Question

For the multiple regression model in Exercise \(14.4\), the value of \(R^{2}\) was \(.06\) and the adjusted \(R^{2}\) was \(.06 .\) The model was based on a data set with 1136 observations. Perform a model utility test for this regression.

Short answer

Without a specific significance level or the exact number of predictors, an exact conclusion cannot be stated here. However, the procedure outlined above can be followed once these are known. In general, the null hypothesis is rejected if the calculated F-value is greater than the table value, indicating model utility.

Step-by-step solution

Determine the hypotheses

The null hypothesis \(H_{0}\) is that the model has no utility, implying that all of the regression coefficients are zero. The alternative hypothesis \(H_{1}\) is that at least one of the regression coefficients is not zero, thus indicating the model may be significant.

Compute the F-value

The observed F-value can be found using the formula: \(F = \frac{(R^{2} / k)}{(1 - R^{2}) / (N - k - 1)}\) where \(R^{2}\) is the coefficient of determination, \(k\) is the number of predictors, and \(N\) is the total number of observations. We have 1136 total observations and the \(R^{2} = 0.06\). However, since the number of predictors \(k\) is not given in the exercise, it will be assumed to be 1. Hence, \(F = \frac{(0.06 / 1)}{(1 - 0.06) / (1136 - 1 -1)}\)

Compare with critical F-value

After calculating the observed F-value, compare it with the critical F-value from the F-distribution table with degrees of freedom df1 = \(k\) and df2 = \(N - k - 1\) at a certain level of significance, generally 0.05. This will determine the applicability of the null hypothesis.

Conclude the result

If the calculated F-value is greater than the table value, reject the null hypothesis, implying that the model has utility to predict the response variable. If the calculated F-value is less than the table value, do not reject the null hypothesis, indicating that the model might not provide any meaningful insights.

Key concepts

Null Hypothesis in Regression

When working with regression analysis, understanding the null hypothesis is crucial to interpreting the results. The null hypothesis (\( H_0 \)) in the context of regression is a statement assuming that the model has no explanatory power. More precisely, it posits that all the coefficients representing the relationship between the independent variables and the dependent variable are equal to zero, meaning that the independent variables do not explain any variance in the dependent variable.

This provides a baseline for comparison. When performing a regression analysis, we typically want to disprove this hypothesis in favor of the alternative hypothesis (\( H_1 \)), which suggests that at least one coefficient is not zero, and therefore at least one independent variable does have a significant effect on the dependent variable. By conducting a model utility test, which usually involves the F-test, we gather evidence against the null hypothesis to support the potential utility of the regression model.

F-value Computation

The F-value in regression analysis measures how well the regression model fits the data compared to a model with no predictive capability. The computation of the F-value is an essential part of hypothesis testing in regression. To calculate the F-value, we use the formula:
\[F = \frac{(R^{2} / k)}{(1 - R^{2}) / (N - k - 1)}\]
where \(R^{2}\) is the coefficient of determination, \(k\) is the number of predictors in the model, and \(N\) is the total number of observations.

In the given scenario, we assume there's only one predictor (\(k=1\)), and we know that \(R^{2} = 0.06\) and there are 1136 observations (\(N\)). Inserting these values in the formula results in the computation of an actual F-value. This F-value is then compared to critical values from the F-distribution to decide whether the observed relations in the model are statistically significant.

R-squared Coefficient

The R-squared coefficient, denoted as \(R^{2}\), is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variables. It is a number between 0 and 1, where 0 indicates no explanatory power and 1 indicates perfect prediction.

An \(R^{2}\) of 0.06, as noted in the exercise, implies that only 6% of the variance in the response variable can be explained by the regression model. Adjusted \(R^{2}\) is a modification of \(R^{2}\) that adjusts for the number of predictors in the model, and it’s especially useful in the context of multiple regressions where there are several independent variables. Despite its utility, \(R^{2}\) should not be the sole measure for the effectiveness of a regression model, which is why hypothesis testing, such as the F-test, is also performed.

F-distribution

The F-distribution is a probability distribution that is the basis for the F-test, commonly used in ANOVA and regression analysis. It is used to compare variances and determine whether the proposed regression model fits the data better than the simplified model suggested by the null hypothesis.

The F-distribution is characterized by two sets of degrees of freedom, df1 and df2. These relate, respectively, to the number of predictors in the model and the number of observations. After computing the F-value from the regression analysis, it is compared to critical values from the F-distribution table, which are based on a specific significance level (often 0.05).

If the computed F-value is greater than the critical value from the F-distribution, we reject the null hypothesis, which supports the alternative view that the regression model has utility. Otherwise, there is insufficient evidence to suggest that the model is significantly better than a model that assumes no relationship between the variables.