Question

Question 3. Multiple linear regression [6 marks] Create a multiple linear regression model, including as explanatory variables wt, am and qsec. To run multiple linear regression to predict variable A based on variables B, C and D you need to use R’s linear model command, Im as follows, storing the results in an object I'll call regm. regm <- lm (A B + C + D) summary(regm) Report the output from the relevant summary() command. Explain why the R2 and estimated slope might be different for this model compared to the simple linear regression model (with just wt as a predictor).

Answer 1

R-Script:

"gear" "carb" 3Q > rm(list=1s() > #attach(mtcars) > names (mtcars) [1] "mpg" "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" > mult. reg.cars <- Im(mpg ~ wt + am + osec) > summary(mult.reg.cars) Call: Im(formula = mpg wt + am + osec) Residuals: Min 1Q Median Max -3.4811 -1.5555 -0.7257 1.4110 4.6610 Coefficients: Estimate Std. Error t value Pr(>1t|) (Intercept) 9.6178 6.9596 1.382 0.177915 wt -3.9165 0.7112 -5.507 6.95e-06 *** 2.9358 1.4109 2.081 0.046716 * qsec 1.2259 0.2887 4.247 0.000216 *** Signif. codes: 0 ***** 0.001 ***' 0.01 *' 0.05 '.' 0.1''1 Residual standard error: 2.459 on 28 degrees of freedom Multiple R-squared: 0.8497, Adjusted R-squared: 0.8336 F-statistic: 52.75 on 3 and 28 DF, p-value: 1.21e-11 > reg.cars < Im(mpg - wt) > summary(reg.cars) Call: 1m(formula = mpg wt) Residuals: Min 1Q Median 3Q Max -4.5432 -2.3647 -0.1252 1.4096 6.8727 am --- Coefficients: Estimate Std. Error t value Pr>|t|) (Intercept) 37.2851 1.8776 19.858 < 2e-16 *** wt -5.3445 0.5591 -9.559 1.29e-10 *** Signif. codes: 0 0.001 ***' 0.01 0.05 '.' 0.1 Residual standard error: 3.046 on 30 degrees of freedom Multiple R-squared: 0.7528, Adjusted R-squared: 0.7446 F-statistic: 91.38 on 1 and 30 DF, p-value: 1.294e-10 '1

Interpretation:

1. R2 for Multiple Linear Regression of mpg on wt, am and qsec is 0.8497 while that of Simple Linear Regression of mpg on wt is 0.7528. This means that wt, am and qsec collectively explains 84.97% of the total variability in mpg while wt alone explains 75.28% of the total variability in mpg.

Adding more predictors makes the model better as it increases R2.

2. All the slope coefficients in Multiple Regression Model are signicant at 5% level since all the p-values < 0.05.

Thus, Model with all three predictors is better than SImple Linear Model with just wt as predictor.

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