The t-test is a special case of regression, so I'll discuss this in that context. The assumption of independence isn't that it's not possible to predict the observations, but that you can't predict the residuals. Strictly speaking this isn't true either, because with finite degrees of freedom in any model, the last few residuals can always be predicted from the rest of the information available. However, outside of that, the residuals should be impossible to predict above chance. The paired samples t-test, and mixed-effects models more generally, are required because otherwise the residuals would be able to be predicted to a large degree.
The expectation is that there will be dependence within pairs $(x_i,y_i)$, but this is not actually a requirement -- the test will work correctly whether this is true or not. The test is applied to the pair-differences $d_i =y_i-x_i$; if there's positive dependence, taking account of this pairing by taking differences is helpful in reducing variation.
There is assumed to be independence between those differences $d_i$ is independent of $d_j$. This is unlikely to be true of time series.
Continuous dependent variable – Although the Wilcoxon signed rank test ranks the differences according to their size and is therefore a non-parametric test, it assumes that the measurements are continuous
If they're not, the tabled distribution doesn't apply and the test will depend on the pattern of ties.
To account for the fact that in most cases the dependent variable is binomially distributed, a continuity correction is applied.
This makes no sense to me. How would a continuity correction deal with the problem? In large samples you could retain a normal approximation but use a variance that takes account of the pattern of ties, and in smaller samples you'd attempt to compute or simulate from the permutation distributon.
See also the discussion here
Some articles says that " the Paired differences should be Symmetrical".
The signed rank test is a permutation test on the signed ranks (the ranks of the absolute differences), so if we look at it in that way, then for the signs to be exchangeable under the null (in the sense that every rank would be as likely to have come from a positive as a negative difference), it would seem to require symmetry.
(If you don't have symmetry, then it's not generally the case that under the null you could legitimately reallocate the signs like that - for a given rank one sign would typically be more likely than the other.)
Best Answer
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:
In contrast, when your 2 groups are paired (related) you normally enter your data as if you have just one group, two measures:
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).