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I' conducting a Normality Test for a numerical variable (level of a substance in blood) in a sample of 368 subjects.
As I will need to run analysis of covariance (ANCOVA) and partial correlations, I find myself a little bit in doubt with this result. According to the p value of the Kolmogorov-Smirnov (0.200), my variable has a normal distibution (while according for Shapiro-Wilk it hasn't). Anyway, I'm wondering what "This is a lower bound of true significance" means, referring to the p = 0.200. Should I care about it? To note, histogram, skewness and kurtosis suggest a normal distribution of the data. Outliers have been excluded with the z score method (>3). If I transform the values in log10, the distribution results non/normal.

Thank you in advance for your replies!

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    $\begingroup$ Part of this is a FAQ: ANOVA does not care how your response variable is distributed. Thus, although your question about "lower bound" has a (simple) answer, it might not be helpful to you. $\endgroup$
    – whuber
    Dec 4, 2018 at 17:40
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    $\begingroup$ In addition "According to the p value of the Kolmogorov-Smirnov (0.200), my variable has a normal distibution" is not a correct interpretation of a failure to reject the null". $\endgroup$
    – Glen_b
    Dec 5, 2018 at 12:23
  • $\begingroup$ Make sure that you understand the meaning of p-values. A non-significant p-value doesn't tell you that the null hypothesis is true. It tells you that you have failed to find evidence that it is false. It's still probably false. $\endgroup$ Aug 29, 2021 at 3:32

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The Kolmogorov-Smirnov test is a test of the null hypothesis that your data come from some prespecified distribution, $F$. Coming from a Normal distribution is a much wider null hypothesis; there are lots of Normal distributions.

At one time it was common to find the best-fitting Normal distribution to your data (by estimating the mean and variance from the data), then pretend that was a prespecified distribution and do a Kolmogorov-Smirnov test. It's pretty clear that you will be less likely to reject the null that way -- the fitted normal distribution will fit your data better, because it has been fitted to your data. So, if you take the test statistic $D$ and your observed value $d$, and calculate the $p$-value $P(D\geq d)$ as if you had a prespecified distribution, the number you get will be too large. It will be larger than the true $P(D>d) for the testing procedure you actually used.

Lilliefors's test fixes this up. It uses the same test statistic $D$ (the maximum vertical difference between cumulative distribution functions), but it calculates the null distribution of $D$ taking account that you have estimated the mean and variance from the same data. How does it do that? By brute-force simulation. Someone (Hubert Lilliefors, or perhaps his research assistants) generated lots of sets of data from Normal distributions, did the KS test on each set, and made a big table of the null sampling distribution. They did this in 1967 when it was a big deal; you could do the same thing in a few minutes now with R but we've gotten used to using the stored tables.

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It is a lower bound. For example, 2 minutes ago, I noticed that we can test normality with lillie.test() function in R software. Then, I used my normal distributed data to get K-S and Shapiro-Wilk test results. I had the statistics from SPSS. Shapiro-Wilk stats were same. But K-S test results were different. R said that p value of K-S was 0.178 and SPSS said that p value of K-S was 0.200 (and This is a lower bound of true significance). I got that my data is normal. It's still normal but SPSS gave us a lower bound.

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