My data consist of compaction measurements from 3 different cell types (X,Y, and Z). My goal is to know whether there are "significant" differences between these measurements, so I have tested for:
Whether my samples are normally distributed
- using the Shapiro–Wilk test
- using the Jarque-Bera test
- plotting qqnorm graphs
- plotting histograms
Whether the samples come from the same distribution
- using two-sample Kolmogorov–Smirnov test (K–S test) and comparing X vs Y, X vs Z, and Y vs Z
- using Kruskal–Wallis comparing X, Y, and Z together
My data consist of 232 measurements for X, 284 for Y, and 124 for Z. The Shapiro-Wilk and Jarque-Bera tests in R always give me p<0.05, which I accept as not being normally distributed. However, when I plot histograms I get a normal-like distribution.
The qqnorm plots also don't look that skewed, but maybe this is just my inexperience in interpreting qqnorm graphs (this is my first time making them).
Because of the supposedly non-normal distribution, I compared my data using KS test and Kruskal-Wallis, which always give me the result that my population Z is drawn from a different distribution compared to X and Y. However, I do not know if this is true, as R always reports for my two-sample Kolmogorov–Smirnov tests:
Warning message: In ks.test(dataX, dataY) : cannot compute correct p-values with ties Warning message: In ks.test(dataX, dataZ) : cannot compute correct p-values with ties Warning message: In ks.test(dataY, dataZ) : cannot compute correct p-values with ties
probably because the samples have different sizes.
I'd like to know what you think about it, and whether I should consider using more parametric tests rather than the non-parametric ones I've used, or whether the tests I've used are valid regardless of the normality of the data. Also, my measurements seem to differ very little among themselves, for example:
---Summary stats for WT cells Min. 1st Qu. Median Mean 3rd Qu. Max. 0.1450 0.3720 0.5000 0.5598 0.7102 1.9290 ---Summary stats for Df cells Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0550 0.4030 0.5445 0.5857 0.7210 1.5350 ---Summary stats for Dp cells Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0670 0.4790 0.6255 0.6782 0.7897 2.0160
Here's a boxplot of the data:
Green=X, Blue=Y, Red=Z
So I'm unsure about the conclusions I may derive from them.