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I have GLS regression coefficients of the 'best' model from four different islands in my study system. I would like to compare how the 'best' model from island 1 predicts the response variable on island 2, 3 and 4. What would be a suitable method to do this?

I'm unsure as to whether I should compare the slopes of each island to see whether they differ significantly from each other (and if so, how I would even go about doing this) or whether the best approach would be to use predict() and proceed via this route (assuming I can use predict() when using ML GLS).

EDIT - I'm unable to reply to Silverfish's comment (below) as my reputation is too low. I have instead updated my question here to address his response.

In short, I would like to see how well/poorly the island 1 model applies to island 2/3/4. In theory, I expect the 'best' models for each island to transfer well to every other island but I would like to formally test this.

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  • $\begingroup$ It isn't quite clear to me what you're trying to do here. When you write " I would like to compare how the 'best' model from island 1 predicts the response variable on island 2, 3 and 4" - what comparison are you hoping to make? How well/poorly the island 1 model applies to island 2? How different/similar it is to the model you constructed for island 2? $\endgroup$ – Silverfish Oct 4 '16 at 10:44
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There are a few ways to judge how well your model for Island $i$ performs on Island $j$, but the simplest might be to calculate and compare the $R^2$, which tells you the ratio of the variance explained by your model. Using the fitted values from Island $i$ to explain Island $j$ will produce a smaller $R^2$ than if you were to use the Island $j$'s actual model. The difference will tell you how well the model from one island extends to the others.

If the fitted values for Island $i$ $$\hat{y}_{i,n}= f_{i,n}(X) \quad n = 0, ... , N$$

Then the $R^2$ can be calculated as $$R^2 = 1-\frac{SSR}{SST}$$ where $SST = \sum_n (y_{j,n} - \overline{y_j})$ and $SSR = \sum_n(y_{j,n} - \hat{y}_{i,n})^2$

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