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Chapter 14: Problem 4

True or False: The value of \(R^{2}\) never decreases as more explanatory variables are added to a regression model.

Short Answer

Expert verified
True

Step by step solution

01

Understand the Concept of R-Squared (\r\(R^{2}\)\r)

R-squared (\r\(R^{2}\)\r) is a statistical measure that represents the proportion of the variance for a dependent variable that is explained by an independent variable or variables in a regression model. It is a value between 0 and 1, where 0 means no explanatory power and 1 means perfect explanatory power.
02

Effect of Adding More Explanatory Variables

When more explanatory variables are added to a regression model, \r\(R^{2}\)\r generally increases or at least stays the same. This is because adding more variables can generally help better explain the variance in the dependent variable, even if the new variables don't add much explanatory power. However, \r\(R^{2}\)\r typically does not decrease unless there is some sort of computational or model specification error.
03

Inclusion of Irrelevant Variables

Even if the added explanatory variables are irrelevant (i.e., they do not actually have a relationship with the dependent variable), \r\(R^{2}\)\r will generally not decrease. It may remain the same or increase very slightly because \r\(R^{2}\)\r measures the proportion of variance explained and having extra variables does not take away the explanatory power of already existing variables in the model.
04

Conclusion: True or False

Given that \r\(R^{2}\)\r does not decrease when more explanatory variables are added, we can conclude that the statement is True.

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Key Concepts

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

Regression Model
A regression model is a statistical tool used to understand relationships between dependent and independent variables. In simpler terms, it helps us predict the value of one variable based on the value(s) of one or more other variables.

Imagine you want to predict a student's exam score based on the hours they studied. Here, the exam score is the dependent variable, and the hours studied is the independent variable. A regression model will help you find the mathematical relationship between these variables.

This model is very useful in various fields like economics, engineering, and social sciences because it allows us to make predictions and understand the underlying patterns in data.
Explanatory Variables
Explanatory variables, also known as independent variables or predictors, are the variables in a regression model that attempt to explain the changes in the dependent variable. They provide the input or cause, and the dependent variable represents the output or effect.

For example, if you're studying the effect of hours studied and attendance on exam scores, both 'hours studied' and 'attendance' would be your explanatory variables. They help us understand and predict the dependent variable, which is the exam score.

Adding more explanatory variables to a regression model can help improve its accuracy. However, they should be relevant to the context to provide meaningful explanations.
Variance Explanation
Variance explanation is a core concept in regression analysis. It refers to how much of the variability in the dependent variable is accounted for by the independent variables. This is where R-squared (\r\( R^{2} \)\r) becomes crucial.

\r\( R^{2} \)\r) measures the proportion of variance in the dependent variable that can be predicted from the independent variables. If \r\( R^{2} \)\r) is 0.7, it means that 70% of the variance in the dependent variable is explained by the model. The closer \r\( R^{2} \)\r) is to 1, the better the model explains the variance.

This metric helps us understand the effectiveness of our regression model in explaining the data.
Statistical Measure in Regression
R-squared (\r\( R^{2} \)\r) is a key statistical measure in regression analysis. It quantifies how well the independent variables explain the variability in the dependent variable. Another important aspect is how R-squared behaves when more explanatory variables are added.

Usually, \r\( R^{2} \)\r) increases or stays the same as more explanatory variables are added. This is because more variables can help the model capture more variance in the dependent variable. However, if these additional variables are irrelevant, the increase might be very slight or even negligible.

Remember, while a high \r\( R^{2} \)\r) value indicates a good fit, it does not guarantee that the model is appropriate. Always validate the relevance of explanatory variables and check for other statistics like the adjusted \r\( R^{2} \)\r).

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Most popular questions from this chapter

What is multicollinearity? How can we check for it? What are the consequences of multicollinearity?

You obtain the multiple regression equation \(\hat{y}=5+3 x_{1}-4 x_{2}\) from a set of sample data. (a) Interpret the slope coefficients for \(x_{1}\) and \(x_{2}\). (b) Determine the regression equation with \(x_{1}=10\). Graph the regression equation with \(x_{1}=10\). (c) Determine the regression equation with \(x_{1}=15 .\) Graph the regression equation with \(x_{1}=15\) (d) Determine the regression equation with \(x_{1}=20 .\) Graph the regression equation with \(x_{1}=20 .\) (e) What is the effect of changing the value \(x_{1}\) on the graph of the regression equation?

A researcher wants to determine a model that can be used to predict the 28 -day strength of a concrete mixture. The following data represent the 28 -day and 7 -day strength (in pounds per square inch) of a certain type of concrete along with the concrete's slump. Slump is a measure of the uniformity of the concrete, with a higher slump indicating a less uniform mixture. $$ \begin{array}{ccc} \text { Slump (inches) } & \text { 7-Day psi } & \text { 28-Day psi } \\ \hline 4.5 & 2330 & 4025 \\ \hline 4.25 & 2640 & 4535 \\\ \hline 3 & 3360 & 4985 \\ \hline 4 & 1770 & 3890 \\ \hline 3.75 & 2590 & 3810 \\ \hline 2.5 & 3080 & 4685 \\ \hline 4 & 2050 & 3765 \\ \hline 5 & 2220 & 3350 \\ \hline 4.5 & 2240 & 3610 \\ \hline 5 & 2510 & 3875 \\ \hline 2.5 & 2250 & 4475 \end{array} $$ (a) Construct a correlation matrix between slump, 7 -day psi, and 28 -day psi. Is there any reason to be concerned with multicollinearity based on the correlation matrix? (b) Find the least-squares regression equation \(\hat{y}=b_{0}+b_{1} x_{1}+b_{2} x_{2},\) where \(x_{1}\) is slump, \(x_{2}\) is 7 -day strength, and \(y\) is the response variable, 28 -day strength. (c) Draw residual plots and a boxplot of the residuals to assess the adequacy of the model. (d) Interpret the regression coefficients for the least-squares regression equation. (e) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\). (f) Test \(H_{0}: \beta_{1}=\beta_{2}=0\) versus \(H_{1}:\) at least one of the \(\beta_{1} \neq 0\) at the \(\alpha=0.05\) level of significance. (g) Test the hypotheses \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\) and \(H_{0}: \beta_{2}=0\) versus \(H_{1}: \beta_{2} \neq 0\) at the \(\alpha=0.05\) level of significance. (h) Predict the mean 28 -day strength of all concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (i) Predict the 28 -day strength of a specific sample of concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (j) Construct \(95 \%\) confidence and prediction intervals for concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. Interpret the results.

More Age Estimation In the article "Bigger Teeth for Longer Life? Longevity and Molar Height in Two Roe Deer Populations" (Biology Letters [June, 2007\(]\) vol. 3 no. 3 \(268-270)\), researchers developed a model to predict the tooth height (in \(\mathrm{mm}\) ), \(y\), of roe deer based on their age, \(x_{1}\), gender, \(x_{2}(0=\) female \(, 1=\) male \(),\) and location, \(x_{3}\) (Trois Fontaines deer, which have a shorter life expectancy, and Chiz茅, which have a longer life expectancy, \(x_{3}=0\) for Trois Fontaines, \(x_{3}=1\) for Chiz茅). The model is $$ \hat{y}=7.790-0.382 x_{1}-0.587 x_{2}-0.925 x_{3}+0.091 x_{2} x_{3} $$ (a) What is the expected tooth length of a female roe deer who is 12 years old and lives in Trois Fontaines? (b) What is the expected tooth length of a male roe deer who is 8 years old and lives in Chiz茅? (c) What is the interaction term? What does the coefficient of the interaction term imply about tooth length?

Kepler's Law of Planetary Motion The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year. Johann Kepler studied the relation between the sidereal year of a planet and its distance from the sun in 1618 . The following data show the distances that the planets are from the sun and their sidereal years. $$ \begin{array}{lcc} \text { Planet } & \begin{array}{l} \text { Distance from Sun, } x \\ \text { (millions of miles) } \end{array} & \text { Sidereal Year, } \boldsymbol{y} \\ \hline \text { Mercury } & 36 & 0.24 \\ \hline \text { Venus } & 67 & 0.62 \\ \hline \text { Earth } & 93 & 1.00 \\ \hline \text { Mars } & 142 & 1.88 \\ \hline \text { Jupiter } & 483 & 11.9 \\ \hline \text { Saturn } & 887 & 29.5 \\ \hline \text { Uranus } & 1785 & 84.0 \\ \hline \text { Neptune } & 2797 & 165.0 \\ \hline \text { Pluto* } & 3675 & 248.0 \end{array} $$ (a) Determine the least-squares regression equation, treating distance from the sun as the explanatory variable. (b) A normal probability plot of the residuals indicates that the residuals are approximately normally distributed. Test whether a linear relation exists between distance from the sun and sidereal year. (c) Draw a scatter diagram, treating distance from the sun as the explanatory variable. (d) Plot the residuals against the explanatory variable, distance from the sun. (e) Does a linear model seem appropriate based on the scatter diagram and residual plot? (Hint: See Section 4.3.) (f) What is the moral?
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