Normality testing with very large sample size?

Question

Hypothesis testing such as Anderson-Darling or Shapiro-Wilk's test check normality of a distribution. However, if the sample size is very large, the test is extremely "accurate" but practically useless because the confidence interval is too small. They will always reject the null, even if the distribution is reasonably normal enough.

How should I test normality when sample size is very large, other than visualizing histograms?

The motivation is that I want to automate checking normality of large data set in a software platform, where everything needs to be automated, not manually visualized and inspected by humans.

One thing that came across me is that instead of using Shapiro-Wilk test, I calculate kurtosis and skewness of the distribution, and if they are $\pm 1.0$, I can assume that my large dataset is "reasonably" normally distributed.

Is my approach correct, or is there any other alternatives?

Don't know about just looking at skewness and kurtosis. Seems the main issue is to be realistic about what closeness to normality is needed for the application at hand. — BruceET, Jun 24 '19 at 06:41
What's your motivation to check normality in the first place? Why is normality important in your application? — COOLSerdash, Jun 24 '19 at 07:43
Shapiro, not Saphiro. Hypothesis tests of assumptions answer the wrong question (e.g. see here); when it comes to assumptions of tests, I suggest avoiding them at any sample size. — Glen_b, Jun 24 '19 at 08:44
@Glen_b I came up with this question after reading the link you put. In the link, I read that hypothesis testing for normality in very large sample size is useless. My question is, how do I know that my distribution is normal "enough", and what technique i should use, other than visualizations? What techniques are realistic about measuring closeness to normality, as Bruce says? — Eric Kim, Jun 24 '19 at 16:46

BruceET · Accepted Answer · 2019-06-24T06:57:10.350

8

Continuation from comment: If you are using simulated normal data from R, then you can be quite confident that what purport to be normal samples really are. So there shouldn't be 'quirks' for the Shapio-Wilk test to detect.

Checking 100,000 standard normal samples of size 1000 with the Shapiro-Wilk test, I got rejections just about 5% of the time, which is what one would expect from a test at the 5% level.

set.seed(2019)
pv = replicate( 10^5, shapiro.test(rnorm(1000))$p.val )
mean(pv <= .05)
[1] 0.05009

Addendum. By contrast, the distribution $\mathsf{Beta}(20,20)$ "looks" very much like a normal distribution, but isn't exactly normal. If I do the same simulation for this approximate model, Shapiro-Wilk rejects about 7% of the time. Looked at from the perspective of power, that's not great. But it seems Shapiro-Wilk is sometimes able to detect that the data aren't exactly normal.

This is a long way from "always," but I think $\mathsf{Beta}(20,20)$ is closer to normal than a lot of real-life "normal" data are. (And the link says always may be "a bit strongly stated." I suspect the greatest trouble may come with samples a lot bigger than 1000, and for some normal approximations that are quite useful--even if imperfect.) "Not every statistically significant difference is a difference of practical importance." Sometimes, people who should know better seem to forget that when doing goodness-of-fit tests.

set.seed(2019)
pv = replicate( 10^5, shapiro.test(rbeta(1000, 20,20))$p.val )
mean(pv <= .05)
[1] 0.07152

edited Jun 24 '19 at 06:57

answered Jun 24 '19 at 05:47

BruceET

56,185

2

you get that value because the 100,000 random samples you randomly generated are based on "perfect" normal distribution. If there is any extra noise to the random samples you generated, saphiro-wilk test will always reject the null, as shown in here. And this is very relative to the real-life applications since no populations are perfectly normal – Eric Kim Jun 24 '19 at 06:04
so, going back to the original question, how do I detect normal-enough nature of the distribution, even if its not perfectly normal? What method should I use to automate detecting near-normality without visually inspecting it? – Eric Kim Jun 24 '19 at 14:04
First, you'd need to decide in quantitative terms what 'normal enough means. One idea. The $D$ stat in 1-sample Kolmogorov-Smirnov test is max dif btw empirical CDF of the sample and CDF of normal. According to a fixed standard, use small enough $D$ as criterion, regardless of P-val. You'd have to check this idea out to see if it does what you really want--once you decide what that is. – BruceET Jun 24 '19 at 14:47
All this discussion is completely moot without a clear definition of "normal enough" or "looks like normal" or "reasonably normal". A test makes only sense if it is clear what you are testing against. Distributions are objects with infinite degrees of freedom and each test can only fix one degree of freedom. The OP needs to make up his mind what he requires from his "normal" samples and what deviations he expects or want to detect. – g g Jun 25 '19 at 07:15
@BruceET I would use the following criteria: 1) Skewness close to 0, maybe a (-1,1) range, or that you feel more comfortable with depending on "how normal-like is normal enough". 2) Kurtosis close to 3 (or excess kurtosis close to 0) High kurtosis is often a greater issue than low kurtosis as it leads to more outliers – David Jun 25 '19 at 08:07

score 2 · Answer 2 · answered Jun 25 '19 at 08:10

As @gg pointed out in a comment, this entire discussion in pointless without defining how normal-like does data have to be for us to consider it "normal enough". In practice, I often like the following criteria:

Skewness close to 0, maybe a (-1,1) range, or that you feel more comfortable with depending on "how normal-like is normal enough".
Kurtosis close to 3 (or excess kurtosis close to 0) High kurtosis is often a greater issue than low kurtosis as it leads to more outliers.
Median not far away from the mean
QQ-plots are your friends!

Albeit, high kurtosis (counterintuitively) tends to lead to conservative tests, and low kurtosis to higher false positives (since variances tend to be overestimated in the former and underestimated in the latter). — justme, May 07 '20 at 12:02

horv77 · Answer 3 · 2020-11-20T17:56:34.003

0

...However, if the sample size is very large, the test is extremely "accurate" but practically useless because the confidence interval is too small. They will always reject the null, even if the distribution is reasonably normal enough...

What if you take a sub-sample of size 100 or 300 from the large sample consisting of several thousands or more. If I'm not mistaken then taking the sub-samples will reflect the same distribution but will work better with the common normality tests.

edited Nov 20 '20 at 17:56

answered Nov 20 '20 at 17:46

horv77

1

Welcome to the site. Was this intended as an answer to the OP's question, a comment requesting clarification from the OP or one of the answerers, or a new question of your own? Please only use the "Your Answer" field to provide answers to the original question. You will be able to comment anywhere when your reputation is >50. If you have a new question, click the blue ASK QUESTION at the top of the page & ask it there, then we can help you properly. Since you're new here, you may want to take our [tour], which has information for new users. – gung - Reinstate Monica Nov 20 '20 at 17:50
It was intended as an answer. – horv77 Nov 20 '20 at 17:57
1

This makes no sense to me. The solution to a significance test being hopeless for very large sample sizes because everything fails is not to find a different problem where significance tests sometimes don't fail. It's better to wonder (1) how to plot and measure departure from normality rather than test for it (2) whether departure from normality matters for what else you intend to do. – Nick Cox Nov 20 '20 at 18:54
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Taking this advice to its logical conclusion -- since there's no basis for choosing 300 or 100 or even fewer elements in the subsample -- we might as well choose the minimum of 2, in which case any normality test will fail to reject the hypothesis. If your objective is not to reject the hypothesis (as suggested by the question), then I suppose in that sense this "works better!" – whuber Nov 20 '20 at 23:36
1

Than what if a normal curve is fit on the histogram of the data with the same mean and sd and then sum the squared vertical differences and test a limit then? There would need to be a sophisticated method to try and find the best bin size for the histogram of the data as well. – horv77 Dec 05 '20 at 08:49
Actually if a human can decide normality by looking at the histogram and we agree that this is the most reliable method, then I believe that a method could be constructed to do the same in algorithmic way. For instance fit the distribution of log normal, normal, exponential etc data with different parameters, that we could find using a gradient descent or other methods, Even training a statistical or machine learning model for it if there is enough time window for the task. And then an evaluation can be carried out after the best fit is found. – horv77 Dec 05 '20 at 14:11

Normality testing with very large sample size?

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