I received this elementary question by email:

In a regression equation am I correct

in thinking that if the beta value is

positive the dependent variable has

increased in response to greater use

of the independent variable, and if

negative the dependent variable has

decreased in response to an increase

in the independent variable – similar

to the way you read correlations?

**Answer**

In explaining the meaning of regression coefficient I found that the following explanation very useful. Suppose we have the regression

$$Y=a+bX$$

Say $X$ changes by $\Delta X$ and $Y$ changes by $\Delta Y$. Since we have the linear relationship we have

$$Y+\Delta Y= a+ b(X+\Delta X)$$

Since $Y=a+bX$ we get that

$$\Delta Y = b \Delta X.$$

Having this is easy to see that if $b$ positive, then positive change in $X$ will result in positive change in $Y$. If $b$ is negative then positive change in $X$ will result in negative change in $Y$.

**Note:** I treated this question as a pedagogical one, i.e. provide simple explanation.

**Note 2:** As pointed out by @whuber this explanation has an important assumption that

the relationship holds for all possible values of $X$ and $Y$. In reality this is a very restricting assumption, on the other hand the the explanation is valid for small values of $\Delta X$, since Taylor theorem says that relationships which can be expressed as differentiable functions (and this is a reasonable assumption to make) are linear locally.

**Attribution***Source : Link , Question Author : Jeromy Anglim , Answer Author : mpiktas*