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I have a data set where I measured the number of molecules (M) present in cells as a function of drug (with or without) and days of treatment (5 timepoints). I repeated the experiment 3 times, with cells from a separate donor each time. I am currently trying to compare the means between groups. However, the data are not normal and heteroskedastic and I'm in the process of figuring out how to best deal with this.

Transforming the data makes the dataset normal, but the heteroskedacity remains. I'm a stats novice, but my reading over the past several days suggests that a linear mixed model should be able to deal with this. Based primarily on the Pinhiero/Bates book, I have cobbled together the following models using lme in R; they are the same except for the varPower statement.

model1 <- lme(sqrt(M) ~ drug + Days + drug*Days, 
            random = ~ 1+drug+Days+drug*Days|Donor, data=D)

model2 <- lme(sqrt(M) ~ drug + Days + drug*Days,
            random = ~ 1+drug+Days+drug*Days|Donor, data=D, varPower(form = ~fitted(.)) )

When I compare these two models using anova(), model2 has a significantly increased log likelihood. However, when I examine the standardized residuals plotted against either fitted values or the independent variables, the graphs for the two models have identical shape. Note that the magnitude of the residuals is slightly greater with varPower() included:

plot(model1, resid(., type="p") ~ fitted(.), abline=0)
plot(model1, resid(., type="p") ~ Days|drug, abline=0)

residual plots

Does the similarity between these plots mean that I have failed to sufficiently account for the unequal variances between groups?

If so, what approaches might yield sufficient correction? Additionally, if you have any general comments about the suitability of lme here, or the structure of the model, those would be welcome as well!

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    $\begingroup$ It seems like you are trying to model count data. If that's the case you should do poisson regression, i.e., use a glm or better a glmm. $\endgroup$ – Roland Oct 9 '14 at 18:40
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    $\begingroup$ I may have missed something here. But it appears you plotted the same model twice. $\endgroup$ – Dan Ayers Jun 9 '17 at 15:07
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    $\begingroup$ @Dan Thank you for your comment. Indeed, those graphs look identical: but check out the scales on the y-axes. $\endgroup$ – whuber Jun 9 '17 at 15:53
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    $\begingroup$ What happens if you specify type="n" in the residual plot? $\endgroup$ – llewmills Oct 6 '18 at 2:48
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A couple of points

  • The use of varPower() seems incorrect. You need to pass it to the weights argument of lme().
  • The shape of the residuals suggests that you have a bounded outcome, with many values at or near the boundary. You could instead consider a Beta mixed effects model for a bounded outcome or mixed model for semi-continuous data.
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Accounting for heteroskedacity gives you valid SE's. It should not change your estimates enough, to actually remove the heteroskedacity.

However, the heteroskedacity from those plots seem to systematic (considering the sharp downward edge on the lower half of the plots) so I'm guessing this is a modeling/data issue which should be fixed prior to trying to account for it.

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