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Chapter 13: Problem 76

Multiple regression is used to model \(y=\) annual income using \(x_{1}=\) number of years of education and \(x_{2}=\) number of years employed in current job. a. It is possible that the coefficient of \(x_{2}\) is positive in a bivariate regression but negative in multiple regression. b. It is possible that the correlation between \(y\) and \(x_{1}\) is 0.30 and the multiple correlation between \(y\) and \(x_{1}\) and \(x_{2}\) is 0.26 . c. If the \(F\) statistic for \(\mathrm{H}_{0}: \beta_{1}=\beta_{2}=0\) has a \(\mathrm{P}\) -value \(=0.001,\) then we can conclude that both predictors have an effect on annual income. d. If \(\beta_{2}=0,\) then annual income is independent of \(x_{2}\) in bivariate regression.

Short Answer

Expert verified
a) True; b) True; c) False (could be only one predictor); d) False.

Step by step solution

01

Understanding Coefficient Sign Change

In a bivariate regression, each predictor is considered individually. When another variable is added in multiple regression, it can reveal hidden relationships, such as multicollinearity. This can cause a coefficient to change sign. Therefore, the coefficient of \(x_{2}\) might be positive in bivariate regression but negative in multiple regression due to the inclusion of \(x_{1}\) affecting the relationship.
02

Analyzing Correlation and Multiple Correlation

The correlation between \(y\) and \(x_{1}\) is the strength of their linear relationship alone, whereas multiple correlation includes \(x_{2}\). A lower multiple correlation (0.26) compared to the simple correlation (0.30) indicates that \(x_{2}\) may be adding noise or not contributing valuable information when added to the model, but it is possible mathematically.
03

Evaluating F-test and P-value

The \(F\)-test evaluates if at least one predictor explains variation in the response variable. A \(P\)-value of 0.001 is very low, suggesting that we reject the null hypothesis \(\mathrm{H}_{0}: \beta_{1}=\beta_{2}=0\). Thus, at least one of the predictors significantly contributes to explaining annual income, but not necessarily both.
04

Understanding Zero Coefficient in Bivariate Regression

If \(\beta_{2}=0\) in the multiple regression context, it suggests no linear relationship between \(x_{2}\) and \(y\) after accounting for \(x_{1}\). However, in bivariate regression, \(x_{2}\) could still have a relationship with \(y\) independent of \(x_{1}\). Thus, \(\beta_{2}=0\) in bivariate regression implies independence only if \(x_{2}\) is considered alone.

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Key Concepts

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

Bivariate Regression
When we deal with bivariate regression, we focus on analyzing the relationship between two variables alone. In our exercise, this refers to observing how a single predictor like the number of years employed (x_{2}) impacts the annual income (y).
The equation in bivariate regression is generally of the form y = b_{0} + b_{1}x, where b_{1} represents the relationship strength.
This analysis is straightforward as it does not account for interactions with any other variables.
  • Simplicity: Modeling only two variables, making it easier to conceptualize.
  • Focus: Isolates the effect of one predictor on the response variable.
Understanding bivariate regression is crucial, as it lays the foundation for grasping more complex relationships in multiple regression.
Correlation
Correlation measures how two variables move together. It is expressed as a number between -1 and 1.
A correlation of 0.30 between y and x_{1} indicates a mild positive linear relationship.
If the multiple correlation (considering both x_{1} and x_{2}) is lower, as seen with a value of 0.26, it suggests not all predictors add useful predictive power to the model.
  • Positive Correlation: Both variables increase or decrease together.
  • Negative Correlation: One variable decreases as the other increases.
  • Zero Correlation: No linear relationship is present.
Low multiple correlations can indicate that additional variables might interfere with the model's predictive strength, introducing complexity without benefit.
F-test
The F-test is a pivotal tool for assessing the overall significance of a multiple regression model. It tests the null hypothesis that none of the independent variables have an effect on the dependent variable.
In our context, having an F-test with a P-value of 0.001 means this null hypothesis can be confidently rejected.
This indicates that at least one independent variable (either years of education or years employed, or both) plays a significant role in predicting annual income.
  • Low P-value: Strong evidence against the null hypothesis.
  • High confidence in rejecting H_0.
  • Interpreting F-statistics: Helps in determining if predictors are effective.
In essence, the F-test evaluates the collective importance of the variables included in the model.
Coefficient Sign Change
In statistical modeling, seeing a sign change in coefficients when moving from bivariate to multiple regression is not uncommon.
This happens due to multicollinearity, which occurs when predictors have some correlation among themselves.
In a simple bivariate model, x_{2} might show a positive effect on y, but in multiple regression, with x_{1} added, this might change.
  • Added Variables: They can unveil hidden relationships.
  • Complex Interactions: Additional variables may alter effects.
  • Multicollinearity: When predictors are not independent.
Interpreting a coefficient's sign changes requires caution; they reflect how one variable's contribution can be dependent on the presence of others.

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Most popular questions from this chapter

In the previous exercise, \(r^{2}=0.88\) when \(x_{1}\) is the predictor and \(R^{2}=0.914\) when both \(x_{1}\) and \(x_{2}\) are predictors. Why do you think that the predictions of \(y\) don't improve much when \(x_{2}\) is added to the model? (The association of \(x_{2}\) with \(y\) is \(r=0.5692 .)\)

Suppose you fit a straight-line regression model to \(y=\) amount of time sleeping per day and \(x=\) age of subject. Values of \(y\) in the sample tend to be quite large for young children and for elderly people, and they tend to be lower for other people. Sketch what you would expect to observe for (a) the scatterplot of \(x\) and \(y\) and (b) a plot of the residuals against the values of age.

For binary response variables, one reason that logistic regression is usually preferred over straight-line regression is that a fixed change in \(x\) often has a smaller impact on a probability \(p\) when \(p\) is near 0 or near 1 than when \(p\) is near the middle of its range. Let \(y\) refer to the decision to rent or to buy a home, with \(p=\) the probability of buying, and let \(x=\) weekly family income. In which case do you think an increase of \(\$ 100\) in \(x\) has greater effect: when \(x=50,000\) (for which \(p\) is near 1 ), when \(x=0\) (for which \(p\) is near 0 ), or when \(x=500\) ? Explain how your answer relates to the choice of a linear versus logistic regression model.

For all students at Walden University, the prediction equation for \(y=\) college GPA (range \(0-4.0\) ) and \(x_{1}=\) high school GPA (range \(0-4.0\) ) and \(x_{2}=\) college board score (range \(200-800\) ) is \(\hat{y}=0.20+0.50 x_{1}+0.002 x_{2}\) a. Find the predicted college GPA for students having (i) high school GPA \(=4.0\) and college board score \(=800\) and (ii) \(x_{1}=2.0\) and \(x_{2}=200\). b. For those students with \(x_{2}=500\), show that \(\hat{y}=1.20+0.50 x_{1}\) c. For those students with \(x_{2}=600\), show that \(\hat{y}=1.40+0.50 x_{1}\). Thus, compared to part \(b\), the slope for \(x_{1}\) is still 0.50 , and increasing \(x_{2}\) by 100 (from 500 to 600 ) shifts the intercept upward by \(100 \times\left(\right.\) slope for \(\left.x_{2}\right)=100(0.002)=0.20\) units.

You own a gift shop that has a campus location and a shopping mall location. You want to compare the regressions of \(y=\) daily total sales on \(x=\) number of people who enter the shop, for total sales listed by day at the campus location and at the mall location. Explain how you can do this using regression modeling a. With a single model, having an indicator variable for location, that assumes the slopes are the same for each location. b. With separate models for each location, permitting the slopes to be different.
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