Question about independence assumption for ANOVA, t-test, and non-parametric tests

Question

I'm a novice in statistics and I have some confusion about the assumption of independence for statistical tests.

1. I searched the Internet and some information says that for the t-test, the observations in the two groups should be independent (that is, measurements in sample 1 and measurements in sample 2 should be different). Some other information says that all observations (even in the same group) should be independent. Which one is correct?
2. Is the independence assumption for ANOVA and the independence assumption for the t-test the same?
3. Do non-parametric tests, such as the Wilcoxon signed rank test, also need to satisfy the independence assumption?

Answer 1

First. Observations in two groups should be independent means that the two groups consist of different individuals, not the same individuals measured twice or specially matched individuals (such as siblings). When you have two independent groups, your data look as follows:

id group characteristic
1     1       3.4
2     1       1.6
3     1       2.8
4     2       0.9
5     2       5.3
6     2       5.0

In contrast, when your 2 groups are paired (related) you normally enter your data as if you have just one group, two measures:

id   characteristic
    measure1  measure2
1     3.4        0.9
2     1.6        5.3
3     2.8        5.0

All observations (even in the same group) should be independent. This is also true and it means that each row of the data (see above data examples) was included in the sample independently of other rows: observation with id=1 is sampled independently from observation id=2 or id=3.

Second. They are the same. T-test for independent groups can be treated as a particular case of one-way ANOVA for independent groups.

Third. There are many different nonparametric tests. The Wilcoxon test you are talking about is a two paired-samples test, thus, it needs non-independent groups (with independent observations within groups). The non-parametric test for two independent groups is called Mann-Whitney test (and rarely called Wilcoxon test, too).