We've covered quite a bit of ground. Let's review the analysis of variance table for the example concerning skin cancer mortality and latitude(skincancer.txt).

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Recall that there were 49 states in the data set.

• The degrees of freedom associated with SSR will always be 1 for the simple linear regression model. The degrees of freedom associated with SSTO is n-1 = 49-1 = 48. The degrees of freedom associated with SSE is n-2 = 49-2 = 47. And the degrees of freedom add up: 1 + 47 = 48.
• The sums of squares add up: SSTO = SSR + SSE. That is, here: 53637 = 36464 + 17173.

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The next column contains the degrees of freedom (df). SSTO has n-1 degrees of freedom, SSE has n-2 degrees of freedom, and SSR has 1 degree of freedom. The degrees of freedom for SSR and SSE add to the degrees of freedom for SSTO: The table now looks like this.

Let's tackle a few more columns of the analysis of variance table, namely the 'mean square' column, labled MS, and the F-statistic column, labeled F.

Definitions of mean squares

We already know the 'mean square error (MSE)' is defined as:

[MSE=frac{sum(y_i-hat{y}_i)^2}{n-2}=frac{SSE}{n-2}.]

That is, we obtain the mean square error by dividing the error sum of squares by its associated degrees of freedom n-2. Similarly, we obtain the 'regression mean square (MSR)' by dividing the regression sum of squares by its degrees of freedom 1:

[MSR=frac{sum(hat{y}_i-bar{y})^2}{1}=frac{SSR}{1}.]

Of course, that means the regression sum of squares (SSR) and the regression mean square (MSR) are always identical for the simple linear regression model.

Now, why do we care about mean squares? Because their expected values suggest how to test the null hypothesis H₀: β₁ = 0 against the alternative hypothesis H_A: β₁ ≠ 0.

Expected mean squares

Imagine taking many, many random samples of size n from some population, and estimating the regression line and determining MSR and MSE for each data set obtained. It has been shown that the average (that is, the expected value) of all of the MSRs you can obtain equals:

[E(MSR)=sigma^2+beta_{1}^{2}sum_{i=1}^{n}(X_i-bar{X})^2]

Similarly, it has been shown that the average (that is, the expected value) of all of the MSEs you can obtain equals:

[E(MSE)=sigma^2]

These expected values suggest how to test H₀: β₁ = 0 versus H_A: β₁ ≠ 0:

• If β₁ = 0, then we'd expect the ratio MSR/MSE to equal 1.
• If β₁ ≠ 0, then we'd expect the ratio MSR/MSE to be greater than 1.

These two facts suggest that we should use the ratio, MSR/MSE, to determine whether or not β₁ = 0.

Ssto

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• to test H₀: β₁ = 0 versus H_A: β₁ < 0
• or to test H₀: β₁ = 0 versus H_A: β₁ > 0.

We can only use MSR/MSE to test H₀: β₁ = 0 versus H_A: β₁ ≠ 0.

We have now completed our investigation of all of the entries of a standard analysis of variance table for simple linear regression. The formula for each entry is summarized for you in the following analysis of variance table:

However, we will always let statistical software do the dirty work of calculating the values for us. Why is the ratio MSR/MSE labeled F* in the analysis of variance table? That's because the ratio is known to follow an F distribution with 1 numerator degree of freedom and n-2 denominator degrees of freedom. For this reason, it is often referred to as the analysis of variance F-test. The following section summarizes the ANOVA F-test.

The ANOVA F-test for the slope parameter β₁

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The null hypothesis is H₀: β₁ = 0.

The alternative hypothesis is H_A: β₁ ≠ 0.

The test statistic is (F^*=frac{MSR}{MSE}).

As always, the P-value is obtained by answering the question: 'What is the probability that we’d get an F* statistic as large as we did, if the null hypothesis is true?'

The P-value is determined by comparing F* to an F distribution with 1 numerator degree of freedom and n-2 denominator degrees of freedom.

In reality, we are going to let statistical software calculate the F* statistic and the P Youtube to mp3 converter mac no download. -value for us. Let's try it out on some new examples!