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I have read similar posts in this website to help me assess whether my diagnostic plots are too far away from normal and if they are showing heteroscedasticity (Interpretation of residuals vs fitted plot, Interpretation of plot(glm.model)) and I researched other sources as well. I haven't been able to find plots that might resemble mine.

I have concerns about my residuals not having a normal distribution (qqplot) as they depart from the line quite a bit, but I am even more concerned about my data showing heteroscedasticity. Are my concerns well founded here?[1]

The plots belong to this linear mixed model with one random effect (individual id) and a fixed effect that is a three way interaction:

lmer(log.prop.out ~ 1 + time*season*sex + (1|id), REML=FALSE, data=in.out)

where log.prop.out is the log of a proportion, whereas time, season and sex are all categorical predictors. Should I be concerned about the log transformation not being enough to normalize my response variable?

I am struggling to decide how bad can these plots look without raising concern.

Thank you very much for your kind guidance!

enter image description here

Without the transformation, using directly the proportions, this is how the residuals vrs fitted plot looks like:

enter image description here

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  • $\begingroup$ Look at the right tail in the second plot. There's definitely something going on. $\endgroup$ – shadowtalker Apr 29 '16 at 12:38
  • $\begingroup$ Thank you ssdecontrol, it did look to me like a funnel-shaped pattern that I have seen in other plots showing heteroscedasticity. I was unsure though about this one, as it did not look as bad as the examples I have seen before showing this pattern. $\endgroup$ – AnnK Apr 29 '16 at 13:25
  • $\begingroup$ What do the results look like without the log transformation? $\endgroup$ – shadowtalker Apr 29 '16 at 14:38
  • $\begingroup$ I have edited my question to add the graph of how the residuals vrs fitted plot without the transformation. I don't know what other possibilities could there be for analyzing this data. I already tried: a) fitting the model with the untransformed variable, which results in the plot above and shows that the residual variance increases as fitted values increase, decreasing again at high fitted values; $\endgroup$ – AnnK Apr 30 '16 at 22:04
  • $\begingroup$ b) fitting the model with the log transformation; and c) using a GLMM with a binomial family. This last option resulted in a model with huge overdispersion! (Residual deviance = 9237.581 on 93 degrees of freedom and a ratio of: 99.329). Would you have any suggestions as to what other options I could try that might fit the data better? I don't have any additional covariates that I could incorporate to the model. $\endgroup$ – AnnK Apr 30 '16 at 22:04
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You should not be concerned about the quality of the transformation, because there's no reason to have done that transformation in the first place. Hopefully it's obvious from the plots that your log-transformed model is in fact worse than the original model.

That said, your original model still looks heteroskedastic. This could be due to the fact that youre fitting a model with an unbounded response to data where the response is in fact bounded (a proportion). These slides could be helpful in describing some of the issues with using linear models to analyze proportion data.

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  • $\begingroup$ Thank you! I needed a transformation that would allow predicted values to be bounded between 0 and 1. I see my mistake now. The logit transformation was what I needed. $\endgroup$ – AnnK May 4 '16 at 13:18

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