[R-sig-eco] Using residuals as dependent variables

[Dixon, Philip M [STAT]]

1.       Regress a suite of ecological and socioeconomic variables against
the residuals from the oceanographic model to determine which factors cause some countries to be above and some below. I.E as trophic level increase the residuals become increasingly negative.

All my answers assume that all variables are measured at the same scale, so a multi-level model is not appropriate.

Absolutely no problem.  This is essentially equivalent to fitting a larger model with both oceanographic and ecological/socioeconomic variables.  If you go one step further, and first calculate residuals of the ecol/socio variables when regressed against oceanographic variables, the regression of the oceanographic residuals on the ecol/socio residuals is exactly the same as fitting the larger model.  The theory supporting this leads to added variable (or partial regression residual) plots.  

2.       Ideally ( and I am unsure how or if it is possible) work out for
each country which variables/s cause the poor fit of that country to the oceanographic model.

This may be a bit tricky.  The problem is potential confounding.  The regression of two sets of residuals suggested above tells you that a particular ecol/socio variable contributes to the larger only to the extent that that ecol/socio variable contains information not present in the set of oceanographic variables.  If an ecol/socio variable is causally important but confounded with some oceanographic variables, the approach will not detect the role of that ecol/socio variable.  If two ecol/socio variables are confounded, a regression model can not distinguish their contributions.  You have to think about your variables and their relationships to assess this.   

Assuming confounding is not an issue, fit the larger model with both oceanographic and ecol/socio variables.  For each country, calculate X\hat{\beta} for each ecol/socio variable.  That's the contribution of that variable for that country to the model prediction.  Compare those components to their sum for the country.
 
Philip Dixon