The Jensen et al. [1980] data of 19 subjects were used in...

The Jensen et al. [1980] data of 19 subjects were used in Problems 9.23 to 9.29. Here we consider the data before training. The exercise $\mathrm{VO}_2, \mathrm{MAX}$ is to be regressed upon three variables. $$ \begin{aligned} Y & =\mathrm{VO}_2, \text { max } \\ X_1 & =\text { maximal ejection fraction } \\ X_2 & =\text { maximal heart rate } \\ X_3 & =\text { maximal systolic blood pressure } \end{aligned} $$ The residual mean square with all three variables in the model is 73.40 . The residual sums of squares are: $$ \begin{aligned} & \operatorname{SSRESID}\left(X_1, X_2\right)=1101.58 \\ & \operatorname{SS}_{\text {RESID }}\left(X_1, X_3\right)=1839.80 \\ & \operatorname{SS}_{\text {RESID }}\left(X_2, X_3\right)=1124.78 \\ & \operatorname{SS}_{\text {RESID }}\left(X_1\right)=1966.32 \\ & \operatorname{SS}_{\text {RESID }}\left(X_2\right)=1125.98 \\ & \operatorname{SS}_{\text {RESID }}\left(X_3\right)=1885.98 \\ & \end{aligned} $$ (a) For each model, compute $C_p$. (b) Plot $C_p$ vs. $p$ and select the best model. (c) Compute and plot the average mean square residual vs. $p$.

The Jensen et al. [1980] data of 19 subjects were used in Problems 9.23 to 9.29. Here we consider the data before training. The exercise $\mathrm{VO}_2, \mathrm{MAX}$ is to be regressed upon three variables.
$$
\begin{aligned}
Y & =\mathrm{VO}_2, \text { max } \\
X_1 & =\text { maximal ejection fraction } \\
X_2 & =\text { maximal heart rate } \\
X_3 & =\text { maximal systolic blood pressure }
\end{aligned}
$$
The residual mean square with all three variables in the model is 73.40 . The residual sums of squares are:
$$
\begin{aligned}
& \operatorname{SSRESID}\left(X_1, X_2\right)=1101.58 \\
& \operatorname{SS}_{\text {RESID }}\left(X_1, X_3\right)=1839.80 \\
& \operatorname{SS}_{\text {RESID }}\left(X_2, X_3\right)=1124.78 \\
& \operatorname{SS}_{\text {RESID }}\left(X_1\right)=1966.32 \\
& \operatorname{SS}_{\text {RESID }}\left(X_2\right)=1125.98 \\
& \operatorname{SS}_{\text {RESID }}\left(X_3\right)=1885.98 \\
&
\end{aligned}
$$
(a) For each model, compute $C_p$.
(b) Plot $C_p$ vs. $p$ and select the best model.
(c) Compute and plot the average mean square residual vs. $p$.

Key Concepts

Multiple Linear Regression

Multiple linear regression is a statistical technique used to model the relationship between one dependent variable and two or more independent variables. It estimates the coefficients of the predictors by minimizing the sum of squared differences between observed and predicted values, thereby providing a framework for understanding how changes in the predictors are associated with changes in the response variable.

Residual Sum of Squares (RSS)

Residual Sum of Squares represents the sum of the squared differences between the observed values and those predicted by a model. It quantifies the discrepancy between the data and the estimation provided by the model, serving as a key metric in evaluating model fit. A lower RSS indicates a model that more closely approximates the data.

Mallows' Cp Statistic

Mallows' Cp is a model selection criterion used in regression analysis to assess the trade-off between complexity and goodness of fit. It helps in identifying a model that neither underfits nor overfits the data by comparing the model's RSS with an estimate of the error variance. A smaller Cp value, particularly one close to the number of predictors plus an intercept, suggests a good model choice.

Model Selection

Model selection involves evaluating a set of candidate models based on criteria that balance model complexity and fit. The goal is to select a model that provides an adequate fit to the data without including unnecessary predictors that might lead to overfitting. Techniques such as Mallows' Cp, adjusted R-squared, and AIC are commonly employed to guide this process.

Average Mean Square Residual

The average mean square residual is computed by dividing the residual sum of squares by its corresponding degrees of freedom. It serves as an estimate of the variance of the error term in the regression model and is useful in comparing models of different complexities. This measure aids in assessing the predictive accuracy and reliability of the model across different subsets of predictors.

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