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Since the residuals and the fitted values are independent, we would expect the residuals to be distributed evenly around some horizontal line, if the model is linear. But does this horizontal line have to be the x-axis? I think in the cases when the expected value of the residuals is not zero, we can have that the fitted values are distribued evenly around some lines above or below the x-axis?

Here is an example to illustrate. The x-axis is the residuals and the y-axis is the fitted values. All the points are above the x-axis, but they seem to be evenly distributed around some horizontal line. Can we claim that the model is linear? enter image description here

XXX
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  • It is more usual to plot $\hat y$ horizontally and residuals vertically. The residuals having a mean of $0$ is what you would expect from ordinary least squares linear regression (otherwise you can reduce the sum of squares by shifting all the $\hat y$s by the same amount), and something similar to that often happens in other modelling – Henry Dec 09 '21 at 09:21
  • @ Henry Thanks for your clarification. To summarize, we generally put residuals on the y-axis so that if the model is linear, the data points should fluctuate around the x-axis? – XXX Dec 09 '21 at 09:31

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It's more conventional to have the fitted values on the x-axis and residuals on the y-axis. Here's a typical plot (from R).

A usual assumption in simple linear modelling is that the error terms are normally distributed with mean zero and constant variance. Given you've interchanged the axes, I think you need to examine the spread of residuals about the vertical line x=0. If you see strong clear patterns then it can be indicative of deviation from the modelling assumptions, which can suggest transformations of the response variable. Box-cox transformations can be helpful if you suspect non-constant variance.

enter image description here

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