Regression is a supervised learning method, which is employed to model and analyze the relationship between a dependent (response) variable and one or more independent (predictor) variables.
Linear Regression is based on minimum squared errors of the fitted values.
The fitted model can then be applied to data for continuous value predictions.
Function Used : lm
head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1fit_lm <- lm(mpg~., data=mtcars)summary(fit_lm)##
## Call:
## lm(formula = mpg ~ ., data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4506 -1.6044 -0.1196 1.2193 4.6271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337 18.71788 0.657 0.5181
## cyl -0.11144 1.04502 -0.107 0.9161
## disp 0.01334 0.01786 0.747 0.4635
## hp -0.02148 0.02177 -0.987 0.3350
## drat 0.78711 1.63537 0.481 0.6353
## wt -3.71530 1.89441 -1.961 0.0633 .
## qsec 0.82104 0.73084 1.123 0.2739
## vs 0.31776 2.10451 0.151 0.8814
## am 2.52023 2.05665 1.225 0.2340
## gear 0.65541 1.49326 0.439 0.6652
## carb -0.19942 0.82875 -0.241 0.8122
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared: 0.869, Adjusted R-squared: 0.8066
## F-statistic: 13.93 on 10 and 21 DF, p-value: 3.793e-07In Call section, the function called to generate the fitted model is displayed.
In Residuals section, it provides a quick summary (min, 1Q, median, 3Q, max) of the residual distribution which would be same as summary(fit_lm$residuals)
In Coefficients section, each coefficient is a Gaussian random variable.
Estimate represents the mean distribution of the variableStd.Error displays the standard error of the variable;t value is Estimate divided by Std.Errorp value indicates the probability of getting a value larger than the t value.Residual standard error outputs the standard deviation of residuals
degree of freedom indicates the differences between the observation in training samples and the number used in the model
Multiple R-squared is obtained by dividing the sum of squares. One can use R-squared to measure how close the data is to fit into the regression line.
Adjusted R-squared uses an unbiased estimate, and will be slightly less than multiple R-squared. It’s better to use Adjusted R-squared value for model interpretation.
F-statistic is retrieved by performing an f-test on the model.
fit_lm$coefficients## (Intercept) cyl disp hp drat wt
## 12.30337416 -0.11144048 0.01333524 -0.02148212 0.78711097 -3.71530393
## qsec vs am gear carb
## 0.82104075 0.31776281 2.52022689 0.65541302 -0.19941925fit_lm$residuals## Mazda RX4 Mazda RX4 Wag Datsun 710
## -1.599505761 -1.111886079 -3.450644085
## Hornet 4 Drive Hornet Sportabout Valiant
## 0.162595453 1.006565971 -2.283039036
## Duster 360 Merc 240D Merc 230
## -0.086256253 1.903988115 -1.619089898
## Merc 280 Merc 280C Merc 450SE
## 0.500970058 -1.391654392 2.227837890
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 1.700426404 -0.542224699 -1.634013415
## Lincoln Continental Chrysler Imperial Fiat 128
## -0.536437711 4.206370638 4.627094192
## Honda Civic Toyota Corolla Toyota Corona
## 0.503261089 4.387630904 -2.143103442
## Dodge Challenger AMC Javelin Camaro Z28
## -1.443053221 -2.532181498 -0.006021976
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 2.508321011 -0.993468693 -0.152953961
## Lotus Europa Ford Pantera L Ferrari Dino
## 2.763727417 -3.070040803 0.006171846
## Maserati Bora Volvo 142E
## 1.058881618 -2.968267683fit_lm$effects## (Intercept) cyl disp hp drat
## -1.136497e+02 -2.859568e+01 6.131391e+00 -3.061197e+00 -4.058009e+00
## wt qsec vs am gear
## -8.802034e+00 -1.987287e+00 3.602342e-01 3.804502e+00 -9.857538e-01
## carb
## -6.377059e-01 2.679169e+00 2.072950e+00 -6.318915e-03 -1.254563e+00
##
## 2.453586e-02 4.730750e+00 5.654391e+00 -9.629859e-01 5.285476e+00
##
## -1.333950e+00 -9.287503e-01 -2.204646e+00 -7.723442e-01 3.228137e+00
##
## -2.849364e-01 2.549272e+00 4.116721e+00 -1.854365e+00 4.859340e-02
##
## 9.356676e-01 -2.205753e+00fit_lm$rank## [1] 11fit_lm$fitted.values## Mazda RX4 Mazda RX4 Wag Datsun 710
## 22.59951 22.11189 26.25064
## Hornet 4 Drive Hornet Sportabout Valiant
## 21.23740 17.69343 20.38304
## Duster 360 Merc 240D Merc 230
## 14.38626 22.49601 24.41909
## Merc 280 Merc 280C Merc 450SE
## 18.69903 19.19165 14.17216
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 15.59957 15.74222 12.03401
## Lincoln Continental Chrysler Imperial Fiat 128
## 10.93644 10.49363 27.77291
## Honda Civic Toyota Corolla Toyota Corona
## 29.89674 29.51237 23.64310
## Dodge Challenger AMC Javelin Camaro Z28
## 16.94305 17.73218 13.30602
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 16.69168 28.29347 26.15295
## Lotus Europa Ford Pantera L Ferrari Dino
## 27.63627 18.87004 19.69383
## Maserati Bora Volvo 142E
## 13.94112 24.36827fit_lm$assign## [1] 0 1 2 3 4 5 6 7 8 9 10fit_lm$qr## $qr
## (Intercept) cyl disp hp
## Mazda RX4 -5.6568542 -35.00178567 -1.305160e+03 -8.297898e+02
## Mazda RX4 Wag 0.1767767 9.94359090 6.224581e+02 3.177801e+02
## Datsun 710 0.1767767 0.21715832 -2.978770e+02 -3.542113e+01
## Hornet 4 Drive 0.1767767 0.01602374 1.633784e-01 2.085379e+02
## Hornet Sportabout 0.1767767 -0.18511084 5.218988e-02 1.565722e-01
## Valiant 0.1767767 0.01602374 5.259438e-02 1.610903e-01
## Duster 360 0.1767767 -0.18511084 5.218988e-02 -1.790983e-01
## Merc 240D 0.1767767 0.21715832 2.433459e-01 8.682158e-02
## Merc 230 0.1767767 0.21715832 2.235391e-01 -7.669726e-02
## Merc 280 0.1767767 0.01602374 -1.401026e-01 2.346322e-02
## Merc 280C 0.1767767 0.01602374 -1.401026e-01 2.346322e-02
## Merc 450SE 0.1767767 -0.18511084 -2.304771e-01 5.732646e-02
## Merc 450SL 0.1767767 -0.18511084 -2.304771e-01 5.732646e-02
## Merc 450SLC 0.1767767 -0.18511084 -2.304771e-01 5.732646e-02
## Cadillac Fleetwood 0.1767767 -0.18511084 4.281840e-01 1.128340e-01
## Lincoln Continental 0.1767767 -0.18511084 3.878989e-01 5.415388e-02
## Chrysler Imperial 0.1767767 -0.18511084 3.207571e-01 -3.565418e-02
## Fiat 128 0.1767767 0.21715832 1.506376e-02 6.852906e-03
## Honda Civic 0.1767767 0.21715832 4.992493e-03 7.130519e-02
## Toyota Corolla 0.1767767 0.21715832 -1.045012e-02 4.854299e-03
## Toyota Corona 0.1767767 0.21715832 1.540473e-01 -1.047923e-01
## Dodge Challenger 0.1767767 -0.18511084 -8.880792e-02 2.389092e-01
## AMC Javelin 0.1767767 -0.18511084 -1.358072e-01 2.263942e-01
## Camaro Z28 0.1767767 -0.18511084 1.861898e-02 -1.880376e-01
## Pontiac Firebird 0.1767767 -0.18511084 1.864735e-01 1.923295e-01
## Fiat X1-9 0.1767767 0.21715832 1.607089e-02 7.121086e-03
## Porsche 914-2 0.1767767 0.21715832 1.547187e-01 -7.584174e-02
## Lotus Europa 0.1767767 0.21715832 7.012005e-02 -2.038653e-01
## Ford Pantera L 0.1767767 -0.18511084 2.197607e-02 -2.782542e-01
## Ferrari Dino 0.1767767 0.01602374 -2.159729e-01 -2.460949e-01
## Maserati Bora 0.1767767 -0.18511084 -1.458785e-01 -6.634167e-01
## Volvo 142E 0.1767767 0.21715832 1.570687e-01 -1.615312e-01
## drat wt qsec vs
## Mazda RX4 -20.34522986 -18.199514334 -1.009678e+02 -2.474873734
## Mazda RX4 Wag -2.08369142 4.262896616 -5.882432e+00 -2.275334960
## Datsun 710 0.54376131 -2.298719924 -2.296134e+00 -0.136279198
## Hornet 4 Drive 0.82206584 -0.317037533 -4.324799e+00 -0.270435586
## Hornet Sportabout -1.88390110 0.616380940 2.522421e+00 0.406895941
## Valiant -0.37171623 2.396228056 3.089107e+00 0.236291724
## Duster 360 -0.08793686 0.242105871 -4.950134e+00 -0.841231300
## Merc 240D -0.09193016 -0.156492420 -5.882167e-02 -1.295019386
## Merc 230 -0.07434489 -0.173333156 6.988638e-01 -0.313628990
## Merc 280 0.08588521 -0.329915020 3.843920e-02 0.503331940
## Merc 280C 0.08588521 -0.329915020 1.596480e-01 0.426004609
## Merc 450SE -0.13528319 -0.194006658 7.369100e-03 -0.046238983
## Merc 450SL -0.13528319 -0.052116991 1.158834e-01 -0.119567690
## Merc 450SLC -0.13528319 -0.072983119 1.866729e-01 -0.164126166
## Cadillac Fleetwood 0.08582482 -0.185328012 -5.580502e-02 -0.111184502
## Lincoln Continental 0.07225159 -0.290847599 -7.269231e-02 -0.061093725
## Chrysler Imperial 0.11451002 -0.343953463 -3.967092e-02 0.001431647
## Fiat 128 -0.02318037 0.009749353 1.212245e-01 0.076616858
## Honda Civic 0.46207265 0.011214818 1.474821e-01 0.221423374
## Toyota Corolla 0.03977957 0.107873288 3.030896e-01 -0.009673993
## Toyota Corona -0.23543047 0.125758311 2.213640e-01 -0.049602398
## Dodge Challenger -0.13607129 0.173819622 -1.587122e-01 -0.084462039
## AMC Javelin 0.04481216 0.077514554 1.602960e-02 -0.102454264
## Camaro Z28 0.16941914 -0.027039115 6.319892e-03 0.142138620
## Pontiac Firebird 0.11603386 0.167065328 -4.218267e-02 -0.100977417
## Fiat X1-9 -0.02262036 0.121005670 5.914552e-02 0.112835483
## Porsche 914-2 0.16943455 0.066251081 -2.733329e-01 -0.358482199
## Lotus Europa -0.29026866 0.468372846 -1.429286e-01 0.225327670
## Ford Pantera L 0.37755782 0.150686316 1.149901e-01 0.198757109
## Ferrari Dino -0.26286111 0.038374156 -2.530123e-01 -0.068492316
## Maserati Bora -0.27787174 0.132176080 1.987287e-01 0.133647943
## Volvo 142E -0.05011268 -0.095405901 -8.574182e-03 0.207441849
## am gear carb
## Mazda RX4 -2.298097039 -20.85965005 -15.90990258
## Mazda RX4 Wag -1.451940264 -2.02391673 4.73923359
## Datsun 710 0.771165266 1.05774493 1.67466143
## Hornet 4 Drive 1.106636950 2.31853284 5.40625729
## Hornet Sportabout -0.817576764 -0.98585837 -1.10916641
## Valiant -0.694550593 -0.27191345 2.91869977
## Duster 360 0.834360111 0.97278576 1.98609700
## Merc 240D 0.299692488 0.13528414 0.36213558
## Merc 230 1.361790581 0.65748269 0.29305961
## Merc 280 -0.007366376 -1.95999774 -1.49862291
## Merc 280C -0.070042669 0.17981601 3.19781483
## Merc 450SE -0.010689498 -0.10814794 0.42855210
## Merc 450SL 0.006616335 -0.04158895 0.24945286
## Merc 450SLC -0.040785203 0.03372733 0.25962216
## Cadillac Fleetwood -0.126327613 0.21197339 -0.35158231
## Lincoln Continental -0.126681411 0.12051264 -0.12657909
## Chrysler Imperial -0.075012790 -0.03719881 0.08421755
## Fiat 128 -0.237474396 0.05867150 0.27814023
## Honda Civic -0.097907766 -0.14606322 -0.15649490
## Toyota Corolla -0.246703517 0.14798929 0.14737805
## Toyota Corona 0.451448479 -0.48302966 0.01523328
## Dodge Challenger 0.102533702 -0.01135677 0.03264996
## AMC Javelin 0.056440332 0.01839298 0.13933456
## Camaro Z28 0.280035716 -0.54610130 0.09743337
## Pontiac Firebird 0.083604994 0.05867197 -0.09418390
## Fiat X1-9 -0.148072534 -0.03280582 0.16415590
## Porsche 914-2 0.525907698 -0.01710709 0.20104607
## Lotus Europa 0.151941269 0.08406615 -0.03694607
## Ford Pantera L -0.278388308 0.43150754 0.37393255
## Ferrari Dino 0.105071636 0.09937047 -0.23377891
## Maserati Bora -0.285847708 0.26904434 -0.20724412
## Volvo 142E -0.169093778 -0.17183832 0.25595958
## attr(,"assign")
## [1] 0 1 2 3 4 5 6 7 8 9 10
##
## $qraux
## [1] 1.176777 1.016024 1.113427 1.166614 1.078523 1.067422 1.053314
## [8] 1.068987 1.073454 1.060903 1.075256
##
## $pivot
## [1] 1 2 3 4 5 6 7 8 9 10 11
##
## $tol
## [1] 1e-07
##
## $rank
## [1] 11
##
## attr(,"class")
## [1] "qr"fit_lm$df.residual## [1] 21fit_lm$xlevels## named list()fit_lm$call## lm(formula = mpg ~ ., data = mtcars)fit_lm$terms## mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
## attr(,"variables")
## list(mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb)
## attr(,"factors")
## cyl disp hp drat wt qsec vs am gear carb
## mpg 0 0 0 0 0 0 0 0 0 0
## cyl 1 0 0 0 0 0 0 0 0 0
## disp 0 1 0 0 0 0 0 0 0 0
## hp 0 0 1 0 0 0 0 0 0 0
## drat 0 0 0 1 0 0 0 0 0 0
## wt 0 0 0 0 1 0 0 0 0 0
## qsec 0 0 0 0 0 1 0 0 0 0
## vs 0 0 0 0 0 0 1 0 0 0
## am 0 0 0 0 0 0 0 1 0 0
## gear 0 0 0 0 0 0 0 0 1 0
## carb 0 0 0 0 0 0 0 0 0 1
## attr(,"term.labels")
## [1] "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" "gear" "carb"
## attr(,"order")
## [1] 1 1 1 1 1 1 1 1 1 1
## attr(,"intercept")
## [1] 1
## attr(,"response")
## [1] 1
## attr(,".Environment")
## <environment: R_GlobalEnv>
## attr(,"predvars")
## list(mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb)
## attr(,"dataClasses")
## mpg cyl disp hp drat wt qsec
## "numeric" "numeric" "numeric" "numeric" "numeric" "numeric" "numeric"
## vs am gear carb
## "numeric" "numeric" "numeric" "numeric"fit_lm$model## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4
## Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3
## Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3
## Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3
## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4
## Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4
## Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4
## Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1
## Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2
## Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1
## Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1
## Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2
## AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2
## Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4
## Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2
## Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1
## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2
## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2
## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4
## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6
## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8
## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2step <- step(lm(mpg ~ ., mtcars), trace = 0);
summary(step)##
## Call:
## lm(formula = mpg ~ wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4811 -1.5555 -0.7257 1.4110 4.6610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.6178 6.9596 1.382 0.177915
## wt -3.9165 0.7112 -5.507 6.95e-06 ***
## qsec 1.2259 0.2887 4.247 0.000216 ***
## am 2.9358 1.4109 2.081 0.046716 *
## ---
## 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-11Also refer the document Effect of Transmission Type on Car’s Mileage which finds the best fit model using Backward Elimination (p-value) Method