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I have a set of experimental parameters and my task it to find reasonable descriptors to describe them (chemistry).

Since I've got descriptors, I checked Pearson correlation for each of experimental parameter and then chose descriptors with P<0.05 for further investigation. But at this step I did not check normality of my data. After that I proceeded with multiple linear regression and found suitable models. For each model I checked normality by Kolmogorov-Smirnov test with Lilliefors correction and also by Shapiro-Wilk test (o.5 from 2). All models passed these tests, however visual diagnostics of normal Q-Q plots showed some non-normal distribution in my opinion. So, my questions are:

(1) Is it enough to rely only on Kolmogorov-Smirnov and Shapiro-Wilk tests to check normality of models?

(2) Is it correct that I did not check normality of my descriptors before doing MLR or it is a serious mistake? If it's not OK, does it mean that I have to transform non-normal distributed descriptors and then repeat Pearson correlation once more?

(3) Say, I have non-normal distribution for one descriptor in model, whereas two others do not have such problem. Can I do transformation only for this problematic descriptor or it should be done for all of them? I supposed it is very simple question and answer is yes, but anyway.

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First, normality of model residuals is not needed for OLS estimates to be BLUE (best linear unbiased estimators, where best means minimum variance) asymptotically. Unless your sample is small (where asymptotic results do not kick in yet), non-normality is not a problem.

Second, selecting descriptors based on correlations with the response variable need not be a good strategy. Often descriptors are jointly significant when they are not significant individually. You may want to study strategies for selection of descriptors more before proceeding (look for tags feature-selection, model-selection, regression-strategies).

Now regarding your questions:

(1) I do not have a strong opinion. These tests should be OK.

(2) Descriptors need not be normal for OLS estimates to be BLUE. Non-normality of descriptors is not a problem. For example, dummy variables are frequently used in multiple linear regression, and clearly they are not normal.

(3) As I said in (2), non-normality of descriptors is not a problem.

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  • $\begingroup$ Thank you. Sorry, I was unclear. Obviously, my choice was not based only on correlation between the response variable and descriptors. I typically relied on cross-validation, bootstrap, and AIC. I think I should consider my dataset as small (17). However, the comments about normality is a bit unclear for me. Let's say, my model demonstrates normal Q-Q plot quite far from normal distribution, but it should not be a problem and I still can go with that? $\endgroup$
    – Mikhail
    Mar 24, 2015 at 11:45
  • $\begingroup$ It is not the model that is normal or non-normal but rather variables in it or its residuals. So I assume you are talking about the residuals here. But what do you mean by a small dataset? Is 17 the number of observations in your sample? That would be very, very few! Very little could be done when it comes to regression modelling when the sample is so small... $\endgroup$ Mar 24, 2015 at 11:48
  • $\begingroup$ Yes, I am talking about residuals of variables. I have dataset with only 17 structures, right. I know that is very limited, but that's all what I can operate with. The main idea was is to find suitable descriptors and then I decided to try multiple linear regression rather than just simple linear regression. $\endgroup$
    – Mikhail
    Mar 24, 2015 at 12:04
  • $\begingroup$ I think the most results for linear regression and OLS estimation need to be taken with a few grains of salt when the sample is so small. I suspect there could be alternative strategies to the standard regression modelling when the sample is so small. I do not have experience with such small samples, but I suggest you look for some specific literature. Perhaps you could do something with bootstrap (just wondering)... By the way, residuals of variables also does not make much sense. Model residuals/regression residuals is perhaps what you have in mind. $\endgroup$ Mar 24, 2015 at 12:11
  • $\begingroup$ OK, I will think about it. Thanks. I am chemist, so statistics is really new for me. That's why I'm confused with terminology. $\endgroup$
    – Mikhail
    Mar 24, 2015 at 12:18
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(1) Use these tests and various graphs for residuals from the models which you have done

(2) I think that it is normality of residuals which is of more interest. Various test statistics are valid if results hold. I do not think it is much use to transform predictors just because you want their distribution to resemble normal distribution. Usually variable transforms are done because you might have problem with heteroskedasticity, which is non constant variance of residuals.

(3) As in previous question, check residual normality and transform only if you have problem with heteroskedasticity.

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  • $\begingroup$ Thank you. So, I can use models if there is no problem with heteroskedasticity, right? $\endgroup$
    – Mikhail
    Mar 24, 2015 at 11:47
  • $\begingroup$ @Mikhail Yes, but of course you must test other things also besides residual normality and/or homoskedasticity. Your sample size is small though.. $\endgroup$
    – Analyst
    Mar 25, 2015 at 6:18

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