I have 3 variables that have been measured on 4 occasions and a sample size of 16. I was considering a repeated measures MANOVA to ascertain if the changes over time were statistically significant. However, I am thinking that my sample size is too small to use a MANOVA in this way. I am considering using 3 separate repeated measures ANOVAs (one for each variable) instead. Is this acceptable and/or justifiable?
As so often, this depends on your purpose and the questions you want to answer.
You sacrifice some information about the correlation of your 4 variables. I say some because part it is usually captured in the estimated coefficients. But you will miss the correlation in the error terms. If you want to capture the relationship between the dependents and the independents you may not miss that, if you want to simulate new data based on your model you might be off by a lot.
It's a choice one often has to make when dealing with many variables and relatively (to the variables) few observations. It's frequently done in -omics research. The usual and ethical way would be state your assumptions clearly (no correlation in the error terms of the 4 variables). Then people can find fault with your assumption but they can't call you dishonest.