Question

Discuss the application of multiple regression model using a real-life example. [Hint: You are supposed to examine a possible relationship between a dependent and at least three important independent variables in the example which you have chosen. Identify dependent variable and independent variable and write the mathematical regression equation for the example and explain each component of the equation.]

Answer 1

Let us consider a popular dataset mtcars for Motor Trend Car in R as follows:

A data frame with 32 observations on 4 (numeric) variables.

[, 1]   mpg   Miles/(US) gallon
[, 2]   disp   Displacement (cu.in.)
[, 3]   hp   Gross horsepower
[, 4]   wt   Weight (1000 lbs)

Dataset:

	mpg	disp	hp	wt
Mazda RX4	21	160	110	2.62
Mazda RX4 Wag	21	160	110	2.875
Datsun 710	22.8	108	93	2.32
Hornet 4 Drive	21.4	258	110	3.215
Hornet Sportabout	18.7	360	175	3.44
Valiant	18.1	225	105	3.46
Duster 360	14.3	360	245	3.57
Merc 240D	24.4	146.7	62	3.19
Merc 230	22.8	140.8	95	3.15
Merc 280	19.2	167.6	123	3.44
Merc 280C	17.8	167.6	123	3.44
Merc 450SE	16.4	275.8	180	4.07
Merc 450SL	17.3	275.8	180	3.73
Merc 450SLC	15.2	275.8	180	3.78
Cadillac Fleetwood	10.4	472	205	5.25
Lincoln Continental	10.4	460	215	5.424
Chrysler Imperial	14.7	440	230	5.345
Fiat 128	32.4	78.7	66	2.2
Honda Civic	30.4	75.7	52	1.615
Toyota Corolla	33.9	71.1	65	1.835
Toyota Corona	21.5	120.1	97	2.465
Dodge Challenger	15.5	318	150	3.52
AMC Javelin	15.2	304	150	3.435
Camaro Z28	13.3	350	245	3.84
Pontiac Firebird	19.2	400	175	3.845
Fiat X1-9	27.3	79	66	1.935
Porsche 914-2	26	120.3	91	2.14
Lotus Europa	30.4	95.1	113	1.513
Ford Pantera L	15.8	351	264	3.17
Ferrari Dino	19.7	145	175	2.77
Maserati Bora	15	301	335	3.57
Volvo 142E	21.4	121	109	2.78

Then, linear regression :

mpg is dependent variable

and other 3 'disp', 'hp' and 'wt' are independent variables

Now,

> model = lm(mpg~disp+hp+wt, data = dataset)
> summary(model)

Call:
lm(formula = mpg ~ disp + hp + wt, data = dataset)

Residuals:
Min 1Q Median 3Q Max
-3.891 -1.640 -0.172 1.061 5.861

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 37.105505 2.110815 17.579 < 2e-16 ***
disp -0.000937 0.010350 -0.091 0.92851
hp -0.031157 0.011436 -2.724 0.01097 *
wt -3.800891 1.066191 -3.565 0.00133 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.639 on 28 degrees of freedom
Multiple R-squared: 0.8268,   Adjusted R-squared: 0.8083
F-statistic: 44.57 on 3 and 28 DF, p-value: 8.65e-11

So, equation:

mpg = 37.105 - 0.0009 * disp - 0.031 * hp -3.80 * wt

Also, p-value of model = 8.65e-11 < 0.05 , so model is significant

Also, R-squared: 0.8268 or these 3 variables explain 82.68% variability in mpg.

Explanation:

• With 1 unit of increase in displacement, mileage decreases by 0.0009 units
• With 1 unit of increase in horsepower, mileage decreases by 0.031 units
• With 1 unit of increase in weight, mileage decreases by 3.80 units

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