Checking Multiple Regression Assumptions
1. We will examine in detail the process of checking multiple regression assumptions next week. This week we’ll assume the assumptions hold for this data set
Identifying Confounding Variables
1. Open the Excel data set for Week 3. Use New Capital Expenses (X1) and Cost of Materials (X2) as the explanatory variables and End of Year Inventory as your response variable (Y)
2. Generate a simple regression equation relating New Capital Expenses to End of Year Inventory. Report the regression equation and the associated Excel regression table output.
3. Interpret the regression coefficient for New Capital Expenses including significance information. Use complete sentences.
4. Assess if the Cost of Materials variable is a potential confounder variable by first generating a multiple regression equation that includes both explanatory variables in the equation. Report the multiple regression equation
5. Examine the new regression slope coefficient for the New Capital Expense variable with the Cost of Materials variable now entered into the equation. Interpret this coefficient including significance information. Use complete sentences. Write up as appropriate (check the note pages for this week)
6. Calculate the percent of change observed in the simple regression slope coefficient to the multiple regression slope coefficient, for the New Capital Expense variable. Report the percentage and show your calculations. If the percentage is larger than 10% (in either direction), then report the Cost of Materials variable as confounding.
Relative Importance of Explanatory Variables
1. Consider the multiple regression equation generated in #3 above. Create a table which lists just the t statistics for each explanatory variable. Ensure that your table is formatted to class expectations
2. Use the magnitude of the t statistic to determine which explanatory variable is the most significant. Report that variable
Using Multiple Regression Model for Prediction
3. Consider the multiple regression equation generated in #3 above. Predict the value of the End of Year Inventory when the Cost of Materials is $725 and New Capital Expenses are $1369. Show your calculations.
Simple linear regression model
Regression Statistics
Multiple R
0.821384888
R Square
0.674673134
Adjusted R Square
0.672245321
Standard Error
1586.927704
Observations
136
ANOVA
df
SS
MS
F
Significance F
Regression
1
7E+08
7E+08
277.8934
1.77E-34
Residual
134
3.37E+08
2518340
Total
135
1.04E+09
Coefficients
Standard Error
t Stat
P-value
Lower 95.0%
Upper 95.0%
Intercept
798.7778605
173.5187
4.603411
9.54E-06
455.5881
1141.967622
New Capital Expenses
2.293338698
0.137572
16.67014
1.77E-34
2.021246
2.56543152
The coefficient for the new capital expenses [2.2933] implies that the value for the end of year inventory is expected to increase by 2.2933 as a result of a unit increase in new capital expenses. The P-value for this coefficient (1.77E-34 is less than significance level (5%) and hence we conclude that the coefficient is statistically significant.
Multiple linear regression model.
Regression Statistics
Multiple R
0.82948959
R Square
0.68805297
Adjusted R Square
0.68336204
Standard Error
1559.78299
Observations
136
ANOVA
df
SS
MS
F
Significance F
Regression
2
713708763.6
3.57E+08
146.6772
2.27183E-34
Residual
133
323578756.1
2432923
Total
135
1037287520
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
751.017158
171.7189238
4.373526
2.45E-05
411.3637782
1090.670538
Cost of Materials
0.03286509
0.013760177
2.388421
0.018326
0.005647998
0.06008219
New Capital Expenses
1.85477276
0.22803722
8.13364
2.58E-13
1.403723969
2.305821546
The coefficient for the new capital expenses [1.8548] in the multiple linear model implies that the value for the end of year inventory is expected to increase by 1.8548 as a result of a unit increase in new capital expenses when Cost of materials is controlled or held constant. The P-value for this coefficient (2.58E-13) is less than significance level (5%) and hence we conclude that the coefficient is statistically significant in the model.
Percent of change observed
=100-80.9
=19.1%
Since the percentage is larger than 10%, then Cost of Materials is a confounding variable.
The larger the t-value, the larger the differences between the groups under consideration. From the table above, New Capital Expenses has a larger t score and hence it is most significant.
Coefficients
Standard Error
t Stat
P-value
Intercept
751.017158
171.7189238
4.373526
2.45E-05
Cost of Materials
0.03286509
0.013760177
2.388421
0.018326
New Capital Expenses
1.85477276
0.22803722
8.13364
2.58E-13
Prediction
End of year inventory =751.017158+0.03286509*Cost of Materials+1.854772768New Capital Expenses
End of year inventory =751.017158+0.03286509*725+1.854772768*1369
End of year inventory=3314.028268