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Estatística Aplicada à Ciência do Solo
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Análise da Produção de Teca em Função de Variáveis do Solo
Análise Exploratória
#-----------------------------------------------------------------------
# Carrega os pacotes necessários.
library(lattice)
library(latticeExtra)
library(plyr)
library(reshape2)
library(car)
library(nFactors)
library(rpart)
library(EACS)#-----------------------------------------------------------------------
# Análise exploratório dos dados.
str(teca_qui)## 'data.frame': 150 obs. of 15 variables:
## $ loc: int 1 1 1 2 2 2 3 3 3 4 ...
## $ cam: Factor w/ 3 levels "[0, 5)","[5, 40)",..: 1 2 3 1 2 3 1 2 3 1 ...
## $ ph : num 6.8 6.7 6.7 4.7 4.7 4.9 7.6 6.8 6.9 6.6 ...
## $ p : num 22.51 0.83 0.01 3.89 0.69 ...
## $ k : num 72.24 13.42 7.23 48.13 12.34 ...
## $ ca : num 8.27 2.91 2.33 0.97 0.76 0.21 7.63 3.22 2.22 5.54 ...
## $ mg : num 1.7 1.77 0.51 0.16 0.14 0 1.53 1.31 1.09 1.77 ...
## $ al : num 0 0 0 0.3 0.6 0.6 0 0 0 0 ...
## $ ctc: num 12.47 6.57 4.52 5.3 4.17 ...
## $ sat: num 81.4 71.7 63.2 23.7 22.4 ...
## $ mo : num 72.2 25.6 9.7 34.4 8.7 9.7 50.4 15.6 9.5 50.2 ...
## $ arg: num 184 215 286 232 213 ...
## $ are: num 770 750 674 741 775 ...
## $ cas: num 1.8 2.2 3.7 0.4 1.1 1.7 8.4 19 14.3 5.5 ...
## $ acc: num 770 750 676 741 775 ...str(teca_crapar)## 'data.frame': 144 obs. of 10 variables:
## $ loc: int 1 1 1 2 2 2 3 3 3 4 ...
## $ cam: Factor w/ 3 levels "[0, 5)","[5, 40)",..: 1 2 3 1 2 3 1 2 3 1 ...
## $ Ur : num 0.223 0.178 0.188 0.131 0.165 ...
## $ Us : num 0.648 0.58 0.647 0.697 0.573 ...
## $ alp: num -0.724 -0.833 -0.722 -0.548 -0.547 ...
## $ n : num 3.59 3.11 3.25 4.01 2.92 ...
## $ cad: num 0.227 0.217 0.247 0.3 0.222 ...
## $ Ui : num 0.45 0.396 0.435 0.431 0.387 ...
## $ I : num 0.815 0.957 0.835 0.62 0.691 ...
## $ S : num -0.341 -0.273 -0.329 -0.515 -0.257 ...
## - attr(*, "na.action")=Class 'omit' Named int 88
## .. ..- attr(*, "names")= chr "40.[5, 40)"str(teca_arv)## 'data.frame': 50 obs. of 5 variables:
## $ loc : int 1 2 3 4 5 6 7 8 9 10 ...
## $ alt : num 12.9 11.9 22.5 20.4 15.6 ...
## $ dap : num 0.194 0.182 0.348 0.304 0.227 ...
## $ vol : num 0.2707 0.0963 0.4974 0.4074 0.2165 ...
## $ prod: num 68.2 24.3 125.3 102.7 54.6 ...# Juntar as inforções químicas, físico-hídricas e de produção.
db <- merge(teca_qui, teca_crapar, all = TRUE)
str(db)## 'data.frame': 150 obs. of 23 variables:
## $ loc: int 1 1 1 2 2 2 3 3 3 4 ...
## $ cam: Factor w/ 3 levels "[0, 5)","[5, 40)",..: 1 2 3 1 2 3 1 2 3 1 ...
## $ ph : num 6.8 6.7 6.7 4.7 4.7 4.9 7.6 6.8 6.9 6.6 ...
## $ p : num 22.51 0.83 0.01 3.89 0.69 ...
## $ k : num 72.24 13.42 7.23 48.13 12.34 ...
## $ ca : num 8.27 2.91 2.33 0.97 0.76 0.21 7.63 3.22 2.22 5.54 ...
## $ mg : num 1.7 1.77 0.51 0.16 0.14 0 1.53 1.31 1.09 1.77 ...
## $ al : num 0 0 0 0.3 0.6 0.6 0 0 0 0 ...
## $ ctc: num 12.47 6.57 4.52 5.3 4.17 ...
## $ sat: num 81.4 71.7 63.2 23.7 22.4 ...
## $ mo : num 72.2 25.6 9.7 34.4 8.7 9.7 50.4 15.6 9.5 50.2 ...
## $ arg: num 184 215 286 232 213 ...
## $ are: num 770 750 674 741 775 ...
## $ cas: num 1.8 2.2 3.7 0.4 1.1 1.7 8.4 19 14.3 5.5 ...
## $ acc: num 770 750 676 741 775 ...
## $ Ur : num 0.223 0.178 0.188 0.131 0.165 ...
## $ Us : num 0.648 0.58 0.647 0.697 0.573 ...
## $ alp: num -0.724 -0.833 -0.722 -0.548 -0.547 ...
## $ n : num 3.59 3.11 3.25 4.01 2.92 ...
## $ cad: num 0.227 0.217 0.247 0.3 0.222 ...
## $ Ui : num 0.45 0.396 0.435 0.431 0.387 ...
## $ I : num 0.815 0.957 0.835 0.62 0.691 ...
## $ S : num -0.341 -0.273 -0.329 -0.515 -0.257 ...# Todos os locais tem pelo menos um registro. Usar o registro que
# estiver disponível na ordem de prioridade das camadas: II, III e I.
xtabs(~loc, data = na.omit(db))## loc
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3
## 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
## 3 3 3 3 3 3 3 3 2 3 3 2 2 3 3 3 3 3 3 1 3 3
## 45 46 47 48 49 50
## 3 3 3 3 3 3# Número de missings por variável separado por camada.
by(data = is.na(db[, -(1:2)]), INDICES = db$cam, FUN = colSums)## INDICES: [0, 5)
## ph p k ca mg al ctc sat mo arg are cas acc Ur Us alp n
## 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
## cad Ui I S
## 1 1 1 1
## ---------------------------------------------------
## INDICES: [5, 40)
## ph p k ca mg al ctc sat mo arg are cas acc Ur Us alp n
## 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3
## cad Ui I S
## 3 3 3 3
## ---------------------------------------------------
## INDICES: [40, 80)
## ph p k ca mg al ctc sat mo arg are cas acc Ur Us alp n
## 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2
## cad Ui I S
## 2 2 2 2#-----------------------------------------------------------------------
# Fazendo a seleção de um valor para cada local.
# Será mantido os valores na camada II de cada local. Caso a camada II
# esteja incompleta, será usada a camada III e por fim a camada I.
# Remove linhas com registros ausentes.
dc <- na.omit(db)
attr(dc, "na.action") <- NULL
# Reordena os níveis (para usar na seleção).
dc$cam <- factor(dc$cam, levels = levels(db$cam)[c(2, 3, 1)])
# Reordena.
dc <- arrange(dc, loc, cam)
# Pega o primeiro registro de cada local.
dd <- do.call(rbind,
by(dc, INDICES = dc$loc,
FUN = function(x) {
x[1, ]}
))
str(dd)## 'data.frame': 50 obs. of 23 variables:
## $ loc: int 1 2 3 4 5 6 7 8 9 10 ...
## $ cam: Factor w/ 3 levels "[5, 40)","[40, 80)",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ ph : num 6.7 4.7 6.8 6.2 5.1 5.2 5.1 6.4 4.9 5 ...
## $ p : num 0.83 0.69 1.66 1.17 0.49 0.69 0.21 1.14 1.36 1.04 ...
## $ k : num 13.4 12.3 25.8 11.1 22.2 ...
## $ ca : num 2.91 0.76 3.22 2.25 0.93 0.7 0.24 2.39 1.15 0.72 ...
## $ mg : num 1.77 0.14 1.31 0.21 0.52 0.36 0.14 0.99 0.05 0.37 ...
## $ al : num 0 0.6 0 0 0.2 0.5 0.6 0 0.3 0.6 ...
## $ ctc: num 6.57 4.17 6.26 4.56 4.1 5.16 5.02 5.28 4.86 3.86 ...
## $ sat: num 71.7 22.4 73.4 54.5 36.8 ...
## $ mo : num 25.6 8.7 15.6 10 11.2 5.8 8.5 11.8 6.3 6.9 ...
## $ arg: num 215 213 234 169 304 ...
## $ are: num 750 775 698 788 665 ...
## $ cas: num 2.2 1.1 19 3.1 7.2 23.4 2.9 2.1 2.5 1.6 ...
## $ acc: num 750 775 704 788 667 ...
## $ Ur : num 0.1785 0.1649 0.3699 0.0372 0.1902 ...
## $ Us : num 0.58 0.573 0.659 0.521 0.608 ...
## $ alp: num -0.833 -0.547 -0.552 -0.478 -0.671 ...
## $ n : num 3.11 2.92 1.96 2.21 2.93 ...
## $ cad: num 0.217 0.222 0.168 0.274 0.228 ...
## $ Ui : num 0.396 0.387 0.538 0.311 0.418 ...
## $ I : num 0.957 0.691 0.918 0.751 0.813 ...
## $ S : num -0.273 -0.257 -0.108 -0.214 -0.265 ...# Deixar o S positivo para facilitar a interpretação.
dd$S <- -1 * dd$SAnálise Fatorial
Na análise fatorial, a variável acc (areia + cascalho + calhau) não foi usada por ter alta correlação, por construção, com areia e cascalho. A variável da curva de água do solo Ur foi deixada de fora para ser usado o cad = Ui - Ur.
#-----------------------------------------------------------------------
# Em cada linha o grupo de variáveis químicas, físicas e hídricas.
j <- c("ph", "p", "k", "ca", "mg", "al", "ctc", "sat", "mo",
"arg", "are", "cas",
"alp", "n", "I", "Us", "Ui", "S", "cad")
X <- dd[, j]
fit0 <- factanal(X,
factors = 4,
# covmat = cov2cor(var(X)),
# covmat = var(X),
rotation = "varimax")
print(fit0, digits = 2, cutoff = 0.5, sort = TRUE)##
## Call:
## factanal(x = X, factors = 4, rotation = "varimax")
##
## Uniquenesses:
## ph p k ca mg al ctc sat mo arg are cas alp
## 0.12 0.72 0.72 0.19 0.37 0.36 0.20 0.02 0.55 0.08 0.04 0.89 0.00
## n I Us Ui S cad
## 0.11 0.08 0.41 0.54 0.00 0.19
##
## Loadings:
## Factor1 Factor2 Factor3 Factor4
## ph 0.93
## k 0.52
## ca 0.79
## mg 0.66
## al -0.80
## ctc 0.69 0.56
## sat 0.98
## mo 0.63
## alp 0.95
## I -0.96
## Us 0.74
## Ui 0.51
## cad 0.89
## arg 0.90
## are -0.92
## n 0.89
## S 0.99
## p
## cas
##
## Factor1 Factor2 Factor3 Factor4
## SS loadings 4.99 3.65 2.57 2.19
## Proportion Var 0.26 0.19 0.14 0.12
## Cumulative Var 0.26 0.45 0.59 0.71
##
## Test of the hypothesis that 4 factors are sufficient.
## The chi square statistic is 535.95 on 101 degrees of freedom.
## The p-value is 1.84e-60Com 4 fatores chegou-se a uma explicação superior à 70% da variância total. Baseado nos carregamentos de valor superior absoluto maior que 0.5, foi possível interpretar cada um dos fatores em termos de índices:
- Índice de cátions do solo. Contraste entre cátions desejáveis ou variáveis que favorecem/resultam de cátions vs. o alumínio (al) do solo.
- Índice de disponibilidade de água. Contraste entre parâmetros que aumentam a disponibilidade de água (alp, Us, Ui, cad) vs. que diminuem (I).
- Índice de porosidade do solo. Contraste entre conteúdo de argila vs. areia. O aumento da argila aumenta a microporosidade que é responsável por maior armazenamento de água.
- Índice de inclinação da curva de retenção de água. Função linear do parâmetro S (slope) e n (parâmetro de forma) que são ambos relacionados à inclinação da curva de retenção e da CRA.
Algumas variáveis não apresentaram carregamento alto e por isso alta singularidade (uniquenesses), sendo por isso variáveis pouco explicadas pelos fatores. Essas variáveis serão removidas e o ajuste será refeito.
#-----------------------------------------------------------------------
# Removendo variáveis de pouca importância ou sem explicação (com
# alta singularidade/unicidade).
# str(fit0)
# Variáveis abandonadas.
j[fit0$uniquenesses > 0.7]## [1] "p" "k" "cas"# Reajuste.
X <- dd[, j[fit0$uniquenesses <= 0.7]]
fit1 <- factanal(X,
factors = 4,
rotation = "varimax",
scores = "regression")
colnames(fit1$loadings) <- c("cation", "agua", "poros", "inclin")
print(fit1, digits = 2, cutoff = 0.5, sort = TRUE)##
## Call:
## factanal(x = X, factors = 4, scores = "regression", rotation = "varimax")
##
## Uniquenesses:
## ph ca mg al ctc sat mo arg are alp n I Us
## 0.13 0.20 0.38 0.35 0.21 0.01 0.56 0.08 0.04 0.00 0.11 0.08 0.41
## Ui S cad
## 0.54 0.00 0.19
##
## Loadings:
## cation agua poros inclin
## ph 0.93
## ca 0.79
## mg 0.66
## al -0.80
## ctc 0.68 0.57
## sat 0.99
## mo 0.62
## alp 0.95
## I -0.96
## Us 0.74
## Ui 0.52
## cad 0.89
## arg 0.89
## are -0.92
## n 0.89
## S 0.99
##
## cation agua poros inclin
## SS loadings 4.56 3.65 2.38 2.12
## Proportion Var 0.28 0.23 0.15 0.13
## Cumulative Var 0.28 0.51 0.66 0.79
##
## Test of the hypothesis that 4 factors are sufficient.
## The chi square statistic is 505.08 on 62 degrees of freedom.
## The p-value is 1.06e-70# Singularidade das variáveis.
sort(fit1$uniquenesses)## alp S sat are I arg
## 0.00500000 0.00500000 0.01005537 0.04343385 0.07668692 0.08386209
## n ph cad ca ctc al
## 0.10625758 0.12559882 0.19047046 0.19832549 0.20827617 0.34872158
## mg Us Ui mo
## 0.37793727 0.40720590 0.53886744 0.56185743# Carregamentos do fator 1 contra 2.
load <- fit1$loadings[, 1:2]
plot(load, type = "n",
xlab = "Cátions", ylab = "Água")
abline(v = 0, h = 0, lty = 2)
text(load, labels = names(X), cex = 0.8)
# Pares de gráficos dos escores.
sc <- fit1$scores
pairs(sc)
Os resultados com o abandono das variáveis de alta singularidade não sofreu modificações substanciais. A interpretação dos fatores foi mantida. Com estes 4 fatores foi obtido uma explicação de 80% da variância total.
Apenas por precaução, foi confirmado por simulação que o número de fatores é de fato 4.
#-----------------------------------------------------------------------
# Determinação por simulação do número de fatores.
ev <- eigen(cor(X))
ap <- parallel(subject = nrow(X),
var = ncol(X),
rep = 100,
cent = 0.05)
nS <- nScree(x = ev$values, aparallel = ap$eigen$qevpea)
plotnScree(nS)
Ajuste de Modelo de Regressão com os Escores
Os escores ou índices definidos pela análise fatorial serão usados como variáveis explicativas da produção de madeira.
#-----------------------------------------------------------------------
# Justar os escores com a variável de produção de madeira.
de <- merge(cbind(loc = dd[, "loc"], as.data.frame(sc)),
teca_arv[, c("loc", "prod")])
names(de)[2:5] <- colnames(fit1$loadings)
str(de)## 'data.frame': 50 obs. of 6 variables:
## $ loc : int 1 2 3 4 5 6 7 8 9 10 ...
## $ cation: num 1.031 -1.421 1.214 0.281 -0.813 ...
## $ agua : num -0.6673 -0.0692 -0.7716 -0.125 -0.3489 ...
## $ poros : num -1.688 -1.17 -1.327 -2.005 -0.363 ...
## $ inclin: num 0.865 0.935 -1.304 0.225 0.887 ...
## $ prod : num 68.2 24.3 125.3 102.7 54.6 ...xyplot(sqrt(prod) ~ cation + agua + poros + inclin,
data = de,
outer = TRUE,
type = c("p", "r"),
as.table = TRUE,
scales = list(x = list(relation = "free")),
xlab = "Escores da análise fatorial",
ylab = "Produção de madeira")
#-----------------------------------------------------------------------
# Ajuste do modelo de regressão.
# ATTENTION: Existe um maior atendimento dos pressupostos usando sqrt do
# que a variável original.
# m0 <- lm(prod ~ . - loc, data = de)
m0 <- lm(sqrt(prod) ~ . - loc, data = de)
summary(m0)##
## Call:
## lm(formula = sqrt(prod) ~ . - loc, data = de)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5113 -1.0517 -0.1062 1.1450 3.2344
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.7856 0.1913 45.931 < 2e-16 ***
## cation 1.7240 0.1942 8.878 1.89e-11 ***
## agua -0.3954 0.1938 -2.040 0.0472 *
## poros 0.4150 0.1967 2.111 0.0404 *
## inclin 0.2030 0.1938 1.048 0.3004
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.353 on 45 degrees of freedom
## Multiple R-squared: 0.6636, Adjusted R-squared: 0.6337
## F-statistic: 22.19 on 4 and 45 DF, p-value: 3.598e-10par(mfrow = c(2, 2))
plot(m0)
layout(1)
# MASS::boxcox(m0)
# abline(v = c(0.5, 1), col = 2)
residualPlots(m0)
## Test stat Pr(>|t|)
## cation -2.456 0.018
## agua -1.025 0.311
## poros -0.003 0.998
## inclin -0.200 0.842
## Tukey test -3.110 0.002im <- influence.measures(m0)
summary(im)## Potentially influential observations of
## lm(formula = sqrt(prod) ~ . - loc, data = de) :
##
## dfb.1_ dfb.catn dfb.agua dfb.pors dfb.incl dffit cov.r
## 12 -0.05 -0.01 -0.03 0.02 -0.18 -0.19 1.59_*
## 36 -0.19 0.05 -0.94 0.28 0.10 -1.01_* 2.47_*
## 38 0.41 -0.44 -0.18 0.54 -0.47 0.95 0.58_*
## 43 -0.21 -0.30 -0.45 -0.41 0.38 -0.81 1.36_*
## 48 -0.13 -0.12 -0.01 -0.31 -0.25 -0.44 1.34_*
## 49 0.05 0.02 0.01 0.09 0.11 0.15 1.37_*
## cook.d hat
## 12 0.01 0.30_*
## 36 0.20 0.58_*
## 38 0.16 0.11
## 43 0.13 0.30_*
## 48 0.04 0.22
## 49 0.00 0.19# Remover observação com maior alavancagem.
which(im$is.inf[, "hat"])## 12 36 43
## 12 36 43r <- which.max(im$infmat[, "hat"])
m0 <- lm(prod ~ . - loc, data = de[-r, ])
# m0 <- lm(sqrt(prod) ~ . - loc, data = de[, 35])
summary(m0)##
## Call:
## lm(formula = prod ~ . - loc, data = de[-r, ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -42.427 -14.990 -4.932 18.135 55.991
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 82.492 3.381 24.399 < 2e-16 ***
## cation 28.434 3.359 8.465 8.76e-11 ***
## agua -3.865 4.992 -0.774 0.4429
## poros 6.216 3.575 1.739 0.0891 .
## inclin 3.655 3.366 1.086 0.2835
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.35 on 44 degrees of freedom
## Multiple R-squared: 0.6312, Adjusted R-squared: 0.5976
## F-statistic: 18.82 on 4 and 44 DF, p-value: 4.396e-09par(mfrow = c(2, 2))
plot(m0)
layout(1)
residualPlots(m0)
## Test stat Pr(>|t|)
## cation -1.416 0.164
## agua -0.826 0.413
## poros 0.167 0.868
## inclin -0.625 0.536
## Tukey test -1.837 0.066m1 <- update(m0, . ~ cation)
summary(m1)##
## Call:
## lm(formula = prod ~ cation, data = de[-r, ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.42 -16.23 -1.68 19.08 60.84
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 83.110 3.378 24.61 < 2e-16 ***
## cation 28.300 3.397 8.33 8.23e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.64 on 47 degrees of freedom
## Multiple R-squared: 0.5962, Adjusted R-squared: 0.5876
## F-statistic: 69.39 on 1 and 47 DF, p-value: 8.233e-11anova(m0, m1)## Analysis of Variance Table
##
## Model 1: prod ~ (loc + cation + agua + poros + inclin) - loc
## Model 2: prod ~ cation
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 44 23998
## 2 47 26274 -3 -2276.5 1.3913 0.258# Carregamento das variáveis no fator 1.
cbind(fit1$loadings[, "cation"])## [,1]
## ph 0.925604336
## ca 0.788356898
## mg 0.659070902
## al -0.804148698
## ctc 0.678101756
## sat 0.986348560
## mo 0.619513624
## arg 0.255178443
## are -0.295877997
## alp 0.005452939
## n 0.086897134
## I -0.007830131
## Us -0.043650910
## Ui -0.079906747
## S 0.116752464
## cad 0.038783719Ajuste de Modelo de Regressão Múltipla com a Variáveis de Solo
#-----------------------------------------------------------------------
de <- merge(subset(dd, select = -cam),
teca_arv[, c("loc", "prod")])
de <- subset(de, select = -loc)
str(de)## 'data.frame': 50 obs. of 22 variables:
## $ ph : num 6.7 4.7 6.8 6.2 5.1 5.2 5.1 6.4 4.9 5 ...
## $ p : num 0.83 0.69 1.66 1.17 0.49 0.69 0.21 1.14 1.36 1.04 ...
## $ k : num 13.4 12.3 25.8 11.1 22.2 ...
## $ ca : num 2.91 0.76 3.22 2.25 0.93 0.7 0.24 2.39 1.15 0.72 ...
## $ mg : num 1.77 0.14 1.31 0.21 0.52 0.36 0.14 0.99 0.05 0.37 ...
## $ al : num 0 0.6 0 0 0.2 0.5 0.6 0 0.3 0.6 ...
## $ ctc : num 6.57 4.17 6.26 4.56 4.1 5.16 5.02 5.28 4.86 3.86 ...
## $ sat : num 71.7 22.4 73.4 54.5 36.8 ...
## $ mo : num 25.6 8.7 15.6 10 11.2 5.8 8.5 11.8 6.3 6.9 ...
## $ arg : num 215 213 234 169 304 ...
## $ are : num 750 775 698 788 665 ...
## $ cas : num 2.2 1.1 19 3.1 7.2 23.4 2.9 2.1 2.5 1.6 ...
## $ acc : num 750 775 704 788 667 ...
## $ Ur : num 0.1785 0.1649 0.3699 0.0372 0.1902 ...
## $ Us : num 0.58 0.573 0.659 0.521 0.608 ...
## $ alp : num -0.833 -0.547 -0.552 -0.478 -0.671 ...
## $ n : num 3.11 2.92 1.96 2.21 2.93 ...
## $ cad : num 0.217 0.222 0.168 0.274 0.228 ...
## $ Ui : num 0.396 0.387 0.538 0.311 0.418 ...
## $ I : num 0.957 0.691 0.918 0.751 0.813 ...
## $ S : num 0.273 0.257 0.108 0.214 0.265 ...
## $ prod: num 68.2 24.3 125.3 102.7 54.6 ...# Número de observações.
nrow(de)## [1] 50# m0 <- lm(prod ~ ., data = de)
m0 <- lm(sqrt(prod) ~ ., data = de)
anova(m0)## Analysis of Variance Table
##
## Response: sqrt(prod)
## Df Sum Sq Mean Sq F value Pr(>F)
## ph 1 124.347 124.347 68.1921 4.191e-09 ***
## p 1 0.001 0.001 0.0007 0.979655
## k 1 0.434 0.434 0.2382 0.629146
## ca 1 1.488 1.488 0.8162 0.373731
## mg 1 8.249 8.249 4.5236 0.042062 *
## al 1 17.590 17.590 9.6461 0.004216 **
## ctc 1 0.082 0.082 0.0447 0.833953
## sat 1 8.148 8.148 4.4686 0.043244 *
## mo 1 0.258 0.258 0.1413 0.709736
## arg 1 6.899 6.899 3.7835 0.061515 .
## are 1 0.527 0.527 0.2891 0.594907
## cas 1 5.846 5.846 3.2062 0.083807 .
## acc 1 5.766 5.766 3.1623 0.085845 .
## Ur 1 3.191 3.191 1.7501 0.196206
## Us 1 7.614 7.614 4.1757 0.050186 .
## alp 1 0.053 0.053 0.0293 0.865219
## n 1 0.066 0.066 0.0365 0.849914
## cad 1 0.307 0.307 0.1683 0.684638
## I 1 0.356 0.356 0.1953 0.661836
## S 1 0.616 0.616 0.3381 0.565448
## Residuals 29 52.881 1.823
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# im <- influence.measures(m0)
# summary(im)
# residualPlots(m0)
# AIC.
m1 <- step(m0)## Start: AIC=44.8
## sqrt(prod) ~ ph + p + k + ca + mg + al + ctc + sat + mo + arg +
## are + cas + acc + Ur + Us + alp + n + cad + Ui + I + S
##
##
## Step: AIC=44.8
## sqrt(prod) ~ ph + p + k + ca + mg + al + ctc + sat + mo + arg +
## are + cas + acc + Ur + Us + alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - p 1 0.0427 52.924 42.842
## - al 1 0.0872 52.968 42.884
## - ph 1 0.2033 53.085 42.993
## - are 1 0.2328 53.114 43.021
## - mo 1 0.2969 53.178 43.081
## - acc 1 0.3369 53.218 43.119
## - mg 1 0.5174 53.399 43.288
## - S 1 0.6164 53.498 43.381
## - cas 1 0.6218 53.503 43.386
## - n 1 0.6336 53.515 43.397
## - ctc 1 0.7480 53.629 43.504
## - k 1 0.7911 53.672 43.544
## - Us 1 0.8616 53.743 43.609
## - Ur 1 0.8758 53.757 43.623
## - I 1 0.9475 53.829 43.689
## - alp 1 1.0035 53.885 43.741
## - cad 1 1.0595 53.941 43.793
## - arg 1 1.1153 53.997 43.845
## <none> 52.881 44.801
## - ca 1 2.4488 55.330 45.065
## - sat 1 8.7794 61.661 50.481
##
## Step: AIC=42.84
## sqrt(prod) ~ ph + k + ca + mg + al + ctc + sat + mo + arg + are +
## cas + acc + Ur + Us + alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - al 1 0.0865 53.010 40.923
## - are 1 0.1914 53.115 41.022
## - ph 1 0.2304 53.154 41.059
## - acc 1 0.2944 53.218 41.119
## - mo 1 0.3215 53.245 41.144
## - mg 1 0.5317 53.456 41.341
## - cas 1 0.5892 53.513 41.395
## - S 1 0.6139 53.538 41.418
## - n 1 0.6379 53.562 41.441
## - ctc 1 0.7715 53.695 41.565
## - k 1 0.7922 53.716 41.584
## - Us 1 0.8742 53.798 41.661
## - Ur 1 0.8899 53.814 41.675
## - I 1 0.9768 53.901 41.756
## - alp 1 1.0298 53.954 41.805
## - cad 1 1.0789 54.003 41.851
## - arg 1 1.0977 54.022 41.868
## <none> 52.924 42.842
## - ca 1 2.4456 55.369 43.100
## - sat 1 8.8935 61.817 48.608
##
## Step: AIC=40.92
## sqrt(prod) ~ ph + k + ca + mg + ctc + sat + mo + arg + are +
## cas + acc + Ur + Us + alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - are 1 0.2616 53.272 39.169
## - ph 1 0.2923 53.303 39.198
## - acc 1 0.3164 53.327 39.221
## - mo 1 0.3199 53.330 39.224
## - cas 1 0.6116 53.622 39.497
## - k 1 0.9357 53.946 39.798
## - mg 1 0.9716 53.982 39.831
## - arg 1 1.0188 54.029 39.875
## - S 1 1.1091 54.119 39.959
## - n 1 1.1852 54.196 40.029
## - ctc 1 1.2844 54.295 40.120
## - Us 1 1.4264 54.437 40.251
## - Ur 1 1.4465 54.457 40.269
## - I 1 1.5461 54.557 40.361
## - alp 1 1.6041 54.614 40.414
## - cad 1 1.6910 54.701 40.493
## <none> 53.010 40.923
## - ca 1 4.1608 57.171 42.701
## - sat 1 26.5823 79.593 59.245
##
## Step: AIC=39.17
## sqrt(prod) ~ ph + k + ca + mg + ctc + sat + mo + arg + cas +
## acc + Ur + Us + alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - acc 1 0.0560 53.328 37.222
## - ph 1 0.2146 53.487 37.370
## - mo 1 0.2332 53.505 37.388
## - cas 1 0.4941 53.766 37.631
## - k 1 0.8939 54.166 38.001
## - S 1 0.9161 54.188 38.022
## - n 1 0.9801 54.252 38.081
## - mg 1 1.0533 54.325 38.148
## - Us 1 1.2436 54.516 38.323
## - Ur 1 1.2582 54.530 38.337
## - I 1 1.3342 54.606 38.406
## - ctc 1 1.3380 54.610 38.410
## - alp 1 1.3933 54.665 38.460
## - cad 1 1.5018 54.774 38.559
## <none> 53.272 39.169
## - arg 1 2.6118 55.884 39.563
## - ca 1 3.9088 57.181 40.710
## - sat 1 27.9644 81.236 58.267
##
## Step: AIC=37.22
## sqrt(prod) ~ ph + k + ca + mg + ctc + sat + mo + arg + cas +
## Ur + Us + alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - mo 1 0.1962 53.524 35.406
## - ph 1 0.2685 53.596 35.473
## - cas 1 0.9740 54.302 36.127
## - k 1 0.9870 54.315 36.139
## - mg 1 1.0026 54.331 36.153
## - S 1 1.2724 54.600 36.401
## - ctc 1 1.3131 54.641 36.438
## - n 1 1.3539 54.682 36.475
## - Us 1 1.5355 54.863 36.641
## - Ur 1 1.5611 54.889 36.665
## - I 1 1.5947 54.923 36.695
## - alp 1 1.6080 54.936 36.707
## - cad 1 1.7923 55.120 36.875
## <none> 53.328 37.222
## - ca 1 3.9231 57.251 38.771
## - arg 1 6.9778 60.306 41.370
## - sat 1 28.1281 81.456 56.402
##
## Step: AIC=35.41
## sqrt(prod) ~ ph + k + ca + mg + ctc + sat + arg + cas + Ur +
## Us + alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - ph 1 0.3272 53.851 33.710
## - k 1 0.7978 54.322 34.145
## - mg 1 0.9349 54.459 34.271
## - cas 1 1.1881 54.712 34.503
## - ctc 1 1.2617 54.786 34.570
## - S 1 1.3558 54.880 34.656
## - n 1 1.4404 54.965 34.733
## - Us 1 1.6370 55.161 34.912
## - Ur 1 1.6624 55.187 34.935
## - alp 1 1.6899 55.214 34.960
## - I 1 1.6986 55.223 34.968
## - cad 1 1.8906 55.415 35.141
## <none> 53.524 35.406
## - ca 1 3.7273 57.251 36.771
## - arg 1 6.7825 60.307 39.371
## - sat 1 28.0329 81.557 54.464
##
## Step: AIC=33.71
## sqrt(prod) ~ k + ca + mg + ctc + sat + arg + cas + Ur + Us +
## alp + n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - k 1 1.025 54.876 32.653
## - mg 1 1.188 55.040 32.801
## - S 1 1.282 55.133 32.886
## - cas 1 1.372 55.223 32.968
## - n 1 1.393 55.244 32.987
## - ctc 1 1.599 55.450 33.173
## - Us 1 1.607 55.458 33.180
## - Ur 1 1.624 55.476 33.196
## - alp 1 1.654 55.505 33.223
## - I 1 1.660 55.511 33.228
## - cad 1 1.843 55.694 33.392
## <none> 53.851 33.710
## - ca 1 4.422 58.273 35.656
## - arg 1 6.899 60.750 37.737
## - sat 1 33.511 87.362 55.902
##
## Step: AIC=32.65
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + Ur + Us + alp +
## n + cad + I + S
##
## Df Sum of Sq RSS AIC
## - S 1 1.010 55.886 31.565
## - n 1 1.189 56.066 31.725
## - Us 1 1.326 56.203 31.847
## - Ur 1 1.354 56.230 31.871
## - alp 1 1.466 56.342 31.971
## - I 1 1.477 56.353 31.981
## - mg 1 1.477 56.353 31.981
## - ctc 1 1.520 56.397 32.019
## - cad 1 1.558 56.434 32.053
## - cas 1 1.684 56.561 32.164
## <none> 54.876 32.653
## - ca 1 4.258 59.135 34.390
## - arg 1 7.764 62.641 37.270
## - sat 1 32.488 87.364 53.903
##
## Step: AIC=31.56
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + Ur + Us + alp +
## n + cad + I
##
## Df Sum of Sq RSS AIC
## - Us 1 0.3267 56.213 29.856
## - Ur 1 0.3640 56.250 29.889
## - n 1 0.4247 56.311 29.943
## - alp 1 0.4576 56.344 29.972
## - I 1 0.4672 56.353 29.981
## - cad 1 0.5830 56.469 30.084
## - ctc 1 1.1039 56.990 30.543
## - mg 1 1.4222 57.308 30.821
## - cas 1 1.7952 57.681 31.146
## <none> 55.886 31.565
## - ca 1 3.5365 59.423 32.633
## - arg 1 9.6406 65.527 37.522
## - sat 1 31.5991 87.485 51.972
##
## Step: AIC=29.86
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + Ur + alp + n +
## cad + I
##
## Df Sum of Sq RSS AIC
## - Ur 1 0.124 56.337 27.966
## - I 1 0.164 56.377 28.002
## - alp 1 0.167 56.380 28.004
## - n 1 0.328 56.540 28.147
## - ctc 1 1.154 57.367 28.872
## - mg 1 1.436 57.649 29.117
## - cas 1 1.740 57.953 29.381
## <none> 56.213 29.856
## - cad 1 3.033 59.246 30.484
## - ca 1 4.028 60.241 31.316
## - arg 1 14.130 70.343 39.068
## - sat 1 33.669 89.882 51.324
##
## Step: AIC=27.97
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + alp + n + cad +
## I
##
## Df Sum of Sq RSS AIC
## - alp 1 0.074 56.410 26.031
## - I 1 0.091 56.427 26.047
## - n 1 0.267 56.603 26.202
## - ctc 1 1.088 57.424 26.922
## - mg 1 1.402 57.738 27.195
## - cas 1 1.961 58.298 27.677
## <none> 56.337 27.966
## - ca 1 3.908 60.245 29.320
## - cad 1 4.553 60.890 29.852
## - arg 1 14.013 70.350 37.073
## - sat 1 33.547 89.883 49.325
##
## Step: AIC=26.03
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + n + cad + I
##
## Df Sum of Sq RSS AIC
## - I 1 0.017 56.427 24.047
## - n 1 0.245 56.655 24.248
## - ctc 1 1.048 57.458 24.952
## - mg 1 1.368 57.778 25.229
## - cas 1 2.128 58.538 25.883
## <none> 56.410 26.031
## - ca 1 3.854 60.264 27.335
## - cad 1 5.516 61.927 28.696
## - arg 1 14.105 70.515 35.190
## - sat 1 34.665 91.075 47.983
##
## Step: AIC=24.05
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + n + cad
##
## Df Sum of Sq RSS AIC
## - n 1 0.257 56.684 22.273
## - ctc 1 1.108 57.536 23.019
## - mg 1 1.445 57.872 23.311
## - cas 1 2.111 58.538 23.883
## <none> 56.427 24.047
## - ca 1 3.838 60.266 25.337
## - cad 1 14.030 70.458 33.149
## - arg 1 14.619 71.046 33.565
## - sat 1 35.038 91.465 46.197
##
## Step: AIC=22.27
## sqrt(prod) ~ ca + mg + ctc + sat + arg + cas + cad
##
## Df Sum of Sq RSS AIC
## - ctc 1 0.979 57.662 21.129
## - mg 1 1.220 57.904 21.338
## - cas 1 2.079 58.763 22.075
## <none> 56.684 22.273
## - ca 1 3.629 60.313 23.376
## - arg 1 14.581 71.264 31.719
## - cad 1 18.580 75.263 34.449
## - sat 1 35.036 91.720 44.336
##
## Step: AIC=21.13
## sqrt(prod) ~ ca + mg + sat + arg + cas + cad
##
## Df Sum of Sq RSS AIC
## - mg 1 0.242 57.904 19.338
## - cas 1 1.633 59.295 20.525
## <none> 57.662 21.129
## - ca 1 8.811 66.473 26.239
## - arg 1 17.335 74.997 32.272
## - cad 1 22.010 79.672 35.295
## - sat 1 60.789 118.451 55.124
##
## Step: AIC=19.34
## sqrt(prod) ~ ca + sat + arg + cas + cad
##
## Df Sum of Sq RSS AIC
## - cas 1 1.766 59.670 18.840
## <none> 57.904 19.338
## - ca 1 8.606 66.510 24.266
## - arg 1 17.810 75.715 30.747
## - cad 1 21.768 79.672 33.295
## - sat 1 78.082 135.986 60.026
##
## Step: AIC=18.84
## sqrt(prod) ~ ca + sat + arg + cad
##
## Df Sum of Sq RSS AIC
## <none> 59.670 18.840
## - ca 1 8.021 67.691 23.146
## - arg 1 16.387 76.057 28.973
## - cad 1 26.462 86.131 35.193
## - sat 1 77.252 136.921 58.369# Resumo do modelo selecionado.
summary(m1)##
## Call:
## lm(formula = sqrt(prod) ~ ca + sat + arg + cad, data = de)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6481 -0.7063 -0.1226 0.7843 2.7592
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.490565 1.002813 4.478 5.11e-05 ***
## ca -0.339112 0.137882 -2.459 0.01782 *
## sat 0.105371 0.013805 7.633 1.19e-09 ***
## arg 0.008467 0.002409 3.515 0.00101 **
## cad -12.887466 2.884897 -4.467 5.29e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.152 on 45 degrees of freedom
## Multiple R-squared: 0.7562, Adjusted R-squared: 0.7345
## F-statistic: 34.89 on 4 and 45 DF, p-value: 2.917e-13# residualPlots(m1)
anova(m1, m0)## Analysis of Variance Table
##
## Model 1: sqrt(prod) ~ ca + sat + arg + cad
## Model 2: sqrt(prod) ~ ph + p + k + ca + mg + al + ctc + sat + mo + arg +
## are + cas + acc + Ur + Us + alp + n + cad + Ui + I + S
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 45 59.670
## 2 29 52.881 16 6.7886 0.2327 0.9983im <- influence.measures(m1)
summary(im)## Potentially influential observations of
## lm(formula = sqrt(prod) ~ ca + sat + arg + cad, data = de) :
##
## dfb.1_ dfb.ca dfb.sat dfb.arg dfb.cad dffit cov.r cook.d
## 4 0.35 0.23 0.01 -0.92 0.51 1.08_* 0.58_* 0.20
## 19 0.41 0.83 -0.87 -0.50 0.32 -1.03_* 0.62_* 0.19
## 21 0.08 0.13 -0.10 0.01 -0.17 -0.42 0.61_* 0.03
## 31 0.06 0.05 -0.04 -0.06 -0.01 -0.07 1.39_* 0.00
## 36 0.16 0.05 -0.05 0.11 -0.35 -0.37 2.24_* 0.03
## 43 -0.10 0.25 -0.17 -0.01 0.21 0.41 2.11_* 0.03
## hat
## 4 0.13
## 19 0.13
## 21 0.03
## 31 0.20
## 36 0.51_*
## 43 0.48_*par(mfrow = c(2, 2))
plot(m1)
layout(1)Árvore de Regressão
#-----------------------------------------------------------------------
# Árvore de regressão.
layout(1)
ar <- rpart(prod ~ ., data = de, method = "anova")
summary(ar)## Call:
## rpart(formula = prod ~ ., data = de, method = "anova")
## n= 50
##
## CP nsplit rel error xerror xstd
## 1 0.57868463 0 1.0000000 1.0363413 0.1730044
## 2 0.10430085 1 0.4213154 0.6039645 0.1362822
## 3 0.07274426 2 0.3170145 0.5898582 0.1304633
## 4 0.01000000 3 0.2442703 0.4832313 0.1032574
##
## Variable importance
## sat ph ca al mg ctc mo Ui
## 20 17 17 16 13 13 3 1
##
## Node number 1: 50 observations, complexity param=0.5786846
## mean=82.08052, MSE=1370.545
## left son=2 (23 obs) right son=3 (27 obs)
## Primary splits:
## sat < 53.17 to the left, improve=0.5786846, (0 missing)
## ph < 5.65 to the left, improve=0.5243626, (0 missing)
## mg < 0.635 to the left, improve=0.5209719, (0 missing)
## al < 0.05 to the right, improve=0.5078733, (0 missing)
## ca < 2.12 to the left, improve=0.4645719, (0 missing)
## Surrogate splits:
## ph < 5.65 to the left, agree=0.96, adj=0.913, (0 split)
## al < 0.05 to the right, agree=0.94, adj=0.870, (0 split)
## ca < 2.245 to the left, agree=0.92, adj=0.826, (0 split)
## mg < 0.635 to the left, agree=0.90, adj=0.783, (0 split)
## ctc < 5.61 to the left, agree=0.86, adj=0.696, (0 split)
##
## Node number 2: 23 observations, complexity param=0.1043009
## mean=51.56746, MSE=658.0742
## left son=4 (10 obs) right son=5 (13 obs)
## Primary splits:
## mo < 9.15 to the left, improve=0.4722245, (0 missing)
## ca < 0.84 to the left, improve=0.3706763, (0 missing)
## sat < 29.435 to the left, improve=0.3378541, (0 missing)
## acc < 641.93 to the right, improve=0.2859104, (0 missing)
## cad < 0.2013866 to the right, improve=0.2581536, (0 missing)
## Surrogate splits:
## sat < 30.435 to the left, agree=0.913, adj=0.8, (0 split)
## ca < 0.84 to the left, agree=0.870, adj=0.7, (0 split)
## al < 0.25 to the right, agree=0.826, adj=0.6, (0 split)
## ph < 5.25 to the left, agree=0.739, adj=0.4, (0 split)
## ctc < 5.2 to the left, agree=0.739, adj=0.4, (0 split)
##
## Node number 3: 27 observations, complexity param=0.07274426
## mean=108.0731, MSE=508.7363
## left son=6 (17 obs) right son=7 (10 obs)
## Primary splits:
## sat < 72.085 to the left, improve=0.3629156, (0 missing)
## cas < 39.85 to the left, improve=0.2433149, (0 missing)
## ca < 3.135 to the left, improve=0.1588320, (0 missing)
## mg < 1.94 to the left, improve=0.1355789, (0 missing)
## k < 33.61 to the left, improve=0.1044933, (0 missing)
## Surrogate splits:
## ph < 6.75 to the left, agree=0.852, adj=0.6, (0 split)
## ca < 6.055 to the left, agree=0.852, adj=0.6, (0 split)
## ctc < 11.025 to the left, agree=0.815, adj=0.5, (0 split)
## mg < 2.66 to the left, agree=0.778, adj=0.4, (0 split)
## Ui < 0.5205158 to the left, agree=0.741, adj=0.3, (0 split)
##
## Node number 4: 10 observations
## mean=31.46805, MSE=151.4165
##
## Node number 5: 13 observations
## mean=67.02855, MSE=498.0069
##
## Node number 6: 17 observations
## mean=97.65175, MSE=159.0397
##
## Node number 7: 10 observations
## mean=125.7894, MSE=604.7241rsq.rpart(ar)##
## Regression tree:
## rpart(formula = prod ~ ., data = de, method = "anova")
##
## Variables actually used in tree construction:
## [1] mo sat
##
## Root node error: 68527/50 = 1370.5
##
## n= 50
##
## CP nsplit rel error xerror xstd
## 1 0.578685 0 1.00000 1.03634 0.17300
## 2 0.104301 1 0.42132 0.60396 0.13628
## 3 0.072744 2 0.31701 0.58986 0.13046
## 4 0.010000 3 0.24427 0.48323 0.10326
plot(ar, uniform = TRUE, main = "Árvore de regressão para produção")
text(ar, use.n = TRUE, all = TRUE, cex = 0.8)
pred <- factor(predict(ar))
plot(sat ~ mo,
data = de,
col = as.integer(pred),
pch = NA)
with(de, text(y = sat,
x = mo,
labels = rownames(de),
col = as.integer(pred)))
Informações da Sessão
## Atualizado em 23 de novembro de 2016.
##
## R version 3.3.2 (2016-10-31)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 16.04.1 LTS
##
## locale:
## [1] LC_CTYPE=pt_BR.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=pt_BR.UTF-8 LC_COLLATE=en_US.UTF-8
## [5] LC_MONETARY=pt_BR.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=pt_BR.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=pt_BR.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods
## [7] base
##
## other attached packages:
## [1] EACS_0.0-2 rpart_4.1-10 nFactors_2.3.3
## [4] boot_1.3-18 psych_1.6.9 MASS_7.3-45
## [7] car_2.1-2 reshape2_1.4.1 captioner_2.2.3
## [10] plyr_1.8.4 nlme_3.1-128 latticeExtra_0.6-28
## [13] RColorBrewer_1.1-2 lattice_0.20-34 knitr_1.13
## [16] wzRfun_0.70 gsubfn_0.6-6 proto_0.3-10
## [19] devtools_1.12.0
##
## loaded via a namespace (and not attached):
## [1] zoo_1.7-13 splines_3.3.2 tcltk_3.3.2
## [4] doBy_4.5-15 htmltools_0.3.5 mgcv_1.8-16
## [7] rpanel_1.1-3 yaml_2.1.13 survival_2.39-5
## [10] nloptr_1.0.4 foreign_0.8-67 withr_1.0.2
## [13] multcomp_1.4-6 stringr_1.0.0 MatrixModels_0.4-1
## [16] mvtnorm_1.0-5 codetools_0.2-15 memoise_1.0.0
## [19] evaluate_0.9 SparseM_1.7 quantreg_5.26
## [22] parallel_3.3.2 pbkrtest_0.4-6 curl_0.9.7
## [25] TH.data_1.0-7 Rcpp_0.12.7 formatR_1.4
## [28] lme4_1.1-12 mnormt_1.5-5 digest_0.6.9
## [31] stringi_1.1.1 grid_3.3.2 tools_3.3.2
## [34] sandwich_2.3-4 magrittr_1.5 Matrix_1.2-7.1
## [37] minqa_1.2.4 rmarkdown_1.0 roxygen2_5.0.1
## [40] httr_1.2.1 R6_2.1.2 nnet_7.3-12
## [43] git2r_0.15.0
Estatística Aplicada à Ciência do Solo |
github.com/walmes/EACS |