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I have an LMER, and I want to check for normality; my plots show this:

enter image description here

enter image description here

However, my Shapiro-Wilk test shows this:

    Shapiro-Wilk normality test

data:  resid(lmerabsolute)
W = 0.97875, p-value = 3.751e-06

What do I do? I would be so grateful for some advice!

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    $\begingroup$ Shapiro Wilk is sensitive to the number of observations. Too many, and it is easy to reject the null even when the data are "normal enough". Forget shapiro-wilk and make a decision based on your plots. From your residual plots, I would think something is up, but I would need to know more about the problem. $\endgroup$ Commented Mar 10, 2020 at 19:05
  • $\begingroup$ @DemetriPananos what information would be useful to you? $\endgroup$
    – user275189
    Commented Mar 10, 2020 at 19:06
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    $\begingroup$ It looks to me like your plots and the Shapiro-Wilk test are saying the same thing - the residuals are not Gaussian. Why do you think otherwise? $\endgroup$
    – jbowman
    Commented Mar 10, 2020 at 19:21
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    $\begingroup$ The assertion in your title is false (they both indicate non-normality), but in any case a potential difference between them (on other data) isn't necessarily relevant, since their purposes are different. The test is not answering a question you should care about. $\endgroup$
    – Glen_b
    Commented Mar 11, 2020 at 5:41
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    $\begingroup$ It looks like your dependent variable is discrete. $\endgroup$
    – Roland
    Commented Mar 11, 2020 at 7:05

1 Answer 1

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The plot and the Shapiro-Wilk test seem totally consistent with each other.

The test gives a tiny p-value, indicating that normality is basically out of the question.

The plot shows deviations from normality, especially at the top right. A normal distribution would give points around the diagonal line, while your points start drifting away from the diagonal line around $x=1$ and higher.

Note, however, that formal normality testing has a tendency to catch differences that, upon visual inspection, are easy to see will be trivial. The test is doing what it is supposed to be doing by flagging a slight deviation from normality as a deviation from normality, and I give a demonstration here. However, most of the time when normality is desired, we just need "close enough" to normality for downstream statistics to work as we want. Hypothesis testing for normality, particularly when sample sizes are large, is likely to catch deviations that have minimal impact on our work, even if the test is correct to notice the deviation.

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