Would paired t-test and repeated measures ANOVA with two level of repeated measures on the same data give the same results? I ran a few tests and realized that their p values are the same. Does mean that the two methods are statistical equivalent? But I also find it strange that their assumptions for normality are slightly different, with paired t-test requiring difference in continuous variable to normal while repeated anova requiring the continuous variable to be normal for each level of within subject factor. Please advise.
yes, they are equivalent. these assumptions question has never been directly addressed, though. it is sometimes indicated, that the assumptions you cite for anova, when met, do cover the normality assumption for paired t-test. however, I still wonder, what when the variables are not normal within each subgroup, but their differences (calculated like for t-test) are normal? This should be enough, so the incongruence between these assumptions (as stated in every major statistics handbook) and in your question, are bothering to me too. ;)
As both are equivalent, "bivariate" repeated measures Anova works as well if only the differences are normally distributed. The more strict repeated measures requirements in the literature are only necessary if more complicated layouts with more factors and other hypotheses are included, in particular between subject factors and unequal sample sizes / heteroskedasticity between them.