Interpreting multivariate tests in repeated measures I have a within subjects design with 4 DVs and I ran a repeated measures analysis on SPSS. My DV is a physiological measurement and is not expected to be linear.
My multivariate Wilks lambda is significant at p = .004 and my Mauchly test approaches significance at p = .054. However, my univariate tests of within subjects is non-significant (uncorrected  p =.07 and greenhouse Geisser p = .082)


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*Given that my sphericity test is almost significant and given that I do not expect linearity in the data, is it reasonable to look at my multivariate test and ignore the non-significant univariate test? 

*Do I even have to look at univariate if the multivariate is significant? I.e., am I justified to look at my pairwise comparisons simply because the multivariate is significant?
 A: Multivariate tests for repeated-measures data are seldom, at least in published results in the experimental psychology I regularly read. Nevertheless there are recommendation for these tests. Most notably (from the top of my head):
O’Brien, R. G., & Kaiser, M. K. (1985). MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin, 97(2), 316–333. doi:10.1037/0033-2909.97.2.316
Algina, J., & Keselman, H. J. (1997). Detecting repeated measures effects with univariate and multivariate statistics. Psychological Methods, 2(2), 208–218. doi:10.1037/1082-989X.2.2.208
Keselman, H. J., Algina, J., & Kowalchuk, R. K. (2001). The analysis of repeated measures designs: A review. British Journal of Mathematical and Statistical Psychology, 54(1), 1–20. doi:10.1348/000711001159357
(I specifically liked the last one, Keselman et al., 2001)
You might also want to have a look at the multivariate statistics book by Tabachnik and Fidell.
It would be surprising if you couldn't find the recommendation for using the multivariate and ignoring the other (unfortunately nonsignificant) tests in your case in these papers.
A: The difference between multivariate and repeated measures tests is that both "favour" different alternatives while both maintaining (approximately) their $\alpha$-level. For "spherical data" i.e. spherical covariance matrices, this difference is rather small. So all your sphericity test tells you is that both tests may differ.
If you introduce a new procedure of picking the smallest p-value, you effectively inflate the type I error. You have to specify which procedure to take prior to analysis or even data gathering. 
As Henrik already wrote, for univariate repeated measures data it is best to take repeated measures analysis since they will work for sure, whereas the multivariate tools will fail e.g. if the data become high dimensional of if by bad luck the empirical covariance matrix becomes singular.
