This is a problem that I think nothing would help but sharing my data here. I have 3 groups (n=7) which are paired. They are concentrations of a chemical compound in different parts fish tissue. For example, the 5th number in G1 and G2 and G3 are 3 different measurements from 3 different parts of the same fish. Thus, We examined 3 different parts of each of our 7 different fishes. Shapiro-Wilk and d'Agostino-Pearson tests are significant (p<0.05) but the Levene's test for means indicates the homogeneity of the data (p=0.57). Because of the outliers, I decided to run Welch's ANOVA instead of normal one-way ANOVA. The result is insignificant.
Some other researches on the same type of data, reported the log(10) values and ran ANOVA and observed significance between the groups of their own data. I did so, and the Log-transformation of my data is normal (Shapiro-Wilk Test) and homogeneous (Leven's test).
One way ANOVA (I'm not sure if it is a good idea to use normal ANOVA when n=7 only), is insignificant but when I use contrast post-hoc, there is a significance between G1 and G3. Welch ANOVA and Kruskal-Wallis tests are still insignificant when I use the Log values. Out of curiosity, I performed t-test for independent samples between all of my Log groups and there are significant differences between G1-G3, G2-G3 but G1-G2 is insignificant. Even the same t-tests on the raw data (before logarithms) shows exactly the same results. These results are in accordance with what other researchers have found with their dataset. Before each t-test, I ran a F-test to make sure my variances are not significant.
So, does it mean that I can accept the paired t-test although it is contrary to the omnibus Welch's ANOVA and other tests?
Update: I ran Mann-Whitney Test for Two Independent Samples since the normality of data was questionable and the results of the t-test were confirmed with non-parametric test too.
My Data are as follows:
G1 G2 G3 37.770 30.94 20.040 22.945 32.55 16.790 53.230 51.83 41.740 21.960 20.99 16.325 25.920 19.11 20.710 40.290 34.74 23.380 28.960 28.21 21.490