing and we will learn more about it 14

15 later. Also, you should note that because these tests are very powerful, if they don t find a difference between the groups, no other of the post-hoc tests will find one. So if you have for example 100 groups, you do not have to do 1000s of post-hoc tests if one of the big multiple group tests does not find any differences. Try as you might, you will not find any differences with the 2 group tests. Therefore, if you do have multiple groups, it is good practice to always do a multiple group test first before the 2 group tests, as this may save you a lot of time. 15

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