I have just read this very interesting paper "Interpreting the Results from Multiple Regression and Structural Equation Models" by Grace and Bollen and wanted to record what I took from it. The figures and a lot of the text is taken straight from this paper - you can find it here http://goo.gl/qgX7El
The main message I took away was that, when looking at multiple regressions, you cannot answer the question "What is the relative importance of a set of causes controlling some observed phenomenon?" if the predictor variables are correlated for unknown reasons.
If the correlations are small however, the coefficients can provide some insight. However, without a theory to guide the analysis, a meaningful answer to the question of relative importance of factors is usually not possible with multiple regressions.
"A multiple regression represents a particular model of relationships in which all potential explanatory variables (predictors) are treated as coequal and their interrelations are unanalyzed."
Here, they consider an example where shrubland plant cover is affected by wildfire. They look at many different plots to determine how elevation, the age of the plot, and the severity of the fire all affect how the fire effects plant cover.
Below, there are three figures that present diagrammatic representations of a multiple regression model in which fire severity, stand age, and elevation are related to vegetation cover. Figure A uses unstandardized parameters, B Standardized parameters and C Semipartial coefficients for the directional pathways.
The main message I took away was that, when looking at multiple regressions, you cannot answer the question "What is the relative importance of a set of causes controlling some observed phenomenon?" if the predictor variables are correlated for unknown reasons.
If the correlations are small however, the coefficients can provide some insight. However, without a theory to guide the analysis, a meaningful answer to the question of relative importance of factors is usually not possible with multiple regressions.
"A multiple regression represents a particular model of relationships in which all potential explanatory variables (predictors) are treated as coequal and their interrelations are unanalyzed."
Here, they consider an example where shrubland plant cover is affected by wildfire. They look at many different plots to determine how elevation, the age of the plot, and the severity of the fire all affect how the fire effects plant cover.
Below, there are three figures that present diagrammatic representations of a multiple regression model in which fire severity, stand age, and elevation are related to vegetation cover. Figure A uses unstandardized parameters, B Standardized parameters and C Semipartial coefficients for the directional pathways.