Most data are not perfectly normal. There are now many effective approaches to deal with non-normality in structural equation modeling, for example, the Satorra-Bentler correction, Bollen-Stine bootstrap, and other techniques that provide robust chi square test statistics and parameter standard errors under non-normality. Therefore, having non-normal data doesn't mean you can't use SEM. It's still a good idea to study your variables' distributions, check for outliers, etc of course.

Hello Sarah Bacso. I suspect that you want to use some kind of ordinary least squares (OLS) model. If so, please note that the normality assumption applies to the errors[1], not to the dependent variable. Note as well that normality of the errors is less important than independence and homscedasticity of the errors (i.e., the other assumptions included in the IID, N(0,sigma2) trio of assumptions). And finally, note that normality of the errors is a sufficient condition, but not a necessary condition. The necessary (normality) condition is that the sampling distributions of the parameter estimates (i.e., the coefficients) be approximately normal. As n increases, those sampling distributions can become approximately normal even when the error distribution is not normal. (See the attached PDF if you need references concerning this point.)

With all that in mind, I believe that testing for normality as a precursor to using a t-test (or any other kind of OLS model) is both "silly and pointless":

Presentation Silly or Pointless Things People Do When Analyzing Data: 1. ...

If you want to assess normality, estimate your model, save the residuals, and then examine various kinds of residual plots. (And remember that independence and homscedasticity of the errors are more important than normality, especially as n increases.)

HTH.

[1] Note the important distinction between errors and residuals (https://en.wikipedia.org/wiki/Errors_and_residuals). The normality assumption for OLS models applies to the former, but can (usually) only be assessed using the latter (e.g., in residual plots).

Please note that the S-W test is one of the most incorrectly interpreted tests in statistics. Because of the nature of the hypotheses p greater than 0.05 favors the hypotheses of a normal distribution. Please read the attached information carefully to be sure that your interpretation is correct. Best wishes David Booth

Dear Sir, Statistical tests of normality of the data should not be the condition for any model. The condition is that the Residual must be normally distributed. But there should not be an outlier in your data set. Identify the outlier in the data set by using the Box- Plot. If any outliers are there trimming the outlier.

There is actually a lot. There are two "camps": one sais that normality is a practical assumption, so the data must be normal distributed and that testing for normality is a must, and the other sais that normality is a mathematical assumption, so it's nice to have data that has an empirical distribution that may be described more or less well by the normal distribution model, and that formal tests of normaliy are nonsense if not harmful.

Statements of camp #1 are found in many places, books, courses, and manuscripts. The problem is that they often rigorousley state the importance of the assumption and the tests, but they don't show what the actual consequences are. This is all like "Believe me, because I am saying this, and many others say this, too". But, as W. Edwards Deming coined: ”In God we trust. All others must bring data.”.

Statements of camp #2 are also not rare, but they are less prominent than those of camp #1, because they need some context and some thinking and understanding. Nothing that is as quickly digested as a general, rather non-mathematical statement. These authors often do provide evidence (by mathematical derivation and, often, by extensive simulation) to show how much particular violations of the normality assumption affect the conclusions of the parametric test. (Dear reader: please feel free to provide conuter-examples if you have)

Here are some manuscripts from camp #2:

Boneau, C. A. (1960). The effects of violations of assumptions underlying the t test. Psychological Bulletin, 57(1), 49–64. https://doi.org/10.1037/h0041412

Curran-Everett, D. (2017). Explorations in statistics: The assumption of normality. Advances in Physiology Education, 41(3), 449–453. https://doi.org/10.1152/advan.00064.2017

Huang, S., & Qu, Y. (2006). The loss in power when the test of differential expression is performed under a wrong scale. Journal of Computational Biology: A Journal of Computational Molecular Cell Biology, 13(3), 786–797. https://doi.org/10.1089/cmb.2006.13.786

Knief, U., & Forstmeier, W. (2021). Violating the normality assumption may be the lesser of two evils. Behavior Research Methods, 53(6), 2576–2590. https://doi.org/10.3758/s13428-021-01587-5

le Cessie, S., Goeman, J. J., & Dekkers, O. M. (2020). Who is afraid of non-normal data? Choosing between parametric and non-parametric tests. European Journal of Endocrinology, 182(2), E1–E3. https://doi.org/10.1530/EJE-19-0922

Rasch, D., Teuscher, F., & Guiard, V. (2007). How robust are tests for two independent samples? Journal of Statistical Planning and Inference, 137(8), 2706–2720. https://doi.org/10.1016/j.jspi.2006.04.011

Rochon, J., Gondan, M., & Kieser, M. (2012). To test or not to test: Preliminary assessment of normality when comparing two independent samples. BMC Medical Research Methodology, 12(1), 81. https://doi.org/10.1186/1471-2288-12-81

Rochon, J., & Kieser, M. (2011). A closer look at the effect of preliminary goodness-of-fit testing for normality for the one-sample t-test. The British Journal of Mathematical and Statistical Psychology, 64(3), 410–426. https://doi.org/10.1348/2044-8317.002003

Shuster, J. J. (2009). Student t-tests for potentially abnormal data. Statistics in Medicine, 28(16), 2170–2184. https://doi.org/10.1002/sim.3581

----

PS:

As a disturbing factor in the discussion I see the ill-defined NHST procedure that is often the seen as the framework in which the (non-)parametric tests are performed. That NHST is a flawed procedure is another topic. However, to understand the arguments from the camps you need to set your mind clear about how we usually do tests: We have some data and we then evaluate the statistical significance of the data relative to some (null-)hypothesis within some statistical model. This is done by calculating a p-value, that gives us the empirical significance of the observed data under the tested hypothesis (in that statistical model). Assuming the tested hypothesis was "true", the p-value is a random value from a standard uniform distribution. Under this assumption, it's unexpected to get values very close to zero (e.g.

IMHO it would still be nice to know if the test was correctly interpreted in the first place. Then the camps can go at it as much as they wish. David Booth

Overall, KS is more conservative than skewness or/and kurtosis tests. Also, the normality of the variables is more conservative than the normality of residuals. You can be easygoing while it is needed.

The short answer is:

I partly agree with you (symmetry and kurtosis values), but plot it show that the distribution is at least symmetric and unimodal.

*****

Now consider this

your committee wants you to perform a statistical test, but which one? there are hundreds or normality tests! That is not a very smart advice and I will add that that committee member does not have a deep understanding of statistics if they "think" in such a dichotomous way. Also tests can detect deviations FROM normality not confirm it!

You should take a step back and reconsider why you need normality. If you have a symmetric distribution around a single mode, which is also close to the mean, then I would agree with you. If mean mode and median are grossly different (plot your variable), then it is hard to justify.

People here keep insisting that "residuals" should be normal despite the fact that we are not doing OLS by hand as if it was 1851 (there are plenty if ways around nonnormality of residuals), remember that if you model a variable's (conditional) mean, but the mean is not a useful summary of your (dependent) variable, you can still "run" a model, but you might be modeling a part of the distribution which is not really useful/meaningful.

You do not provide any plots, pvalues or sample size so it is hard to give a useful answer, but building up on Jochen's comment I will add this:

When the data IS INDEED normally distributed, small and very small pvalues are as likely as big ones (because they are uniform).

Now, statistical tests can be used to REJECT a hypothesis, not to accept it, i.e. if you run a normality test and you get a pvalue of 0.1, what can you say, not much actually (it is inconclusive).

On another note, you can apply one of the hundreds of normality tests available and pick whatever result you prefer! Would this be recommendable or even scientific? Clearly not, but it is pretty much what you are doing: you want to show that your data is normal then you are selecting a method that agrees with you

The point of the comments above is that thresholds and dichotomy are not very useful in general, and NHST additionally complicates the approach.

My advice would be to plot the data (plots are more useful in these cases), then you can assess symmetry and other properties: I would worry more about asymmetry rather than kurtosis.

If you plot your data and anybody can tell there is a big asymmetry or bimodality, then it becomes harder for you to claim normality

The R code below shows you that

- you can have 1000 samples from a normal distribution and obtain a pvalue (0.03526) smaller than 0.05

- you can have a small sample from a very skewed distribution and still obtain a pvalue (0.6807) larger than 0.05

set.seed(1234758)

x=rnorm(1000)

shapiro.test(x)

plot(density(x))

set.seed(1234758)

x=rbeta(10,1,10)

shapiro.test(x)

plot(density(x))

Babak Jamshidi what you say about KS is actually very misleading.

KS for normality needs a "pre-defined" set of parameters (e.g. mean=0 and sd=1), in which case its pvalues are uniform under the null.

It is not meant to be compared to a distribution which parameters you estimate from the same variable you are testing, which makes KS-test for normality useless.

Also, that "residuals" need to be normal is also debatable, yes if we use a naive estimator of the standard error I agree, but who does that anymore...

If the population from which the sample is chosen if not-normal.. then using parametric statistics is valid... and offcourse the data should be skewed.

Testing the normality of data does not stop anyone from further analysis. Finding normal data is rare. If you find perfect data there is no need for further analysis. Normality is primarily checked to see the deviation in distribution. In case mean and median are not close to each other, one can transform the data to some extent. If you have picked the sample from the same population any sample size above 30 is assumed to be normal.

It depends on the statistical test you are going to use. It may be robust to non-normality. There is a lot of evidence to support this in the literature.

Before normality test, check the outliers in your data.

It’s true it’s so easy to reject normality when you have even a moderate sample size… when the sample size increases, even small deviations from normal are detected as significant. Anyway, try using the Jacque Bera test as it relies upon skewness and kurtosis

Sarah Bacso Unfortunately, most of the data obtained is not distributed normally. Normal distribution is usually the case for naturally occurring data. For example, height, weight, shoe size. If you are going to use parametric tests in the analysis, the data should be distributed normally. If you don't follow this rule, you'll get criticism. There are various methods to normalize data, but these methods can corrupt the data. My advice is to use non-parametric methods. There are studies that show that if the skewness and kurtosis (e.g., Tabachnick & Fidell, 2013; Curran et al., 1996) are not too high, you can bypass the assumption of normality. You have already specified this.