class: center, middle, inverse, title-slide # Bases du modèle linéaire ## Chapitre II - Régression linéaire multiple ### Antoine Bichat - Émilie Lebarbier<br>AgroParisTech --- # Chargement des données ```r chenilles <- read.table("chenilles.txt", sep = "", header = TRUE) ``` --- # Statistiques descriptives ```r summary(chenilles) ``` ``` Altitude Pente NbPins Hauteur Min. :1075 Min. :15.00 Min. : 0.00 Min. :2.400 1st Qu.:1228 1st Qu.:24.00 1st Qu.: 4.00 1st Qu.:3.700 Median :1309 Median :28.00 Median : 8.00 Median :4.400 Mean :1315 Mean :29.03 Mean :11.45 Mean :4.452 3rd Qu.:1396 3rd Qu.:34.00 3rd Qu.:18.00 3rd Qu.:5.300 Max. :1575 Max. :46.00 Max. :32.00 Max. :6.500 Diametre Densite Orient HautMax Min. : 5.80 Min. :1.000 Min. :1.100 Min. : 3.600 1st Qu.:11.50 1st Qu.:1.200 1st Qu.:1.600 1st Qu.: 5.900 Median :15.70 Median :1.500 Median :1.700 Median : 7.200 Mean :15.25 Mean :1.791 Mean :1.658 Mean : 7.539 3rd Qu.:18.30 3rd Qu.:2.400 3rd Qu.:1.800 3rd Qu.: 9.100 Max. :21.80 Max. :3.300 Max. :1.900 Max. :13.700 NbStrat Melange NbNids LogNids Min. :1.100 Min. :1.300 Min. :0.0300 Min. :-3.5066 1st Qu.:1.500 1st Qu.:1.600 1st Qu.:0.1800 1st Qu.:-1.7148 Median :2.000 Median :1.800 Median :0.6700 Median :-0.4005 Mean :1.982 Mean :1.761 Mean :0.8112 Mean :-0.8133 3rd Qu.:2.500 3rd Qu.:2.000 3rd Qu.:1.1300 3rd Qu.: 0.1222 Max. :2.900 Max. :2.000 Max. :3.0000 Max. : 1.0986 ``` --- # Nuages de points ```r plot(chenilles) ``` <img src="chap2_files/figure-html/plotall-1.png" width="648" style="display: block; margin: auto;" /> --- # Corrélation ```r cor(chenilles) ``` ``` Altitude Pente NbPins Hauteur Diametre Altitude 1.0000000 0.1205209 0.5375645 0.32105284 0.28377386 Pente 0.1205209 1.0000000 0.3219411 0.13668772 0.11341631 NbPins 0.5375645 0.3219411 1.0000000 0.41443493 0.29492039 Hauteur 0.3210528 0.1366877 0.4144349 1.00000000 0.90465522 Diametre 0.2837739 0.1134163 0.2949204 0.90465522 1.00000000 Densite 0.5146683 0.3006666 0.9795521 0.43929974 0.30623032 Orient 0.2684928 -0.1522218 0.1284678 0.05810293 -0.07870719 HautMax 0.3601466 0.2619061 0.7589609 0.77192534 0.59619783 NbStrat 0.3637243 0.3256728 0.8767888 0.45958851 0.26746450 Melange -0.1276449 0.1280044 0.1870038 -0.12111138 -0.09063278 NbNids -0.5302188 -0.4554582 -0.5639008 -0.35790141 -0.15777120 LogNids -0.5336138 -0.4294396 -0.5177880 -0.42529455 -0.20093786 Densite Orient HautMax NbStrat Melange Altitude 0.5146683 0.26849282 0.36014656 0.36372429 -0.12764490 Pente 0.3006666 -0.15222176 0.26190610 0.32567283 0.12800444 NbPins 0.9795521 0.12846781 0.75896092 0.87678882 0.18700384 Hauteur 0.4392997 0.05810293 0.77192534 0.45958851 -0.12111138 Diametre 0.3062303 -0.07870719 0.59619783 0.26746450 -0.09063278 Densite 1.0000000 0.15067805 0.81021606 0.90853190 0.10829434 Orient 0.1506780 1.00000000 0.06000991 0.06324509 0.13084916 HautMax 0.8102161 0.06000991 1.00000000 0.85363911 0.00327079 NbStrat 0.9085319 0.06324509 0.85363911 1.00000000 0.14782424 Melange 0.1082943 0.13084916 0.00327079 0.14782424 1.00000000 NbNids -0.5702405 -0.21174829 -0.55113165 -0.63587161 -0.11276133 LogNids -0.5282782 -0.22968178 -0.54137095 -0.59398965 -0.03644182 NbNids LogNids Altitude -0.5302188 -0.53361382 Pente -0.4554582 -0.42943963 NbPins -0.5639008 -0.51778795 Hauteur -0.3579014 -0.42529455 Diametre -0.1577712 -0.20093786 Densite -0.5702405 -0.52827817 Orient -0.2117483 -0.22968178 HautMax -0.5511316 -0.54137095 NbStrat -0.6358716 -0.59398965 Melange -0.1127613 -0.03644182 NbNids 1.0000000 0.87615265 LogNids 0.8761526 1.00000000 ``` --- # Corrélation partielle ```r library(ppcor) pcor(chenilles)$estimate ``` ``` Altitude Pente NbPins Hauteur Diametre Altitude 1.000000000 -0.26009632 0.31794782 -0.22941607 0.32040785 Pente -0.260096325 1.00000000 0.14816951 -0.19967515 0.21217545 NbPins 0.317947820 0.14816951 1.00000000 0.34335104 -0.25143160 Hauteur -0.229416069 -0.19967515 0.34335104 1.00000000 0.87615184 Diametre 0.320407851 0.21217545 -0.25143160 0.87615184 1.00000000 Densite -0.111376894 -0.07426884 0.91506636 -0.41859082 0.31040664 Orient 0.212124771 -0.16077342 -0.32917969 0.38567779 -0.40828972 HautMax -0.009889088 0.05751548 -0.35693150 0.60343420 -0.24086221 NbStrat -0.213668900 -0.09051463 -0.02545177 0.01211458 -0.18978280 Melange -0.375307807 -0.04227593 0.53213399 -0.35008751 0.34547134 NbNids -0.230113948 -0.19928141 0.01490487 0.04215984 0.02941938 LogNids -0.243445082 -0.22084912 0.11066058 -0.36813458 0.31372475 Densite Orient HautMax NbStrat Melange Altitude -0.11137689 0.212124771 -0.009889088 -0.21366890 -0.37530781 Pente -0.07426884 -0.160773423 0.057515481 -0.09051463 -0.04227593 NbPins 0.91506636 -0.329179687 -0.356931501 -0.02545177 0.53213399 Hauteur -0.41859082 0.385677793 0.603434199 0.01211458 -0.35008751 Diametre 0.31040664 -0.408289722 -0.240862212 -0.18978280 0.34547134 Densite 1.00000000 0.402217980 0.431984664 0.28316189 -0.49084273 Orient 0.40221798 1.000000000 -0.177011779 -0.25228882 0.38605261 HautMax 0.43198466 -0.177011779 1.000000000 0.39653322 0.11460045 NbStrat 0.28316189 -0.252288821 0.396533218 1.00000000 0.16560578 Melange -0.49084273 0.386052611 0.114600446 0.16560578 1.00000000 NbNids 0.04227516 -0.051324563 -0.088471934 -0.14808280 -0.11651361 LogNids -0.07389864 0.002488219 0.182970387 -0.13201643 -0.03694391 NbNids LogNids Altitude -0.23011395 -0.243445082 Pente -0.19928141 -0.220849123 NbPins 0.01490487 0.110660581 Hauteur 0.04215984 -0.368134583 Diametre 0.02941938 0.313724754 Densite 0.04227516 -0.073898643 Orient -0.05132456 0.002488219 HautMax -0.08847193 0.182970387 NbStrat -0.14808280 -0.132016433 Melange -0.11651361 -0.036943905 NbNids 1.00000000 0.641194407 LogNids 0.64119441 1.000000000 ``` --- # Facteur confondant ```r var <- c("NbNids", "NbStrat", "Densite") cor(chenilles)[var, var] ``` ``` NbNids NbStrat Densite NbNids 1.0000000 -0.6358716 -0.5702405 NbStrat -0.6358716 1.0000000 0.9085319 Densite -0.5702405 0.9085319 1.0000000 ``` -- ```r pcor(chenilles)$estimate[var, var] ``` ``` NbNids NbStrat Densite NbNids 1.00000000 -0.1480828 0.04227516 NbStrat -0.14808280 1.0000000 0.28316189 Densite 0.04227516 0.2831619 1.00000000 ``` --- # Regression linéaire multiple ```r modele_log <- lm(log(NbNids) ~ Altitude + Pente + NbPins + Hauteur + Diametre + Densite + Orient + HautMax + NbStrat + Melange, data = chenilles) modele_log ``` ``` Call: lm(formula = log(NbNids) ~ Altitude + Pente + NbPins + Hauteur + Diametre + Densite + Orient + HautMax + NbStrat + Melange, data = chenilles) Coefficients: (Intercept) Altitude Pente NbPins Hauteur 11.300912 -0.004505 -0.053606 0.074581 -1.328277 Diametre Densite Orient HautMax NbStrat 0.236101 -0.451118 -0.187810 0.185636 -1.266028 Melange -0.537203 ``` --- # Graphes de diagnostic ```r par(mfrow=c(2,2)) plot(modele_log) ``` <img src="chap2_files/figure-html/diag-1.png" width="504" style="display: block; margin: auto;" /> --- ```r summary(modele_log) # Ne s'affiche pas entièrement ici ``` ``` Call: lm(formula = log(NbNids) ~ Altitude + Pente + NbPins + Hauteur + Diametre + Densite + Orient + HautMax + NbStrat + Melange, data = chenilles) Residuals: Min 1Q Median 3Q Max -1.69082 -0.27556 -0.02122 0.31358 1.74750 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 11.300912 3.156550 3.580 0.00167 ** Altitude -0.004505 0.001563 -2.882 0.00865 ** Pente -0.053606 0.021843 -2.454 0.02250 * NbPins 0.074581 0.100233 0.744 0.46470 Hauteur -1.328277 0.570061 -2.330 0.02938 * Diametre 0.236101 0.104611 2.257 0.03428 * Densite -0.451118 1.572916 -0.287 0.77695 Orient -0.187810 1.007950 -0.186 0.85389 HautMax 0.185636 0.236344 0.785 0.44057 NbStrat -1.266028 0.861235 -1.470 0.15572 Melange -0.537203 0.773372 -0.695 0.49456 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.8268 on 22 degrees of freedom Multiple R-squared: 0.6968, Adjusted R-squared: 0.5589 F-statistic: 5.055 on 10 and 22 DF, p-value: 0.0007441 ``` --- # Modèle nul ```r modele_log_0 <- lm(log(NbNids) ~ 1, data = chenilles) anova(modele_log_0, modele_log) ``` ``` Analysis of Variance Table Model 1: log(NbNids) ~ 1 Model 2: log(NbNids) ~ Altitude + Pente + NbPins + Hauteur + Diametre + Densite + Orient + HautMax + NbStrat + Melange Res.Df RSS Df Sum of Sq F Pr(>F) 1 32 49.596 2 22 15.039 10 34.557 5.0553 0.0007441 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- # Anova de type 1 ```r anova(modele_log) ``` ``` Analysis of Variance Table Response: log(NbNids) Df Sum Sq Mean Sq F value Pr(>F) Altitude 1 14.1222 14.1222 20.6589 0.0001593 *** Pente 1 6.7095 6.7095 9.8152 0.0048376 ** NbPins 1 1.4175 1.4175 2.0736 0.1639516 Hauteur 1 1.8035 1.8035 2.6383 0.1185567 Diametre 1 8.0480 8.0480 11.7732 0.0023866 ** Densite 1 0.1353 0.1353 0.1979 0.6608026 Orient 1 0.0385 0.0385 0.0563 0.8146664 HautMax 1 0.0001 0.0001 0.0001 0.9910625 NbStrat 1 1.9528 1.9528 2.8567 0.1051153 Melange 1 0.3298 0.3298 0.4825 0.4945619 Residuals 22 15.0389 0.6836 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- # Anova de type 2 ```r library(car) Anova(modele_log) ``` ``` Anova Table (Type II tests) Response: log(NbNids) Sum Sq Df F value Pr(>F) Altitude 5.6794 1 8.3082 0.008648 ** Pente 4.1173 1 6.0231 0.022502 * NbPins 0.3785 1 0.5537 0.464703 Hauteur 3.7113 1 5.4292 0.029376 * Diametre 3.4820 1 5.0938 0.034281 * Densite 0.0562 1 0.0823 0.776946 Orient 0.0237 1 0.0347 0.853895 HautMax 0.4217 1 0.6169 0.440567 NbStrat 1.4772 1 2.1609 0.155715 Melange 0.3298 1 0.4825 0.494562 Residuals 15.0389 22 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- # Stepwise selection ```r modele_log_stepwise <- step(modele_log_0, data = chenilles, scope =~ Altitude + Pente + NbPins + Hauteur + Diametre + Densite + Orient + HautMax + NbStrat + Melange) ``` ``` Start: AIC=15.44 log(NbNids) ~ 1 Df Sum of Sq RSS AIC + NbStrat 1 17.4987 32.097 3.0848 + HautMax 1 14.5358 35.060 5.9985 + Altitude 1 14.1222 35.474 6.3855 + Densite 1 13.8412 35.755 6.6459 + NbPins 1 13.2969 36.299 7.1444 + Pente 1 9.1464 40.450 10.7171 + Hauteur 1 8.9707 40.625 10.8602 <none> 49.596 15.4443 + Orient 1 2.6164 46.980 15.6558 + Diametre 1 2.0025 47.594 16.0842 + Melange 1 0.0659 49.530 17.4004 Step: AIC=3.08 log(NbNids) ~ NbStrat Df Sum of Sq RSS AIC + Altitude 1 5.7642 26.333 -1.4474 + Pente 1 3.0899 29.007 1.7445 <none> 32.097 3.0848 + Orient 1 1.8378 30.260 3.1390 + Hauteur 1 1.4585 30.639 3.5501 + HautMax 1 0.2153 31.882 4.8627 + Melange 1 0.1338 31.964 4.9470 + Diametre 1 0.0945 32.003 4.9874 + Densite 1 0.0368 32.061 5.0469 + NbPins 1 0.0020 32.095 5.0828 - NbStrat 1 17.4987 49.596 15.4443 Step: AIC=-1.45 log(NbNids) ~ NbStrat + Altitude Df Sum of Sq RSS AIC + Pente 1 3.0701 23.263 -3.5382 + Densite 1 2.2734 24.060 -2.4269 + NbPins 1 1.9416 24.392 -1.9749 <none> 26.333 -1.4474 + Hauteur 1 0.6000 25.733 -0.2081 + Orient 1 0.5599 25.773 -0.1566 + HautMax 1 0.0482 26.285 0.4922 + Diametre 1 0.0383 26.295 0.5046 + Melange 1 0.0119 26.321 0.5376 - Altitude 1 5.7642 32.097 3.0848 - NbStrat 1 9.1407 35.474 6.3855 Step: AIC=-3.54 log(NbNids) ~ NbStrat + Altitude + Pente Df Sum of Sq RSS AIC + NbPins 1 2.4280 20.835 -5.1757 + Densite 1 2.3402 20.923 -5.0370 <none> 23.263 -3.5382 + Orient 1 1.2150 22.048 -3.3084 + Hauteur 1 0.6449 22.618 -2.4660 + HautMax 1 0.0771 23.186 -1.6477 + Diametre 1 0.0608 23.202 -1.6246 + Melange 1 0.0020 23.261 -1.5410 - Pente 1 3.0701 26.333 -1.4474 - NbStrat 1 5.5013 28.764 1.4668 - Altitude 1 5.7445 29.008 1.7445 Step: AIC=-5.18 log(NbNids) ~ NbStrat + Altitude + Pente + NbPins Df Sum of Sq RSS AIC + Orient 1 1.3806 19.454 -5.4382 <none> 20.835 -5.1757 + Hauteur 1 0.4769 20.358 -3.9398 - NbPins 1 2.4280 23.263 -3.5382 + Melange 1 0.1239 20.711 -3.3725 + Densite 1 0.1029 20.732 -3.3391 + HautMax 1 0.0717 20.763 -3.2895 + Diametre 1 0.0391 20.796 -3.2377 - Pente 1 3.5565 24.392 -1.9749 - NbStrat 1 6.5118 27.347 1.7991 - Altitude 1 8.1380 28.973 3.7054 Step: AIC=-5.44 log(NbNids) ~ NbStrat + Altitude + Pente + NbPins + Orient Df Sum of Sq RSS AIC <none> 19.454 -5.4382 - Orient 1 1.3806 20.835 -5.1757 + Hauteur 1 0.5057 18.949 -4.3074 + Densite 1 0.2830 19.172 -3.9218 + HautMax 1 0.0867 19.368 -3.5855 + Melange 1 0.0138 19.441 -3.4616 + Diametre 1 0.0000 19.454 -3.4382 - NbPins 1 2.5935 22.048 -3.3084 - Pente 1 4.3177 23.772 -0.8237 - Altitude 1 6.4076 25.862 1.9570 - NbStrat 1 6.6742 26.129 2.2954 ``` --- # Graphes de diagnostic ```r par(mfrow=c(2,2)) plot(modele_log_stepwise) ``` <img src="chap2_files/figure-html/diaglog-1.png" width="504" style="display: block; margin: auto;" /> --- # Résultat de l'analyse avec {broom} ```r library(broom) tidy(modele_log_stepwise) ``` ``` # A tibble: 6 x 5 term estimate std.error statistic p.value <chr> <dbl> <dbl> <dbl> <dbl> 1 (Intercept) 11.1 2.48 4.48 0.000122 2 NbStrat -1.75 0.573 -3.04 0.00516 3 Altitude -0.00437 0.00147 -2.98 0.00600 4 Pente -0.0544 0.0222 -2.45 0.0212 5 NbPins 0.0715 0.0377 1.90 0.0685 6 Orient -1.18 0.849 -1.38 0.178 ``` ```r glance(modele_log_stepwise) ``` ``` # A tibble: 1 x 11 r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC <dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> 1 0.608 0.535 0.849 8.37 7.03e-5 6 -38.1 90.2 101. # … with 2 more variables: deviance <dbl>, df.residual <int> ``` --- class: center, middle, inverse # Des questions ? .footnote[Slides créées avec le package <b><a href="https://github.com/yihui/xaringan" target="_blank">xaringan</a></b>.]