Factorial Completo en Bloques Completos y al Azar

FCBCA + COVARIABLE

Creacion de datos

set.seed(123)

# Variable respuesta: Conductancia estomatica

ce = c(
  rnorm(15,100,10),
  rnorm(15,120,12)
)

# Factor 1 : Hora de evaluacion 
hora = gl(2,15,30, labels = c(6,12))

# Factor 2 : Iluminacion 
ilum = gl(3,5,30, c('IB','IM','IA'))

# Bloqueo : Parcela 
parc = gl(5,1,30, paste0('B', 1:5))

datos = data.frame(hora, ilum, parc, ce)
head(datos)
##   hora ilum parc        ce
## 1    6   IB   B1  94.39524
## 2    6   IB   B2  97.69823
## 3    6   IB   B3 115.58708
## 4    6   IB   B4 100.70508
## 5    6   IB   B5 101.29288
## 6    6   IM   B1 117.15065

Analisis Descriptivo

library(ggplot2)

ggplot(datos)+
  aes(y=ce)+
  geom_boxplot()

ggplot(datos)+
  aes(hora, ce)+
  geom_boxplot()

ggplot(datos)+
  aes(ilum, ce)+
  geom_boxplot()

ggplot(datos)+
  aes(parc, ce)+
  geom_boxplot()

ggplot(datos)+
  aes(ilum, ce, fill=hora)+
  geom_boxplot()

ggplot(datos)+
  aes(parc, ce, color=hora)+
  geom_point(size=5)+
  facet_wrap(~ilum)

Modelo

\[y_{ijk} = \mu + \tau_i + \beta_j + (\tau\beta)_{ij} + \delta_k + \epsilon_{ijk}\]

Analisis de varianza

mod1 = aov(ce ~ parc + hora + ilum + hora:ilum, datos)
summary(mod1)
##             Df Sum Sq Mean Sq F value  Pr(>F)   
## parc         4  262.4    65.6   0.462 0.76262   
## hora         1 1805.8  1805.8  12.725 0.00193 **
## ilum         2  231.5   115.8   0.816 0.45651   
## hora:ilum    2   74.7    37.4   0.263 0.77120   
## Residuals   20 2838.4   141.9                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • la interacción hora:iluminación = p.valor 0.7712 >5%, no rechaza la hipotesis nula. La interacción es nula. (NOTA: Solamente cuando existe interacción se grafica).

Hipotesis de variable hora:

\[H_0{2}: \t_{6} = \t_{12} = 0\]

Hipotesis de variable iluminación:

\[H_0{2}: \beta_{IB} = \beta_{IM} = \beta_{IA} = 0\]

mod2 = aov(ce ~ hora + ilum + hora:ilum +  parc, datos) 
summary(mod2)
##             Df Sum Sq Mean Sq F value  Pr(>F)   
## hora         1 1805.8  1805.8  12.725 0.00193 **
## ilum         2  231.5   115.8   0.816 0.45651   
## parc         4  262.4    65.6   0.462 0.76262   
## hora:ilum    2   74.7    37.4   0.263 0.77120   
## Residuals   20 2838.4   141.9                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
resid = mod1$residuals
plot(resid, pch=16, cex=2)
grid()

#normalidad
shapiro.test(resid)
## 
##  Shapiro-Wilk normality test
## 
## data:  resid
## W = 0.96166, p-value = 0.3413
datos$trt = interaction(datos$hora,
                        datos$ilum)

ggplot(datos)+
  aes(trt, ce)+
  geom_boxplot()

tapply(datos$ce, datos$trt, var)
##      6.IB     12.IB      6.IM     12.IM      6.IA     12.IA 
##  65.77564 286.70940 135.37080  17.09574  40.89132 229.33821
datos$ce_log = log(datos$ce)
ggplot(datos)+
  aes(trt, ce_log)+
  geom_boxplot()

ce_log_var = tapply(datos$ce_log, datos$trt,
                    var)
sort(ce_log_var, decreasing = 
       T)/min(ce_log_var)
##     12.IB     12.IA      6.IM      6.IB      6.IA     12.IM 
## 15.473325 12.433759  9.559637  4.346150  2.824160  1.000000
modlog = aov(ce_log ~ parc + hora + ilum + hora:ilum, datos)
summary(modlog)
##             Df  Sum Sq Mean Sq F value  Pr(>F)   
## parc         4 0.02122 0.00530   0.467 0.75939   
## hora         1 0.14615 0.14615  12.862 0.00185 **
## ilum         2 0.01722 0.00861   0.758 0.48172   
## hora:ilum    2 0.00357 0.00178   0.157 0.85574   
## Residuals   20 0.22726 0.01136                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
shapiro.test(modlog$residuals)
## 
##  Shapiro-Wilk normality test
## 
## data:  modlog$residuals
## W = 0.96786, p-value = 0.4824
TukeyHSD(modlog, 'hora')
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = ce_log ~ parc + hora + ilum + hora:ilum, data = datos)
## 
## $hora
##           diff        lwr       upr     p adj
## 12-6 0.1395957 0.05840189 0.2207896 0.0018458