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I have conducted a parametric test in a study, n=290. I want to assess whether the residuals of this test are normally distributed.

  • The skewness and kurtosis of the residuals are -0.017 and -0.438 respectively. I think this is considered as normal.
  • Unfortunately, the Kolmogorov-Smirnov of the residuals has a p-value of 0.021. I think this is considered not normal.

Question

What should I do when skewness and kurtosis look normal but Kolmogorov-Smirnov is significant?

Jeromy Anglim
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user25551
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    What "parametric test" are you anticipating using? Many (if not most) of them do not require normality of the residuals, but only need behavior that suggests the sampling distributions of appropriate statistics (such as the variance of the residuals) are sufficiently close to the distributions of Normal residuals to allow accurate computation with a Normal approximation. – whuber May 12 '13 at 17:25
  • multiple regression and spearman correlation – user25551 May 12 '13 at 23:20
  • "Multiple regression" is a large body of related procedures--a model, really--and "spearman correlation" is a statistic. Neither is a statistical hypothesis test (although various tests are commonly associated with them). – whuber May 13 '13 at 12:51

2 Answers2

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In order to make sure that I can use parametric test, I need to make sure that my residual distribution is normal.

There is really no way to demonstrate that you have exact normality, but that's okay because approximate normality will generally be sufficient for hypothesis tests in regression to work the way you want.

However, when I refer to the value of skewness and kurtosis of the residual, it is -0.017 and -0.438 respectively, where i think this is considered as normal.

You can obtain values like that with residuals from a simple regression on normal data, but the kurtosis is just significant at the 5% level.

(Technical aside: I used simulation to assess the significance of the kurtosis of residuals here; not knowing the number of predictors, I did it for both independent normals and for one predictor at the given sample size, both showed essentially the same p-value; results should be similar for regression with small numbers of predictors.)

This doesn't actually suggest a problem with the inference when doing a regression or correlation, however. Your data won't be exactly normal; the essential question is 'are the data so badly non-normal that the inference no longer has the properties you wish?'

Unfortunately, when i do kolmogorov-smirnov, the significant value is 0.021, which indicates the residual is not normal.

What were the specified population mean and variance of the residuals for your KS test and how did you get such population values?

Could anybody please explain to me what to do.

I suggest you don't do a hypothesis test to assess the suitability of the assumption of normality, but instead to look at diagnostic displays that show you how badly non-normal the data are.

Some pointers -

See the points here

Also see the discussion on this question

See the comments under this answer, and the advice in this answer

Consider this advice

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Glen_b
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  • Most of this is great (+1). But concerning the first part of this answer, you can be pretty confident that no null hypothesis is correct, either. Although this is a point worth making in other contexts, when pressed early and hard in an answer to an elementary straightforward question it seems it might be just a little disingenuous and counterproductive. – whuber May 13 '13 at 12:56
  • @whuber I have tried to amend it as best I can. – Glen_b May 13 '13 at 13:25
  • Thank you for that. I didn't intend anything other than to draw your attention to the possibility you might have been making a legitimate point in a way that could be seen as extreme. – whuber May 13 '13 at 13:25
  • The K-S test needs to be modified when applied to situations with unknown parameters [Durbin 1975](http://www.jstor.org/stable/2334482). So I would not put that much trust on it here. – StasK Jul 06 '13 at 13:17
  • @StasK Indeed, that was the purpose behind my question "*What were the specified population mean and variance of the residuals for your KS test and how did you get such population values?*" – Glen_b Jul 07 '13 at 01:25
  • I am not sure this is a meaningful question. Unlike say the likelihood ratio that you can take a square root of to produce a signed ratio, and get some sort of residuals, K-S does not have an asymptotically normal distribution, or a derivative of the normal such as chi-square. I am not convinced that analyzing the residuals from K-S test is a fruitful idea. – StasK Jul 08 '13 at 13:31
  • @StasK We are having a failure to communicate. The 'residuals' being referred to are not deviations in a KS test -- they're the input *to* the KS test - what is being tested for normality in the first place. See the second sentence of the original poster's question. – Glen_b Jul 08 '13 at 22:21
  • OMFG. Somebody needs to learn to write their questions. I am not going to learn to read between the lines. – StasK Jul 08 '13 at 22:34
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  1. If you are running one or more tests of normality, a rejection by one of the tests is enough to reject the normality. You can't cherry pick the normality test suite looking for a desired results, it's a dangerous approach.

  2. Look at this answer Kolmogorov-Smirnov test - reliability on other tests of normality. Basically, KS test is not good to test normality. The tests like Jarque-Bera will use the moments and are better than eye balling the skewness/kurtosis or KS test.

Let's see what is JB test statistics (mentioned in my answer in the above link): $jb=290/6*(0.017^2+0.438^2/4)=2.33$, the critical value at 300 observations and 5% confidence is 3.68, so JB test does not reject normality.

CORRECTION: My conclusion: assuming that Kurtosis is corrected for 3, there is not enough evidence to suggest that the data is not normal based on JB test alone, but since KS test is rejecting it I'd still reject normality.

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Aksakal
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    A slightly different interpretation of this question presents an interesting counterpoint. The question is of the form "I conducted *one* hypothesis test (KS) that finds a significant difference, but the *effect size* (skewness and kurtosis) is not large. What do I do?" In this interpretation, a good answer to the question would discuss the sensitivity of the (unnamed) analysis to nonzero skewness and kurtosis in the residuals. Unfortunately the OP did not respond to a request for information about that original analysis. IMHO, that makes the question unanswerable--but Glen_b did a good job. – whuber Apr 30 '14 at 20:57