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As we know, plotting residuals against fitted values may indicate whether the model is misspecified or the variance is not constant, or both. Let's focus on model misspecification only and set aside heteroskedasticity for the moment.

Residuals vs fitted values

For example, example (b) shows that residuals increase with fitted values. What kind of model misspecification does example (b) imply? Might it indicate that some important predictor is not included at the right-hand side of the equation? Or might it indicate that some of the predictors need to be transformed to a higher order, for example, from its original scale to square?

Also what might example (c) say about model misspecification?

These examples seem to be pretty common in textbooks or online tutorial but it appears to me that few have tried to figure out what the underlying model misspecification contributes to these different types of residuals-fitted values plots.

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  • $\begingroup$ The explanation below the plots doesn't help???? $\endgroup$ – Florian Hartig Mar 21 '17 at 11:37
  • $\begingroup$ It helps but it doesn't say what kind of model misspecication may generate this pattern of bias. There could be multiple scenarios that result in this kind of bias. As I dig deeper in my data, turns out it was something other than those mentioned in the original post that caused this form of bias. $\endgroup$ – TCL Mar 21 '17 at 19:30
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    $\begingroup$ most likely scenario for bias = model formula is wrong. Missing quadratic effect, missing predictor ... $\endgroup$ – Florian Hartig Mar 22 '17 at 10:01
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Both (b) and (c) most likely just mean your model is missing a predictor, or else some of its coefficients deviate from their optimal values under the data. E.g. the scenario in (b) could happen if the data truly lay around a line, but the line you fitted had a slope that was too small. This could be the result of an iterative optimization algorithm that was terminated too soon (before it found the optimum).

The scenario in (c) is more likely to reflect a missing predictor. Let's say you again fitted a line, but the true function by which the data were generated was quadratic, with a negative coefficient in the quadratic term. You subtracted out the linear part, so you're left with the (concave) parabola of the quadratic part that you missed out.

These kinds of biases only happen if the model term(s) you left out is/are somehow linked with those that you did include (e.g. $x$ is correlated with $x^2$). If you left out some relevant effect that is completely independent of the things you did model, that won't show up in the residuals as a bias (it will just be part of the apparently random noise).

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