I wish to plot a continuous response variable (richness) against categorical predictors, but in doing so I need to control for a continuous 'nuisance' predictor (depth). My solution is to run a linear model of richness~depth, and then plot the residuals from the model instead of the raw data. However, in order to improve the model fit, I log-transformed the response variable. What implications does this have for plotting the residuals? Can I back-transform them? Would I be better to use an untransformed response and put up with the poorer fit?
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Right, I think I’ve got this. My aim is to de-trend the data. In a model without transformation, this is response value – fitted value = response residuals. When the model is fitted with the response variable natural-log transformed, this is equivalent to response residuals = exp(response value) / exp(fitted value). Therefore, in an untransformed model, the residuals are the difference between response and fitted values; in the transformed model, the residuals are the ratio between the back-transformed response and fitted values. This means that raw data values – exp(fitted values from log model) = difference between raw and fitted values. This is the de-trended data that I’ve been after: the value that remains once the influence of this predictor has been removed.