# Multiple linear regression, partial F-Test

I had seen many formulas out there and I hardly knew how to solve this exercise. Where it consists of: Enter link data set for the question I use $$X_2,X_7\; and \;X_8$$

• Using the partial F test, determine the contribution of $$x_7$$ to the model. How is this partial F statistic related to the t test for $$β_7$$ calculated in part c above?

and well, as a first step I first tried to calculate the analysis of variance for $$X_7$$ and Y

Now I return to the full model and the calculations are as follows In the Montgomery book Introduction to Linear Regression Analysis CHAPTER 3, problem 3.1

It results in this way:

This is the result and I don't know how he did it $$F_o = \frac{(257.094 - 243.03)}{2.911} = 4.84$$

It is here that I cannot understand. What is the formula to calculate?

1) The partial F-test gives you a way of comparing two models. In particular, it gives you a way of comparing nested models. For example, model (2) is nested inside of model (1). However, model (3) is not nested within either model (1) or model (2).

\begin{align} Y &= \beta_0 + \beta_1x_1 + \beta_2x_2 &&(1)\\ Y &= \beta_0 + \beta_1x_1 && (2)\\ Y &= \beta_3x_3 && (3)\\ \end{align}

The partial F-test tests tells you whether the full model is significantly better than some nested model. In particular, the partial F-test can be used to determine whether the inclusion of extra variables provides additional explanatory power to warrant their inclusion into the regression.

Say you want to compare models (1) and (2) above. You could set up a partial F - test to test the hypotheses

\begin{align} H_0:& \beta_{2} = 0 \\ H_1:& \beta_{2} \neq 0 \end{align}

Just compute your F - statistic and compare it to an F - distribution with the appropriate degrees of freedom.

2) This website explains how to calculate regression sum of squares (SSR) and error sum of squares (SSE).

• I just saw the website you provided and good if I understand that part. My difficulty is that I don't know how to use the formula to test if $X_7$ contributes to the model Dec 2 '19 at 11:45