91Ó°ÊÓ

Predicting weight Let's use multiple regression to predict total body weight (TBW, in pounds) using data from a study of female college athletes. Possible predictors are \(\mathrm{HGT}=\) height (in inches), \(\% \mathrm{BF}=\) percent body fat, and age. The display shows the correlation matrix for these variables. a. Which explanatory variable gives by itself the best predictions of weight? Explain. b. With height as the sole predictor, \(\hat{y}=-106+3.65\) (HGT) and \(r^{2}=0.55\). If you add \%BF as a predictor, you know that \(R^{2}\) will be at least \(0.55 .\) Explain why. c. When you add \% body fat to the model, \(\hat{y}=-121+\) \(3.50(\mathrm{HGT})+1.35(\% \mathrm{BF})\) and \(R^{2}=0.66 .\) When you add age to the model, \(\hat{y}=-97.7+3.43(\mathrm{HGT})+\) \(1.36(\% \mathrm{BF})-0.960(\mathrm{AGE})\) and \(R^{2}=0.67\). Once you know height and \% body fat, does age seem to help you in predicting weight? Explain, based on comparing the \(R^{2}\) values.

Step by step solution

01

Determine the Best Predictor

To find the best single predictor of weight, look for the variable with the highest correlation with TBW. The correlation matrix will show the correlation coefficients between weight and other variables. The variable with the highest absolute correlation value with weight is the best predictor.

02

Understand Multiple Regression Basics

In multiple regression, adding a new predictor to the model will never decrease the \(R^2\) value. This is because \(R^2\) is the proportion of the variance for a dependent variable that's explained by independent variables in a regression model. When you add predictors, even if they are not very helpful, they can't decrease the \(R^2\) because it is always adjusted to reflect any new information.

03

Analyze Improvement with Added Predictors

When \,%BF\ is added, the model changes to \(\hat{y} = -121 + 3.50(HGT) + 1.35(%BF)\) with an increased \(R^2\) of 0.66. The increase in \(R^2\) from 0.55 to 0.66 indicates that \,%BF\ is a valuable predictor, as it explains additional variance in weight.

04

Evaluate the Contribution of Age

Adding \(AGE\) changes the model to \(\hat{y} = -97.7 + 3.43(HGT) + 1.36( \% \,BF) - 0.960(AGE)\) with \(R^2 = 0.67\). The \(R^2\) only increases marginally from 0.66 to 0.67. This suggests that while \(AGE\) contributes some additional explanation of variance, it provides minimal additional predictive power beyond \(HGT\) and \%BF\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

In the context of multiple regression analysis, the correlation coefficient is a key statistic that measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where a value close to 1 or -1 indicates a strong linear relationship, and a value close to 0 indicates a weak linear relationship.
For example, if we look at the correlation between height (HGT) and total body weight (TBW) among female college athletes, we can determine how effectively height alone can predict weight. If the correlation coefficient between these two variables is close to 1 or -1, height is a good predictor on its own.
However, if you're working with multiple predictors, a single high correlation coefficient might not tell the full story. It's crucial to see how it relates to others in a multiple regression model because added variables might better capture or explain variations together, even if their individual correlation coefficients are lower.

• The closer the correlation coefficient to ±1, the stronger the relationship.
• A positive coefficient indicates both variables increase together.
• A negative coefficient shows that one variable decreases as the other increases.

Coefficient of Determination (R^2)

The coefficient of determination, denoted as \(R^2\), tells us how well the predictor variables can explain the variance in the dependent variable (in our case, total body weight).
In simple terms, \(R^2\) is the proportion of the variance in the dependent variable that is predictable from the independent variables. For example, if \(R^2 = 0.55\), it means 55% of the variance in weight can be explained by height alone. This provides a clear picture of the prediction power of the model with just height.
As more predictors are added, \(R^2\) never decreases. In our exercise, adding \%BF increases \(R^2\) to 0.66, showing enhanced prediction ability with the inclusion of another variable.
Yet, when age is added, the increase from 0.66 to 0.67 suggests minimal added value, signifying age does not offer substantial additional information after considering height and body fat.

• \(R^2\) values range from 0 to 1.
• Higher \(R^2\) values mean a better fit of the model to data.
• \(R^2\) will not decrease with the addition of more predictors.

Predictor Variables

Predictor variables are the independent variables used in a regression model to predict the outcome of the dependent variable. In our exercise, the predictor variables include height (\(HGT\)), percent body fat (%BF), and age. These are evaluated to see how well they predict total body weight among female college athletes.
Each predictor variable contributes to the predictive power of the regression model. For instance, height, when used alone, proved to be a significant predictor of weight but became even more powerful when combined with percent body fat information. This exemplifies the synergy of multiple predictors.
When choosing predictor variables:

• Select variables based on strong theoretical backgrounds or previous research.
• Be cautious of multicollinearity, which occurs when predictor variables are highly correlated amongst themselves, leading to less reliable predictions.
• Evaluate the incremental value of each predictor using the change in \(R^2\) and other statistical tests.

Understanding and choosing the right set of predictor variables is integral to building a robust multiple regression model that accurately forecasts the dependent variable.