Regression line
Data Cars2004nh
data(Cars2004nh, package = "mffSM")
head(Cars2004nh)
## vname type drive price.retail price.dealer price cons.city cons.highway
## 1 Chevrolet.Aveo.4dr 1 1 11690 10965 11327.5 8.4 6.9
## 2 Chevrolet.Aveo.LS.4dr.hatch 1 1 12585 11802 12193.5 8.4 6.9
## 3 Chevrolet.Cavalier.2dr 1 1 14610 13697 14153.5 9.0 6.4
## 4 Chevrolet.Cavalier.4dr 1 1 14810 13884 14347.0 9.0 6.4
## 5 Chevrolet.Cavalier.LS.2dr 1 1 16385 15357 15871.0 9.0 6.4
## 6 Dodge.Neon.SE.4dr 1 1 13670 12849 13259.5 8.1 6.5
## consumption engine.size ncylinder horsepower weight iweight lweight wheel.base length width
## 1 7.65 1.6 4 103 1075 0.0009302326 6.980076 249 424 168
## 2 7.65 1.6 4 103 1065 0.0009389671 6.970730 249 389 168
## 3 7.70 2.2 4 140 1187 0.0008424600 7.079184 264 465 175
## 4 7.70 2.2 4 140 1214 0.0008237232 7.101676 264 465 173
## 5 7.70 2.2 4 140 1187 0.0008424600 7.079184 264 465 175
## 6 7.30 2.0 4 132 1171 0.0008539710 7.065613 267 442 170
## ftype fdrive
## 1 personal front
## 2 personal front
## 3 personal front
## 4 personal front
## 5 personal front
## 6 personal front
dim(Cars2004nh)
## [1] 425 20
summary(Cars2004nh)
## vname type drive price.retail price.dealer
## Length:425 Min. :1.000 Min. :1.000 Min. : 10280 Min. : 9875
## Class :character 1st Qu.:1.000 1st Qu.:1.000 1st Qu.: 20370 1st Qu.: 18973
## Mode :character Median :1.000 Median :1.000 Median : 27905 Median : 25672
## Mean :2.219 Mean :1.692 Mean : 32866 Mean : 30096
## 3rd Qu.:3.000 3rd Qu.:2.000 3rd Qu.: 39235 3rd Qu.: 35777
## Max. :6.000 Max. :3.000 Max. :192465 Max. :173560
##
## price cons.city cons.highway consumption engine.size ncylinder
## Min. : 10078 Min. : 6.20 Min. : 5.100 Min. : 5.65 Min. :1.300 Min. :-1.000
## 1st Qu.: 19600 1st Qu.:11.20 1st Qu.: 8.100 1st Qu.: 9.65 1st Qu.:2.400 1st Qu.: 4.000
## Median : 26656 Median :12.40 Median : 9.000 Median :10.70 Median :3.000 Median : 6.000
## Mean : 31481 Mean :12.36 Mean : 9.142 Mean :10.75 Mean :3.208 Mean : 5.791
## 3rd Qu.: 37514 3rd Qu.:13.80 3rd Qu.: 9.800 3rd Qu.:11.65 3rd Qu.:3.900 3rd Qu.: 6.000
## Max. :183012 Max. :23.50 Max. :19.600 Max. :21.55 Max. :8.300 Max. :12.000
## NA's :14 NA's :14 NA's :14
## horsepower weight iweight lweight wheel.base length
## Min. :100.0 Min. : 923 Min. :0.0003067 Min. :6.828 Min. :226.0 Min. :363.0
## 1st Qu.:165.0 1st Qu.:1412 1st Qu.:0.0005542 1st Qu.:7.253 1st Qu.:262.0 1st Qu.:450.0
## Median :210.0 Median :1577 Median :0.0006341 Median :7.363 Median :272.0 Median :472.0
## Mean :216.8 Mean :1626 Mean :0.0006412 Mean :7.373 Mean :274.9 Mean :470.6
## 3rd Qu.:255.0 3rd Qu.:1804 3rd Qu.:0.0007082 3rd Qu.:7.498 3rd Qu.:284.0 3rd Qu.:490.0
## Max. :500.0 Max. :3261 Max. :0.0010834 Max. :8.090 Max. :366.0 Max. :577.0
## NA's :2 NA's :2 NA's :2 NA's :2 NA's :26
## width ftype fdrive
## Min. :163.0 personal:242 front:223
## 1st Qu.:175.0 wagon : 30 rear :110
## Median :180.0 SUV : 60 4x4 : 92
## Mean :181.1 pickup : 24
## 3rd Qu.:185.0 sport : 49
## Max. :206.0 minivan : 20
## NA's :28
In the following, we will use a subset of the dataset Cars2004nh where all needed variables
(consumption, lweight, engine.size) are known (not missing).
isComplete <- complete.cases(Cars2004nh[, c("consumption", "lweight", "engine.size")])
sum(!isComplete)
## [1] 16
CarsNow <- subset(Cars2004nh, isComplete)
dim(CarsNow)
## [1] 409 20
summary(CarsNow)
## vname type drive price.retail price.dealer
## Length:409 Min. :1.000 Min. :1.000 Min. : 10280 Min. : 9875
## Class :character 1st Qu.:1.000 1st Qu.:1.000 1st Qu.: 20585 1st Qu.: 19261
## Mode :character Median :1.000 Median :1.000 Median : 27905 Median : 25672
## Mean :2.232 Mean :1.699 Mean : 32819 Mean : 30052
## 3rd Qu.:3.000 3rd Qu.:2.000 3rd Qu.: 39235 3rd Qu.: 35688
## Max. :6.000 Max. :3.000 Max. :192465 Max. :173560
##
## price cons.city cons.highway consumption engine.size ncylinder
## Min. : 10078 Min. : 6.20 Min. : 5.100 Min. : 5.65 Min. :1.300 Min. :-1.000
## 1st Qu.: 19912 1st Qu.:11.20 1st Qu.: 8.100 1st Qu.: 9.65 1st Qu.:2.400 1st Qu.: 4.000
## Median : 26656 Median :12.40 Median : 9.000 Median :10.70 Median :3.000 Median : 6.000
## Mean : 31435 Mean :12.36 Mean : 9.142 Mean :10.75 Mean :3.178 Mean : 5.763
## 3rd Qu.: 37514 3rd Qu.:13.80 3rd Qu.: 9.800 3rd Qu.:11.65 3rd Qu.:3.800 3rd Qu.: 6.000
## Max. :183012 Max. :23.50 Max. :19.600 Max. :21.55 Max. :6.000 Max. :12.000
##
## horsepower weight iweight lweight wheel.base length
## Min. :100.0 Min. : 923 Min. :0.0003445 Min. :6.828 Min. :226.0 Min. :363
## 1st Qu.:165.0 1st Qu.:1415 1st Qu.:0.0005543 1st Qu.:7.255 1st Qu.:262.0 1st Qu.:450
## Median :210.0 Median :1577 Median :0.0006341 Median :7.363 Median :272.0 Median :472
## Mean :215.8 Mean :1622 Mean :0.0006416 Mean :7.371 Mean :274.6 Mean :470
## 3rd Qu.:250.0 3rd Qu.:1804 3rd Qu.:0.0007067 3rd Qu.:7.498 3rd Qu.:284.0 3rd Qu.:490
## Max. :493.0 Max. :2903 Max. :0.0010834 Max. :7.973 Max. :366.0 Max. :561
## NA's :23
## width ftype fdrive
## Min. :163.0 personal:231 front:212
## 1st Qu.:175.0 wagon : 29 rear :108
## Median :180.0 SUV : 59 4x4 : 89
## Mean :181.1 pickup : 23
## 3rd Qu.:185.0 sport : 47
## Max. :206.0 minivan : 20
## NA's :25
In the following, we will try to model dependence of the car consumption (consumption)
on its weight (weight) or possibly its logarithmic transformation (lweight).
weight as a covariate)par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ weight, data = CarsNow, pch = PCH, col = COL, bg = BGC, xlab = "Weight [kg]", ylab = "Consumption [l/100 km]")
#lines(lowess(CarsNow[, "weight"], CarsNow[, "consumption"]), col = "blue", lwd = 2)
lweight as a covariate)par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ lweight, data = CarsNow, pch = PCH, col = COL, bg = BGC, xlab = "Log(weight) [log(kg)]", ylab = "Consumption [l/100 km]")
#lines(lowess(CarsNow[, "lweight"], CarsNow[, "consumption"]), col = "blue", lwd = 2)
(meanY <- with(CarsNow, mean(consumption)))
## [1] 10.75134
(sdY <- with(CarsNow, sd(consumption)))
## [1] 2.133556
m0 <- lm(consumption ~ 1, data = CarsNow)
summary(m0)
##
## Call:
## lm(formula = consumption ~ 1, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1013 -1.1013 -0.0513 0.8987 10.7987
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.7513 0.1055 101.9 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.134 on 408 degrees of freedom
lweight as a covariate)m1 <- lm(consumption ~ lweight, data = CarsNow)
summary(m1)
##
## Call:
## lm(formula = consumption ~ lweight, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.6544 -0.7442 -0.1526 0.5160 5.1616
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -58.2480 1.8941 -30.75 <2e-16 ***
## lweight 9.3606 0.2569 36.44 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.035 on 407 degrees of freedom
## Multiple R-squared: 0.7654, Adjusted R-squared: 0.7648
## F-statistic: 1328 on 1 and 407 DF, p-value: < 2.2e-16
par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ lweight, data = CarsNow, pch = PCH, col = COL2, bg = BGC2, xlab = "Log(weight) [log(kg)]", ylab = "Consumption [l/100 km]")
abline(h = coef(m0), col = "blue", lty = 5, lwd = 2)
abline(m1, col = "red2", lwd = 2)
legend(6.9, 20, legend = c("m0", "m1"), col = c("blue", "red2"), lty = c(5, 1), lwd = 2)
m1m1names(m1)
## [1] "coefficients" "residuals" "effects" "rank" "fitted.values" "assign"
## [7] "qr" "df.residual" "xlevels" "call" "terms" "model"
m1[["coefficients"]]
## (Intercept) lweight
## -58.24803 9.36056
coef(m1) ## hat{beta}
## (Intercept) lweight
## -58.24803 9.36056
m1[["fitted.values"]]
## 1 2 3 4 5 6 7 8 9 10
## 7.089395 7.001912 8.017106 8.227639 8.017106 7.890073 8.048596 8.001320 7.977593 7.977593
## 11 12 13 14 15 16 17 18 19 20
## 8.281458 7.330084 7.589535 7.638931 6.625287 6.770556 6.966689 8.079981 8.079981 8.304429
## 21 22 23 24 25 26 28 31 32 33
## 10.132628 7.219104 7.431371 8.173510 8.258431 8.296778 7.679896 7.638931 7.890073 8.553455
## 34 35 36 37 38 39 40 41 42 43
## 8.281458 8.281458 8.281458 8.486193 8.486193 6.966689 8.227639 8.227639 8.312073 8.501182
## 44 45 46 47 48 49 50 51 52 53
## 7.597786 7.679896 7.679896 5.662408 5.892803 5.753239 10.337952 10.646670 9.825697 10.176560
## 54 55 56 57 58 59 60 61 62 63
## 10.300954 9.608688 9.615338 9.819194 9.961219 10.658572 9.845178 9.825697 8.478689 10.207814
## 64 65 66 67 68 71 72 73 75 76
## 10.226516 9.276886 9.440871 7.961741 8.590614 9.948396 9.948396 10.126335 9.427315 10.207814
## 77 80 81 82 83 84 85 86 87 88
## 8.235347 9.413739 8.516147 9.124001 9.655142 10.676396 8.553455 9.890476 9.186847 10.145201
## 89 90 91 92 93 95 96 97 98 99
## 10.411513 9.562002 10.176560 9.825697 10.514745 8.968577 9.088903 9.304417 8.715866 8.575768
## 100 101 102 103 104 105 106 107 108 109
## 9.986812 10.050489 9.954810 10.916649 10.634753 10.835302 10.676396 11.020208 10.232742 10.562933
## 110 111 112 113 114 115 116 117 118 119
## 10.951296 10.682329 10.864436 9.948396 10.350253 10.706027 12.120180 12.120180 10.325636 10.170296
## 120 121 122 123 124 126 127 128 129 130
## 11.133947 11.133947 10.288589 10.508704 10.544891 10.044141 10.550909 12.110000 12.110000 10.232742
## 131 132 133 134 135 136 137 138 139 140
## 10.870252 11.128293 9.890476 10.664517 10.676396 9.555313 10.694186 10.454158 10.723761 11.025928
## 141 142 143 144 145 146 147 148 149 150
## 10.514745 10.362537 10.574941 9.838689 9.548620 10.018705 8.538549 10.939761 11.099970 10.634753
## 151 152 153 154 155 156 157 158 159 160
## 10.957058 11.071562 10.899277 11.704087 9.890476 10.899277 10.634753 10.145201 10.145201 10.694186
## 161 162 163 164 165 166 167 168 169 170
## 10.544891 10.980071 11.456182 11.246320 11.133947 10.598912 12.120180 11.201532 10.207814 10.782633
## 171 172 173 174 175 176 177 178 179 181
## 10.628789 10.056833 10.145201 11.212749 11.212749 10.356397 10.356397 10.550909 10.604895 12.110000
## 182 184 185 186 187 188 189 190 191 192
## 12.435313 9.825697 9.825697 10.658572 10.658572 11.082935 10.574941 11.313102 11.877972 11.757121
## 193 194 195 196 197 198 199 200 201 202
## 11.423357 10.928212 10.939761 11.235143 11.704087 11.735944 11.543156 12.017878 11.597108 12.043559
## 203 204 205 206 207 208 209 210 211 212
## 12.877248 11.570171 11.043067 10.664517 11.543156 12.830210 13.016961 11.772973 11.950777 12.089607
## 213 214 215 216 217 218 219 220 221 222
## 11.971475 11.634690 11.935224 11.450719 11.688118 12.094709 11.516064 11.516064 11.867525 11.128293
## 223 224 225 226 227 228 229 230 231 232
## 11.296451 11.966305 11.428836 11.428836 12.816052 12.816052 13.035432 10.846967 12.186082 13.035432
## 233 234 235 236 237 238 239 240 243 244
## 11.434311 10.962817 11.094295 11.543156 12.356279 12.858461 10.688259 11.257483 10.604895 10.604895
## 245 246 247 248 249 250 251 252 253 254
## 11.139598 9.760466 12.043559 9.694778 9.046611 10.331796 10.508704 11.461642 9.081868 9.290662
## 255 256 257 258 260 261 262 263 264 265
## 11.122635 10.031432 10.037789 9.481422 10.157757 10.319472 11.461642 8.766957 9.366068 11.456182
## 266 267 268 269 270 271 272 273 274 275
## 11.940411 11.666783 12.084502 11.607861 7.158797 7.158797 9.461169 9.386528 12.140507 12.523436
## 276 277 278 279 280 281 282 283 285 286
## 12.942707 9.467925 9.961219 10.018705 10.176560 10.082167 9.864618 10.544891 9.707952 10.018705
## 287 288 289 290 291 292 293 294 295 296
## 9.661760 9.694778 8.686545 9.011222 9.555313 10.082167 7.589535 6.374916 14.738988 12.668489
## 297 298 299 301 302 303 304 305 306 307
## 13.978199 14.172229 14.052984 14.077781 13.974027 14.155871 15.989878 16.388400 14.015666 15.734063
## 308 309 310 311 312 313 314 315 316 317
## 14.102511 14.568227 12.989184 12.043559 13.741638 13.299256 12.043559 12.933384 13.012337 12.853759
## 318 319 320 321 322 323 324 325 326 327
## 11.597108 11.483448 12.246506 14.762034 13.577641 15.123258 12.140507 13.763004 14.838447 13.839522
## 328 329 330 331 332 333 334 335 336 337
## 12.825493 12.296562 13.534002 11.682789 11.456182 13.982370 10.417617 11.212749 12.069169 11.836114
## 338 339 340 341 342 343 344 345 346 347
## 14.781196 14.237374 13.375195 8.868309 10.319472 10.069509 10.652623 10.870252 11.570171 10.939761
## 348 349 350 351 352 353 354 355 356 357
## 13.249788 10.939761 9.575364 10.018705 11.412390 9.359238 9.661760 12.069169 11.836114 10.985816
## 358 359 360 361 362 363 364 365 366 367
## 10.628789 13.450523 8.319711 10.729665 12.120180 12.687661 7.389301 10.496610 10.658572 11.909244
## 368 369 371 372 373 374 375 376 377 378
## 12.513686 10.706027 11.510636 8.312073 11.054475 9.628625 7.304590 9.575364 10.550909 9.081868
## 379 380 381 382 383 384 385 386 387 388
## 8.235347 9.400143 10.294774 12.145582 8.723182 11.564774 13.308222 11.257483 12.145582 12.735419
## 389 390 391 392 393 394 395 396 397 398
## 11.661442 12.965975 12.610737 12.687661 12.687661 12.806602 13.698759 11.537744 12.754454 12.017878
## 399 400 401 402 403 404 405 406 407 408
## 12.390938 11.867525 11.516064 12.947365 12.266560 12.366195 15.594714 15.266114 11.060174 12.316510
## 409 410 411 412 413 414 415 416 417 418
## 13.703056 13.616743 11.296451 11.580955 13.177366 13.672936 14.906690 9.379713 10.331796 13.190988
## 420 421 422 423 424 425 426 427 428
## 12.181029 9.172918 10.928212 11.830868 14.599507 10.700108 8.478689 11.809857 12.956674
fitted(m1) ## hat{Y}
## 1 2 3 4 5 6 7 8 9 10
## 7.089395 7.001912 8.017106 8.227639 8.017106 7.890073 8.048596 8.001320 7.977593 7.977593
## 11 12 13 14 15 16 17 18 19 20
## 8.281458 7.330084 7.589535 7.638931 6.625287 6.770556 6.966689 8.079981 8.079981 8.304429
## 21 22 23 24 25 26 28 31 32 33
## 10.132628 7.219104 7.431371 8.173510 8.258431 8.296778 7.679896 7.638931 7.890073 8.553455
## 34 35 36 37 38 39 40 41 42 43
## 8.281458 8.281458 8.281458 8.486193 8.486193 6.966689 8.227639 8.227639 8.312073 8.501182
## 44 45 46 47 48 49 50 51 52 53
## 7.597786 7.679896 7.679896 5.662408 5.892803 5.753239 10.337952 10.646670 9.825697 10.176560
## 54 55 56 57 58 59 60 61 62 63
## 10.300954 9.608688 9.615338 9.819194 9.961219 10.658572 9.845178 9.825697 8.478689 10.207814
## 64 65 66 67 68 71 72 73 75 76
## 10.226516 9.276886 9.440871 7.961741 8.590614 9.948396 9.948396 10.126335 9.427315 10.207814
## 77 80 81 82 83 84 85 86 87 88
## 8.235347 9.413739 8.516147 9.124001 9.655142 10.676396 8.553455 9.890476 9.186847 10.145201
## 89 90 91 92 93 95 96 97 98 99
## 10.411513 9.562002 10.176560 9.825697 10.514745 8.968577 9.088903 9.304417 8.715866 8.575768
## 100 101 102 103 104 105 106 107 108 109
## 9.986812 10.050489 9.954810 10.916649 10.634753 10.835302 10.676396 11.020208 10.232742 10.562933
## 110 111 112 113 114 115 116 117 118 119
## 10.951296 10.682329 10.864436 9.948396 10.350253 10.706027 12.120180 12.120180 10.325636 10.170296
## 120 121 122 123 124 126 127 128 129 130
## 11.133947 11.133947 10.288589 10.508704 10.544891 10.044141 10.550909 12.110000 12.110000 10.232742
## 131 132 133 134 135 136 137 138 139 140
## 10.870252 11.128293 9.890476 10.664517 10.676396 9.555313 10.694186 10.454158 10.723761 11.025928
## 141 142 143 144 145 146 147 148 149 150
## 10.514745 10.362537 10.574941 9.838689 9.548620 10.018705 8.538549 10.939761 11.099970 10.634753
## 151 152 153 154 155 156 157 158 159 160
## 10.957058 11.071562 10.899277 11.704087 9.890476 10.899277 10.634753 10.145201 10.145201 10.694186
## 161 162 163 164 165 166 167 168 169 170
## 10.544891 10.980071 11.456182 11.246320 11.133947 10.598912 12.120180 11.201532 10.207814 10.782633
## 171 172 173 174 175 176 177 178 179 181
## 10.628789 10.056833 10.145201 11.212749 11.212749 10.356397 10.356397 10.550909 10.604895 12.110000
## 182 184 185 186 187 188 189 190 191 192
## 12.435313 9.825697 9.825697 10.658572 10.658572 11.082935 10.574941 11.313102 11.877972 11.757121
## 193 194 195 196 197 198 199 200 201 202
## 11.423357 10.928212 10.939761 11.235143 11.704087 11.735944 11.543156 12.017878 11.597108 12.043559
## 203 204 205 206 207 208 209 210 211 212
## 12.877248 11.570171 11.043067 10.664517 11.543156 12.830210 13.016961 11.772973 11.950777 12.089607
## 213 214 215 216 217 218 219 220 221 222
## 11.971475 11.634690 11.935224 11.450719 11.688118 12.094709 11.516064 11.516064 11.867525 11.128293
## 223 224 225 226 227 228 229 230 231 232
## 11.296451 11.966305 11.428836 11.428836 12.816052 12.816052 13.035432 10.846967 12.186082 13.035432
## 233 234 235 236 237 238 239 240 243 244
## 11.434311 10.962817 11.094295 11.543156 12.356279 12.858461 10.688259 11.257483 10.604895 10.604895
## 245 246 247 248 249 250 251 252 253 254
## 11.139598 9.760466 12.043559 9.694778 9.046611 10.331796 10.508704 11.461642 9.081868 9.290662
## 255 256 257 258 260 261 262 263 264 265
## 11.122635 10.031432 10.037789 9.481422 10.157757 10.319472 11.461642 8.766957 9.366068 11.456182
## 266 267 268 269 270 271 272 273 274 275
## 11.940411 11.666783 12.084502 11.607861 7.158797 7.158797 9.461169 9.386528 12.140507 12.523436
## 276 277 278 279 280 281 282 283 285 286
## 12.942707 9.467925 9.961219 10.018705 10.176560 10.082167 9.864618 10.544891 9.707952 10.018705
## 287 288 289 290 291 292 293 294 295 296
## 9.661760 9.694778 8.686545 9.011222 9.555313 10.082167 7.589535 6.374916 14.738988 12.668489
## 297 298 299 301 302 303 304 305 306 307
## 13.978199 14.172229 14.052984 14.077781 13.974027 14.155871 15.989878 16.388400 14.015666 15.734063
## 308 309 310 311 312 313 314 315 316 317
## 14.102511 14.568227 12.989184 12.043559 13.741638 13.299256 12.043559 12.933384 13.012337 12.853759
## 318 319 320 321 322 323 324 325 326 327
## 11.597108 11.483448 12.246506 14.762034 13.577641 15.123258 12.140507 13.763004 14.838447 13.839522
## 328 329 330 331 332 333 334 335 336 337
## 12.825493 12.296562 13.534002 11.682789 11.456182 13.982370 10.417617 11.212749 12.069169 11.836114
## 338 339 340 341 342 343 344 345 346 347
## 14.781196 14.237374 13.375195 8.868309 10.319472 10.069509 10.652623 10.870252 11.570171 10.939761
## 348 349 350 351 352 353 354 355 356 357
## 13.249788 10.939761 9.575364 10.018705 11.412390 9.359238 9.661760 12.069169 11.836114 10.985816
## 358 359 360 361 362 363 364 365 366 367
## 10.628789 13.450523 8.319711 10.729665 12.120180 12.687661 7.389301 10.496610 10.658572 11.909244
## 368 369 371 372 373 374 375 376 377 378
## 12.513686 10.706027 11.510636 8.312073 11.054475 9.628625 7.304590 9.575364 10.550909 9.081868
## 379 380 381 382 383 384 385 386 387 388
## 8.235347 9.400143 10.294774 12.145582 8.723182 11.564774 13.308222 11.257483 12.145582 12.735419
## 389 390 391 392 393 394 395 396 397 398
## 11.661442 12.965975 12.610737 12.687661 12.687661 12.806602 13.698759 11.537744 12.754454 12.017878
## 399 400 401 402 403 404 405 406 407 408
## 12.390938 11.867525 11.516064 12.947365 12.266560 12.366195 15.594714 15.266114 11.060174 12.316510
## 409 410 411 412 413 414 415 416 417 418
## 13.703056 13.616743 11.296451 11.580955 13.177366 13.672936 14.906690 9.379713 10.331796 13.190988
## 420 421 422 423 424 425 426 427 428
## 12.181029 9.172918 10.928212 11.830868 14.599507 10.700108 8.478689 11.809857 12.956674
m1[["residuals"]]
## 1 2 3 4 5 6 7
## 0.560605107 0.648087614 -0.317105535 -0.527639294 -0.317105535 -0.590073084 -0.748596095
## 8 9 10 11 12 13 14
## 0.048679561 -0.377592770 0.072407230 -0.231457858 -0.530083712 -1.689534710 -0.838930920
## 15 16 17 18 19 20 21
## 0.974712673 0.829444033 0.633310836 -0.129981071 -0.129981071 -0.354428569 -1.132627888
## 22 23 24 25 26 28 31
## 0.830895501 0.968628806 0.426490490 0.341569362 0.303222073 -0.279896237 -0.088930920
## 32 33 34 35 36 37 38
## -0.340073084 -0.103454529 -0.431457858 -0.431457858 -0.431457858 -0.636192729 -0.636192729
## 39 40 41 42 43 44 45
## -0.166689164 0.272360706 0.272360706 0.387927037 0.748818380 -0.947785534 -1.029896237
## 46 47 48 49 50 51 52
## -1.029896237 0.437591665 0.657197067 0.346760719 -0.537952493 -1.346670138 -1.475696780
## 53 54 55 56 57 58 59
## -0.926559528 -1.000954184 -0.208687774 -0.215338261 -0.569194133 -0.161219149 -1.008571679
## 60 61 62 63 64 65 66
## -0.045177661 -0.025696780 1.321310726 0.042186486 0.473484062 -1.326886349 -1.490871158
## 67 68 71 72 73 75 76
## -1.061740844 -0.190614403 0.601603538 0.601603538 0.123664926 -0.827314950 0.042186486
## 77 80 81 82 83 84 85
## -0.085346631 0.686260919 0.783853453 -0.524000751 0.294857738 -0.876395664 -0.103454529
## 86 87 88 89 90 91 92
## 0.209523999 0.363153029 -0.345200838 -0.161513266 -1.112001592 -0.526559528 -1.375696780
## 93 95 96 97 98 99 100
## -0.564744622 -0.268576560 -0.388902997 -3.654417428 -0.015865797 0.124231849 -0.586811947
## 101 102 103 104 105 106 107
## -0.900489472 -0.004810001 -0.966648580 -0.834753447 -0.085302418 -0.876395664 -0.270208384
## 108 109 110 111 112 113 114
## -1.432741784 0.187067060 -0.051295973 -1.032329458 -0.314435838 0.001603538 -1.100252837
## 115 116 117 118 119 120 121
## 0.193972909 -0.520180285 -0.520180285 -0.825635965 -0.670296189 0.266053156 0.266053156
## 122 123 124 126 127 128 129
## 0.761411155 0.541296398 0.505108621 -0.794141178 0.149090903 -0.510000230 -0.510000230
## 130 131 132 133 134 135 136
## 0.467258216 0.379748343 -0.078292625 0.209523999 -0.564516779 -0.576395664 0.394686911
## 137 138 139 140 141 142 143
## -0.894185774 -0.654158261 -0.473761028 -0.325928250 -0.864744622 -0.712537038 -0.624941357
## 144 145 146 147 148 149 150
## -0.338688538 -0.748619803 -0.868704857 0.861450825 -0.839761086 -2.099970182 -0.534753447
## 151 152 153 154 155 156 157
## 0.442941909 0.178438216 -0.649276673 -0.454086889 0.059523999 -0.349276673 -0.084753447
## 158 159 160 161 162 163 164
## -0.345200838 -0.345200838 -0.744185774 -0.144891379 -1.030071158 -1.506181919 0.003680136
## 165 166 167 168 169 170 171
## -0.233946844 -0.798912075 -0.520180285 -0.151532409 0.492186486 0.467367331 -0.678789407
## 172 173 174 175 176 177 178
## 1.193166536 1.304799162 -0.812749379 -0.812749379 0.243603047 0.543603047 0.149090903
## 179 181 182 184 185 186 187
## -0.204895175 -0.510000230 -0.435313059 0.274303220 0.274303220 -1.008571679 -1.008571679
## 188 189 190 191 192 193 194
## -0.382935491 -0.924941357 -0.613101789 -0.627972263 -1.507121366 -1.323357011 0.321788034
## 195 196 197 198 199 200 201
## -0.839761086 -0.985143074 -0.254086889 -0.285943619 -1.293156391 -0.767878064 -0.347107895
## 202 203 204 205 206 207 208
## -0.243558629 -1.077248148 2.729828987 -0.643066906 -0.864516779 -0.493156391 -1.780210059
## 209 210 211 212 213 214 215
## -1.966960594 -1.022973292 -0.900776987 -1.039606845 -0.921474751 0.365310224 0.064776426
## 216 217 218 219 220 221 222
## -0.400719089 -0.938117775 -0.294709358 -0.766063578 -0.766063578 -0.067525209 0.121707375
## 223 224 225 226 227 228 229
## 0.353549078 -0.716304599 0.371164171 0.371164171 -1.216052415 -1.216052415 -1.435432352
## 230 231 232 233 234 235 236
## 2.103033331 0.063917855 2.214567648 -1.034311442 1.287183336 -0.544295396 1.706843609
## 237 238 239 240 243 244 245
## -1.306278836 -0.608461268 -1.038259493 -1.757483324 -0.804895175 -0.204895175 -0.439597650
## 246 247 248 249 250 251 252
## 2.039533868 1.156441371 0.405222279 1.053388762 -0.681796255 1.741296398 0.988358436
## 253 254 255 256 257 258 260
## 1.018132374 0.359337989 0.477365012 1.218568342 1.212211425 2.118577615 -0.207756923
## 261 262 263 264 265 266 267
## 1.280528381 0.338358436 1.833043299 1.333932118 -0.406181919 -0.890410917 0.783216844
## 268 269 270 271 272 273 274
## 0.365498451 0.042139019 2.141203046 2.141203046 1.788831269 2.063471869 0.309492757
## 275 276 277 278 279 280 281
## 1.476563862 2.307293123 0.182075181 2.288780851 -0.218704857 -0.376559528 0.967833475
## 282 283 285 286 287 288 289
## 0.535381916 -0.144891379 1.342047667 1.781295143 1.388240162 2.105222279 1.263454697
## 290 291 292 293 294 295 296
## 2.038778446 0.694686911 1.367833475 0.860465290 1.825084333 0.211011850 0.281510958
## 297 298 299 301 302 303 304
## 0.971800799 0.777771217 -0.602984367 -0.027780567 0.075973102 -0.605871313 -0.039878167
## 305 306 307 308 309 310 311
## 5.161599809 -0.265666469 -0.134062644 1.147488745 0.731773072 -0.989184431 0.406441371
## 312 313 314 315 316 317 318
## -1.041638160 0.150744312 -1.343558629 0.016616391 0.737663046 -0.603758650 0.902892105
## 319 320 321 322 323 324 325
## 1.466551647 1.303494449 2.387965732 0.472359250 0.826741743 -0.690507243 1.836995567
## 326 327 328 329 330 331 332
## 2.611552572 1.460478300 0.124506777 0.203437858 0.515998304 1.267211323 -0.756181919
## 333 334 335 336 337 338 339
## 0.967630356 -0.317617316 0.687250621 0.080831068 -0.386113904 1.168803883 -0.487374310
## 340 341 342 343 344 345 346
## 0.374805207 2.681690537 1.330528381 0.230491435 -0.152622800 -0.470251657 -0.770171013
## 347 348 349 350 351 352 353
## 2.610238914 3.900211716 1.210238914 0.474635722 -0.068704857 1.387610255 2.190762151
## 354 355 356 357 358 359 360
## 0.038240162 -0.819168932 1.613886096 -0.285815595 -1.378789407 -1.450522784 -0.269711120
## 361 362 363 364 365 366 367
## -0.029664882 0.579819715 1.362339297 0.660698810 0.953390147 0.241428321 -1.359243651
## 368 369 371 372 373 374 375
## -0.263685553 -0.006027091 -0.910635593 -1.012072963 -0.804475268 -1.278625077 -0.154589660
## 376 377 378 379 380 381 382
## 0.224635722 -0.750909097 -0.131867626 -0.935346631 -0.600143494 -1.144773556 -0.895582092
## 383 384 385 386 387 388 389
## 0.676818407 -1.314774322 1.991778257 -0.557483324 -1.445582092 -1.485418766 -1.261441895
## 390 391 392 393 394 395 396
## -1.715974501 -0.610737331 0.562339297 -1.437660703 -1.556602076 -0.998758744 -0.287744098
## 397 398 399 400 401 402 403
## -0.304453974 -1.317878064 -1.140938448 -1.167525209 -0.816063578 -1.497365031 -1.716560321
## 404 405 406 407 408 409 410
## -1.816194685 0.355285819 -0.316114423 0.589825761 1.133489934 2.246944464 -0.066742706
## 411 412 413 414 415 416 417
## 1.403549078 1.119044931 -0.227365795 0.377063507 0.043310033 -0.429713016 0.768203745
## 418 420 421 422 423 424 425
## -0.390987762 1.868970789 -0.222917563 3.121788034 0.969131577 0.350492976 -0.900108305
## 426 427 428
## 1.221310726 1.440142944 2.343325609
residuals(m1) ## U = Y - hat{Y}
## 1 2 3 4 5 6 7
## 0.560605107 0.648087614 -0.317105535 -0.527639294 -0.317105535 -0.590073084 -0.748596095
## 8 9 10 11 12 13 14
## 0.048679561 -0.377592770 0.072407230 -0.231457858 -0.530083712 -1.689534710 -0.838930920
## 15 16 17 18 19 20 21
## 0.974712673 0.829444033 0.633310836 -0.129981071 -0.129981071 -0.354428569 -1.132627888
## 22 23 24 25 26 28 31
## 0.830895501 0.968628806 0.426490490 0.341569362 0.303222073 -0.279896237 -0.088930920
## 32 33 34 35 36 37 38
## -0.340073084 -0.103454529 -0.431457858 -0.431457858 -0.431457858 -0.636192729 -0.636192729
## 39 40 41 42 43 44 45
## -0.166689164 0.272360706 0.272360706 0.387927037 0.748818380 -0.947785534 -1.029896237
## 46 47 48 49 50 51 52
## -1.029896237 0.437591665 0.657197067 0.346760719 -0.537952493 -1.346670138 -1.475696780
## 53 54 55 56 57 58 59
## -0.926559528 -1.000954184 -0.208687774 -0.215338261 -0.569194133 -0.161219149 -1.008571679
## 60 61 62 63 64 65 66
## -0.045177661 -0.025696780 1.321310726 0.042186486 0.473484062 -1.326886349 -1.490871158
## 67 68 71 72 73 75 76
## -1.061740844 -0.190614403 0.601603538 0.601603538 0.123664926 -0.827314950 0.042186486
## 77 80 81 82 83 84 85
## -0.085346631 0.686260919 0.783853453 -0.524000751 0.294857738 -0.876395664 -0.103454529
## 86 87 88 89 90 91 92
## 0.209523999 0.363153029 -0.345200838 -0.161513266 -1.112001592 -0.526559528 -1.375696780
## 93 95 96 97 98 99 100
## -0.564744622 -0.268576560 -0.388902997 -3.654417428 -0.015865797 0.124231849 -0.586811947
## 101 102 103 104 105 106 107
## -0.900489472 -0.004810001 -0.966648580 -0.834753447 -0.085302418 -0.876395664 -0.270208384
## 108 109 110 111 112 113 114
## -1.432741784 0.187067060 -0.051295973 -1.032329458 -0.314435838 0.001603538 -1.100252837
## 115 116 117 118 119 120 121
## 0.193972909 -0.520180285 -0.520180285 -0.825635965 -0.670296189 0.266053156 0.266053156
## 122 123 124 126 127 128 129
## 0.761411155 0.541296398 0.505108621 -0.794141178 0.149090903 -0.510000230 -0.510000230
## 130 131 132 133 134 135 136
## 0.467258216 0.379748343 -0.078292625 0.209523999 -0.564516779 -0.576395664 0.394686911
## 137 138 139 140 141 142 143
## -0.894185774 -0.654158261 -0.473761028 -0.325928250 -0.864744622 -0.712537038 -0.624941357
## 144 145 146 147 148 149 150
## -0.338688538 -0.748619803 -0.868704857 0.861450825 -0.839761086 -2.099970182 -0.534753447
## 151 152 153 154 155 156 157
## 0.442941909 0.178438216 -0.649276673 -0.454086889 0.059523999 -0.349276673 -0.084753447
## 158 159 160 161 162 163 164
## -0.345200838 -0.345200838 -0.744185774 -0.144891379 -1.030071158 -1.506181919 0.003680136
## 165 166 167 168 169 170 171
## -0.233946844 -0.798912075 -0.520180285 -0.151532409 0.492186486 0.467367331 -0.678789407
## 172 173 174 175 176 177 178
## 1.193166536 1.304799162 -0.812749379 -0.812749379 0.243603047 0.543603047 0.149090903
## 179 181 182 184 185 186 187
## -0.204895175 -0.510000230 -0.435313059 0.274303220 0.274303220 -1.008571679 -1.008571679
## 188 189 190 191 192 193 194
## -0.382935491 -0.924941357 -0.613101789 -0.627972263 -1.507121366 -1.323357011 0.321788034
## 195 196 197 198 199 200 201
## -0.839761086 -0.985143074 -0.254086889 -0.285943619 -1.293156391 -0.767878064 -0.347107895
## 202 203 204 205 206 207 208
## -0.243558629 -1.077248148 2.729828987 -0.643066906 -0.864516779 -0.493156391 -1.780210059
## 209 210 211 212 213 214 215
## -1.966960594 -1.022973292 -0.900776987 -1.039606845 -0.921474751 0.365310224 0.064776426
## 216 217 218 219 220 221 222
## -0.400719089 -0.938117775 -0.294709358 -0.766063578 -0.766063578 -0.067525209 0.121707375
## 223 224 225 226 227 228 229
## 0.353549078 -0.716304599 0.371164171 0.371164171 -1.216052415 -1.216052415 -1.435432352
## 230 231 232 233 234 235 236
## 2.103033331 0.063917855 2.214567648 -1.034311442 1.287183336 -0.544295396 1.706843609
## 237 238 239 240 243 244 245
## -1.306278836 -0.608461268 -1.038259493 -1.757483324 -0.804895175 -0.204895175 -0.439597650
## 246 247 248 249 250 251 252
## 2.039533868 1.156441371 0.405222279 1.053388762 -0.681796255 1.741296398 0.988358436
## 253 254 255 256 257 258 260
## 1.018132374 0.359337989 0.477365012 1.218568342 1.212211425 2.118577615 -0.207756923
## 261 262 263 264 265 266 267
## 1.280528381 0.338358436 1.833043299 1.333932118 -0.406181919 -0.890410917 0.783216844
## 268 269 270 271 272 273 274
## 0.365498451 0.042139019 2.141203046 2.141203046 1.788831269 2.063471869 0.309492757
## 275 276 277 278 279 280 281
## 1.476563862 2.307293123 0.182075181 2.288780851 -0.218704857 -0.376559528 0.967833475
## 282 283 285 286 287 288 289
## 0.535381916 -0.144891379 1.342047667 1.781295143 1.388240162 2.105222279 1.263454697
## 290 291 292 293 294 295 296
## 2.038778446 0.694686911 1.367833475 0.860465290 1.825084333 0.211011850 0.281510958
## 297 298 299 301 302 303 304
## 0.971800799 0.777771217 -0.602984367 -0.027780567 0.075973102 -0.605871313 -0.039878167
## 305 306 307 308 309 310 311
## 5.161599809 -0.265666469 -0.134062644 1.147488745 0.731773072 -0.989184431 0.406441371
## 312 313 314 315 316 317 318
## -1.041638160 0.150744312 -1.343558629 0.016616391 0.737663046 -0.603758650 0.902892105
## 319 320 321 322 323 324 325
## 1.466551647 1.303494449 2.387965732 0.472359250 0.826741743 -0.690507243 1.836995567
## 326 327 328 329 330 331 332
## 2.611552572 1.460478300 0.124506777 0.203437858 0.515998304 1.267211323 -0.756181919
## 333 334 335 336 337 338 339
## 0.967630356 -0.317617316 0.687250621 0.080831068 -0.386113904 1.168803883 -0.487374310
## 340 341 342 343 344 345 346
## 0.374805207 2.681690537 1.330528381 0.230491435 -0.152622800 -0.470251657 -0.770171013
## 347 348 349 350 351 352 353
## 2.610238914 3.900211716 1.210238914 0.474635722 -0.068704857 1.387610255 2.190762151
## 354 355 356 357 358 359 360
## 0.038240162 -0.819168932 1.613886096 -0.285815595 -1.378789407 -1.450522784 -0.269711120
## 361 362 363 364 365 366 367
## -0.029664882 0.579819715 1.362339297 0.660698810 0.953390147 0.241428321 -1.359243651
## 368 369 371 372 373 374 375
## -0.263685553 -0.006027091 -0.910635593 -1.012072963 -0.804475268 -1.278625077 -0.154589660
## 376 377 378 379 380 381 382
## 0.224635722 -0.750909097 -0.131867626 -0.935346631 -0.600143494 -1.144773556 -0.895582092
## 383 384 385 386 387 388 389
## 0.676818407 -1.314774322 1.991778257 -0.557483324 -1.445582092 -1.485418766 -1.261441895
## 390 391 392 393 394 395 396
## -1.715974501 -0.610737331 0.562339297 -1.437660703 -1.556602076 -0.998758744 -0.287744098
## 397 398 399 400 401 402 403
## -0.304453974 -1.317878064 -1.140938448 -1.167525209 -0.816063578 -1.497365031 -1.716560321
## 404 405 406 407 408 409 410
## -1.816194685 0.355285819 -0.316114423 0.589825761 1.133489934 2.246944464 -0.066742706
## 411 412 413 414 415 416 417
## 1.403549078 1.119044931 -0.227365795 0.377063507 0.043310033 -0.429713016 0.768203745
## 418 420 421 422 423 424 425
## -0.390987762 1.868970789 -0.222917563 3.121788034 0.969131577 0.350492976 -0.900108305
## 426 427 428
## 1.221310726 1.440142944 2.343325609
m1[["rank"]] ## r
## [1] 2
m1[["df.residual"]] ## n - r
## [1] 407
summary methodMany other useful quantities can be extracted from the object returned by the summary method.
sm1 <- summary(m1)
print(sm1)
##
## Call:
## lm(formula = consumption ~ lweight, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.6544 -0.7442 -0.1526 0.5160 5.1616
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -58.2480 1.8941 -30.75 <2e-16 ***
## lweight 9.3606 0.2569 36.44 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.035 on 407 degrees of freedom
## Multiple R-squared: 0.7654, Adjusted R-squared: 0.7648
## F-statistic: 1328 on 1 and 407 DF, p-value: < 2.2e-16
names(sm1)
## [1] "call" "terms" "residuals" "coefficients" "aliased" "sigma"
## [7] "df" "r.squared" "adj.r.squared" "fstatistic" "cov.unscaled"
sm1[["sigma"]] ## sqrt(MS_e)
## [1] 1.034628
sm1[["r.squared"]] ## R^2
## [1] 0.7654182
sm1[["adj.r.squared"]] ## R^2_{adj}
## [1] 0.7648418
sm1[["fstatistic"]] ## Overall F-statistic
## value numdf dendf
## 1328.002 1.000 407.000
lmdeviance(m1) ## SS_e
## [1] 435.6751
deviance(m0) ## SS_T
## [1] 1857.242
anova(m0, m1)
## Analysis of Variance Table
##
## Model 1: consumption ~ 1
## Model 2: consumption ~ lweight
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 408 1857.24
## 2 407 435.68 1 1421.6 1328 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Matrix ^var(ˆβ)=MSe(X⊤X)−1:
vcov(m1)
## (Intercept) lweight
## (Intercept) 3.5876356 -0.48634907
## lweight -0.4863491 0.06597886
vcov(m1)cov2cor(vcov(m1))
## (Intercept) lweight
## (Intercept) 1.0000000 -0.9996352
## lweight -0.9996352 1.0000000
confint(m1, level = 0.95)
## 2.5 % 97.5 %
## (Intercept) -61.971477 -54.524575
## lweight 8.855615 9.865505
The code below calculates a series of confidence intervals for l⊤β, where l=(1,x)⊤, x=6.8,6.9,…,9. That is, it calculates a series of confidence intervals for E(Y|X=x).
print(lweight.grid <- seq(6.8, 9, by = 0.1))
## [1] 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0
pdata <- data.frame(lweight = lweight.grid)
predict(m1, newdata = pdata)
## 1 2 3 4 5 6 7 8 9 10
## 5.403783 6.339839 7.275895 8.211951 9.148007 10.084063 11.020119 11.956175 12.892231 13.828287
## 11 12 13 14 15 16 17 18 19 20
## 14.764343 15.700399 16.636455 17.572511 18.508567 19.444623 20.380679 21.316735 22.252791 23.188847
## 21 22 23
## 24.124903 25.060959 25.997015
predict(m1, newdata = pdata, se.fit = TRUE, interval = "confidence", level = 0.95)
## $fit
## fit lwr upr
## 1 5.403783 5.098287 5.709280
## 2 6.339839 6.081488 6.598191
## 3 7.275895 7.063145 7.488645
## 4 8.211951 8.042013 8.381889
## 5 9.148007 9.015362 9.280652
## 6 10.084063 9.977247 10.190880
## 7 11.020119 10.918511 11.121728
## 8 11.956175 11.836433 12.075918
## 9 12.892231 12.739092 13.045370
## 10 13.828287 13.634215 14.022360
## 11 14.764343 14.525646 15.003040
## 12 15.700399 15.415114 15.985685
## 13 16.636455 16.303441 16.969470
## 14 17.572511 17.191055 17.953967
## 15 18.508567 18.078197 18.938937
## 16 19.444623 18.965012 19.924235
## 17 20.380679 19.851590 20.909769
## 18 21.316735 20.737992 21.895478
## 19 22.252791 21.624261 22.881322
## 20 23.188847 22.510425 23.867270
## 21 24.124903 23.396506 24.853301
## 22 25.060959 24.282520 25.839399
## 23 25.997015 25.168479 26.825552
##
## $se.fit
## 1 2 3 4 5 6 7 8 9
## 0.15540481 0.13142251 0.10822512 0.08644682 0.06747597 0.05433726 0.05168797 0.06091249 0.07790118
## 10 11 12 13 14 15 16 17 18
## 0.09872403 0.12142426 0.14512355 0.16940312 0.19404527 0.21892760 0.24397663 0.26914583 0.29440437
## 19 20 21 22 23
## 0.31973109 0.34511097 0.37053309 0.39598931 0.42147346
##
## $df
## [1] 407
##
## $residual.scale
## [1] 1.034628
predict(m1, newdata = pdata, interval = "confidence", level = 0.95)
## fit lwr upr
## 1 5.403783 5.098287 5.709280
## 2 6.339839 6.081488 6.598191
## 3 7.275895 7.063145 7.488645
## 4 8.211951 8.042013 8.381889
## 5 9.148007 9.015362 9.280652
## 6 10.084063 9.977247 10.190880
## 7 11.020119 10.918511 11.121728
## 8 11.956175 11.836433 12.075918
## 9 12.892231 12.739092 13.045370
## 10 13.828287 13.634215 14.022360
## 11 14.764343 14.525646 15.003040
## 12 15.700399 15.415114 15.985685
## 13 16.636455 16.303441 16.969470
## 14 17.572511 17.191055 17.953967
## 15 18.508567 18.078197 18.938937
## 16 19.444623 18.965012 19.924235
## 17 20.380679 19.851590 20.909769
## 18 21.316735 20.737992 21.895478
## 19 22.252791 21.624261 22.881322
## 20 23.188847 22.510425 23.867270
## 21 24.124903 23.396506 24.853301
## 22 25.060959 24.282520 25.839399
## 23 25.997015 25.168479 26.825552
p1p1 <- predict(m1, newdata = pdata, interval = "confidence", level = 0.95)
par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ lweight, data = CarsNow, pch = PCH, col = COL2, bg = BGC2, xlab = "Log(weight) [log(kg)]", ylab = "Consumption [l/100 km]")
abline(m1, col = "red2", lwd = 2)
lines(pdata[, "lweight"], p1[, "lwr"], col = "blue", lwd = 2, lty = 5)
lines(pdata[, "lweight"], p1[, "upr"], col = "blue", lwd = 2, lty = 5)
par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ weight, data = CarsNow, pch = PCH, col = COL2, bg = BGC2, xlab = "Weight [kg]", ylab = "Consumption [l/100 km]")
lines(exp(pdata[, "lweight"]), p1[, "fit"], col = "red2", lwd = 2)
lines(exp(pdata[, "lweight"]), p1[, "lwr"], col = "blue", lwd = 2, lty = 5)
lines(exp(pdata[, "lweight"]), p1[, "upr"], col = "blue", lwd = 2, lty = 5)
