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If given the choice between two statistical models (for argument's sake, let's say Model 1 is $y = \beta_0 + \beta_1 x_1 + \epsilon$ and Model 2 is $y = \beta_0 + \beta_1 x^2_1 + \epsilon$), is there a way to select which model is more appropriate based upon an analysis of the residuals?

Let's stipulate that the models have the same number of parameters, $k$, and that $SSE_{Model 1} = SSE_{Model 2}$. The only difference is that residuals from Model 1 'appear' to be more systematic than those in Model 2. That is, upon examining the residuals for Model 1 along the dimension $x$, it becomes clearly apparent that there are areas along $x$ where $Prob(\epsilon_i \ge 0 | x_i) > .5$ - and there are areas along $x$ where $Prob(\epsilon_j \ge 0 | x_j) < .5$.

Since a test of just $SSE$ would rule that the two models are equivalent (as both models have the same overall error measurement) - and since the number of parameters, $k$, is the same for both models, the usual tests, Mallow's CP, Akaike's AIC & Schwarz's BIC wouldn't discriminate. Yet, I can see with my eyes that one model has more (or at least a less-random pattern of) systematic error than the other.

Yes, I know that this probably means that neither model is the 'correct' model, but I'm constrained to select between the given choices of models.

I don't know of a test (other than just 'eyeballing' the residuals) that would allow me to decide between the alternatives.

Is there a formal test, perhaps one based upon the pattern of residuals, that would allow me to discriminate? I could see something that perhaps stepped through the various values of $x$ and tested as the null hypothesis that the residuals are such that $Prob(\epsilon_{segment} \ge 0|x_{segment}) = .5$. Perhaps the model where the cumulative stepwise test is most false would be the model that I should select against.

Any ideas?

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Informally you could treat the model residuals as the dependent variable, and fit a regression model with the variables that the residuals appear to have a relationship with as the explanatory variables. Say the estimated parameters are significant in one model but not the other, then a pattern has been detected in one model but not the other. This style of approach can be formalised in hypothesis tests. See section 10.4.1 of the book "Linear Regression Analysis" by Seber and Lee. The section is about using variance functions to construct tests for homogeneity of variances of model residuals. – dandar Sep 25 '12 at 10:34
This would allow tests for certain types of systematic variation in the residuals - e.g., significant non-zero slope implies one type of model bias, or significant quadratic trend implies another type of systematic model error, etc. I think that this could work - thanks. It is much simpler than the sequential tests of distributions that I was originally thinking. Quite simple after all... – Craig Allen Sep 27 '12 at 5:21

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