Practica Analisis Factorial
Oscar BolaƱos Flores
2022-10-27
library(haven) #LibrerĆa para leer archivos .sav
library(corrplot) #LibrerĆa para el grĆ”fico de correlaciones## corrplot 0.92 loadedlibrary(corrr) #Otra opciĆ³n de librerĆa para el cĆ”lculo y grĆ”fico de correlaciones
library(ggplot2)
library(grid)
library(gridExtra)
library(GPArotation)setwd("~/ESPOL/5TO/EstadĆstica Multivariante/Practica 3 Estadistica Multivariante")
datos <- read_sav("Ciencias_Letras.sav") #Lee archivo de datos
datos$Alumno<-as.factor(datos$Alumno) # Convierte variable Alumno en factorlibrary(psych) #LibrerĆa para realizar anĆ”lisis descriptivos##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphadescribeBy(datos[,2:6],skew=FALSE,range=FALSE)## Warning in describeBy(datos[, 2:6], skew = FALSE, range = FALSE): no grouping
## variable requested## vars n mean sd median trimmed mad min max range skew kurtosis se
## CNa 1 20 5.80 1.01 6.0 5.81 1.48 4 8 4 0.08 -0.42 0.22
## Mat 2 20 5.70 1.26 6.0 5.75 1.48 3 8 5 -0.21 -0.75 0.28
## Fra 3 20 5.60 1.10 5.5 5.56 0.74 4 8 4 0.33 -0.73 0.24
## Lat 4 20 6.05 1.15 6.0 6.06 1.48 4 8 4 -0.09 -0.88 0.26
## Lit 5 20 5.65 1.04 6.0 5.62 1.48 4 8 4 0.15 -0.42 0.23d2 <- outlier(datos[,2:6],cex=1)
datos.d2 <- data.frame(datos[,2:6],d2)
pairs.panels(datos[,2:6],bg=c("yellow","blue")[(d2>10)+1],pch=21,stars=TRUE)
r.datos <- cor(datos[,2:6])
cor.plot(r.datos, numbers = TRUE)
# Matriz de correlaciones y letras
(coor.ci.le=round(cor(datos[,2:6]),3))## CNa Mat Fra Lat Lit
## CNa 1.000 0.656 0.497 0.420 0.584
## Mat 0.656 1.000 0.099 0.230 0.317
## Fra 0.497 0.099 1.000 0.813 0.841
## Lat 0.420 0.230 0.813 1.000 0.766
## Lit 0.584 0.317 0.841 0.766 1.000PROCESO DEL ANALISIS FACTORIAL
Aqui vamos a empezar con las pruebas estadisticas antes de empezar con el AF:
# Se puede conocer la presencia de multicolinealidad al evaluar el
# Determinante de la matriz de correlaciones de las variables
# ingresadas al estudio:
R=cor(datos[,2:6])
det(R)## [1] 0.02585506EXPLICACION
Un determinante bajo, es decir, cercano a 0, indica alta multicolinealidad entre las variables. De ser igual a cero(matriz no singular), esto indicaria que algunas de las variables son linealmente dependientes y no se podrian realizar ciertos calculos necesarios para la modelizacion multivariante. En este caso observamos que es muy cercano a 0, lo que sugiere un alto nivel de colinealidad en el conjunto de variables involucradas en la matriz.
# Por consiguiente vamos a ubicar el calculo de los estimadores del Test de Barlett y el MSA(KMO)
bartlett.test(datos[,2:6])##
## Bartlett test of homogeneity of variances
##
## data: datos[, 2:6]
## Bartlett's K-squared = 1.1957, df = 4, p-value = 0.8788Ahora bien la explicacion del Test de Bartlett es muy similar a la prueba de Chi-cuadrado, es decir a valores pequeƱos de K-squared expresa que existe una homogeneidad en las varianzas y por lo consiguiente existen un buen numero de correlaciones positivas entre las variables y el p-value y se puede corroborarlo fijando en H0: Las varianzas son homogeneas(correlaciones aceptables), con una alfa = 0.05. Otra explicacion es que dicho test proporciona la probabilidad de que la matriz de correlacion de las variables sea una matriz de identidad.
# Test MSA o KMO:
KMO(datos[,2:6])## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = datos[, 2:6])
## Overall MSA = 0.67
## MSA for each item =
## CNa Mat Fra Lat Lit
## 0.65 0.42 0.63 0.76 0.81La medida de adecuacion de la muestra MSA o KMO (Kaiser -Meyer-Oikin) contrasta si las correlaciones parciales entre las variables son suficientemente pequeƱas. El estadistico KMO varia entre 0 y 1. Los valores pequeƱos indican que en analisis factorial puede ser no una buena idea, dado que las correlaciones entre los pares de variables no pueden ser explicadas por otras variables. Los menores que 0.5 indican que no debe utilizarse el AF con la matriz de datos que se estan analizando
fit.pa.none=fa(R,nfactors=2,fm="pa", rotate="none",n.obs=220)## maximum iteration exceededfit.pa.rot.vari=fa(R,nfactors=2,fm="pa", rotate="varimax",n.obs=220)## maximum iteration exceededfit.pa.rot.pro=fa(R,nfactors=2,fm="pa", rotate="quartimax",n.obs=220)## maximum iteration exceededsort(fit.pa.none$communality,decreasing=T) -> c1
sort(fit.pa.rot.vari$communality,decreasing=T) -> c2
sort(fit.pa.rot.pro$communality,decreasing=T) -> c3
fit.ml.none=fa(R,nfactors=2,fm="ml", rotate="none",n.obs=220)
fit.ml.rot.vari=fa(R,nfactors=2,fm="ml", rotate="varimax",n.obs=220)
fit.ml.rot.pro=fa(R,nfactors=2,fm="ml", rotate="quartimax",n.obs=220)
sort(fit.ml.none$communality,decreasing=T) -> c4
sort(fit.ml.rot.vari$communality,decreasing=T) -> c5
sort(fit.ml.rot.pro$communality,decreasing=T) -> c6
(cbind(c1,c2,c3,c4,c5,c6))## c1 c2 c3 c4 c5 c6
## Fra 0.9616610 0.9616610 0.9616610 0.9950018 0.9950018 0.9950018
## Mat 0.8755444 0.8755444 0.8755444 0.9601570 0.9601570 0.9601570
## Lit 0.8249298 0.8249298 0.8249298 0.7948098 0.7948098 0.7948098
## Lat 0.6933034 0.6933034 0.6933034 0.7089913 0.7089913 0.7089913
## CNa 0.6429004 0.6429004 0.6429004 0.6219474 0.6219474 0.6219474#Normalidad de los datos
par(mfrow=c(2,3))
qqnorm(datos$CNa)
qqline(datos$CNa)
qqnorm(datos$Mat)
qqline(datos$Mat)
qqnorm(datos$Fra)
qqline(datos$Fra)
qqnorm(datos$Lat)
qqline(datos$Lat)
qqnorm(datos$Lit)
qqline(datos$Lit)
#Realizamos un test de normalidad univariante (Shapiro-Wilk):
shapiro.test(datos$CNa)##
## Shapiro-Wilk normality test
##
## data: datos$CNa
## W = 0.91452, p-value = 0.07777shapiro.test(datos$Mat)##
## Shapiro-Wilk normality test
##
## data: datos$Mat
## W = 0.94253, p-value = 0.2675shapiro.test(datos$Fra)##
## Shapiro-Wilk normality test
##
## data: datos$Fra
## W = 0.91746, p-value = 0.08855shapiro.test(datos$Lat)##
## Shapiro-Wilk normality test
##
## data: datos$Lat
## W = 0.92895, p-value = 0.1474shapiro.test(datos$Lit)##
## Shapiro-Wilk normality test
##
## data: datos$Lit
## W = 0.90352, p-value = 0.04803#En este caso podriamos decir que todas nuestras vairables son normales.#Test de Normalidad Multivariante
library(MVN)## Warning: package 'MVN' was built under R version 4.2.2(mnvdemisdatos <- mvn(datos[,2:6],mvnTest="mardia"))## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 19.3219379270703 0.985316618918034 YES
## 2 Mardia Kurtosis -1.67841282727946 0.0932665352017992 YES
## 3 MVN <NA> <NA> YES
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling CNa 0.9161 0.0158 NO
## 2 Anderson-Darling Mat 0.5960 0.1055 YES
## 3 Anderson-Darling Fra 0.7535 0.0415 NO
## 4 Anderson-Darling Lat 0.6461 0.0784 YES
## 5 Anderson-Darling Lit 0.9553 0.0126 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew Kurtosis
## CNa 20 5.80 1.005249 6.0 4 8 5 6 0.08269092 -0.4194141
## Mat 20 5.70 1.260743 6.0 3 8 5 7 -0.20659525 -0.7518840
## Fra 20 5.60 1.095445 5.5 4 8 5 6 0.32863353 -0.7283333
## Lat 20 6.05 1.145931 6.0 4 8 5 7 -0.09120884 -0.8782383
## Lit 20 5.65 1.039990 6.0 4 8 5 6 0.15268958 -0.4181932#Matriz de correlaciones8ciencias y letras
(cor.ci.le=round(cor(datos[,2:6]),3))## CNa Mat Fra Lat Lit
## CNa 1.000 0.656 0.497 0.420 0.584
## Mat 0.656 1.000 0.099 0.230 0.317
## Fra 0.497 0.099 1.000 0.813 0.841
## Lat 0.420 0.230 0.813 1.000 0.766
## Lit 0.584 0.317 0.841 0.766 1.000#Extraccion por componente principales
R=cor(datos[,2:6])
(fit.pca=principal(R,nfactors=2,rotate="none"))## Principal Components Analysis
## Call: principal(r = R, nfactors = 2, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 h2 u2 com
## CNa 0.76 0.50 0.83 0.169 1.7
## Mat 0.50 0.81 0.90 0.098 1.7
## Fra 0.87 -0.40 0.92 0.075 1.4
## Lat 0.85 -0.32 0.83 0.169 1.3
## Lit 0.92 -0.17 0.87 0.125 1.1
##
## PC1 PC2
## SS loadings 3.17 1.20
## Proportion Var 0.63 0.24
## Cumulative Var 0.63 0.87
## Proportion Explained 0.73 0.27
## Cumulative Proportion 0.73 1.00
##
## Mean item complexity = 1.4
## Test of the hypothesis that 2 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0.06
##
## Fit based upon off diagonal values = 0.99plot(fit.pca,labels=row.names(R),cex=0.7,xlim=c(-2,2),ylim=c(-0.8,0.8))
INTERPRETACIĆN
UNICIDAD(Uniquenesses): Es el porcentaje de varianza que no ha sido explicada por el facotr y es igual a: 1 - Comunalidad.
COMUNALIDAD(loadings-Saturaciones): Porcentaje de la variablidad de la variable explicada por ese Factor.
FACTOR1-FACTOR2: Algebraicamente, un facot se estima mediante una combinacion lineal de variables ovservadas. Cuando se encuentran los factores de āmejor ajuste2ā, debe recordarse que estos factores no son Ćŗnicos. Se puede demostrar que cualquier rotaciĆ³n de los factores que mejor se ajuste es tambiĆ©n el mejor factor. La rotaciĆ³n de factores se utiliza para ajustar la varianza que explicarĆ” el factor
Si todos los factores explican conjuntamente un gran porcentaje de varianza en una variable dada, esa variable tiene una alta comunalidad(y por lo tanto una singularidad baja)
SS loadings: La saturaciĆ³n acumulada
Proportion Var: ProporciĆ³n de la varianza
Cumulative Var: Varianza acumulada
Las cargas de patrĆ³n son los coeficientes de regresiĆ³n de la ecuaciĆ³n del modelo factorial. En el modelo, la variable que se predice se entiende como un rasgo observado estandarizado (en un FA de correlaciones) o centrado(en un FA de covarianzas), mientras que los factores se entienden como rasgos latentes estandarizados (con varianza 1). Los coeficientes de esa combinaciĆ³n lineal son los valores de la matriz de patrones. COmo se desprende claramente, los coeficientes de patrĆ³n nunca son mayores que los coeficientes de estructura, que son correlaciones o covarianzas entre la variable prevista y los factores normalizados.
plot(fit.pa.none,labels=row.names(R),cex=0.7,xlim=c(-2,2),ylim=c(-0.8,0.8))
fit.pa=fa(datos[,2:6], nfactors=2,fm="pa", rotate = "none", n.obs=220,scores="regression")## maximum iteration exceeded(fit.pa$scores)## PA1 PA2
## [1,] 0.06447078 1.33146498
## [2,] -0.08873103 -0.74854046
## [3,] -0.42494511 0.25493202
## [4,] 0.32416622 1.67395867
## [5,] 0.42179630 0.16510852
## [6,] -0.16092353 -1.58222764
## [7,] -0.50276841 -0.22511928
## [8,] -0.43901941 0.36352134
## [9,] 0.74252973 -1.02911585
## [10,] 0.13113956 -0.55888895
## [11,] -0.14836267 1.08751882
## [12,] -1.52293847 0.24730839
## [13,] 0.19488856 0.02975168
## [14,] 2.34401328 -0.05354858
## [15,] 0.05602728 1.08641843
## [16,] -2.08173416 -1.06422932
## [17,] 0.69285503 -1.72634580
## [18,] 1.22209577 -0.49697066
## [19,] -1.51449497 0.49235495
## [20,] 0.68993525 0.75264876Fp = fit.pa$scores
grap.fact=data.frame(y1=Fp[,1],y2=Fp[,2],lab=1:20,grupo=datos$Alumno)
ggplot(grap.fact,aes(y1,y2,label=lab))+geom_point()+geom_text(vjust=2)+xlab("Fact1")+ylab("Fact 2") + geom_hline(yintercept=0)+geom_vline(xintercept=0,size=1)
fit.ml=fa(datos[,2:6], nfactors=2,fm="ml", rotate = "none", n.obs=220,scores="regression")
(fit.ml$scores)## ML2 ML1
## [1,] -0.6281014 1.0173812
## [2,] 0.2825004 -0.5540239
## [3,] -0.5145675 0.2173476
## [4,] -0.7388025 1.7945847
## [5,] 0.4032203 0.2572924
## [6,] 0.6035472 -1.3332075
## [7,] -0.3923255 -0.5643957
## [8,] -0.6707367 0.2098258
## [9,] 1.1648392 -0.5228020
## [10,] 0.4430236 -0.5373570
## [11,] -0.7105400 1.0044752
## [12,] -1.3642898 -0.5948386
## [13,] 0.1646125 0.2368646
## [14,] 2.0571969 1.0969395
## [15,] -0.5854657 1.0124192
## [16,] -1.1711497 -2.1716534
## [17,] 1.5994196 -1.2895019
## [18,] 1.2146613 0.2710683
## [19,] -1.4069255 -0.5898765
## [20,] 0.2498834 1.0394579plot(fit.ml,labels=row.names(R),cex=0.7,xlim=c(-1,1),ylim=c(-1,1))
QUINTO PASO
Calcular la matriz inicial de factores no rotados para que nos dĆ© una indicaciĆ³n preliminar acerca del nĆŗmero de factores a extraer. La matriz de factores contiene las cargas factoriales para cada variable sobre cada factor. Considerando que matriz de factores no ha sido aĆŗn rotada, el investigador deberĆ” estar interesado en la mejor combinaciĆ³n lineal de variables, es decir, en encontrar aquella combinaciĆ³n particular de las variables originales que cuenta con el mayor porcentaje de varianza de los datos. En consecuencia, el primer factor puede contemplarse como el mejor resumen de las relaciones lineales que los datos manifiestan. El segundo factor se define como la segunda mejor combinaciĆ³n lineal de las variables, los factores subsiguientes se definen de forma anĆ”loga hasta haber agotado la varianza de los datos.
NOTA IMPORTANTE: las cargas factoriales son las correlaciones entre cada variable y el factor.
Es por esto que la soluciĆ³n factorial no rotada puede no suministrar un patrĆ³n significativo de cargas de las variables, razĆ³n por la cuĆ”l el siguiente paso es:
ā¢ Hacer uso de un mĆ©todo de rotaciĆ³n para lograr soluciones factoriales mĆ”s simples y teĆ³ricamente mĆ”s significativas
En una tercera etapa, el investigador valora la necesidad de especificar de nuevo el modelo de factores debido a:
ā¢ EliminaciĆ³n de variables
ā¢ Cambiar el mĆ©todo de rotaciĆ³n
ā¢ Extraer un nĆŗmero diferente de factores, o
ā¢ Cambiar el mĆ©todo de extracciĆ³n (AF o ACP)
ROTACION DE FACTORES
Concretamente, se giran los ejes a partir del origen, en referencia a los factores, hasta alcanzar una determinada posiciĆ³n. Como se indicĆ³ previamente, el primer factor tiende a ser un factor general dado que cuenta del mayor porcentaje de varianza. El segundo y los siguientes factores se basan en la varianza residual(cada uno explica porcentajes de varianza cada vez menores). El efecto de rotar la matriz de factores es redistribuir la varianza de los primeros factores a los Ćŗltimos para lograr un patrĆ³n de factores mĆ”s simple y teĆ³ricamente mĆ”s significativo.
Existen dos tipos de rotaciĆ³n y son:
ā¢ Ortogonal: Los ejes de rotacion forman un angulod e 90 grados
ā¢ Oblicua: Los ejes de rotacion forman distintos angulo
METODOS
Varimax: Metodo de rotacion ortogonal que minimiza el nĆŗmero de variables que tienen saturaciones altas en cada factor. Simplifica la interpretaciĆ³n de los factores.
Metodo Oblimin directo: Metodo para la rotacion oblicua(no ortogonal). El metodo necesita un valor delta servira para ajustar los ejes de las saturaciones buscan una mejor aproximacion, pero considerando que la varianza se distribuira entre todos los factores y no habremos logrado el objetivo de reducir la dimensionalidad.
Metodo quartimax: Metodo de rotacion que minimiza el numero de factores necesarios para explicar cada variable.
Metodo equamax: Metodo de rotacion que es combinacion del metodo varimax, que simplifica los factores, y el metodo de quartimax, que simplifica las variables. Se minimiza tanto el numero de variables que saturan alto en un factor como el numero de factores necesarios para explicar una variable
Rotacion Promax: Rotacion oblicua que permite que los factores esten correlacionados. Esta rotacion se puede calcular mas rapidamente que una rotacion oblimin directa, por lo que es util para conjuntos de datos grandes.
El metodo mas usado es varimax
NOTA IMPORTANTE
La opcion rotation de R incluye āvarimaxā, āpromaxā, ānoneā. Se agrega la opcion scores == āregressionā o āBarlettā para generar los factores. Se usa la opcion covmat para ingresar la matriz de correlacion o covarianza directamente.
Determinar el numero de factores
Una decision crucial en el AF es determinar el numero de factores a extraerse. El paquet nFactor ofrece un conjunto de rutinas para direccionar esta decision. Desde luego, cualquier solucion factorial debe ser interpretable para ser de utilidad.
Fp = fit.ml$scores
grap.fact=data.frame(y1=Fp[,1],y2=Fp[,2],lab=1:20,grupo=datos$Alumno)
ggplot(grap.fact,aes(y1,y2,label=lab))+geom_point()+geom_text(vjust=2)+xlab("Fact1")+ylab("Fact 2") + geom_hline(yintercept=0)+geom_vline(xintercept=0,size=1)
library(nFactors)## Warning: package 'nFactors' was built under R version 4.2.2## Loading required package: lattice##
## Attaching package: 'nFactors'## The following object is masked from 'package:lattice':
##
## parallelev <- eigen(cor(datos[2:6]))
ap <- parallel(subject=nrow(datos[,2:6]),var=ncol(datos[,2:6]),rep=100,cent=0.05)
nS <- nScree(x=ev$values,aparallel=ap$eigen$qevpea)
plotnScree(nS,xlab="Numero de componentes", ylab = "Autovalores",main="Solucion por autovalores para determinar el numero de factores o componentes")
(fit.pa=fa(datos[,2:6], nfactors=2,fm="pa", rotate = "none", n.obs=220))## maximum iteration exceeded## Factor Analysis using method = pa
## Call: fa(r = datos[, 2:6], nfactors = 2, n.obs = 220, rotate = "none",
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## CNa 0.70 0.39 0.64 0.357 1.6
## Mat 0.48 0.80 0.88 0.124 1.6
## Fra 0.90 -0.40 0.96 0.038 1.4
## Lat 0.80 -0.23 0.69 0.307 1.2
## Lit 0.90 -0.14 0.82 0.175 1.0
##
## PA1 PA2
## SS loadings 2.97 1.03
## Proportion Var 0.59 0.21
## Cumulative Var 0.59 0.80
## Proportion Explained 0.74 0.26
## Cumulative Proportion 0.74 1.00
##
## Mean item complexity = 1.4
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 10 and the objective function was 3.66 with Chi Square of 60.31
## The degrees of freedom for the model are 1 and the objective function was 0.11
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 20 with the empirical chi square 0.19 with prob < 0.66
## The total number of observations was 20 with Likelihood Chi Square = 1.62 with prob < 0.2
##
## Tucker Lewis Index of factoring reliability = 0.863
## RMSEA index = 0.169 and the 90 % confidence intervals are 0 0.67
## BIC = -1.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.99 0.94
## Multiple R square of scores with factors 0.97 0.89
## Minimum correlation of possible factor scores 0.94 0.78(fit.pa.rot.vari=fa(datos[,2:6], nfactors=2,fm="pa", rotate = "varimax", n.obs=220))## maximum iteration exceeded## Factor Analysis using method = pa
## Call: fa(r = datos[, 2:6], nfactors = 2, n.obs = 220, rotate = "varimax",
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## CNa 0.42 0.68 0.64 0.357 1.7
## Mat 0.04 0.93 0.88 0.124 1.0
## Fra 0.98 0.08 0.96 0.038 1.0
## Lat 0.81 0.18 0.69 0.307 1.1
## Lit 0.85 0.31 0.82 0.175 1.3
##
## PA1 PA2
## SS loadings 2.53 1.47
## Proportion Var 0.51 0.29
## Cumulative Var 0.51 0.80
## Proportion Explained 0.63 0.37
## Cumulative Proportion 0.63 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 10 and the objective function was 3.66 with Chi Square of 60.31
## The degrees of freedom for the model are 1 and the objective function was 0.11
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 20 with the empirical chi square 0.19 with prob < 0.66
## The total number of observations was 20 with Likelihood Chi Square = 1.62 with prob < 0.2
##
## Tucker Lewis Index of factoring reliability = 0.863
## RMSEA index = 0.169 and the 90 % confidence intervals are 0 0.67
## BIC = -1.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.98 0.94
## Multiple R square of scores with factors 0.97 0.89
## Minimum correlation of possible factor scores 0.94 0.78(fit.pa.rot.pro=fa(datos[,2:6], nfactors=2,fm="pa", rotate = "quartimax", n.obs=220))## maximum iteration exceeded## Factor Analysis using method = pa
## Call: fa(r = datos[, 2:6], nfactors = 2, n.obs = 220, rotate = "quartimax",
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## CNa 0.50 0.63 0.64 0.357 1.9
## Mat 0.15 0.92 0.88 0.124 1.0
## Fra 0.98 -0.03 0.96 0.038 1.0
## Lat 0.83 0.09 0.69 0.307 1.0
## Lit 0.88 0.21 0.82 0.175 1.1
##
## PA1 PA2
## SS loadings 2.69 1.30
## Proportion Var 0.54 0.26
## Cumulative Var 0.54 0.80
## Proportion Explained 0.67 0.33
## Cumulative Proportion 0.67 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 10 and the objective function was 3.66 with Chi Square of 60.31
## The degrees of freedom for the model are 1 and the objective function was 0.11
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 20 with the empirical chi square 0.19 with prob < 0.66
## The total number of observations was 20 with Likelihood Chi Square = 1.62 with prob < 0.2
##
## Tucker Lewis Index of factoring reliability = 0.863
## RMSEA index = 0.169 and the 90 % confidence intervals are 0 0.67
## BIC = -1.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.99 0.94
## Multiple R square of scores with factors 0.97 0.89
## Minimum correlation of possible factor scores 0.95 0.77par(mfrow=c(1,3))
plot(fit.pa,labels=row.names(R),cex=0.7,xlim=c(-1,1),ylim=c(-1,1))
plot(fit.pa.rot.vari,labels=row.names(R),cex=0.7,xlim=c(-1,1),ylim=c(-1,1))
plot(fit.pa.rot.pro,labels=row.names(R),cex=0.7,xlim=c(-1,1),ylim=c(-1,1))
par(mfrow=c(1,3)) #3 rows an 2 Columns
plot(fit.pa, labels=row.names(R), cex=.7, xlim=c(-1,1))
plot(fit.pa.rot.pro, labels=row.names(R), cex=.7, xlim=c(-1,1))
plot(fit.pa.rot.vari, labels=row.names(R), cex=.7, xlim=c(-1,1))
(fit.pa = fa(datos[,2:6],nfactors = 2,fm="pa",n.obs=220,scores="regression"))## maximum iteration exceeded## Factor Analysis using method = pa
## Call: fa(r = datos[, 2:6], nfactors = 2, n.obs = 220, scores = "regression",
## fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## CNa 0.36 0.62 0.64 0.357 1.6
## Mat -0.07 0.95 0.88 0.124 1.0
## Fra 1.01 -0.11 0.96 0.038 1.0
## Lat 0.82 0.03 0.69 0.307 1.0
## Lit 0.85 0.15 0.82 0.175 1.1
##
## PA1 PA2
## SS loadings 2.61 1.39
## Proportion Var 0.52 0.28
## Cumulative Var 0.52 0.80
## Proportion Explained 0.65 0.35
## Cumulative Proportion 0.65 1.00
##
## With factor correlations of
## PA1 PA2
## PA1 1.0 0.3
## PA2 0.3 1.0
##
## Mean item complexity = 1.1
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 10 and the objective function was 3.66 with Chi Square of 60.31
## The degrees of freedom for the model are 1 and the objective function was 0.11
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 20 with the empirical chi square 0.19 with prob < 0.66
## The total number of observations was 20 with Likelihood Chi Square = 1.62 with prob < 0.2
##
## Tucker Lewis Index of factoring reliability = 0.863
## RMSEA index = 0.169 and the 90 % confidence intervals are 0 0.67
## BIC = -1.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.99 0.95
## Multiple R square of scores with factors 0.98 0.90
## Minimum correlation of possible factor scores 0.95 0.79(fit.pa.rot.vari=fa(datos[,2:6],nfactors=2,fm="pa",rotate="varimax",n.obs=220,scores="regression"))## maximum iteration exceeded## Factor Analysis using method = pa
## Call: fa(r = datos[, 2:6], nfactors = 2, n.obs = 220, rotate = "varimax",
## scores = "regression", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## CNa 0.42 0.68 0.64 0.357 1.7
## Mat 0.04 0.93 0.88 0.124 1.0
## Fra 0.98 0.08 0.96 0.038 1.0
## Lat 0.81 0.18 0.69 0.307 1.1
## Lit 0.85 0.31 0.82 0.175 1.3
##
## PA1 PA2
## SS loadings 2.53 1.47
## Proportion Var 0.51 0.29
## Cumulative Var 0.51 0.80
## Proportion Explained 0.63 0.37
## Cumulative Proportion 0.63 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 10 and the objective function was 3.66 with Chi Square of 60.31
## The degrees of freedom for the model are 1 and the objective function was 0.11
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 20 with the empirical chi square 0.19 with prob < 0.66
## The total number of observations was 20 with Likelihood Chi Square = 1.62 with prob < 0.2
##
## Tucker Lewis Index of factoring reliability = 0.863
## RMSEA index = 0.169 and the 90 % confidence intervals are 0 0.67
## BIC = -1.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.98 0.94
## Multiple R square of scores with factors 0.97 0.89
## Minimum correlation of possible factor scores 0.94 0.78(fit.pa.rot.pro=fa(datos[,2:6],nfactors=2,fm="pa",rotate= "quartimax",n.obs=220,scores="regression"))## maximum iteration exceeded## Factor Analysis using method = pa
## Call: fa(r = datos[, 2:6], nfactors = 2, n.obs = 220, rotate = "quartimax",
## scores = "regression", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PA1 PA2 h2 u2 com
## CNa 0.50 0.63 0.64 0.357 1.9
## Mat 0.15 0.92 0.88 0.124 1.0
## Fra 0.98 -0.03 0.96 0.038 1.0
## Lat 0.83 0.09 0.69 0.307 1.0
## Lit 0.88 0.21 0.82 0.175 1.1
##
## PA1 PA2
## SS loadings 2.69 1.30
## Proportion Var 0.54 0.26
## Cumulative Var 0.54 0.80
## Proportion Explained 0.67 0.33
## Cumulative Proportion 0.67 1.00
##
## Mean item complexity = 1.2
## Test of the hypothesis that 2 factors are sufficient.
##
## The degrees of freedom for the null model are 10 and the objective function was 3.66 with Chi Square of 60.31
## The degrees of freedom for the model are 1 and the objective function was 0.11
##
## The root mean square of the residuals (RMSR) is 0.02
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 20 with the empirical chi square 0.19 with prob < 0.66
## The total number of observations was 20 with Likelihood Chi Square = 1.62 with prob < 0.2
##
## Tucker Lewis Index of factoring reliability = 0.863
## RMSEA index = 0.169 and the 90 % confidence intervals are 0 0.67
## BIC = -1.37
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## PA1 PA2
## Correlation of (regression) scores with factors 0.99 0.94
## Multiple R square of scores with factors 0.97 0.89
## Minimum correlation of possible factor scores 0.95 0.77Fp1=fit.pa$scores
Fp2=fit.pa.rot.vari$scores
Fp3=fit.pa.rot.pro$scores
grap.fact1=data.frame(y1=Fp1[,1],y2=Fp1[,2],lab=1:20,grupo=datos$Alumno)
grap.fact2=data.frame(y1=Fp2[,1],y2=Fp2[,2],lab=1:20,grupo=datos$Alumno)
grap.fact3=data.frame(y1=Fp3[,1],y2=Fp3[,2],lab=1:20,grupo=datos$Alumno)
p1<-ggplot(grap.fact1,aes(y1,y2,label=lab))+geom_point()+geom_text(vjust = 2)+xlab("Fact 1")+ylab("Fact 2") + geom_hline(yintercept=0,size=1) +geom_vline(xintercept=0,size=1)
p2<-ggplot(grap.fact2,aes(y1,y2,label=lab))+geom_point()+geom_text(vjust = 2)+xlab("Fact 1")+ylab("Fact 2") + geom_hline(yintercept=0,size=1) +geom_vline(xintercept=0,size=1)
p3<-ggplot(grap.fact3,aes(y1,y2,label=lab))+geom_point()+geom_text(vjust = 2)+xlab("Fact 1")+ylab("Fact 2") + geom_hline(yintercept=0,size=1) +geom_vline(xintercept=0,size=1)
grid.arrange(p1, p2,p3, nrow = 1)
par(mfrow=c(1,3))
fa.diagram(fit.pa)
fa.diagram(fit.pa.rot.vari)
fa.diagram(fit.pa.rot.pro)
library(GPArotation)
par(mfrow=c(1,3))
biplot.psych(fa(datos[,2:6], nfactors=2, fm="pa", rotate="none"), main=paste("Biplot con rotaciĆ³n","none"), col=c(2,3,4),pch=c(21,18))## maximum iteration exceeded
biplot.psych(fa(datos[,2:6], nfactors=2, fm="pa", rotate="varimax"), main=paste("Biplot con rotaciĆ³n","varimax"), col=c(2,3,4),pch=c(21,18))## maximum iteration exceeded
biplot.psych(fa(datos[,2:6], nfactors=2, fm="pa", rotate="Promax"), main=paste("Biplot con rotaciĆ³n","quartimax"), col=c(2,3,4),pch=c(21,18))## maximum iteration exceeded
fit.pa=fa(R, nfactors = 2, fm="pa", rotate = "none", n.obs = 220)## maximum iteration exceededfit.pa.rot.vari=fa(R, nfactors = 2, fm="pa", rotate = "varimax", n.obs = 220)## maximum iteration exceededfit.pa.rot.pro=fa(R, nfactors = 2, fm="pa", rotate = "quartimax", n.obs = 220)## maximum iteration exceededcorvar.pa.none<-fit.pa.none$loadings[,1:2]
corvar.pa.rot.vari<-fit.pa.rot.vari$loadings[,1:2]
corvar.pa.rot.pro<-fit.pa.rot.pro$loadings[,1:2]
fit.ml.none=fa(R,nfactors=2, fm="ml",rotate="none",n.obs = 220)
fit.ml.rot.vari=fa(R,nfactors=2, fm="ml",rotate="varimax",n.obs = 220)
fit.pa.rot.pro=fa(R, nfactors=2, fm="ml",rotate="quartimax",n.obs=220)
corvar.ml.none<-fit.ml.none$loadings[,1:2]
corvar.ml.rot.vari<-fit.ml.rot.vari$loadings[,1:2]
corvar.ml.rot.pro<-fit.ml.rot.pro$loadings[,1:2]
p1<-plot(-1:1,-1:1,type="n",asp=1,xlab="F1", ylab="F2")+
abline(h=0, v=0, lty=2, col=8)+
symbols(0,0,1,inches=F,add=T)+
symbols(0,0,sqrt(.5),inches=F,add=T)+
arrows(0,0,corvar.pa.none[,1], corvar.pa.none[,2],length=.1)+
text(corvar.pa.none[,1],corvar.pa.none[,2],colnames(datos[,2:6]),pos=4, offset=.6, col=2, font=2)
p2<-plot(-1:1,-1:1,type="n",asp=1,xlab="F1", ylab="F2")+
abline(h=0, v=0, lty=2, col=8)+
symbols(0,0,1,inches=F,add=T)+
symbols(0,0,sqrt(.5),inches=F,add=T)+
arrows(0,0,corvar.pa.rot.vari[,1], corvar.pa.rot.vari[,2],length=.1)+
text(corvar.pa.rot.vari[,1],corvar.pa.rot.vari[,2],colnames(datos[,2:6]),pos=4, offset=.6, col=2, font=2)
p3<-plot(-1:1,-1:1,type="n",asp=1,xlab="F1", ylab="F2")+
abline(h=0, v=0, lty=2, col=8)+
symbols(0,0,1,inches=F,add=T)+
symbols(0,0,sqrt(.5),inches=F,add=T)+
arrows(0,0,corvar.pa.rot.pro[,1], corvar.pa.rot.pro[,2],length=.1)+
text(corvar.pa.rot.pro[,1],corvar.pa.rot.pro[,2],colnames(datos[,2:6]),pos=4, offset=.6, col=2, font=2)
p4<-plot(-1:1,-1:1,type="n",asp=1,xlab="F1", ylab="F2")+
abline(h=0, v=0, lty=2, col=8)+
symbols(0,0,1,inches=F,add=T)+
symbols(0,0,sqrt(.5),inches=F,add=T)+
arrows(0,0,corvar.ml.none[,1], corvar.ml.none[,2],length=.1)+
text(corvar.ml.none[,1],corvar.ml.none[,2],colnames(datos),pos=4, offset=.6, col=2, font=2)
corvar.ml.none<-fit.ml.none$loadings[,1:2]
corvar.ml.rot.vari<-fit.ml.rot.vari$loadings[,1:2]
corvar.ml.rot.pro## ML2 ML1
## CNa 0.5224070 0.59079462
## Mat 0.1386091 0.98782046
## Fra 0.9791756 -0.03704285
## Lat 0.8341655 0.11471314
## Lit 0.8689721 0.19924189p5<-plot(-1:1,-1:1,type="n",asp=1,xlab="F1", ylab="F2")+
abline(h=0, v=0, lty=2, col=8)+
symbols(0,0,1,inches=F,add=T)+
symbols(0,0,sqrt(.5),inches=F,add=T)+
arrows(0,0,corvar.ml.rot.vari[,1], corvar.ml.rot.vari[,2],length=.1)+
text(corvar.ml.rot.vari[,1],corvar.ml.rot.vari[,2],colnames(datos[,2:6]),pos=4, offset=.6, col=2, font=2)
modelo_varimax<-fa(R,nfactors=2, rotate = "varimax", fa="minres")
fa.diagram(modelo_varimax)