FAQ/ab-a - CBU statistics Wiki

Upload page content

You can upload content for the page named below. If you change the page name, you can also upload content for another page. If the page name is empty, we derive the page name from the file name.

File to load page content from
Page name
Comment
Flind the wroneg tetters tin eaech wrord

Revision 18 as of 2011-02-04 10:30:12

location: FAQ / ab-a

What is the relationship between regressions involving variables A and B to those involving B-A and A+B in predicting an outcome?

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 that if $$B_text{i|i,j}$$ represents the regression coefficient of variable i in a regression with i and j as predictors being used to predict a response, y, that

$$B_text{a|a,b-a}$$ - $$B_text{b-a|a,b-a}$$ = $$B_text{a|a,b}$$ 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) = $$\sqrt{0.446text{2} + 0.399text{2} - 2(0.026)}$$ = $$\sqrt{0.306}$$ = 0.553.

Relationships between a,b and a+b

It also follows Predicted y = $$B_text{a}$$a + $$B_text{a+b}$$(a+b)

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

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

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

So it follows that knowing the relationship between the response with both a+b and a is enough to give the relationship between the response and b and the relationship between the response and both a+b and b is enough to give the relationship with a.

It is also true that unless a and b are highly correlated so a $$\approx$$ b,

$$B_text{a}$$a + $$B_text{b}$$b $$\ne$$ $$B_text{a+b}$$(a+b)

because $$B_text{a}$$ and $$B_text{b}$$ will not in general be equal.

One can also interpret this as knowing the a+b sum does not tell you the numbers (a and b) that were added together to give it if these number were weighted unequally (so that $$B_text{a}$$ is not equal to $$B_text{b}$$).

If a and b are highly correlated then the relationships between y and a and y and b are nearly equal and the relationship between y and a+b will then be equal to the relationship between y and either a or b.