I have recently conducted a study with 5 dependent variables each measured on different scales. The data meets all assumptions for a MANOVA, except the Kolmogorov–Smirnov Test is significant, revealing that my data is not normally distributed. I have read in several academic sources that parametric tests can still be run for non-normal data if the sample size is large. My sample size is roughly N = 1000. I am curious what my fellow researchers on here think?
Runnning the KS test to "decide" if the data normal distributed is nonsense. It's significant. Now what? Yeay, you have demonstrated that you gatherd a large enough sample to reject the (practically non-sensical) hypotehsis that the data is from a variable with a perfect normal distribution. And what if it was non-signficant? All fine because this demonstates that the data are in fact from such a variable (apart from the actually silly conclusion, this would be interpreting a non-significant result, what is a no-go anyway)?
The real question is if the assumption of normally distributed test statistics is reasonable. No formal normality test can answer that question, because it needs to account for the scientific problem, the context of the experiment and the aim of the analysis and a judgement of how large and how serious a bias in your conclusions can reasonably be.
Although formal tests of assumptions are nonsense, it is worth checking the assumptions. It can always be that you missed something relevant and the data literally screams at you that your ides are far from reasonable. The best way to check this is having a look at residual diagnostic plots. If you find particular patterns there, they can give you a hint what you missed (and if you should re-consider your model; be it the functional and/or the stochastic part). This can be meaningful even if the sample size is large. It may show you, for instance, that a multiplicative model might be more appropriate than an additive model - even if the inference on additive effects may not be too bad, given the large sample size. Or it may, possibly, show you that you may have missed an interaction between predictors, etc.
Generally, checking the plot of residuals is helpful. If it does not meet the normality shape, modify it by using a transformation.
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