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Chapter 14
Multiple Regression Analysis
LEARNING OBJECTIVES
This chapter presents the potential of multiple regression analysis as a tool in business decision making and its applications, thereby enabling you to:
1. Develop a multiple regression model.
2. Understand and apply significance tests of the regression model and its coefficients.
3. Compute and interpret residuals, the standard error of the estimate, and the coefficient of determination.
4. Interpret multiple regression computer output.
CHAPTER TEACHING STRATEGY
In chapter 13 using simple regression, the groundwork was prepared for chapter 14 by presenting the regression model along with mechanisms for testing the strength of the model such as se, r2, a t test of the slope, and the residuals. In this chapter, multiple regression is presented as an extension of the simple linear regression case. It is initially pointed out that any model that has at least one interaction term or a variable that represents a power of two or more is considered a multiple regression model. Multiple regression opens up the possibilities of predicting by multiple independent variables and nonlinear relationships. It is emphasized in the chapter that with both simple and multiple regression models there is only one dependent variable. Simple regression utilizes only one independent variable also whereas multiple regression can utilize more than one independent variable.
Presented early in chapter 14 are the simultaneous equations that need to be solved to develop a first-order multiple regression model using two predictors. This should help the student see that there are three equations with three unknowns to be solved. In addition, there are eight values that need to be determined before solving the simultaneous equations ( x1, x2, y, x12, . . .) Suppose there are five predictors. Six simultaneous equations must be solved and the number of sums needed as constants in the equations become overwhelming. At this point, the student will begin to realize that most researchers do not want to take the time nor the effort to solve for multiple regression models by hand. For this reason, much of the chapter is presented using computer printouts. The assumption is that the use of multiple regression analysis is largely from computer analysis.
Topics included in this chapter are similar to the ones in chapter 13 including tests of the slope, R2, and se. In addition, an adjusted R2 is introduced in chapter 14. The adjusted R2 takes into account the degrees of freedom error and total degrees of freedom whereas R2 does not. If there is a significant discrepancy between adjusted R2 and R2, then the regression model may not be as strong as it appears to be with the R2. The gap between R2 and adjusted R2 tends to increase as non significant independent variables are added to the regression model and decreases with increased sample size.
CHAPTER OUTLINE
14.1 The Multiple Regression Model
Multiple Regression Model with Two Independent Variables (First-Order)
Determining the Multiple Regression Equation
A Multiple Regression Model
14.2 Significant Tests of the Regression Model and its Coefficients
Testing the Overall Model
Significance Tests of the Regression Coefficients
14.3 Residuals, Standard Error of the Estimate, and R2
Residuals
SSE and Standard Error of the Estimate
Coefficient of Determination (R2)
Adjusted R2
14.4 Interpreting Multiple Regression Computer Output
KEY TERMS
Adjusted R2 R2
Coefficient of Multiple Determination (R2) Residual
Dependent Variable Response Plane
Independent Variable Response Surface
Least Squares Analysis Response Variable
Multiple Regression Standard Error of the Estimate
Outliers Sum of Squares of Error
Partial Regression Coefficient
SOLUTIONS TO PROBLEMS IN CHAPTER 14
14.1 The regression model is:
EMBED Equation.3 = 25.03 - 0.0497 x1 + 1.928 x2
Predicted value of y for x1 = 200 and x2 = 7 is:
EMBED Equation.3 = 25.03 - 0.0497(200) + 1.928(7) = 28.586
14.2 The regression model is:
EMBED Equation.3 = 118.56 - 0.0794 x1 - 0.88428 x2 + 0.3769 x3
Predicted value of y for x1 = 33, x2 = 29, and x3 = 13 is:
EMBED Equation.3 = 118.56 - 0.0794(33) - 0.88428(29) + 0.3769(13) = 95.19538
14.3 The regression model is:
EMBED Equation.3 = 121.62 - 0.174 x1 + 6.02 x2 + 0.00026 x3 + 0.0041 x4
There are four independent variables. If x2, x3, and x4 are held constant, the predicted y will decrease by - 0.174 for every unit increase in x1. Predicted y will increase by 6.02 for every unit increase in x2 is x1, x3, and x4 are held constant. Predicted y will increase by 0.00026 for every unit increase in x3 holding x1, x2, and x4 constant. If x4 is increased by one unit, the predicted y will increase by 0.0041 if x1, x2, and x3 are held constant.
14.4 The regression model is:
EMBED Equation.3 = 31,409.5 + 0.08425 x1 + 289.62 x2 - 0.0947 x3
For every unit increase in x1, the predicted y increases by 0.08425 if x2 and x3 are held constant. The predicted y will increase by 289.62 for every unit increase in x2 if x1 and x3 are held constant. The predicted y will decrease by 0.0947 for every unit increase in x3 if x1 and x2 are held constant.
14.5 The regression model is:
Per Capita = -538 + 0.23368 Paper Consumption +
18.09 Fish Consumption 0.2116 Gasoline Consumption.
For every unit increase in paper consumption, the predicted per capita consumption increases by 0.23368 if fish and gasoline consumption are held constant. For every unit increase in fish consumption, the predicted per capita consumption increases by 18.09 if paper and gasoline consumption are held constant. For every unit increase in gasoline consumption, the predicted per capita consumption decreases by 0.2116 if paper and fish consumption are held constant.
14.6 The regression model is:
Insider Ownership =
17.68 - 0.0594 Debt Ratio - 0.118 Dividend Payout
The coefficients mean that for every unit of increase in debt ratio there is a predicted decrease of - 0.0594 in insider ownership if dividend payout is held constant. On the other hand, if dividend payout is increased by one unit, there is a predicted drop of insider ownership by 0.118 if debt ratio is held constant.
14.7 There are 9 predictors in this model. The F test for overall significance of the model is 1.99 with a probability of .0825. This model is not significant at ( = .05. Only one of the t values is statistically significant. Predictor x1 has a t of 2.73 which has an associated probability of .011 and this is significant at ( = .05.
14.8 This model contains three predictors. The F test is significant at ( = .05 but not at ( = .01. The t values indicate that only one of the three predictors is significant. Predictor x1 yields a t value of 3.41 with an associated probability of .005. I would recommend rerunning the model using only x1 and then search for other variables besides x2 and x3 to include in future models.
14.9 The regression model is:
Per Capita Consumption = -538 + 0.23368 Paper Consumption +
18.09 Fish Consumption 0.2116 Gasoline Consumption
This model yields an F = 24.63 with p-value = .002. Thus, there is overall significance at ( = .01. One of the three predictors is significant. Paper Consumption has a t = 5.31 with p-value of .003. Paper Consumption is statistically significant at ( = .01. The p-values of the t statistics for the other two predictors are insignificant indicating that a model with just
Paper Consumption as a single predictor might be nearly as strong.
14.10 The regression model is:
Insider Ownership =
17.68 - 0.0594 Debt Ratio - 0.118 Dividend Payout
The overall value of F is only 0.02 with p-value of .982. This model is not significant. Neither of the t values are significant (tDebt = -0.19 with a p-value of .855 and tDividend = -0.11 with a p-value of .913).
14.11 The regression model is:
EMBED Equation.3 = 3.981 + 0.07322 x1 - 0.03232 x2 - 0.003886 x3
The overall F for this model is 100.47 with is significant at ( = .001. Only one of the predictors, x1, has a significant t value (t = 3.50, p-value of .005). The other independent variables have non significant t values
(x2: t = -1.55, p-value of .15 and x3: t = -1.01, p-value of .332). Since x2 and x3 are nonsignificant predictors, the researcher should consider the using a simple regression model with only x1 as a predictor. The R2 would drop some but the model would be much more parsimonious.
14.12 The regression equation for the model using both x1 and x2 is:
EMBED Equation.3 = 243.44 - 16.608 x1 - 0.0732 x2
The overall F = 156.89 with a p-value of .000. x1 is a significant predictor of y as indicated by t = - 16.10 and a p-value of .000.
For x2, t = -0.39 with a p-value of .702. x2 is not a significant predictor of Y when included with x1. Since x2 is not a significant predictor, the researcher might want to rerun the model using justx1 as a predictor.
The regression model using only x1 as a predictor is:
EMBED Equation.3 = 235.143 - 16.7678 x1
There is very little change in the coefficient of x1 from model one (2 predictors) to this model. The overall F = 335.47 with a p-value of .000 is highly significant. By using the one predictor model, we get virtually the same predictability as with the two predictor model and it is more parsimonious.
14.13 There are 3 predictors in this model and 15 observations.
The regression equation is:
EMBED Equation.3 = 657.0534435 + 5.710310868 x1 0.416916682 x2 3.471481072 x3
F = 8.96 with a p-value of .0027
x1 is significant at ( = .01 (t = 3.187, p-value of .009)
x3 is significant at ( = .05 (t = - 2.406, p-value of .035)
The model is significant overall.
14.14 The standard error of the estimate is 3.503. R2 is .408 and the adjusted R2 is only .203. This indicates that there are a lot of insignificant predictors in the model. That is underscored by the fact that eight of the nine predictors have nonsignificant t values.
14.15 se = 9.722, R2 = .515 but the adjusted R2 is only .404. The difference in the two is due to the fact that two of the three predictors in the model are non-significant. The model fits the data only modestly. The adjusted R2 indicates that 40.4% of the variance of y is accounted for by this model, but 59.6% is unaccounted for.
14.16 The standard error of the estimate of 2085 indicates that this model predicts within + 2085 on Per Capita Consumption about 68% of the time. The entire range of Per Capita for the data is slightly more than 19,000. Relative to this range, the standard error of the estimate is small. R2 = .937 and the adjusted value of R2 is .899. Overall, this model is strong.
14.17 se = 6.544. R2 = .005 but the adjusted value of R2 is zero. This model has
no predictability.
14.18 The value of se = 0.2331, R2 = .965, and adjusted R2 = .955. This is a
very strong regression model. However, since X2 and X3 are nonsignificant predictors, the researcher should consider the using a simple regression model with only X1 as a predictor. The R2 would drop some but the model would be much more parsimonious.
14.19 For the regression equation for the model using both x1 and x2, se = 6.333,
R2 = .963 and adjusted R2 = .957. Overall, this is a very strong model. For the regression model using only x1 as a predictor, the standard error of the estimate is 6.124, R2 = .963 and the adjusted R2 = .960. The value of R2 is the same as it was with the two predictors. However, the adjusted R2 is slightly higher because the nonsignificant variable has been removed. In conclusion, by using the one predictor model, we get virtually the same predictability as with the two predictor model and it is more parsimonious.
14.20 R2 = .842, adjusted R2 = .630, se = 109.43. The model is significant
overall. The R2 is relatively high but the adjusted R2 is more than 20%
lower indicating an inflated value of R2. The model is not as strong as the
R2 = .842 might indicate.
14.21 Both the Normal Plot and the Histogram indicate that there may be some
problem with the error terms being normally distributed. Both the I Chart
and the Residuals vs. Fits show that there may be some lack of homogeneity of error variance.
14.22 There are four predictors. The equation of the regression model is:
EMBED Equation.3 = -55.9 + 0.0105 x1 0.107 x2 + 0.579 x3 0.870 x4
The test for overall significance yields an F = 55.52 with a p-value of .000
which is significant at ( = .001. Three of the t tests for regression coefficients are significant at ( = .01 including the coefficients for
x2, x3, and x4. The R2 value of 80.2% indicates strong predictability for the model. The value of the adjusted R2 (78.8%) is close to R2 and se is 9.025.
14.23 There are two predictors in this model. The equation of the regression model is:
EMBED Equation.3 = 203.3937 + 1.1151 x1 2.2115 x2
The F test for overall significance yields a value of 24.55 with an
associated p-value of .0000013 which is significant at ( = .00001. Both
variables yield t values that are significant at a 5% level of significance.
x2 is significant at ( = .001. The R2 is a rather modest 66.3% and the
standard error of the estimate is 51.761.
14.24 The regression model is:
EMBED Equation.3 = 137.27 + 0.0025 x1 + 29.206 x2
F = 10.89 with p = .005, se = 9.401, R2 = .731, adjusted R2 = .664. For x1, t = 0.01 with p = .99 and for x2, t = 4.47 with p = .002. This model has good predictability. The gap between R2 and adjusted R2 indicates that there may be a nonsignificant predictor in the model. The t values show x1 has virtually no predictability and x2 is a significant predictor of y.
14.25 The regression model is:
EMBED Equation.3 = 362 4.75 x1 - 13.9 x2 + 1.87 x3
F = 16.05 with p = .001, se = 37.07, R2 = .858, adjusted R2 = .804. For x1, t = -4.35 with p = .002; for x2, t = -0.73 with p = .483, for x3, t = 1.96 with p = .086. Thus, only one of the three predictors, x1, is a significant predictor in this model. This model has very good predictability (R2 = .858). The gap between R2 and adjusted R2 underscores the fact that there are two nonsignificant predictors in this model.
14.26 The overall F for this model was 12.19 with a p-value of .002 which is significant at ( = .01. The t test for Silver is significant at ( = .01 ( t = 4.94, p = .001). The t test for Aluminum yields a t = 3.30 with a p-value of .016 which is significant at ( = .05. The t test for Copper was insignificant with a p-value of .939. The value of R2 was 82.1% compared
to an adjusted R2 of 75.3%. The gap between the two indicates the presence of some insignificant predictors (Copper). The standard error of the estimate is 53.44.
14.27 The regression model was:
Employment = 71.03 + 0.4620 NavalVessels + 0.02082 Commercial
F = 1.22 with p = .386 (not significant)
R2 = .379 and adjusted R2 = .068
The low value of adjusted R2 indicates that the model has very low predictability. Both t values are not significant (tNavalVessels = 0.67 with
p = .541 and tCommercial = 1.07 with p = .345). Neither predictor is a significant predictor of employment.
14.28 The regression model was:
All = -1.06 + 0.475 Food + 0.250 Shelter 0.008 Apparel +
0.272 Fuel Oil
F = 97.98 with a p-value of .000
se = 0.7472, R2 = .963 and adjusted R2 = .953
One of the predictor variables, Food produces a t value that is significant at ( = .001. Two others are significant at ( = .05: Shelter (t = 2.48, p-value of .025 and Fuel Oil (t = 2.36 with a p-value of .032).
14.29 The regression model was:
Corn = -2718 + 6.26 Soybeans 0.77 Wheat
F = 14.25 with a p-value of .003 which is significant at ( = .01
se = 862.4, R2 = 80.3%, adjusted R2 = 74.6%
One of the two predictors, Soybeans, yielded a t value that was significant at ( = .01 while the other predictor, Wheat was not significant (t = -0.75 with a p-value of .476).
14.30 The regression model was:
Grocery = 76.23 + 0.08592 Housing + 0.16767 Utility
+ 0.0284 Transportation - 0.0659 Healthcare
F = 2.29 with p = .095 which is not significant at ( = .05.
se = 4.416, R2 = .315, and adjusted R2 = .177.
Only one of the four predictors has a significant t ratio and that is Utility with t = 2.57 and p = .018. The ratios and their respective probabilities are:
thousing = 1.68 with p = .109, ttransportation = 0.17 with p = .87, and
thealthcare = - 0.64 with p = .53.
This model is very weak. Only the predictor, Utility, shows much promise in accounting for the grocery variability.
14.31 The regression equation is:
EMBED Equation.3 = 87.9 0.256 x1 2.71 x2 + 0.0706 x3
F = 47.57 with a p-value of .000 significant at ( = .001.
se = 0.8503, R2 = .941, adjusted R2 = .921.
All three predictors produced significant t tests with two of them (x2 and x3
significant at .01 and the other, x1 significant at ( = .05. This is a very
strong model.
14.32 Two of the diagnostic charts indicate that there may be a problem with the
error terms being normally distributed. The I Chart shows some slight tendency for the error terms to be nonindependent. In addition, the residuals vs. fits chart indicates a potential heteroscadasticity problem with residuals for higher values of x producing more variability that those for lower values of x.
PAGE
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