![]() Alternatively, Table 4 of Appendix B shows that with two degrees of freedom in the numerator and seven degrees of freedom in the denominator, F. 000 in the last column of the analysis of variance table (Figure 15.6) indicates that we can reject H 0: β 1 = β 2 = 0 because the p-value is less than a =. Using equation (15.14), we obtain the test statistic. In the analysis of variance part of the output, we see that MSR = 10.8003 and MSE =. With two independent variables, the hypotheses are written as follows:įigure 15.6 is the output for the multiple regression model with miles traveled (x 1) and number of deliveries (x 2) as the two independent variables. Let us apply the F test to the Butler Trucking Company multiple regression problem. A summary of the F test for significance in multiple regression follows. To determine how large the value of MSR/MSE must be to reject H 0, we make use of the fact that if H 0 is true and the assumptions about the multiple regression model are valid, the sampling distribution of MSR/MSE is an F distribution with p degrees of freedom in the numerator and n – p – 1 in the denominator. However, if H 0 is false, MSR overestimates s 2 and the value of MSR/MSE becomes larger. = β p = 0 is true, MSR also provides an unbiased estimate of s 2, and the value of MSR/MSE should be close to 1. Hence, the mean square due to regression (MSR) is SSR/p and the mean square due to error (MSE) is SSE/(n – p – 1).Īs discussed in Chapter 14, MSE provides an unbiased estimate of s 2, the variance of the error term e. In the multiple regression case, the total sum of squares has n – 1 degrees of freedom, the sum of squares due to regression (SSR) has p degrees of freedom, and the sum of squares due to error has n – p – 1 degrees of freedom. A mean square is a sum of squares divided by its corresponding degrees of freedom. However, if H 0 cannot be rejected, we do not have sufficient evidence to conclude that a significant relationship is present.īefore describing the steps of the F test, we need to review the concept of mean square. If H 0 is rejected, the test gives us sufficient statistical evidence to conclude that one or more of the parameters are not equal to zero and that the overall relationship between y and the set of independent variables x 1, x 2. The hypotheses for the F test involve the parameters of the multiple regression model. The multiple regression model as defined in Section 15.4 is In the material that follows, we will explain the F test and the t test and apply each to the Butler Trucking Company example. A separate t test is conducted for each of the independent variables in the model we refer to each of these t tests as a test for individual significance. If the F test shows an overall significance, the t test is used to determine whether each of the individual independent variables is significant.The F test is used to determine whether a significant relationship exists between the dependent variable and the set of all the independent variables we will refer to the F test as the test for overall significance. ![]() In multiple regression, the t test and the F test have different purposes. In simple linear regression, both tests provide the same conclusion that is, if the null hypothesis is rejected, we conclude that b 1 A 0. The significance tests we used in simple linear regression were a t test and an F test. In this section we show how to conduct significance tests for a multiple regression relationship.
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