# How to interpret r's output for a multiple linear regression model

Context: Two properties that may influence the pull strength of a wire bond in a semiconductor manufacturing process are the wire length and the die height. A scatterplot matrix of the pull strengths, wire lengths and die heights is given below:

A multiple linear regression model fit to these data is summarised in the computer output:

I know how to interpret the intercept, but not sure about the rest. Like what is the length and the height estimate means (2.744 and 0.013). Are these the slopes? How do you interpret slope when I have two predictors? Is this interpretation correct: A unit increase in length is associated with a 2.744 unit increase in strength for some fixed height? Like I don't know how to interpret the second predictor when talking about the slope of the first predictor.

Many thanks.

Here is your regression equation: $$\hat y = \hat\beta_0 + \hat\beta_{L} x_{L} + \hat\beta_{H} x_{H}$$.

Let's look at the partial derivatives.

$$\dfrac{\partial \hat y}{\partial x_{L}} = \hat\beta_{L}\\ \dfrac{\partial \hat y}{\partial x_{H}} = \hat\beta_{H}$$

Therefore, in the equation that lacks interactions and high-order terms (quadratic, etc), the slope coefficients are the partial derivatives.

Partial derivatives mean that, holding all other variables constant, the slope in the direction of the denominator variable is the value of the partial derivative. Since these partial derivatives are constants, then we can interpret the regression to mean that, holding height constant, increasing length by $$1$$ results in a change in $$\hat y$$ of $$\hat\beta_{L}$$; and, holding length constant, increasing height by $$1$$ results in a change in $$\hat y$$ of $$\hat\beta_{H}$$.

• Hi Dave, thanks for that,. So I can only interpret the a slope when the other slope is held constant? Also what about the residuals. any comments? Many thanks Dave. Commented Jun 7, 2022 at 14:11
• @CountDOOKU You can do any of the tricks that you would do in multivariable calculus. You can take gradients and directional derivatives, for instance. // I do not follow what you mean about the residuals, but if you have a new question, please post it as a new question.
– Dave
Commented Jun 7, 2022 at 14:14