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I have done a data analysis and am interested whether my data is normally distributed to know whether I can apply two-way ANOVA.

As a result I have plotted a Q-Q plot but I am unsure how to interpret it.

enter image description here

My intuition is that it is normally distributed, however what do the edge effects imply? Are they still implying normality?

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    $\begingroup$ stats.stackexchange.com/questions/101274/… $\endgroup$ – SmallChess Mar 11 '17 at 11:18
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    $\begingroup$ You have edited the question and removed the graph which 4 people discussed. That weakens the value of the thread to others unless people dive into the edit history, which should not be necessary. It would be a much better idea to leave the original question in place and add your new graph, and also to explain it! $\endgroup$ – Nick Cox Mar 11 '17 at 17:39
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    $\begingroup$ As you have not reacted to my comment, I have taken the liberty of doing the edit myself. I can't explain the new graph because only you know how it differs. $\endgroup$ – Nick Cox Mar 12 '17 at 8:36
  • $\begingroup$ I already answered you in my answer. $\endgroup$ – SmallChess Mar 12 '17 at 11:25
  • $\begingroup$ @StudentT I take it you're addressing the OP, not me. (Similarly, I am addressing the OP, and not you.) $\endgroup$ – Nick Cox Mar 13 '17 at 12:12
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A main idea with plots of this kind is that data points from a normal distribution should plot along the diagonal line of equality. In practice, even if you call up a random sample generator that draws samples from a normal, there will always be wiggle and waggle around a straight line.

What you refer to as "edge effects" are small but apparently systematic departures from a normal distribution. So, what should you think?

  1. An ideal condition for two-way ANOVA is that the data are normally distributed conditional on the groups concerned; you're plotting a graph for residuals, which is a guess at that distribution. But any wild values in your data might have thrown off the group means, so that the residuals might give an over-optimistic picture of the suitability of the data. Without the raw data, it is hard to say more.

  2. Nevertheless a guess from experience is that your graph is consistent with an idea that your data are not especially problematic for twoway ANOVA. If in doubt, you could and should cross-check with a quite different analysis, e.g. Kruskal-Wallis.

EDIT: These comments were on a different graph, which can be seen by examining the edit history.

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I'd look at @Glen_b excellent answer at:

How to interpret a QQ plot

Your plot looks very close to the "light tailed" version. Do you agree?

While your data is not exactly normal, it's not very far away. If you are happy with the discrepancies in your modelling assumptions, you may assume normality in your model. But I wouldn't say this is "normally distributed" (your word). It's definitely not normal, but it's good assumption.

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I don't know what software you're using, but you might draw several samples (of the same size as your data) from an actual normal distribution, make a Q-Q plot for each sample, and compare this ensemble to the Q-Q plot of your data. I think you'll see that the edge effects are normal (no pun intended) because of the small number of values that come from the tails of the distribution.

There are also formal tests for normality, such as Anderson-Darling.

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  • $\begingroup$ The test doesn't tell you if the data come from normal. $\endgroup$ – SmallChess Mar 11 '17 at 11:35
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As other answers mention, while your QQ plot is not fully normal due to deviations from the regression line at the beginning and end points, it is not too far away.

One option for a formal test could be to apply the Shapiro-Wilk normality test, whereby:

Null Hypothesis: Assumption of normality cannot be rejected

Alternative Hypothesis: Assumption of normality is rejected

From the plot shown I assume you are using R. Therefore, you can test the relevant variable by inputting:

shapiro.test(variable)

This will yield your p-value and W statistic, from which you can then determine whether your hypothesis is significant at a specified level.

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  • $\begingroup$ The problem is that. If I don't reject my H0, I can't conclude it's normal. $\endgroup$ – SmallChess Mar 11 '17 at 11:36
  • $\begingroup$ The test can only tell me if the data come from non-normal population, it doesn't tell me if it come from normal. $\endgroup$ – SmallChess Mar 11 '17 at 11:36
  • $\begingroup$ The point of QQ is to ensure the normality assumption is okay, but the test is not doing it. I can only conclude "I fail to reject ....". $\endgroup$ – SmallChess Mar 11 '17 at 11:37
  • $\begingroup$ Well yes, with any hypothesis you cannot conclude with 100% certainty. If you wanted to ensure normalisation your data, you could look into eliminating the top and bottom 10% of values if you want to try and normalise your data by eliminating outliers (since these portions of the data seem to be deviating from the regression line). Other options are to scale your data or transform using a method such as Box-Cox. $\endgroup$ – Michael Grogan Mar 11 '17 at 11:37
  • $\begingroup$ Which regression line are you referring to? The OP is interested in two-way ANOVA. The only line in evidence is that on the QQ plot, which isn't a regression line. Eliminating outliers because they are awkward is usually a bad idea. $\endgroup$ – Nick Cox Mar 11 '17 at 11:43

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