Question

Fill in the missing entries in the analysis of variance table for a simple linear regression analysis and test for a significant regression with \(\alpha=.05\) in Exercises \(3-4 .\) Calculate the coefficient of determination, \(r^{2},\) and interpret its significance. $$ \begin{array}{lclll} \hline \text { Source } & d f & \text { SS } & \text { MS } & F \\ \hline \text { Regression } & & 3 & & \\ \text { Error } & 14 & & 2 & \\ \hline \text { Total } & & & & \\ \hline \end{array} $$

Short answer

And what percentage of the variability in the response variable can be explained by the predictor variable? Answer: There is no significant linear relationship between the predictor variable and the response variable at the 0.05 significance level since we fail to reject the null hypothesis. Approximately 9.68% of the variability in the response variable can be explained by the predictor variable in this simple linear regression model.

Step-by-step solution

Fill in the missing degrees of freedom (df) values

For a simple linear regression, the degrees of freedom (df) for regression are always 1. So, the df value for regression is 1. The total df equals the sum of the regression df and the error df. Therefore, the total df is \((1+14)=15\). Now, our table looks like: $$ \begin{array}{lclll} \hline \text { Source } & d f & \text { SS } & \text { MS } & F \\\ \hline \text { Regression } & 1 & 3 & & \\\ \text { Error } & 14 & & 2 & \\\ \hline \text { Total } & 15 & & & \\\ \hline \end{array} $$

Fill in the missing Sum of Squares (SS) values

To find the total SS value, we will use the equation: \(SS_{Total} = SS_{Regression} + SS_{Error}\), where \(SS_{Regression} = 3\). Thus, the total SS = \((3+SS_{Error})\). The error SS value is the product of the error df and the error MS (\(14 \times 2\)). To find the total SS value, we compute: $$SS_{Total} = 3 + (14 \times 2) = 3+28 = 31$$ Now, our table looks like: $$ \begin{array}{lclll} \hline \text { Source } & d f & \text { SS } & \text { MS } & F \\\ \hline \text { Regression } & 1 & 3 & & \\\ \text { Error } & 14 & 28 & 2 & \\\ \hline \text { Total } & 15 & 31 & & \\\ \hline \end{array} $$

Calculate the F-value

F-value is computed by dividing the regression MS by the error MS: $$F = \frac{MS_{Regression}}{MS_{Error}}$$ First, we need to find the \(MS_{Regression}\) value by dividing the SS value by its corresponding df value. So, $$MS_{Regression} = \frac{SS_{Regression}}{df_{Regression}} = \frac{3}{1} = 3$$. Next, we compute the F-value: $$F = \frac{MS_{Regression}}{MS_{Error}} = \frac{3}{2} = 1.5$$ Now, our ANOVA table looks like: $$ \begin{array}{lclll} \hline \text { Source } & d f & \text { SS } & \text { MS } & F \\\ \hline \text { Regression } & 1 & 3 & 3 & 1.5 \\\ \text { Error } & 14 & 28 & 2 & \\\ \hline \text { Total } & 15 & 31 & & \\\ \hline \end{array} $$

Perform hypothesis test for significant regression

We will use the F-test for the hypothesis test to determine if the linear regression is significant at the \(\alpha = 0.05\) significance level. Hypotheses: - \(H_0: \beta_1 = 0\) (no significant linear relationship) - \(H_1: \beta_1 \neq 0\) (significant linear relationship) Using a standard F-distribution table or an F-distribution calculator, we find the critical F-value for df = \((1,14)\) and \(\alpha = 0.05\). The critical F-value is approximately 4.67. Since the computed F-value (1.5) is less than the critical F-value (4.67), we fail to reject the null hypothesis (\(H_0\)). There is no significant linear relationship between the predictor variable and the response variable at the 0.05 significance level.

Calculate and interpret the coefficient of determination (\(r^2\))

The coefficient of determination (\(r^2\)) is calculated using: $$r^2 = \frac{SS_{Regression}}{SS_{Total}}$$ We plug in the values from the table: $$r^2 = \frac{3}{31} = 0.0968$$ The coefficient of determination, \(r^2\), is equal to 0.0968, which means that approximately 9.68% of the variability in the response variable can be explained by the predictor variable in this simple linear regression model.

Key concepts

Analysis of Variance (ANOVA)

Analysis of Variance, commonly known as ANOVA, is a statistical technique used to compare means of three or more samples to ascertain if at least one sample mean is significantly different from the others. When applied in the context of simple linear regression, ANOVA helps to test if there is a significant linear relationship between the independent variable and the dependent variable.

The essence of ANOVA in regression is to partition the total variation observed in the dependent variable into two parts: variation due to the regression (explained by the model) and the residual variation (error). In the ANOVA table, the 'Sum of Squares' column reflects these variations, and it does so by squaring the differences from the mean to avoid canceling out negative deviations. The 'Degrees of Freedom (df)' are indicative of the number of values that are free to vary, with one deducted for the estimated mean.

By comparing the mean squares, which is Sum of Squares divided by respective degrees of freedom, we obtain the F-value. If the F-value is significantly high, it suggests that the regression model explains a substantial portion of the variability in the data, hence affirming a significant relationship between variables.

Coefficient of Determination

The coefficient of determination, denoted as \(r^2\), is a key measure in regression analysis that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. In simple terms, it measures how well the regression model fits the data. An \(r^2\) value closer to 1 indicates that the model's predictions closely match the actual data, while an \(r^2\) value near 0 means the model does not explain the variability in the outcome well.

To calculate \(r^2\), we divide the regression sum of squares (SS) by the total sum of squares: \[r^2 = \frac{SS_{Regression}}{SS_{Total}}\]. In the context of our exercise, with an \(r^2\) of 0.0968, it implies that only about 9.68% of the data’s variability is explained by the model, indicating a relatively weak predictive power.

F-test

The F-test in regression analysis is a hypothesis test that compares the fits of different linear models. In the context of our example, the F-test evaluates whether the regression model fits the data better than a model with no independent variables. The null hypothesis \(H_0\) of the F-test is that the independent variables do not explain any of the variation in the dependent variable; in other words, the regression model is not statistically significant.

To perform the F-test, we calculate the F-value by dividing the mean square due to regression (MS Regression) by the mean square due to error (MS Error). If the calculated F-value is greater than a critical F-value from the F-distribution table at a specified confidence level, we reject the null hypothesis, suggesting that the model provides a better fit to the data than one without the independent variable. However, in our exercise example, the F-value of 1.5 did not exceed the critical value of 4.67, therefore we fail to reject the null hypothesis, indicating an insignificant regression model.

Hypothesis Testing

Hypothesis testing in the context of regression analysis is a process used to determine if there is enough statistical evidence to infer that a certain condition is true for the entire population. Using the F-test as part of hypothesis testing, we set up two hypotheses: the null hypothesis \(H_0\) suggests that there is no effect or no relationship, and the alternative hypothesis \(H_1\) suggests that there is an effect or a relationship.

In hypothesis testing, we use a p-value to weigh the strength of the evidence. If the p-value is less than our chosen significance level (α), typically 0.05, we reject the null hypothesis in favor of the alternative. By adhering to this method, we guard against incorrectly asserting that a relationship exists when in reality, it does not—an error known as a Type I error. In our exercise example, the lack of significance in the F-test leads us to maintain the null hypothesis, meaning that the evidence does not warrant a conclusion of a significant linear relationship between the independent and dependent variables.