Question

Going for it on fourth down in the NFL. Refer to the Chance (Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.69 (p. 679). Recall that statisticians at California State University, Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goal line. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model, $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}$.The regression yielded the following results: $\mathbf{y}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{13}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{141}\mathbf{x}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0009}{\mathbf{x}}^{\mathbf{2}}$,${\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{226}$.

a) If possible, give a practical interpretation of each of the b estimates in the model.

b) Give a practical interpretation of the coefficient of determination,${\mathbf{R}}^{\mathbf{2}}$.

c) In Exercise 11.63, the coefficient of correlation for the straight-line model was reported as${\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{18}$. Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.

d) What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?

Short answer

a.${\mathbf{\beta }}_{\mathbf{0}}$ indicates the y-intercept term of the curve. It means it gives the value of E(y) when${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$

${\mathbf{\beta }}_{\mathbf{1}}$indicates the magnitude of the shift in parabola due to changes in the value of x (shift parameter)

_{${\mathbf{\beta }}_{\mathbf{2}}$}indicates the rate of curvature of the parabola. (shape parameter).

b. Here, 23% is a very low value for ${\mathbf{R}}^{\mathbf{2}}$meaning the model is not a good fit for the data.

c. When a straight-line model was fitted to the data, the value of ${\mathbf{R}}^{\mathbf{2}}$was 18% while when a quadratic model is fitted to the data, the value of ${\mathbf{R}}^{\mathbf{2}}$increases to 23%. This means that the quadratic model is a better fit for the data than a straight-line model. However, 23% is still a lower value meaning a better quadratic model can be used to fit the data.

d. To test whether a quadratic model is a good fit for the data, F-test needs to be done.

Step-by-step solution

Interpretation of beta estimates

${\mathbf{\beta }}_{\mathbf{0}}$indicates the y-intercept term of the curve. It means it gives the value of E(y) when${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$

${\mathbf{\beta }}_{\mathbf{1}}$indicates the magnitude of the shift in parabola due to changes in the value of x (shift parameter)

${\mathbf{\beta }}_{\mathbf{2}}$indicates the rate of curvature of the parabola. (Shape parameter).

Simplification of R2

The value of${\mathbf{R}}^{\mathbf{2}}$given here is 0.226 which denotes that about 23% of the variation in the variables can be explained by the model. A higher value of${\mathbf{R}}^{\mathbf{2}}$means that the model is a good fit for the data while a lower value suggests otherwise.

Here, 23% is a very low value for${\mathbf{R}}^{\mathbf{2}}$meaning the model is not a good fit for the data.

Analysis of R2

When a straight-line model was fitted to the data, the value of ${\mathbf{R}}^{\mathbf{2}}$was 18% while when a quadratic model is fitted to the data, the value of ${\mathbf{R}}^{\mathbf{2}}$increases to 23%. This means that the quadratic model is a better fit for the data than a straight-line model. However, 23% is still a lower value meaning a better quadratic model can be used to fit the data.

Significance of the model

To test whether a quadratic model is a good fit for the data, F-test needs to be done wherethe null hypothesis is whether the model parameters are explaining the model where the beta values are zero and the alternate hypothesis is whether the beta values are non-zero.

Mathematically,

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

${\mathbf{H}}_{\mathbf{a}}$:At least one of the parameters${\mathbf{\beta }}_{\mathbf{1}}$or${\mathbf{\beta }}_{\mathbf{2}}$is non zero