# Do the residual plot and QQ plot look normal?

I am doing linear mixed model and would like to check the assumptions using residual plot and QQ plot. Here is my code:

data1.frame <- read.delim("height.txt", fileEncoding="UTF-16")
lmer50      <- lmer(response ~ (1|jumper) + group*gender, data=data1.frame, REML=FALSE,
na.action=na.omit)
plot(fitted(lmer50), residuals(lmer50))


For the residual plot, here is the output:

qqnorm(data1.frame$response) qqline(data1.frame$response)


For the qq plot, this is the output:

Could I ask if both of them look normal? Can I apply linear mixed model in my rating scale data?

Pretty obviously not normal. A step function is not a straight line. However, you also seem to be checking (unconditional) normality of the response, which is not assumed to be normal in a mixed model (you'd have some mixture of normals, depending on the fixed effects)

You clearly have discrete data. So your response's conditional distribution will be discrete, not normal.

This is what a Q-Q plot of normal residuals with sample size close to yours tends to look like:

The non-normality you will have in your conditional response is not automatically a big problem - it may or may not be. You don't seem to have strong skewness, for example, nor heavy tails and your sample size is largish.

Please describe your response in more detail. What is this "rating scale data"? (It sounds like it might be an ordinal scale.)

One thing you can do is use simulation to investigate the effect of this discrete scale on the inferences you wish to perform compared to having an actually normal error term.

• Hi Glen_b, my DV is 9-scale rating, hence it is indeed ordinal scale, rather than interval. So I shouldn't use linear mixed model for this data? – user3288202 Dec 15 '14 at 10:02
• 9-scale rating, hence it is indeed ordinal scale That moves and makes weep :-) – ttnphns Dec 15 '14 at 10:35
• should I use poisson model instead of linear mixed model then? – user3288202 Dec 15 '14 at 11:22
• Hang on, if you're saying it's a Likert-like scale composed of 9 items, you already assumed the items were interval when you added them. If you already assume it for all of them before you added them, why would you not assume it after? That would be bizarre. – Glen_b -Reinstate Monica Dec 15 '14 at 11:22
• Something that may matter a good deal more than the normality assumption is the boundedness of the response variable; that doesn't tend to go so well with models that are purely linear in predictors. – Glen_b -Reinstate Monica Dec 17 '14 at 13:50

Likert data simply cannot be normal. Although in some cases it is safe enough to treat it as normal, it isn't actually ever normal and treating it as such is potentially dangerous.

In addition to the points @Glen_b has made, your residual plot doesn't look good. The residuals should be symmetrical (vertically) around the 0 line. Either there is something seriously wrong with your model or you have strong skew in opposite directions at the two ends (this seems more likely). This means that your model will be attenuated through much of its range (as you get closer to the ends, the predicted values will be closer to the grand mean than they should be), but then will overshoot the possible values if you were to move far enough out from the mean of X. So in addition to your interval estimates being distorted, the model isn't even picking out your means correctly.

You would do best to use a mixed ordinal logistic regression.

• Although the main point here is surely correct, I'd add a little caution about the residual plot. For all we know many, many points are being over-plotted. The eye can be hung up on the few data points with large residuals, but any apparent tilt from those extremes may well be balanced by points nearer the middle of the distribution. Can we see the raw data? – Nick Cox Dec 15 '14 at 20:33
• @NickCox, that is what I mean by "strong skew in opposite directions at the two ends". There are a lot of min values & only a few max values on the right, & vice versa on the left. The regression line may give the actual means of those numbers, but is likely to be missing the mean underlying values. – gung - Reinstate Monica Dec 15 '14 at 20:38
• Pleased to infer that our interpretations are close. From my reading of the qqplot, there are more 9s than 1s and more 8s than 2s. But I'd rather see the data. – Nick Cox Dec 15 '14 at 20:57
• Hi Nick, I'm willing to give you my data. Could I ask how I can give you my data set? – user3288202 Dec 17 '14 at 7:20
• You're welcome, @user3288202. – gung - Reinstate Monica Dec 18 '14 at 14:36