1. Is that correct and reliable to use a parametric test when the hypothesis of normality is not approved by Kolmogorov–Smirnov test? (Sample size greater than 100)
2. Can we assume our data is normal and apply post_hoc or ANCOVA If our sample size is greater than 100?
1. Often but not always. It depends on the actual distribution. Side-note: normality tests never "approve normality". They can only show that your sample size is large enough to statistically recognize deifferences between the frequency distribution of your sample an the idealized model of the normal distribution. A significant normality test shows that the sample size is large enough but no indication if the deviation is of any relevance; a non-significant normality test only says that the sample size is too small and nothing else - particularily not that there would be no deviation or that a possible deviation would be irrelevant.
2. If your variable is not normal, it won't in some woundrous way become normal distributed just because you analyze it with some statistical model. Side-note: with "a variable (Y) is normal", is meant that the mean of Y may depend on other variables (X1, X2, ...). If the mean does depend on other variables, then you would "see" the normal distribution only when you have many values of Y obtained under the same values of the predictors (X1, X2, ...). After correctly modelling the dependency structure (by an ANOVA, regression or ANCOVA), then the residuals of the fitted model will be normal distributed with mean 0. This is easier to check. So in this way, the distribution of the residuals does in fact depend on the model. Finding non-normal residuals can indicate that the variable is not normal distributed, but it can also indicate that the model is mis-specified (or both).
KS-test is only appropriate if you know the parameters beforehand, but not if they are estimated the data. There is more appropriate tools to test normality, but you could compute a histogram of your variable you should be able to gauge symmetry/unimodality etc, pvalues do not mean much when you test normality with a large sample size
Inspect the histogram of the distribution. If not heavily skewed e.g., run the ANOVA. Note that the Quade RANCOVA can have more power than the parametric ANCOVA. See attachment and:
Poster rANCOVA - A Robust Method for Covariate Adjustment in Observ...
https://www.researchgate.net/post/Is-it-acceptable-to-use-Quades-test-for-non-parametric-ANCOVA
Many thanks for all your answers; it was so helpful to me. Jochen Wilhelm Stefano Nembrini Jos Feys Y. Luo
Did you try Anderson–Darling test or Shapiro to test the normality https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test
As the sample size increases the parametric test gets more sensitive/ powerful to departures from normality. You may use visual techniques in this case.
Best !!
AN
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