Métodos de seleção de variáveis
##=============================================================================
## Aplicação de modelos de regressão linear e
## não linear em ciências agrárias
##
## 09 à 11 de Dezembro de 2014 - Goiânia/GO
## Embrapa Arroz e Feijão
##
## Prof. Dr. Walmes M. Zeviani
## LEG/DEST/UFPR
##=============================================================================
##-----------------------------------------------------------------------------
## Definições da sessão.
pkg <- c("lattice", "latticeExtra", "gridExtra", "car", "alr3",
"plyr", "reshape", "doBy", "multcomp", "asbio", "wzRfun")
sapply(pkg, require, character.only=TRUE)
## lattice latticeExtra gridExtra car alr3 plyr
## TRUE TRUE TRUE TRUE TRUE TRUE
## reshape doBy multcomp asbio wzRfun
## TRUE TRUE TRUE TRUE TRUE
trellis.device(color=FALSE)
Stepwise
##-----------------------------------------------------------------------------
## Dados.
url <- "http://www.leg.ufpr.br/~walmes/data/areafoliarUva.txt"
uva <- read.table(url, header=TRUE, sep="\t", stringsAsFactors=FALSE)
uva$cult <- factor(uva$cult)
str(uva)
## 'data.frame': 300 obs. of 9 variables:
## $ id : chr "malbec_1.jpg" "malbec_10.jpg" "malbec_11.jpg" "malbec_12.jpg" ...
## $ cult: Factor w/ 3 levels "malbec","merlot",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ area: num 100.8 85.8 119.5 137 84.7 ...
## $ mc : num 12 11.5 12.5 15.5 10 12 15.5 17.5 13.5 13.3 ...
## $ nc : num 7.5 9 8.5 10 7 8.5 11 13 10 9.5 ...
## $ ml : num 12.8 10.5 13 14.4 11 12 14 14 12 15 ...
## $ nld : num 6.4 8.5 8.6 9 6.5 9 10 9.5 9 9.2 ...
## $ nle : num 7.5 7 9 10 7 8.2 11 11 8.5 8.3 ...
## $ cll : num 9.5 9.5 10.2 12 7.5 8.9 13.5 10.8 9.7 10.3 ...
##-----------------------------------------------------------------------------
## Comprimento da nervura lateral: média dos lados direito e esquerdo.
uva$nl <- with(uva, apply(cbind(nld, nle), 1, mean))
uva <- subset(uva, select=-c(nld, nle))
str(uva)
## 'data.frame': 300 obs. of 8 variables:
## $ id : chr "malbec_1.jpg" "malbec_10.jpg" "malbec_11.jpg" "malbec_12.jpg" ...
## $ cult: Factor w/ 3 levels "malbec","merlot",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ area: num 100.8 85.8 119.5 137 84.7 ...
## $ mc : num 12 11.5 12.5 15.5 10 12 15.5 17.5 13.5 13.3 ...
## $ nc : num 7.5 9 8.5 10 7 8.5 11 13 10 9.5 ...
## $ ml : num 12.8 10.5 13 14.4 11 12 14 14 12 15 ...
## $ cll : num 9.5 9.5 10.2 12 7.5 8.9 13.5 10.8 9.7 10.3 ...
## $ nl : num 6.95 7.75 8.8 9.5 6.75 ...
##-----------------------------------------------------------------------------
## Ver.
## splom(uva[,3:8], groups=uva$cult)
Uva <- split(uva, f=uva$cult)
## str(Uva)
with(Uva[[1]], splom(cbind(area,mc,nc,ml,nl,cll)))
## with(Uva[[2]], splom(cbind(area,mc,nc,ml,nl,cll)))
## with(Uva[[3]], splom(cbind(area,mc,nc,ml,nl,cll)))
##-----------------------------------------------------------------------------
## Malbec.
mal <- subset(uva, cult=="malbec")
##-----------------------------------------------------------------------------
## Modelos.
## Apenas efeitos aditivos.
m0 <- lm(area~mc+ml+nc+nl+cll, data=mal)
## Diagnóstico.
par(mfrow=c(2,2)); plot(m0); layout(1)
## Transformação.
MASS::boxcox(m0); abline(v=0.5, col=2)
m1 <- update(m0, sqrt(area)~.)
par(mfrow=c(2,2)); plot(m1); layout(1)
## Inferência.
summary(m1)
##
## Call:
## lm(formula = sqrt(area) ~ mc + ml + nc + nl + cll, data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.84138 -0.23439 -0.02851 0.14774 1.87394
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.20824 0.16965 1.227 0.223
## mc 0.27039 0.04612 5.862 6.73e-08 ***
## ml 0.35067 0.03092 11.340 < 2e-16 ***
## nc 0.02091 0.04581 0.456 0.649
## nl 0.32234 0.07408 4.351 3.43e-05 ***
## cll -0.03684 0.03926 -0.938 0.350
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3698 on 94 degrees of freedom
## Multiple R-squared: 0.9784, Adjusted R-squared: 0.9773
## F-statistic: 853.3 on 5 and 94 DF, p-value: < 2.2e-16
vif(m1)
## mc ml nc nl cll
## 17.848533 5.518162 10.653027 15.858220 8.085096
##-----------------------------------------------------------------------------
## Não poderia tentar um modelo maior? Com interações e termos
## quadráticos, por exemplo?
## Modelo quadrático completo.
m2 <- update(m1, .~(mc+nc+ml+nl+cll)^2+
I(mc^2)+I(nc^2)+I(ml^2)+I(nl^2)+I(cll^2))
## Diagnóstico.
par(mfrow=c(2,2)); plot(m2); layout(1)
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Altos leverages porque o modelo tem muitos termos. Simplificar o
## modelo.
summary(m2)
##
## Call:
## lm(formula = sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) +
## I(nc^2) + I(ml^2) + I(nl^2) + I(cll^2) + mc:nc + mc:ml +
## mc:nl + mc:cll + nc:ml + nc:nl + nc:cll + ml:nl + ml:cll +
## nl:cll, data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54580 -0.17053 0.01312 0.13645 0.56176
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.455695 0.310257 1.469 0.14587
## mc 0.182184 0.264190 0.690 0.49247
## nc 0.226102 0.348965 0.648 0.51891
## ml 0.366351 0.148907 2.460 0.01607 *
## nl 1.226574 0.378141 3.244 0.00173 **
## cll -0.902841 0.201932 -4.471 2.57e-05 ***
## I(mc^2) 0.072435 0.039310 1.843 0.06913 .
## I(nc^2) 0.001509 0.023232 0.065 0.94838
## I(ml^2) 0.058711 0.018965 3.096 0.00272 **
## I(nl^2) -0.009540 0.080695 -0.118 0.90619
## I(cll^2) 0.032119 0.025333 1.268 0.20856
## mc:nc -0.041382 0.040880 -1.012 0.31450
## mc:ml -0.091415 0.037340 -2.448 0.01657 *
## mc:nl 0.003009 0.109233 0.028 0.97809
## mc:cll -0.030896 0.053601 -0.576 0.56598
## nc:ml 0.059610 0.048743 1.223 0.22499
## nc:nl -0.083565 0.119891 -0.697 0.48784
## nc:cll 0.018013 0.059472 0.303 0.76277
## ml:nl -0.068842 0.064273 -1.071 0.28739
## ml:cll -0.028612 0.028890 -0.990 0.32502
## nl:cll 0.094022 0.062817 1.497 0.13844
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2402 on 79 degrees of freedom
## Multiple R-squared: 0.9924, Adjusted R-squared: 0.9904
## F-statistic: 513.1 on 20 and 79 DF, p-value: < 2.2e-16
##-----------------------------------------------------------------------------
## Stepwise.
## Critério de AIC.
m3 <- step(m2, k=2)
## Start: AIC=-266.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:nl + mc:cll + nc:ml +
## nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nl 1 0.00004 4.5570 -268.85
## - I(nc^2) 1 0.00024 4.5572 -268.85
## - I(nl^2) 1 0.00081 4.5578 -268.83
## - nc:cll 1 0.00529 4.5623 -268.73
## - mc:cll 1 0.01917 4.5762 -268.43
## - nc:nl 1 0.02802 4.5850 -268.24
## - ml:cll 1 0.05658 4.6136 -267.62
## - mc:nc 1 0.05911 4.6161 -267.56
## - ml:nl 1 0.06618 4.6232 -267.41
## - nc:ml 1 0.08627 4.6433 -266.98
## <none> 4.5570 -266.85
## - I(cll^2) 1 0.09273 4.6497 -266.84
## - nl:cll 1 0.12923 4.6862 -266.05
## - I(mc^2) 1 0.19587 4.7529 -264.64
## - mc:ml 1 0.34573 4.9027 -261.54
## - I(ml^2) 1 0.55284 5.1098 -257.40
##
## Step: AIC=-268.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl +
## nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nc^2) 1 0.00022 4.5573 -270.85
## - I(nl^2) 1 0.00076 4.5578 -270.83
## - nc:cll 1 0.00528 4.5623 -270.73
## - mc:cll 1 0.01951 4.5766 -270.42
## - ml:cll 1 0.05870 4.6157 -269.57
## - nc:nl 1 0.06884 4.6259 -269.35
## - ml:nl 1 0.07461 4.6317 -269.23
## <none> 4.5570 -268.85
## - I(cll^2) 1 0.09269 4.6497 -268.84
## - nc:ml 1 0.09674 4.6538 -268.75
## - mc:nc 1 0.10526 4.6623 -268.57
## - nl:cll 1 0.12941 4.6865 -268.05
## - mc:ml 1 0.38703 4.9441 -262.70
## - I(ml^2) 1 0.58958 5.1466 -258.68
## - I(mc^2) 1 0.60915 5.1662 -258.30
##
## Step: AIC=-270.84
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(nl^2) +
## I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll +
## ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nl^2) 1 0.00087 4.5581 -272.82
## - nc:cll 1 0.00613 4.5634 -272.71
## - mc:cll 1 0.02110 4.5784 -272.38
## - ml:cll 1 0.05911 4.6164 -271.56
## - ml:nl 1 0.07439 4.6317 -271.23
## - nc:nl 1 0.07442 4.6317 -271.23
## <none> 4.5573 -270.85
## - I(cll^2) 1 0.09443 4.6517 -270.79
## - mc:nc 1 0.11328 4.6706 -270.39
## - nl:cll 1 0.12983 4.6871 -270.04
## - nc:ml 1 0.13125 4.6885 -270.00
## - mc:ml 1 0.51202 5.0693 -262.20
## - I(ml^2) 1 0.58939 5.1467 -260.68
## - I(mc^2) 1 0.86389 5.4212 -255.49
##
## Step: AIC=-272.83
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll + ml:nl +
## ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:cll 1 0.00738 4.5655 -274.66
## - mc:cll 1 0.02100 4.5791 -274.37
## - ml:cll 1 0.05974 4.6179 -273.52
## <none> 4.5581 -272.82
## - I(cll^2) 1 0.09907 4.6572 -272.68
## - mc:nc 1 0.12869 4.6868 -272.04
## - ml:nl 1 0.13044 4.6886 -272.00
## - nc:ml 1 0.13871 4.6969 -271.83
## - nc:nl 1 0.16108 4.7192 -271.35
## - nl:cll 1 0.17479 4.7329 -271.06
## - mc:ml 1 0.51177 5.0699 -264.19
## - I(ml^2) 1 0.61947 5.1776 -262.08
## - I(mc^2) 1 0.91497 5.4731 -256.53
##
## Step: AIC=-274.66
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + ml:nl + ml:cll +
## nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:cll 1 0.02169 4.5872 -276.19
## - ml:cll 1 0.07384 4.6394 -275.06
## <none> 4.5655 -274.66
## - I(cll^2) 1 0.09969 4.6652 -274.50
## - mc:nc 1 0.12428 4.6898 -273.98
## - ml:nl 1 0.13367 4.6992 -273.78
## - nc:nl 1 0.15570 4.7212 -273.31
## - nl:cll 1 0.16929 4.7348 -273.02
## - nc:ml 1 0.39676 4.9623 -268.33
## - I(ml^2) 1 0.65650 5.2220 -263.23
## - mc:ml 1 1.14985 5.7154 -254.20
## - I(mc^2) 1 1.49542 6.0610 -248.33
##
## Step: AIC=-276.19
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:cll 1 0.07030 4.6575 -276.67
## - I(cll^2) 1 0.07811 4.6653 -276.50
## <none> 4.5872 -276.19
## - ml:nl 1 0.11425 4.7015 -275.73
## - mc:nc 1 0.14213 4.7294 -275.14
## - nl:cll 1 0.14783 4.7351 -275.02
## - nc:nl 1 0.14884 4.7361 -275.00
## - nc:ml 1 0.40464 4.9919 -269.74
## - I(ml^2) 1 0.64508 5.2323 -265.03
## - I(mc^2) 1 1.50089 6.0881 -249.88
## - mc:ml 1 1.51718 6.1044 -249.62
##
## Step: AIC=-276.67
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(cll^2) 1 0.03019 4.6877 -278.02
## <none> 4.6575 -276.67
## - ml:nl 1 0.09566 4.7532 -276.64
## - nl:cll 1 0.12936 4.7869 -275.93
## - nc:nl 1 0.13195 4.7895 -275.88
## - mc:nc 1 0.14474 4.8023 -275.61
## - nc:ml 1 0.34893 5.0065 -271.44
## - I(ml^2) 1 0.63604 5.2936 -265.87
## - I(mc^2) 1 1.54683 6.2044 -249.99
## - mc:ml 1 1.64835 6.3059 -248.37
##
## Step: AIC=-278.02
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:nc +
## mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## <none> 4.6877 -278.02
## - mc:nc 1 0.12074 4.8085 -277.48
## - ml:nl 1 0.16116 4.8489 -276.64
## - nc:nl 1 0.28518 4.9729 -274.12
## - nc:ml 1 0.50310 5.1908 -269.83
## - I(ml^2) 1 0.69800 5.3857 -266.14
## - I(mc^2) 1 1.52137 6.2091 -251.92
## - mc:ml 1 1.66571 6.3534 -249.62
## - nl:cll 1 1.71357 6.4013 -248.87
anova(m3)
## Analysis of Variance Table
##
## Response: sqrt(area)
## Df Sum Sq Mean Sq F value Pr(>F)
## mc 1 547.86 547.86 10050.9224 < 2.2e-16 ***
## nc 1 0.06 0.06 1.0534 0.307599
## ml 1 33.08 33.08 606.9500 < 2.2e-16 ***
## nl 1 2.48 2.48 45.4234 1.721e-09 ***
## cll 1 0.12 0.12 2.2097 0.140806
## I(mc^2) 1 0.40 0.40 7.2614 0.008472 **
## I(ml^2) 1 1.73 1.73 31.7438 2.181e-07 ***
## mc:nc 1 0.59 0.59 10.8010 0.001470 **
## mc:ml 1 2.91 2.91 53.3713 1.313e-10 ***
## nc:ml 1 0.82 0.82 14.9803 0.000211 ***
## nc:nl 1 0.01 0.01 0.1729 0.678578
## ml:nl 1 0.01 0.01 0.1131 0.737496
## nl:cll 1 1.71 1.71 31.4370 2.448e-07 ***
## Residuals 86 4.69 0.05
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Critério de BIC.
m4 <- step(m2, k=log(nrow(mal)))
## Start: AIC=-212.14
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:nl + mc:cll + nc:ml +
## nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nl 1 0.00004 4.5570 -216.75
## - I(nc^2) 1 0.00024 4.5572 -216.74
## - I(nl^2) 1 0.00081 4.5578 -216.73
## - nc:cll 1 0.00529 4.5623 -216.63
## - mc:cll 1 0.01917 4.5762 -216.33
## - nc:nl 1 0.02802 4.5850 -216.13
## - ml:cll 1 0.05658 4.6136 -215.51
## - mc:nc 1 0.05911 4.6161 -215.46
## - ml:nl 1 0.06618 4.6232 -215.31
## - nc:ml 1 0.08627 4.6433 -214.87
## - I(cll^2) 1 0.09273 4.6497 -214.73
## - nl:cll 1 0.12923 4.6862 -213.95
## - I(mc^2) 1 0.19587 4.7529 -212.54
## <none> 4.5570 -212.14
## - mc:ml 1 0.34573 4.9027 -209.43
## - I(ml^2) 1 0.55284 5.1098 -205.30
##
## Step: AIC=-216.75
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl +
## nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nc^2) 1 0.00022 4.5573 -221.35
## - I(nl^2) 1 0.00076 4.5578 -221.34
## - nc:cll 1 0.00528 4.5623 -221.24
## - mc:cll 1 0.01951 4.5766 -220.92
## - ml:cll 1 0.05870 4.6157 -220.07
## - nc:nl 1 0.06884 4.6259 -219.85
## - ml:nl 1 0.07461 4.6317 -219.73
## - I(cll^2) 1 0.09269 4.6497 -219.34
## - nc:ml 1 0.09674 4.6538 -219.25
## - mc:nc 1 0.10526 4.6623 -219.07
## - nl:cll 1 0.12941 4.6865 -218.55
## <none> 4.5570 -216.75
## - mc:ml 1 0.38703 4.9441 -213.20
## - I(ml^2) 1 0.58958 5.1466 -209.19
## - I(mc^2) 1 0.60915 5.1662 -208.81
##
## Step: AIC=-221.35
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(nl^2) +
## I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll +
## ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nl^2) 1 0.00087 4.5581 -225.93
## - nc:cll 1 0.00613 4.5634 -225.82
## - mc:cll 1 0.02110 4.5784 -225.49
## - ml:cll 1 0.05911 4.6164 -224.66
## - ml:nl 1 0.07439 4.6317 -224.33
## - nc:nl 1 0.07442 4.6317 -224.33
## - I(cll^2) 1 0.09443 4.6517 -223.90
## - mc:nc 1 0.11328 4.6706 -223.50
## - nl:cll 1 0.12983 4.6871 -223.14
## - nc:ml 1 0.13125 4.6885 -223.11
## <none> 4.5573 -221.35
## - mc:ml 1 0.51202 5.0693 -215.30
## - I(ml^2) 1 0.58939 5.1467 -213.79
## - I(mc^2) 1 0.86389 5.4212 -208.59
##
## Step: AIC=-225.93
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll + ml:nl +
## ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:cll 1 0.00738 4.5655 -230.38
## - mc:cll 1 0.02100 4.5791 -230.08
## - ml:cll 1 0.05974 4.6179 -229.24
## - I(cll^2) 1 0.09907 4.6572 -228.39
## - mc:nc 1 0.12869 4.6868 -227.75
## - ml:nl 1 0.13044 4.6886 -227.72
## - nc:ml 1 0.13871 4.6969 -227.54
## - nc:nl 1 0.16108 4.7192 -227.06
## - nl:cll 1 0.17479 4.7329 -226.77
## <none> 4.5581 -225.93
## - mc:ml 1 0.51177 5.0699 -219.90
## - I(ml^2) 1 0.61947 5.1776 -217.79
## - I(mc^2) 1 0.91497 5.4731 -212.24
##
## Step: AIC=-230.38
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + ml:nl + ml:cll +
## nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:cll 1 0.02169 4.5872 -234.51
## - ml:cll 1 0.07384 4.6394 -233.38
## - I(cll^2) 1 0.09969 4.6652 -232.82
## - mc:nc 1 0.12428 4.6898 -232.29
## - ml:nl 1 0.13367 4.6992 -232.09
## - nc:nl 1 0.15570 4.7212 -231.63
## - nl:cll 1 0.16929 4.7348 -231.34
## <none> 4.5655 -230.38
## - nc:ml 1 0.39676 4.9623 -226.65
## - I(ml^2) 1 0.65650 5.2220 -221.55
## - mc:ml 1 1.14985 5.7154 -212.52
## - I(mc^2) 1 1.49542 6.0610 -206.65
##
## Step: AIC=-234.51
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:cll 1 0.07030 4.6575 -237.59
## - I(cll^2) 1 0.07811 4.6653 -237.42
## - ml:nl 1 0.11425 4.7015 -236.65
## - mc:nc 1 0.14213 4.7294 -236.06
## - nl:cll 1 0.14783 4.7351 -235.94
## - nc:nl 1 0.14884 4.7361 -235.92
## <none> 4.5872 -234.51
## - nc:ml 1 0.40464 4.9919 -230.66
## - I(ml^2) 1 0.64508 5.2323 -225.95
## - I(mc^2) 1 1.50089 6.0881 -210.81
## - mc:ml 1 1.51718 6.1044 -210.54
##
## Step: AIC=-237.59
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(cll^2) 1 0.03019 4.6877 -241.55
## - ml:nl 1 0.09566 4.7532 -240.16
## - nl:cll 1 0.12936 4.7869 -239.46
## - nc:nl 1 0.13195 4.7895 -239.40
## - mc:nc 1 0.14474 4.8023 -239.14
## <none> 4.6575 -237.59
## - nc:ml 1 0.34893 5.0065 -234.97
## - I(ml^2) 1 0.63604 5.2936 -229.40
## - I(mc^2) 1 1.54683 6.2044 -213.52
## - mc:ml 1 1.64835 6.3059 -211.90
##
## Step: AIC=-241.55
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:nc +
## mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nc 1 0.12074 4.8085 -243.61
## - ml:nl 1 0.16116 4.8489 -242.78
## <none> 4.6877 -241.55
## - nc:nl 1 0.28518 4.9729 -240.25
## - nc:ml 1 0.50310 5.1908 -235.96
## - I(ml^2) 1 0.69800 5.3857 -232.28
## - I(mc^2) 1 1.52137 6.2091 -218.05
## - mc:ml 1 1.66571 6.3534 -215.75
## - nl:cll 1 1.71357 6.4013 -215.00
##
## Step: AIC=-243.61
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:nl 1 0.08114 4.8896 -246.54
## <none> 4.8085 -243.61
## - nc:ml 1 0.46374 5.2722 -239.01
## - I(ml^2) 1 0.63334 5.4418 -235.84
## - nc:nl 1 0.90355 5.7120 -231.00
## - nl:cll 1 1.88232 6.6908 -215.18
## - mc:ml 1 1.95782 6.7663 -214.06
## - I(mc^2) 1 2.32931 7.1378 -208.72
##
## Step: AIC=-246.54
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:ml + nc:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## <none> 4.8896 -246.54
## - nc:ml 1 0.6152 5.5048 -239.30
## - I(ml^2) 1 1.0318 5.9214 -232.00
## - nc:nl 1 1.5783 6.4679 -223.18
## - nl:cll 1 1.8375 6.7271 -219.25
## - mc:ml 1 3.8024 8.6920 -193.62
## - I(mc^2) 1 4.3803 9.2699 -187.18
m4 <- step(m2, k=12)
## Start: AIC=-56.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:nl + mc:cll + nc:ml +
## nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nl 1 0.00004 4.5570 -68.850
## - I(nc^2) 1 0.00024 4.5572 -68.845
## - I(nl^2) 1 0.00081 4.5578 -68.833
## - nc:cll 1 0.00529 4.5623 -68.734
## - mc:cll 1 0.01917 4.5762 -68.431
## - nc:nl 1 0.02802 4.5850 -68.237
## - ml:cll 1 0.05658 4.6136 -67.617
## - mc:nc 1 0.05911 4.6161 -67.562
## - ml:nl 1 0.06618 4.6232 -67.409
## - nc:ml 1 0.08627 4.6433 -66.975
## - I(cll^2) 1 0.09273 4.6497 -66.836
## - nl:cll 1 0.12923 4.6862 -66.054
## - I(mc^2) 1 0.19587 4.7529 -64.642
## - mc:ml 1 0.34573 4.9027 -61.538
## - I(ml^2) 1 0.55284 5.1098 -57.400
## <none> 4.5570 -56.850
##
## Step: AIC=-68.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl +
## nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nc^2) 1 0.00022 4.5573 -80.845
## - I(nl^2) 1 0.00076 4.5578 -80.833
## - nc:cll 1 0.00528 4.5623 -80.734
## - mc:cll 1 0.01951 4.5766 -80.422
## - ml:cll 1 0.05870 4.6157 -79.570
## - nc:nl 1 0.06884 4.6259 -79.350
## - ml:nl 1 0.07461 4.6317 -79.225
## - I(cll^2) 1 0.09269 4.6497 -78.836
## - nc:ml 1 0.09674 4.6538 -78.749
## - mc:nc 1 0.10526 4.6623 -78.566
## - nl:cll 1 0.12941 4.6865 -78.049
## - mc:ml 1 0.38703 4.9441 -72.698
## <none> 4.5570 -68.850
## - I(ml^2) 1 0.58958 5.1466 -68.683
## - I(mc^2) 1 0.60915 5.1662 -68.303
##
## Step: AIC=-80.84
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(nl^2) +
## I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll +
## ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nl^2) 1 0.00087 4.5581 -92.825
## - nc:cll 1 0.00613 4.5634 -92.710
## - mc:cll 1 0.02110 4.5784 -92.383
## - ml:cll 1 0.05911 4.6164 -91.556
## - ml:nl 1 0.07439 4.6317 -91.225
## - nc:nl 1 0.07442 4.6317 -91.225
## - I(cll^2) 1 0.09443 4.6517 -90.794
## - mc:nc 1 0.11328 4.6706 -90.389
## - nl:cll 1 0.12983 4.6871 -90.036
## - nc:ml 1 0.13125 4.6885 -90.005
## - mc:ml 1 0.51202 5.0693 -82.197
## <none> 4.5573 -80.845
## - I(ml^2) 1 0.58939 5.1467 -80.682
## - I(mc^2) 1 0.86389 5.4212 -75.486
##
## Step: AIC=-92.83
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll + ml:nl +
## ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:cll 1 0.00738 4.5655 -104.664
## - mc:cll 1 0.02100 4.5791 -104.366
## - ml:cll 1 0.05974 4.6179 -103.523
## - I(cll^2) 1 0.09907 4.6572 -102.675
## - mc:nc 1 0.12869 4.6868 -102.041
## - ml:nl 1 0.13044 4.6886 -102.004
## - nc:ml 1 0.13871 4.6969 -101.828
## - nc:nl 1 0.16108 4.7192 -101.352
## - nl:cll 1 0.17479 4.7329 -101.062
## - mc:ml 1 0.51177 5.0699 -94.185
## <none> 4.5581 -92.825
## - I(ml^2) 1 0.61947 5.1776 -92.083
## - I(mc^2) 1 0.91497 5.4731 -86.532
##
## Step: AIC=-104.66
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + ml:nl + ml:cll +
## nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:cll 1 0.02169 4.5872 -116.190
## - ml:cll 1 0.07384 4.6394 -115.059
## - I(cll^2) 1 0.09969 4.6652 -114.504
## - mc:nc 1 0.12428 4.6898 -113.978
## - ml:nl 1 0.13367 4.6992 -113.778
## - nc:nl 1 0.15570 4.7212 -113.310
## - nl:cll 1 0.16929 4.7348 -113.023
## - nc:ml 1 0.39676 4.9623 -108.330
## <none> 4.5655 -104.664
## - I(ml^2) 1 0.65650 5.2220 -103.228
## - mc:ml 1 1.14985 5.7154 -94.201
## - I(mc^2) 1 1.49542 6.0610 -88.330
##
## Step: AIC=-116.19
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:cll 1 0.07030 4.6575 -126.669
## - I(cll^2) 1 0.07811 4.6653 -126.501
## - ml:nl 1 0.11425 4.7015 -125.729
## - mc:nc 1 0.14213 4.7294 -125.138
## - nl:cll 1 0.14783 4.7351 -125.018
## - nc:nl 1 0.14884 4.7361 -124.996
## - nc:ml 1 0.40464 4.9919 -119.736
## <none> 4.5872 -116.190
## - I(ml^2) 1 0.64508 5.2323 -115.032
## - I(mc^2) 1 1.50089 6.0881 -99.883
## - mc:ml 1 1.51718 6.1044 -99.616
##
## Step: AIC=-126.67
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(cll^2) 1 0.03019 4.6877 -138.02
## - ml:nl 1 0.09566 4.7532 -136.64
## - nl:cll 1 0.12936 4.7869 -135.93
## - nc:nl 1 0.13195 4.7895 -135.88
## - mc:nc 1 0.14474 4.8023 -135.61
## - nc:ml 1 0.34893 5.0065 -131.44
## <none> 4.6575 -126.67
## - I(ml^2) 1 0.63604 5.2936 -125.87
## - I(mc^2) 1 1.54683 6.2044 -109.99
## - mc:ml 1 1.64835 6.3059 -108.37
##
## Step: AIC=-138.02
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:nc +
## mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nc 1 0.12074 4.8085 -147.48
## - ml:nl 1 0.16116 4.8489 -146.64
## - nc:nl 1 0.28518 4.9729 -144.12
## - nc:ml 1 0.50310 5.1908 -139.83
## <none> 4.6877 -138.02
## - I(ml^2) 1 0.69800 5.3857 -136.14
## - I(mc^2) 1 1.52137 6.2091 -121.92
## - mc:ml 1 1.66571 6.3534 -119.62
## - nl:cll 1 1.71357 6.4013 -118.87
##
## Step: AIC=-147.48
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:nl 1 0.08114 4.8896 -157.81
## - nc:ml 1 0.46374 5.2722 -150.27
## <none> 4.8085 -147.48
## - I(ml^2) 1 0.63334 5.4418 -147.11
## - nc:nl 1 0.90355 5.7120 -142.26
## - nl:cll 1 1.88232 6.6908 -126.44
## - mc:ml 1 1.95782 6.7663 -125.32
## - I(mc^2) 1 2.32931 7.1378 -119.98
##
## Step: AIC=-157.81
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:ml + nc:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:ml 1 0.6152 5.5048 -157.96
## <none> 4.8896 -157.81
## - I(ml^2) 1 1.0318 5.9214 -150.66
## - nc:nl 1 1.5783 6.4679 -141.83
## - nl:cll 1 1.8375 6.7271 -137.90
## - mc:ml 1 3.8024 8.6920 -112.28
## - I(mc^2) 1 4.3803 9.2699 -105.84
##
## Step: AIC=-157.95
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## <none> 5.5048 -157.96
## - nc:nl 1 1.0120 6.5168 -153.08
## - I(ml^2) 1 1.3041 6.8089 -148.69
## - nl:cll 1 1.7846 7.2894 -141.88
## - I(mc^2) 1 3.8417 9.3465 -117.02
## - mc:ml 1 4.2401 9.7449 -112.84
anova(m4)
## Analysis of Variance Table
##
## Response: sqrt(area)
## Df Sum Sq Mean Sq F value Pr(>F)
## mc 1 547.86 547.86 8857.5924 < 2.2e-16 ***
## nc 1 0.06 0.06 0.9283 0.33790
## ml 1 33.08 33.08 534.8878 < 2.2e-16 ***
## nl 1 2.48 2.48 40.0304 9.822e-09 ***
## cll 1 0.12 0.12 1.9473 0.16635
## I(mc^2) 1 0.40 0.40 6.3993 0.01318 *
## I(ml^2) 1 1.73 1.73 27.9749 8.717e-07 ***
## mc:ml 1 3.43 3.43 55.5113 5.737e-11 ***
## nc:nl 1 0.01 0.01 0.1368 0.71239
## nl:cll 1 1.78 1.78 28.8528 6.183e-07 ***
## Residuals 89 5.50 0.06
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m4)
##
## Call:
## lm(formula = sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) +
## I(ml^2) + mc:ml + nc:nl + nl:cll, data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.55490 -0.19004 0.04119 0.16520 0.61451
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.798594 0.285393 2.798 0.006298 **
## mc -0.109223 0.200735 -0.544 0.587723
## nc 0.961161 0.229840 4.182 6.76e-05 ***
## ml 0.642210 0.105488 6.088 2.84e-08 ***
## nl 0.308865 0.155011 1.993 0.049377 *
## cll -0.863242 0.165664 -5.211 1.21e-06 ***
## I(mc^2) 0.052710 0.006688 7.881 7.61e-12 ***
## I(ml^2) 0.036515 0.007952 4.592 1.44e-05 ***
## mc:ml -0.090003 0.010870 -8.280 1.16e-12 ***
## nc:nl -0.103543 0.025598 -4.045 0.000111 ***
## nl:cll 0.098658 0.018367 5.371 6.18e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2487 on 89 degrees of freedom
## Multiple R-squared: 0.9908, Adjusted R-squared: 0.9897
## F-statistic: 955.4 on 10 and 89 DF, p-value: < 2.2e-16
par(mfrow=c(2,2)); plot(m4); layout(1)
## k maior que log(nrow(mal)).
step(m2, k=6)
## Start: AIC=-182.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:nl + mc:cll + nc:ml +
## nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nl 1 0.00004 4.5570 -188.85
## - I(nc^2) 1 0.00024 4.5572 -188.84
## - I(nl^2) 1 0.00081 4.5578 -188.83
## - nc:cll 1 0.00529 4.5623 -188.73
## - mc:cll 1 0.01917 4.5762 -188.43
## - nc:nl 1 0.02802 4.5850 -188.24
## - ml:cll 1 0.05658 4.6136 -187.62
## - mc:nc 1 0.05911 4.6161 -187.56
## - ml:nl 1 0.06618 4.6232 -187.41
## - nc:ml 1 0.08627 4.6433 -186.97
## - I(cll^2) 1 0.09273 4.6497 -186.84
## - nl:cll 1 0.12923 4.6862 -186.05
## - I(mc^2) 1 0.19587 4.7529 -184.64
## <none> 4.5570 -182.85
## - mc:ml 1 0.34573 4.9027 -181.54
## - I(ml^2) 1 0.55284 5.1098 -177.40
##
## Step: AIC=-188.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl +
## nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nc^2) 1 0.00022 4.5573 -194.84
## - I(nl^2) 1 0.00076 4.5578 -194.83
## - nc:cll 1 0.00528 4.5623 -194.73
## - mc:cll 1 0.01951 4.5766 -194.42
## - ml:cll 1 0.05870 4.6157 -193.57
## - nc:nl 1 0.06884 4.6259 -193.35
## - ml:nl 1 0.07461 4.6317 -193.22
## - I(cll^2) 1 0.09269 4.6497 -192.84
## - nc:ml 1 0.09674 4.6538 -192.75
## - mc:nc 1 0.10526 4.6623 -192.57
## - nl:cll 1 0.12941 4.6865 -192.05
## <none> 4.5570 -188.85
## - mc:ml 1 0.38703 4.9441 -186.70
## - I(ml^2) 1 0.58958 5.1466 -182.68
## - I(mc^2) 1 0.60915 5.1662 -182.30
##
## Step: AIC=-194.84
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(nl^2) +
## I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll +
## ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nl^2) 1 0.00087 4.5581 -200.82
## - nc:cll 1 0.00613 4.5634 -200.71
## - mc:cll 1 0.02110 4.5784 -200.38
## - ml:cll 1 0.05911 4.6164 -199.56
## - ml:nl 1 0.07439 4.6317 -199.22
## - nc:nl 1 0.07442 4.6317 -199.22
## - I(cll^2) 1 0.09443 4.6517 -198.79
## - mc:nc 1 0.11328 4.6706 -198.39
## - nl:cll 1 0.12983 4.6871 -198.04
## - nc:ml 1 0.13125 4.6885 -198.00
## <none> 4.5573 -194.84
## - mc:ml 1 0.51202 5.0693 -190.20
## - I(ml^2) 1 0.58939 5.1467 -188.68
## - I(mc^2) 1 0.86389 5.4212 -183.49
##
## Step: AIC=-200.83
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll + ml:nl +
## ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:cll 1 0.00738 4.5655 -206.66
## - mc:cll 1 0.02100 4.5791 -206.37
## - ml:cll 1 0.05974 4.6179 -205.52
## - I(cll^2) 1 0.09907 4.6572 -204.68
## - mc:nc 1 0.12869 4.6868 -204.04
## - ml:nl 1 0.13044 4.6886 -204.00
## - nc:ml 1 0.13871 4.6969 -203.83
## - nc:nl 1 0.16108 4.7192 -203.35
## - nl:cll 1 0.17479 4.7329 -203.06
## <none> 4.5581 -200.82
## - mc:ml 1 0.51177 5.0699 -196.19
## - I(ml^2) 1 0.61947 5.1776 -194.08
## - I(mc^2) 1 0.91497 5.4731 -188.53
##
## Step: AIC=-206.66
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + ml:nl + ml:cll +
## nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:cll 1 0.02169 4.5872 -212.19
## - ml:cll 1 0.07384 4.6394 -211.06
## - I(cll^2) 1 0.09969 4.6652 -210.50
## - mc:nc 1 0.12428 4.6898 -209.98
## - ml:nl 1 0.13367 4.6992 -209.78
## - nc:nl 1 0.15570 4.7212 -209.31
## - nl:cll 1 0.16929 4.7348 -209.02
## <none> 4.5655 -206.66
## - nc:ml 1 0.39676 4.9623 -204.33
## - I(ml^2) 1 0.65650 5.2220 -199.23
## - mc:ml 1 1.14985 5.7154 -190.20
## - I(mc^2) 1 1.49542 6.0610 -184.33
##
## Step: AIC=-212.19
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:cll 1 0.07030 4.6575 -216.67
## - I(cll^2) 1 0.07811 4.6653 -216.50
## - ml:nl 1 0.11425 4.7015 -215.73
## - mc:nc 1 0.14213 4.7294 -215.14
## - nl:cll 1 0.14783 4.7351 -215.02
## - nc:nl 1 0.14884 4.7361 -215.00
## <none> 4.5872 -212.19
## - nc:ml 1 0.40464 4.9919 -209.74
## - I(ml^2) 1 0.64508 5.2323 -205.03
## - I(mc^2) 1 1.50089 6.0881 -189.88
## - mc:ml 1 1.51718 6.1044 -189.62
##
## Step: AIC=-216.67
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(cll^2) 1 0.03019 4.6877 -222.02
## - ml:nl 1 0.09566 4.7532 -220.64
## - nl:cll 1 0.12936 4.7869 -219.93
## - nc:nl 1 0.13195 4.7895 -219.88
## - mc:nc 1 0.14474 4.8023 -219.61
## <none> 4.6575 -216.67
## - nc:ml 1 0.34893 5.0065 -215.44
## - I(ml^2) 1 0.63604 5.2936 -209.87
## - I(mc^2) 1 1.54683 6.2044 -193.99
## - mc:ml 1 1.64835 6.3059 -192.37
##
## Step: AIC=-222.02
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:nc +
## mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nc 1 0.12074 4.8085 -225.48
## - ml:nl 1 0.16116 4.8489 -224.64
## - nc:nl 1 0.28518 4.9729 -222.12
## <none> 4.6877 -222.02
## - nc:ml 1 0.50310 5.1908 -217.83
## - I(ml^2) 1 0.69800 5.3857 -214.14
## - I(mc^2) 1 1.52137 6.2091 -199.92
## - mc:ml 1 1.66571 6.3534 -197.62
## - nl:cll 1 1.71357 6.4013 -196.87
##
## Step: AIC=-225.48
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:nl 1 0.08114 4.8896 -229.81
## <none> 4.8085 -225.48
## - nc:ml 1 0.46374 5.2722 -222.27
## - I(ml^2) 1 0.63334 5.4418 -219.11
## - nc:nl 1 0.90355 5.7120 -214.26
## - nl:cll 1 1.88232 6.6908 -198.44
## - mc:ml 1 1.95782 6.7663 -197.32
## - I(mc^2) 1 2.32931 7.1378 -191.98
##
## Step: AIC=-229.81
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:ml +
## nc:ml + nc:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## <none> 4.8896 -229.81
## - nc:ml 1 0.6152 5.5048 -223.96
## - I(ml^2) 1 1.0318 5.9214 -216.66
## - nc:nl 1 1.5783 6.4679 -207.83
## - nl:cll 1 1.8375 6.7271 -203.90
## - mc:ml 1 3.8024 8.6920 -178.28
## - I(mc^2) 1 4.3803 9.2699 -171.84
##
## Call:
## lm(formula = sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) +
## I(ml^2) + mc:ml + nc:ml + nc:nl + nl:cll, data = mal)
##
## Coefficients:
## (Intercept) mc nc ml nl cll
## 0.69573 0.32544 0.23507 0.47770 0.72972 -0.86188
## I(mc^2) I(ml^2) mc:ml nc:ml nc:nl nl:cll
## 0.05808 0.03283 -0.12890 0.07700 -0.14639 0.10014
##-----------------------------------------------------------------------------
## Qual o melhor modelo partindo das interações duplas?
m5 <- update(m2, .~(mc+nc+ml+nl+cll)^2)
m5 <- step(m5, k=log(nrow(mal)))
## Start: AIC=-210.98
## sqrt(area) ~ mc + nc + ml + nl + cll + mc:nc + mc:ml + mc:nl +
## mc:cll + nc:ml + nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:cll 1 0.00040 5.8042 -215.58
## - nc:cll 1 0.00223 5.8061 -215.55
## - ml:nl 1 0.00504 5.8089 -215.50
## - nl:cll 1 0.02681 5.8306 -215.13
## <none> 5.8038 -210.98
## - ml:cll 1 0.28764 6.0915 -210.75
## - mc:nc 1 0.37047 6.1743 -209.40
## - nc:ml 1 0.56275 6.3666 -206.33
## - mc:nl 1 0.74116 6.5450 -203.57
## - nc:nl 1 0.96897 6.7728 -200.15
## - mc:ml 1 1.99222 7.7960 -186.08
##
## Step: AIC=-215.58
## sqrt(area) ~ mc + nc + ml + nl + cll + mc:nc + mc:ml + mc:nl +
## nc:ml + nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:cll 1 0.0047 5.8089 -220.10
## - ml:nl 1 0.0082 5.8124 -220.04
## - nl:cll 1 0.0285 5.8328 -219.69
## <none> 5.8042 -215.58
## - ml:cll 1 0.3384 6.1426 -214.52
## - mc:nc 1 0.3859 6.1901 -213.75
## - nc:ml 1 0.7306 6.5348 -208.33
## - mc:nl 1 1.3568 7.1611 -199.18
## - nc:nl 1 1.9281 7.7324 -191.50
## - mc:ml 1 3.2034 9.0076 -176.24
##
## Step: AIC=-220.11
## sqrt(area) ~ mc + nc + ml + nl + cll + mc:nc + mc:ml + mc:nl +
## nc:ml + nc:nl + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:nl 1 0.0046 5.8136 -224.63
## - nl:cll 1 0.0974 5.9063 -223.05
## <none> 5.8089 -220.10
## - ml:cll 1 0.3441 6.1531 -218.96
## - mc:nc 1 0.5055 6.3144 -216.37
## - nc:ml 1 0.8787 6.6876 -210.62
## - mc:nl 1 1.4031 7.2120 -203.07
## - nc:nl 1 1.9274 7.7364 -196.06
## - mc:ml 1 3.2269 9.0358 -180.53
##
## Step: AIC=-224.63
## sqrt(area) ~ mc + nc + ml + nl + cll + mc:nc + mc:ml + mc:nl +
## nc:ml + nc:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nl:cll 1 0.0937 5.9073 -227.64
## <none> 5.8136 -224.63
## - ml:cll 1 0.4493 6.2628 -221.79
## - nc:ml 1 0.8893 6.7028 -215.00
## - mc:nc 1 0.8930 6.7066 -214.95
## - mc:nl 1 1.7463 7.5598 -202.97
## - nc:nl 1 1.9757 7.7892 -199.98
## - mc:ml 1 3.6427 9.4563 -180.59
##
## Step: AIC=-227.64
## sqrt(area) ~ mc + nc + ml + nl + cll + mc:nc + mc:ml + mc:nl +
## nc:ml + nc:nl + ml:cll
##
## Df Sum of Sq RSS AIC
## <none> 5.9073 -227.64
## - mc:nc 1 0.8025 6.7098 -219.50
## - nc:ml 1 0.8069 6.7142 -219.44
## - ml:cll 1 1.3442 7.2515 -211.74
## - mc:nl 1 1.8283 7.7355 -205.28
## - nc:nl 1 2.0702 7.9775 -202.20
## - mc:ml 1 3.8203 9.7276 -182.36
m6 <- update(m2, .~mc+nc+ml+nl+cll+
I(mc^2)+I(nc^2)+I(ml^2)+I(nl^2)+I(cll^2))
m6 <- step(m6, k=log(nrow(mal)))
## Start: AIC=-183.95
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2)
##
## Df Sum of Sq RSS AIC
## - mc 1 0.00060 9.5755 -188.54
## - I(mc^2) 1 0.13388 9.7088 -187.16
## - I(ml^2) 1 0.13549 9.7104 -187.15
## - nc 1 0.28772 9.8626 -185.59
## - I(nc^2) 1 0.34887 9.9238 -184.97
## - cll 1 0.36986 9.9448 -184.76
## - I(cll^2) 1 0.38788 9.9628 -184.58
## <none> 9.5749 -183.94
## - ml 1 0.79353 10.3685 -180.59
## - I(nl^2) 1 0.86751 10.4424 -179.88
## - nl 1 1.40576 10.9807 -174.85
##
## Step: AIC=-188.54
## sqrt(area) ~ nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2)
##
## Df Sum of Sq RSS AIC
## - I(ml^2) 1 0.1825 9.7580 -191.26
## - cll 1 0.3701 9.9456 -189.36
## - I(cll^2) 1 0.3885 9.9640 -189.17
## <none> 9.5755 -188.54
## - nc 1 0.5803 10.1559 -187.26
## - I(nc^2) 1 0.6956 10.2712 -186.14
## - I(nl^2) 1 0.8669 10.4425 -184.48
## - ml 1 1.1470 10.7225 -181.84
## - nl 1 1.4063 10.9818 -179.45
## - I(mc^2) 1 5.2718 14.8474 -149.29
##
## Step: AIC=-191.26
## sqrt(area) ~ nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(nl^2) +
## I(cll^2)
##
## Df Sum of Sq RSS AIC
## - I(cll^2) 1 0.4178 10.176 -191.67
## - cll 1 0.4248 10.183 -191.61
## <none> 9.758 -191.26
## - nc 1 0.6649 10.423 -189.28
## - I(nc^2) 1 0.7614 10.520 -188.35
## - I(nl^2) 1 2.7963 12.554 -170.67
## - nl 1 4.1070 13.865 -160.74
## - I(mc^2) 1 5.6184 15.376 -150.39
## - ml 1 9.2839 19.042 -129.01
##
## Step: AIC=-191.67
## sqrt(area) ~ nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(nl^2)
##
## Df Sum of Sq RSS AIC
## - cll 1 0.0075 10.183 -196.21
## <none> 10.176 -191.67
## - nc 1 1.4250 11.601 -183.17
## - I(nc^2) 1 1.5190 11.695 -182.37
## - I(nl^2) 1 2.4805 12.656 -174.47
## - nl 1 3.9786 14.155 -163.28
## - I(mc^2) 1 6.7216 16.898 -145.56
## - ml 1 10.5656 20.741 -125.07
##
## Step: AIC=-196.21
## sqrt(area) ~ nc + ml + nl + I(mc^2) + I(nc^2) + I(nl^2)
##
## Df Sum of Sq RSS AIC
## <none> 10.183 -196.21
## - nc 1 1.5769 11.760 -186.41
## - I(nc^2) 1 1.6390 11.822 -185.89
## - I(nl^2) 1 2.6823 12.866 -177.43
## - nl 1 4.1370 14.320 -166.72
## - I(mc^2) 1 6.7175 16.901 -150.15
## - ml 1 11.5384 21.722 -125.06
summary(m1)
##
## Call:
## lm(formula = sqrt(area) ~ mc + ml + nc + nl + cll, data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.84138 -0.23439 -0.02851 0.14774 1.87394
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.20824 0.16965 1.227 0.223
## mc 0.27039 0.04612 5.862 6.73e-08 ***
## ml 0.35067 0.03092 11.340 < 2e-16 ***
## nc 0.02091 0.04581 0.456 0.649
## nl 0.32234 0.07408 4.351 3.43e-05 ***
## cll -0.03684 0.03926 -0.938 0.350
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3698 on 94 degrees of freedom
## Multiple R-squared: 0.9784, Adjusted R-squared: 0.9773
## F-statistic: 853.3 on 5 and 94 DF, p-value: < 2.2e-16
m7 <- update(m1, .~mc+ml+nl)
## summary(m4)$adj.r.squared
## summary(m5)$adj.r.squared
## summary(m6)$adj.r.squared
## summary(m7)$adj.r.squared
coef(m4)
## (Intercept) mc nc ml nl cll I(mc^2)
## 0.79859412 -0.10922320 0.96116145 0.64221019 0.30886492 -0.86324198 0.05270993
## I(ml^2) mc:ml nc:nl nl:cll
## 0.03651507 -0.09000315 -0.10354332 0.09865800
coef(m7)
## (Intercept) mc ml nl
## 0.2024610 0.2725136 0.3434172 0.3077210
Forward e backward
##-----------------------------------------------------------------------------
## Forward e Backward.
m8 <- step(m7, direction="forward",
scope=list(lower=formula(m7), upper=formula(m2)))
## Start: AIC=-196.11
## sqrt(area) ~ mc + ml + nl
##
## Df Sum of Sq RSS AIC
## + I(mc^2) 1 0.33288 12.655 -196.71
## <none> 12.988 -196.11
## + I(nc^2) 1 0.16166 12.826 -195.37
## + mc:ml 1 0.15677 12.831 -195.33
## + ml:nl 1 0.10368 12.884 -194.91
## + cll 1 0.10219 12.886 -194.90
## + I(ml^2) 1 0.07337 12.915 -194.68
## + mc:nl 1 0.03604 12.952 -194.39
## + nc 1 0.01025 12.978 -194.19
## + I(cll^2) 1 0.00729 12.981 -194.17
## + I(nl^2) 1 0.00014 12.988 -194.11
##
## Step: AIC=-196.71
## sqrt(area) ~ mc + ml + nl + I(mc^2)
##
## Df Sum of Sq RSS AIC
## + mc:ml 1 4.1590 8.4963 -234.55
## + ml:nl 1 2.8171 9.8381 -219.89
## + mc:nl 1 2.0807 10.5746 -212.67
## + I(ml^2) 1 1.7636 10.8917 -209.72
## + I(nl^2) 1 1.4127 11.2425 -206.55
## <none> 12.6553 -196.71
## + cll 1 0.1569 12.4983 -195.96
## + I(cll^2) 1 0.1285 12.5267 -195.73
## + I(nc^2) 1 0.0537 12.6015 -195.13
## + nc 1 0.0114 12.6438 -194.80
##
## Step: AIC=-234.55
## sqrt(area) ~ mc + ml + nl + I(mc^2) + mc:ml
##
## Df Sum of Sq RSS AIC
## + I(ml^2) 1 1.12437 7.3719 -246.75
## + ml:nl 1 0.43171 8.0646 -237.77
## + I(nc^2) 1 0.21154 8.2847 -235.08
## + nc 1 0.18111 8.3152 -234.71
## + I(cll^2) 1 0.17472 8.3216 -234.63
## <none> 8.4963 -234.55
## + cll 1 0.06598 8.4303 -233.33
## + mc:nl 1 0.01145 8.4848 -232.69
## + I(nl^2) 1 0.00698 8.4893 -232.64
##
## Step: AIC=-246.75
## sqrt(area) ~ mc + ml + nl + I(mc^2) + I(ml^2) + mc:ml
##
## Df Sum of Sq RSS AIC
## <none> 7.3719 -246.75
## + I(cll^2) 1 0.138611 7.2333 -246.65
## + I(nc^2) 1 0.087238 7.2847 -245.94
## + nc 1 0.061903 7.3100 -245.59
## + cll 1 0.021427 7.3505 -245.04
## + I(nl^2) 1 0.002835 7.3691 -244.79
## + ml:nl 1 0.001160 7.3707 -244.76
## + mc:nl 1 0.000044 7.3719 -244.75
m9 <- step(m2, direction="backward",
scope=list(lower=formula(m7), upper=formula(m2)))
## Start: AIC=-266.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:nl + mc:cll + nc:ml +
## nc:nl + nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:nl 1 0.00004 4.5570 -268.85
## - I(nc^2) 1 0.00024 4.5572 -268.85
## - I(nl^2) 1 0.00081 4.5578 -268.83
## - nc:cll 1 0.00529 4.5623 -268.73
## - mc:cll 1 0.01917 4.5762 -268.43
## - nc:nl 1 0.02802 4.5850 -268.24
## - ml:cll 1 0.05658 4.6136 -267.62
## - mc:nc 1 0.05911 4.6161 -267.56
## - ml:nl 1 0.06618 4.6232 -267.41
## - nc:ml 1 0.08627 4.6433 -266.98
## <none> 4.5570 -266.85
## - I(cll^2) 1 0.09273 4.6497 -266.84
## - nl:cll 1 0.12923 4.6862 -266.05
## - I(mc^2) 1 0.19587 4.7529 -264.64
## - mc:ml 1 0.34573 4.9027 -261.54
## - I(ml^2) 1 0.55284 5.1098 -257.40
##
## Step: AIC=-268.85
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(nc^2) + I(ml^2) +
## I(nl^2) + I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl +
## nc:cll + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nc^2) 1 0.00022 4.5573 -270.85
## - I(nl^2) 1 0.00076 4.5578 -270.83
## - nc:cll 1 0.00528 4.5623 -270.73
## - mc:cll 1 0.01951 4.5766 -270.42
## - ml:cll 1 0.05870 4.6157 -269.57
## - nc:nl 1 0.06884 4.6259 -269.35
## - ml:nl 1 0.07461 4.6317 -269.23
## <none> 4.5570 -268.85
## - I(cll^2) 1 0.09269 4.6497 -268.84
## - nc:ml 1 0.09674 4.6538 -268.75
## - mc:nc 1 0.10526 4.6623 -268.57
## - nl:cll 1 0.12941 4.6865 -268.05
## - mc:ml 1 0.38703 4.9441 -262.70
## - I(ml^2) 1 0.58958 5.1466 -258.68
## - I(mc^2) 1 0.60915 5.1662 -258.30
##
## Step: AIC=-270.84
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(nl^2) +
## I(cll^2) + mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll +
## ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(nl^2) 1 0.00087 4.5581 -272.82
## - nc:cll 1 0.00613 4.5634 -272.71
## - mc:cll 1 0.02110 4.5784 -272.38
## - ml:cll 1 0.05911 4.6164 -271.56
## - ml:nl 1 0.07439 4.6317 -271.23
## - nc:nl 1 0.07442 4.6317 -271.23
## <none> 4.5573 -270.85
## - I(cll^2) 1 0.09443 4.6517 -270.79
## - mc:nc 1 0.11328 4.6706 -270.39
## - nl:cll 1 0.12983 4.6871 -270.04
## - nc:ml 1 0.13125 4.6885 -270.00
## - mc:ml 1 0.51202 5.0693 -262.20
## - I(ml^2) 1 0.58939 5.1467 -260.68
## - I(mc^2) 1 0.86389 5.4212 -255.49
##
## Step: AIC=-272.83
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + nc:cll + ml:nl +
## ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - nc:cll 1 0.00738 4.5655 -274.66
## - mc:cll 1 0.02100 4.5791 -274.37
## - ml:cll 1 0.05974 4.6179 -273.52
## <none> 4.5581 -272.82
## - I(cll^2) 1 0.09907 4.6572 -272.68
## - mc:nc 1 0.12869 4.6868 -272.04
## - ml:nl 1 0.13044 4.6886 -272.00
## - nc:ml 1 0.13871 4.6969 -271.83
## - nc:nl 1 0.16108 4.7192 -271.35
## - nl:cll 1 0.17479 4.7329 -271.06
## - mc:ml 1 0.51177 5.0699 -264.19
## - I(ml^2) 1 0.61947 5.1776 -262.08
## - I(mc^2) 1 0.91497 5.4731 -256.53
##
## Step: AIC=-274.66
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + mc:cll + nc:ml + nc:nl + ml:nl + ml:cll +
## nl:cll
##
## Df Sum of Sq RSS AIC
## - mc:cll 1 0.02169 4.5872 -276.19
## - ml:cll 1 0.07384 4.6394 -275.06
## <none> 4.5655 -274.66
## - I(cll^2) 1 0.09969 4.6652 -274.50
## - mc:nc 1 0.12428 4.6898 -273.98
## - ml:nl 1 0.13367 4.6992 -273.78
## - nc:nl 1 0.15570 4.7212 -273.31
## - nl:cll 1 0.16929 4.7348 -273.02
## - nc:ml 1 0.39676 4.9623 -268.33
## - I(ml^2) 1 0.65650 5.2220 -263.23
## - mc:ml 1 1.14985 5.7154 -254.20
## - I(mc^2) 1 1.49542 6.0610 -248.33
##
## Step: AIC=-276.19
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + ml:cll + nl:cll
##
## Df Sum of Sq RSS AIC
## - ml:cll 1 0.07030 4.6575 -276.67
## - I(cll^2) 1 0.07811 4.6653 -276.50
## <none> 4.5872 -276.19
## - ml:nl 1 0.11425 4.7015 -275.73
## - mc:nc 1 0.14213 4.7294 -275.14
## - nl:cll 1 0.14783 4.7351 -275.02
## - nc:nl 1 0.14884 4.7361 -275.00
## - nc:ml 1 0.40464 4.9919 -269.74
## - I(ml^2) 1 0.64508 5.2323 -265.03
## - I(mc^2) 1 1.50089 6.0881 -249.88
## - mc:ml 1 1.51718 6.1044 -249.62
##
## Step: AIC=-276.67
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + I(cll^2) +
## mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## - I(cll^2) 1 0.03019 4.6877 -278.02
## <none> 4.6575 -276.67
## - ml:nl 1 0.09566 4.7532 -276.64
## - nl:cll 1 0.12936 4.7869 -275.93
## - nc:nl 1 0.13195 4.7895 -275.88
## - mc:nc 1 0.14474 4.8023 -275.61
## - nc:ml 1 0.34893 5.0065 -271.44
## - I(ml^2) 1 0.63604 5.2936 -265.87
## - I(mc^2) 1 1.54683 6.2044 -249.99
## - mc:ml 1 1.64835 6.3059 -248.37
##
## Step: AIC=-278.02
## sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:nc +
## mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
##
## Df Sum of Sq RSS AIC
## <none> 4.6877 -278.02
## - mc:nc 1 0.12074 4.8085 -277.48
## - ml:nl 1 0.16116 4.8489 -276.64
## - nc:nl 1 0.28518 4.9729 -274.12
## - nc:ml 1 0.50310 5.1908 -269.83
## - I(ml^2) 1 0.69800 5.3857 -266.14
## - I(mc^2) 1 1.52137 6.2091 -251.92
## - mc:ml 1 1.66571 6.3534 -249.62
## - nl:cll 1 1.71357 6.4013 -248.87
anova(m8, m9)
## Analysis of Variance Table
##
## Model 1: sqrt(area) ~ mc + ml + nl + I(mc^2) + I(ml^2) + mc:ml
## Model 2: sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) + I(ml^2) + mc:nc +
## mc:ml + nc:ml + nc:nl + ml:nl + nl:cll
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 93 7.3719
## 2 86 4.6877 7 2.6842 7.0348 1.251e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m8)
##
## Call:
## lm(formula = sqrt(area) ~ mc + ml + nl + I(mc^2) + I(ml^2) +
## mc:ml, data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.84155 -0.16950 0.04039 0.16181 0.75831
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.379139 0.292470 1.296 0.19807
## mc -0.004508 0.123166 -0.037 0.97088
## ml 0.540192 0.112127 4.818 5.63e-06 ***
## nl 0.370740 0.054532 6.799 9.99e-10 ***
## I(mc^2) 0.042726 0.005229 8.170 1.49e-12 ***
## I(ml^2) 0.032688 0.008679 3.766 0.00029 ***
## mc:ml -0.073988 0.011103 -6.664 1.86e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2815 on 93 degrees of freedom
## Multiple R-squared: 0.9876, Adjusted R-squared: 0.9868
## F-statistic: 1239 on 6 and 93 DF, p-value: < 2.2e-16
summary(m9)
##
## Call:
## lm(formula = sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) +
## I(ml^2) + mc:nc + mc:ml + nc:ml + nc:nl + ml:nl + nl:cll,
## data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56377 -0.16721 -0.00008 0.15014 0.54414
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.54902 0.28324 1.938 0.055864 .
## mc 0.18649 0.24126 0.773 0.441641
## nc 0.26781 0.31486 0.851 0.397377
## ml 0.42136 0.13027 3.235 0.001730 **
## nl 1.07455 0.33956 3.165 0.002148 **
## cll -0.89663 0.16679 -5.376 6.45e-07 ***
## I(mc^2) 0.06696 0.01267 5.283 9.46e-07 ***
## I(ml^2) 0.04963 0.01387 3.578 0.000571 ***
## mc:nc -0.03509 0.02358 -1.488 0.140319
## mc:ml -0.10865 0.01965 -5.528 3.41e-07 ***
## nc:ml 0.07250 0.02386 3.038 0.003153 **
## nc:nl -0.09098 0.03977 -2.287 0.024632 *
## ml:nl -0.07131 0.04147 -1.719 0.089125 .
## nl:cll 0.10381 0.01851 5.607 2.45e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2335 on 86 degrees of freedom
## Multiple R-squared: 0.9921, Adjusted R-squared: 0.991
## F-statistic: 835.1 on 13 and 86 DF, p-value: < 2.2e-16
Comparando modelos
##-----------------------------------------------------------------------------
## Medidas de ajuste.
measures <- function(x){
L <- list(logLik=logLik(x),
AIC=AIC(x),
BIC=BIC(x),
PRESS=press(x),
RMSE=summary(x)$sigma,
R2=summary(x)$r.squared,
R2adj=summary(x)$adj.r.squared,
npar=length(coef(x)),
dfres=df.residual(x),
nobs=length(fitted(x)))
unlist(L)
}
modl <- list(m4=m4, m5=m5, m6=m6, m7=m7, m8=m8, m9=m9)
round(t(sapply(modl, measures)), 3)
## logLik AIC BIC PRESS RMSE R2 R2adj npar dfres nobs
## m4 3.084 17.833 49.095 8.131 0.249 0.991 0.990 11 89 100
## m5 -0.444 26.889 60.756 9.425 0.259 0.990 0.989 12 88 100
## m6 -27.673 71.346 92.187 19.104 0.331 0.983 0.982 7 93 100
## m7 -39.837 89.674 102.700 16.323 0.368 0.978 0.978 4 96 100
## m8 -11.519 39.038 59.880 9.003 0.282 0.988 0.987 7 93 100
## m9 11.117 7.765 46.843 6.379 0.233 0.992 0.991 14 86 100
## Eis a diferença entre significância estatística e significância
## prática.
##-----------------------------------------------------------------------------
## Modelo final.
anova(m4)
## Analysis of Variance Table
##
## Response: sqrt(area)
## Df Sum Sq Mean Sq F value Pr(>F)
## mc 1 547.86 547.86 8857.5924 < 2.2e-16 ***
## nc 1 0.06 0.06 0.9283 0.33790
## ml 1 33.08 33.08 534.8878 < 2.2e-16 ***
## nl 1 2.48 2.48 40.0304 9.822e-09 ***
## cll 1 0.12 0.12 1.9473 0.16635
## I(mc^2) 1 0.40 0.40 6.3993 0.01318 *
## I(ml^2) 1 1.73 1.73 27.9749 8.717e-07 ***
## mc:ml 1 3.43 3.43 55.5113 5.737e-11 ***
## nc:nl 1 0.01 0.01 0.1368 0.71239
## nl:cll 1 1.78 1.78 28.8528 6.183e-07 ***
## Residuals 89 5.50 0.06
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(m4)
##
## Call:
## lm(formula = sqrt(area) ~ mc + nc + ml + nl + cll + I(mc^2) +
## I(ml^2) + mc:ml + nc:nl + nl:cll, data = mal)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.55490 -0.19004 0.04119 0.16520 0.61451
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.798594 0.285393 2.798 0.006298 **
## mc -0.109223 0.200735 -0.544 0.587723
## nc 0.961161 0.229840 4.182 6.76e-05 ***
## ml 0.642210 0.105488 6.088 2.84e-08 ***
## nl 0.308865 0.155011 1.993 0.049377 *
## cll -0.863242 0.165664 -5.211 1.21e-06 ***
## I(mc^2) 0.052710 0.006688 7.881 7.61e-12 ***
## I(ml^2) 0.036515 0.007952 4.592 1.44e-05 ***
## mc:ml -0.090003 0.010870 -8.280 1.16e-12 ***
## nc:nl -0.103543 0.025598 -4.045 0.000111 ***
## nl:cll 0.098658 0.018367 5.371 6.18e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2487 on 89 degrees of freedom
## Multiple R-squared: 0.9908, Adjusted R-squared: 0.9897
## F-statistic: 955.4 on 10 and 89 DF, p-value: < 2.2e-16
plot(sqrt(mal$area)~fitted(m4), asp=1)
points(sqrt(mal$area)~fitted(m7), pch=3)
abline(a=0, b=1)
grid()
## identify(y=sqrt(mal$area), x=fitted(m7))
##-----------------------------------------------------------------------------
## Medidas de influência.
im <- influence.measures(m7)
summary(im)
## Potentially influential observations of
## lm(formula = sqrt(area) ~ mc + ml + nl, data = mal) :
##
## dfb.1_ dfb.mc dfb.ml dfb.nl dffit cov.r cook.d hat
## 36 0.05 0.23 0.13 -0.28 0.29 1.13_* 0.02 0.10
## 58 0.61 3.03_* -2.57_* -0.92 4.01_* 0.22_* 2.58_* 0.23_*
## 92 -0.19 -0.07 0.09 0.05 -0.20 1.16_* 0.01 0.11
## 94 -0.10 -0.03 0.06 0.01 -0.11 1.18_* 0.00 0.12
## 96 0.13 0.00 -0.06 0.00 0.13 1.21_* 0.00 0.14_*
## 98 -0.22 -0.45 0.48 0.14 0.70_* 0.81_* 0.11 0.06
## im <- influence.measures(m4)
## summary(im)
## m10 <- update(m7, data=mal[-c(58),])
m10 <- update(m7, data=mal[-c(58,98),])
## m10 <- update(m7, data=mal[-c(58,98,96),])
modl <- list(m4=m4, m5=m5, m6=m6, m7=m7,
m8=m8, m9=m9, m10=m10)
round(t(sapply(modl, measures)), 3)
## logLik AIC BIC PRESS RMSE R2 R2adj npar dfres nobs
## m4 3.084 17.833 49.095 8.131 0.249 0.991 0.990 11 89 100
## m5 -0.444 26.889 60.756 9.425 0.259 0.990 0.989 12 88 100
## m6 -27.673 71.346 92.187 19.104 0.331 0.983 0.982 7 93 100
## m7 -39.837 89.674 102.700 16.323 0.368 0.978 0.978 4 96 100
## m8 -11.519 39.038 59.880 9.003 0.282 0.988 0.987 7 93 100
## m9 11.117 7.765 46.843 6.379 0.233 0.992 0.991 14 86 100
## m10 -13.964 37.927 50.852 8.416 0.285 0.987 0.986 4 94 98
##-----------------------------------------------------------------------------
## Predição por meio do modelo.
coef(m10)
## (Intercept) mc ml nl
## 0.1523111 0.2009084 0.3917068 0.3499388
predict(m10,
newdata=list(
mc=13,
nl=8.3,
ml=12.2))^2
## 1
## 109.1489
Praticar
##-----------------------------------------------------------------------------
## Dados de qualidade de vinho.
url <- "http://www.leg.ufpr.br/~walmes/data/MontgomeryASPE5th/Example12.14.txt"
vin <- read.table(url, header=TRUE, sep="\t")
str(vin)
## 'data.frame': 38 obs. of 6 variables:
## $ Clarity : num 1 1 1 1 1 1 1 1 1 1 ...
## $ Aroma : num 3.3 4.4 3.9 3.9 5.6 4.6 4.8 5.3 4.3 4.3 ...
## $ Body : num 2.8 4.9 5.3 2.6 5.1 4.7 4.8 4.5 4.3 3.9 ...
## $ Flavor : num 3.1 3.5 4.8 3.1 5.5 5 4.8 4.3 3.9 4.7 ...
## $ Oakiness: num 4.1 3.9 4.7 3.6 5.1 4.1 3.3 5.2 2.9 3.9 ...
## $ Quality : num 9.8 12.6 11.9 11.1 13.3 12.8 12.8 12 13.6 13.9 ...
splom(vin)
m0 <- lm(Quality~(.)^2, data=vin)
summary(m0)
##
## Call:
## lm(formula = Quality ~ (.)^2, data = vin)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7336 -0.4046 0.0032 0.4765 1.8250
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -27.75186 31.99982 -0.867 0.395
## Clarity 35.95087 25.66356 1.401 0.175
## Aroma -7.23944 5.62565 -1.287 0.212
## Body 2.62752 4.41221 0.596 0.558
## Flavor 7.36225 5.64433 1.304 0.206
## Oakiness 5.69028 6.41138 0.888 0.384
## Clarity:Aroma 5.58991 4.01478 1.392 0.178
## Clarity:Body -4.40419 3.12289 -1.410 0.172
## Clarity:Flavor -2.79479 4.74991 -0.588 0.562
## Clarity:Oakiness -5.95396 4.72540 -1.260 0.221
## Aroma:Body 0.55192 0.47265 1.168 0.255
## Aroma:Flavor -0.21139 0.33799 -0.625 0.538
## Aroma:Oakiness 0.20448 0.68185 0.300 0.767
## Body:Flavor -0.18409 0.40954 -0.450 0.657
## Body:Oakiness -0.04582 0.65142 -0.070 0.945
## Flavor:Oakiness -0.33247 0.61154 -0.544 0.592
##
## Residual standard error: 1.159 on 22 degrees of freedom
## Multiple R-squared: 0.809, Adjusted R-squared: 0.6788
## F-statistic: 6.212 on 15 and 22 DF, p-value: 7.129e-05
##-----------------------------------------------------------------------------
## Informações da sessão.
Sys.time()
## [1] "2014-12-11 21:01:39 BRST"
sessionInfo()
## R version 3.1.2 (2014-10-31)
## Platform: i686-pc-linux-gnu (32-bit)
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C LC_TIME=pt_BR.UTF-8
## [4] LC_COLLATE=en_US.UTF-8 LC_MONETARY=pt_BR.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=pt_BR.UTF-8 LC_NAME=C LC_ADDRESS=C
## [10] LC_TELEPHONE=C LC_MEASUREMENT=pt_BR.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] tcltk methods splines grid stats graphics grDevices utils
## [9] datasets base
##
## other attached packages:
## [1] wzRfun_0.4 asbio_1.1-1 multcomp_1.3-7 TH.data_1.0-5
## [5] mvtnorm_1.0-1 doBy_4.5-12 survival_2.37-7 reshape_0.8.5
## [9] plyr_1.8.1 alr3_2.0.5 car_2.0-22 gridExtra_0.9.1
## [13] latticeExtra_0.6-26 RColorBrewer_1.0-5 lattice_0.20-29 rmarkdown_0.3.3
## [17] knitr_1.8
##
## loaded via a namespace (and not attached):
## [1] deSolve_1.11 digest_0.6.4 evaluate_0.5.5 formatR_1.0
## [5] htmltools_0.2.6 MASS_7.3-35 Matrix_1.1-4 multcompView_0.1-5
## [9] nnet_7.3-8 pixmap_0.4-11 plotrix_3.5-10 Rcpp_0.11.3
## [13] sandwich_2.3-2 scatterplot3d_0.3-35 stringr_0.6.2 tkrplot_0.0-23
## [17] tools_3.1.2 yaml_2.1.13 zoo_1.7-11