Regresión múltiple
Odacir Padilla S.
2024-05-20
Ejercicio
El diseño eficiente de ciertos tipos de incineradores de desechos
municipales exige que se disponga de información acerca del contenido
energético de los desechos. Los autores del artículo
“Modeling the Energy Content of Municipal Solid Waste Using
Multiple Regression Analysis” (J. of the Air and Waste Mgmt.
Assoc., 1996: 650-656) bondadosamente nos proporcionaron la
información siguiente acerca de:
\(y\)= contenido energético
(kcal/kg),
las tres variables físicas de composición:
\(x_1\)= % de plástico por peso,
\(x_2\) = % de papel por peso
\(x_3\) = % de basura por peso
y la variable próxima de análisis
\(x_4\) = % de humedad por peso para
especímenes de desechos de cierta región.
A continuación se presenta la información para las 12 primeras
observaciones.
plástico <- c(18.69,19.43,19.24,22.64,16.54,21.44,19.53,23.97,21.45,20.34,17.03,21.03)
papel <- c(15.65,23.51,24.23,22.20,23.56,23.65,24.45,19.39,23.84,26.50,23.46,26.99)
basura <- c(45.01,39.69,43.16,35.76,41.20,35.56,40.18,44.11,35.41,34.21,32.45,38.19)
agua <- c(58.21,46.31,46.63,45.85,55.14,54.24,47.20,43.82,51.01,49.06,53.23,51.78)
y <- c(947,1407,1452,1553,989,1162,1466,1656,1254,1336,1097,1266)
data <- data.frame(plástico,papel,basura,agua,y)
data## plástico papel basura agua y
## 1 18.69 15.65 45.01 58.21 947
## 2 19.43 23.51 39.69 46.31 1407
## 3 19.24 24.23 43.16 46.63 1452
## 4 22.64 22.20 35.76 45.85 1553
## 5 16.54 23.56 41.20 55.14 989
## 6 21.44 23.65 35.56 54.24 1162
## 7 19.53 24.45 40.18 47.20 1466
## 8 23.97 19.39 44.11 43.82 1656
## 9 21.45 23.84 35.41 51.01 1254
## 10 20.34 26.50 34.21 49.06 1336
## 11 17.03 23.46 32.45 53.23 1097
## 12 21.03 26.99 38.19 51.78 1266regmult<- lm(y~plástico+papel+basura+agua)
regmult##
## Call:
## lm(formula = y ~ plástico + papel + basura + agua)
##
## Coefficients:
## (Intercept) plástico papel basura agua
## 2830.9739 23.4287 0.8243 2.4007 -42.1352summary(regmult)##
## Call:
## lm(formula = y ~ plástico + papel + basura + agua)
##
## Residuals:
## Min 1Q Median 3Q Max
## -42.579 -15.178 9.568 11.736 49.628
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2830.9739 372.1728 7.607 0.000126 ***
## plástico 23.4287 5.7564 4.070 0.004749 **
## papel 0.8243 4.4053 0.187 0.856874
## basura 2.4007 3.0765 0.780 0.460752
## agua -42.1352 2.9744 -14.166 2.07e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 32.2 on 7 degrees of freedom
## Multiple R-squared: 0.9865, Adjusted R-squared: 0.9788
## F-statistic: 128.2 on 4 and 7 DF, p-value: 1.261e-06confint(regmult)## 2.5 % 97.5 %
## (Intercept) 1950.924992 3711.022858
## plástico 9.816832 37.040503
## papel -9.592539 11.241186
## basura -4.873961 9.675414
## agua -49.168439 -35.101875step(regmult,direction = "backward")## Start: AIC=86.86
## y ~ plástico + papel + basura + agua
##
## Df Sum of Sq RSS AIC
## - papel 1 36 7294 84.918
## - basura 1 631 7889 85.860
## <none> 7257 86.859
## - plástico 1 17174 24431 99.425
## - agua 1 208058 215315 125.539
##
## Step: AIC=84.92
## y ~ plástico + basura + agua
##
## Df Sum of Sq RSS AIC
## - basura 1 767 8061 84.119
## <none> 7294 84.918
## - plástico 1 18814 26108 98.221
## - agua 1 263871 271165 126.307
##
## Step: AIC=84.12
## y ~ plástico + agua
##
## Df Sum of Sq RSS AIC
## <none> 8061 84.119
## - plástico 1 18469 26530 96.414
## - agua 1 268087 276148 124.525##
## Call:
## lm(formula = y ~ plástico + agua)
##
## Coefficients:
## (Intercept) plástico agua
## 2977.40 22.82 -42.58modelo final
modeloF<- lm(formula = y ~ plástico + agua)
summary(modeloF)##
## Call:
## lm(formula = y ~ plástico + agua)
##
## Residuals:
## Min 1Q Median 3Q Max
## -42.115 -17.157 0.957 15.156 52.495
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2977.397 198.761 14.980 1.14e-07 ***
## plástico 22.819 5.025 4.541 0.0014 **
## agua -42.575 2.461 -17.301 3.25e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 29.93 on 9 degrees of freedom
## Multiple R-squared: 0.985, Adjusted R-squared: 0.9817
## F-statistic: 296.4 on 2 and 9 DF, p-value: 6.119e-09Validación de los supuestos
shapiro.test(modeloF$residuals)##
## Shapiro-Wilk normality test
##
## data: modeloF$residuals
## W = 0.96116, p-value = 0.8002plot(modeloF)
Homogeneidad de varianza
library(lmtest)## Warning: package 'lmtest' was built under R version 4.3.3## Loading required package: zoo## Warning: package 'zoo' was built under R version 4.3.3##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericbptest(modeloF)##
## studentized Breusch-Pagan test
##
## data: modeloF
## BP = 1.7069, df = 2, p-value = 0.4259Autocorrelacion
dwtest(modeloF)##
## Durbin-Watson test
##
## data: modeloF
## DW = 2.1502, p-value = 0.6287
## alternative hypothesis: true autocorrelation is greater than 0#datos influyentes
library(car)## Warning: package 'car' was built under R version 4.3.3## Loading required package: carData## Warning: package 'carData' was built under R version 4.3.3influencePlot(modeloF)
## StudRes Hat CookD
## 1 0.9136547 0.3978989 0.187324112
## 2 -1.8055125 0.2399977 0.274271647
## 7 2.3938436 0.1809115 0.276542743
## 8 -0.1108100 0.3987490 0.003049067library(corrplot)## Warning: package 'corrplot' was built under R version 4.3.3## corrplot 0.92 loadedcorrplot(cor(dplyr::select(data, plástico, papel,basura,agua,y)),
method = "number", tl.col = "black")