= How do I interpret a regression involving A and B-A as predictors? =

Suppose we have a response Y and two continuous predictors such as age of onset (A) and duration of hearing deficit (B-A) with B representing the individual's current age. Then there is an equivalence between the coefficients in this regression and the ones associated with the same response,y, being predicted using A and B as predictors. 

In particular if $$B_text{i}$$ represents the regression coefficient for variable i then in a regression using a and b-a as predictors

Predicted y = $$B_text{a}$$a + $$B_text{b-a}$$(b-a)

= $$B_text{a}$$a + $$B_text{b-a}$$b - $$B_text{b-a}$$a

= $$(B_text{a}$$ - $$B_text{b-a}$$)a + $$B_text{b-a}$$b

So it follows for $$B_text{i|i,j)$$ representing the variable i regression coefficient in a regression with i and j as predictors being used to predict a response, y, we have

So $$B_text{a|a,b-a}$$ - $$B_text{b-a|a,b-a}$$ = $$B_text{a|a,b}$$ and

$$B_text{b-a|a,b-a}$$ = $$B_text{b|a,b}$$

In other words subtracting the regression coefficients for a and b-a in a regression using a and b-a as predictor is equivalent to the regression coefficient for a in a regression with a and b as predictors and the regression coefficient for b-a with a and b-a as predictors is the same as the regression coefficient for b in a regression with a as the other predictor. 

It also follows that the standard errors of the regression coefficients for a and b respectively can be derived using the standard errors of the regression coefficients for a and b-a.

se($$B_text{a|a,b}$$) = se($$B_text{a|a,b-a}$$ - $$B_text{b-a|a,b-a}$$) = $$\sqrt{V(B_text{a|a,b-a}) \mbox{ + } V(B_text{b-a|a,b-a}) \mbox{ - } 2\mbox{Cov}(B_text{a|a,b-a},B_text{b-a|a,b-a})}$$ and

se($$B_text{b|a,b}$$) = $$B_text{b-a|a,b-a}$$

__Example__

For one study involving a response y and variables a and b-a we have regression coefficients (s.es) of 1.170 (0.446) for a and 1.023 (0.399) for b-a.

It follows in a regression involving a and b on the same response the regression (s.e.) of b equals that of b-a in the a, b-a regression, namely 1.023 (0.399). 

The regression coefficient for a equals 1.170 - 1.023 = 0.148. Given a covariance of 0.026 between the a and b-a regression coefficients

The se(a) in the regression involving a and b is computed using the s.es and covariance from the regression coefficients in the regression with a and b-a as predictors.

se(a) = $$\sqrt{0.446^text{2} + 0.399^text{2} - 2(0.026)}$$ = $$\sqrt{0.306}$$ = 0.553.