Could an inappropriate link function contribute to non-uniform residuals in a GLM, or could that only be caused by an inappropriate error structure?
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$\begingroup$ Hi there, could you elaborate further on what you mean by "non-uniform" residuals? $\endgroup$– MichelleCommented Feb 12, 2012 at 4:44
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$\begingroup$ It's unusual for the choice of link function to matter much, when fitting GLMs - this is effectively what the Li-Duan theorem says. $\endgroup$– guestCommented Feb 12, 2012 at 5:15
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A brief answer is yes an inappropriate link function can lead to residuals appearing "non-random". however it can also be due to a predictor which needs to be transformed. I am assuming that you are refering to a plot of residuals against the linear predictor.
One way to check that it isn't t a predictor is to create pseudo "partial residuals". These are done by $u_{ij}=\hat{g}(y_{i})-x_{i}^T\hat{\beta}+x_{ij}\hat{\beta}_{j}$ where i indexes observations and j indexes variables. The "hat" on the link function refers to the first order taylor series of g commonly refered to as the working response in a irls algorithm. you plot $u_{ij}$ against $x_{ij}$ (ie one plot per covariate) and it should look like a straight line. If this passes then i would consider changing the link function. Of course this is only feasible when you only have a small number of variables.
Note that there are no hard and fast rules for using diagnostics - this is one approach of many.
answered Feb 12, 2012 at 5:23