INFORME DE ANÁLISIS MULTIVARIADO
ARTURO JOSE BELTRAN MENDOZA Y LAURA ESTEFANIA MORA JOAQUI
22/2/2021
OBJETIVOS
Describir un conjunto de variables por una combinación lineal de factores comunes subyacentes.
Determinar un número reducido de factores que puedan representar a las variables originales.
RESUMEN
En el mundo existe cierta preocupación por el alto índice de mortalidad de las de personas que contiene dentro de su cuerpo una de las enfermedades más difíciles de combatir, esta enfermedad es llamada cáncer y se clasifica en diferentes tipos de cáncer, los cuales tiene ciertas características que la diferencia de acuerdo a la parte del cuerpo en que se desarrolla esta enfermedad.
El siguiente informe realiza una aplicación del método de Análisis Factorial a una base de datos que contiene la información de 569 datos provenientes de un estudio realizado por el Instituto Nacional de Cancerología a 569 mujeres con cancer.
Las variables contenidas en la base de datos que se utilizaran para el siguiente análisis son:
perimeter: Perimetro del cancer.
area: Area del cancer.
smoothness: Suavidad del cancer.
compactness: Compacidad del cancer.
concavidad del cancer.
concave points: puntos cóncavos del cancer.
symmetry: Simetría del cancer.
INTRODUCCIÓN
El cáncer de mama hoy en día es una de las enfermedades mas peligrosas que ataca el cuerpo humano en especial a las mujeres, esta enfermedad se encuentra alrededor del mundo y la población más vulnerable son cierto porcentaje de mujeres que tiene que luchar con esta enfermedad.
Cada día, alrededor del mundo, se publica y se realiza nuevos estudios sobre las causas y tratamientos; sin embargo, todo coinciden que el punto crítico de estos estudios es la detección temprana.
La detección temprana es importante debido a que cuando un tejido anormal o cáncer es encontrado a tiempo, puede ser más fácil de tratar. Si no se detecta temprano, la persona perjudicada no podrá ser tratada y tendrá que sufrir consecuencias severas.
Para este trabajo se realizare una observación de una base de datos de cancer de mama, con 569 datos recogidos por el Instituto Nacional de Cancerologia, se analizara todo los datos adjuntos mediante el método de análisis factorial.
# Filtrando variables
datos<-datos[,c(3:9)]
datos## # A tibble: 569 x 7
## perimeter area smoothness compactness concavity `concave points` symmetry
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 123. 1001 0.118 0.278 0.300 0.147 0.242
## 2 133. 1326 0.0847 0.0786 0.0869 0.0702 0.181
## 3 130 1203 0.110 0.160 0.197 0.128 0.207
## 4 77.6 386. 0.142 0.284 0.241 0.105 0.260
## 5 135. 1297 0.100 0.133 0.198 0.104 0.181
## 6 82.6 477. 0.128 0.17 0.158 0.0809 0.209
## 7 120. 1040 0.0946 0.109 0.113 0.074 0.179
## 8 90.2 578. 0.119 0.164 0.0937 0.0598 0.220
## 9 87.5 520. 0.127 0.193 0.186 0.0935 0.235
## 10 84.0 476. 0.119 0.240 0.227 0.0854 0.203
## # ... with 559 more rows# Resumen estadístico de la base de datos
summary(datos)## perimeter area smoothness compactness
## Min. : 43.79 Min. : 143.5 Min. :0.05263 Min. :0.01938
## 1st Qu.: 75.17 1st Qu.: 420.3 1st Qu.:0.08637 1st Qu.:0.06492
## Median : 86.24 Median : 551.1 Median :0.09587 Median :0.09263
## Mean : 91.97 Mean : 654.9 Mean :0.09636 Mean :0.10434
## 3rd Qu.:104.10 3rd Qu.: 782.7 3rd Qu.:0.10530 3rd Qu.:0.13040
## Max. :188.50 Max. :2501.0 Max. :0.16340 Max. :0.34540
## concavity concave points symmetry
## Min. :0.00000 Min. :0.00000 Min. :0.1060
## 1st Qu.:0.02956 1st Qu.:0.02031 1st Qu.:0.1619
## Median :0.06154 Median :0.03350 Median :0.1792
## Mean :0.08880 Mean :0.04892 Mean :0.1812
## 3rd Qu.:0.13070 3rd Qu.:0.07400 3rd Qu.:0.1957
## Max. :0.42680 Max. :0.20120 Max. :0.3040# Estructura de la base de datos
str(datos)## tibble [569 x 7] (S3: tbl_df/tbl/data.frame)
## $ perimeter : num [1:569] 122.8 132.9 130 77.6 135.1 ...
## $ area : num [1:569] 1001 1326 1203 386 1297 ...
## $ smoothness : num [1:569] 0.1184 0.0847 0.1096 0.1425 0.1003 ...
## $ compactness : num [1:569] 0.2776 0.0786 0.1599 0.2839 0.1328 ...
## $ concavity : num [1:569] 0.3001 0.0869 0.1974 0.2414 0.198 ...
## $ concave points: num [1:569] 0.1471 0.0702 0.1279 0.1052 0.1043 ...
## $ symmetry : num [1:569] 0.242 0.181 0.207 0.26 0.181 ...# Nombre de las variables
colnames(datos)## [1] "perimeter" "area" "smoothness" "compactness"
## [5] "concavity" "concave points" "symmetry"# Dimensíón de la base de datos
dim(datos)## [1] 569 7boxplot(datos, las = 2, col = "red", cex.main=0.1)
title("boxplot de cada variable")
- Se pueden observar la existencia de valores atípicos en las variable area, pero no es muy influyente.
CORRELACIONES
# Matriz de correlación
matriz_correlaciones <- cor(datos)
matriz_correlaciones## perimeter area smoothness compactness concavity
## perimeter 1.0000000 0.9865068 0.2072782 0.5569362 0.7161357
## area 0.9865068 1.0000000 0.1770284 0.4985017 0.6859828
## smoothness 0.2072782 0.1770284 1.0000000 0.6591232 0.5219838
## compactness 0.5569362 0.4985017 0.6591232 1.0000000 0.8831207
## concavity 0.7161357 0.6859828 0.5219838 0.8831207 1.0000000
## concave points 0.8509770 0.8232689 0.5536952 0.8311350 0.9213910
## symmetry 0.1830272 0.1512931 0.5577748 0.6026410 0.5006666
## concave points symmetry
## perimeter 0.8509770 0.1830272
## area 0.8232689 0.1512931
## smoothness 0.5536952 0.5577748
## compactness 0.8311350 0.6026410
## concavity 0.9213910 0.5006666
## concave points 1.0000000 0.4624974
## symmetry 0.4624974 1.0000000# Correlación entre variables
ggpairs(datos) +
labs(title = "Diagrama de dispersión con correlaciones")+
theme_bw() +
theme(plot.title = element_text(hjust = 0.5)) 
# Grafico de las correlaciones
corrplot(cor(datos), order = "hclust", tl.col='black', tl.cex=1) 
Las variables area y perimeter tienen una alta correlación lineal.
Las variables area y symmetry presentan una baja correlación lineal
Las variables area y concavity presentan una alta correlación lineal.
Las variables area y compactness tienen una correlación lineal moderada.
PROCESO PARA VERIFICAR SI ES POSIBLE REALIZAR UN ANÁLISIS FACTORIAL A LA BASE DE DATOS.
det(matriz_correlaciones)## [1] 2.987618e-05El determinante de la matriz de correlación es cercano a cero, lo cual indica multicolinealidad entre las variables.
Algunas variables están correlacionadas entre sí.
Algunas variables no presentan una relación lineal, pero si estan relacionadas.
Prueba de Hipotesis
H0: Las variables no estan correlacionadas
H1: las variables estan correlacionadas
# Calculo del estimador del Test de Bartlett
bartlett.test(datos)##
## Bartlett test of homogeneity of variances
##
## data: datos
## Bartlett's K-squared = 47426, df = 6, p-value < 2.2e-16- Como el p-value < 2.2e-16 < 0.05, rechazamos la hipotesis nula, a favor de la hipotesis alternativa, por lo tanto las variables estan correlacionadas.
# Test MSA o KMO: Medida de adecuación de la muestra MSA o KMO
KMO(datos)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = datos)
## Overall MSA = 0.77
## MSA for each item =
## perimeter area smoothness compactness concavity
## 0.68 0.69 0.72 0.80 0.79
## concave points symmetry
## 0.83 0.94- El anterior indicador nos dice como una variable puede ser explicada apartir de las demás. Puesto que cada variable tiene un valor mayor a 0.5, podemos concluir que es posible realizar el análisis factorial.
ANÁLISIS FACTORIAL:
Supuestos
Los datos no presentan valores atípicos muy influyentes.
El tamaño de la muestra es adecuado.
No hay multicolinealidad perfecta entre las variables.
Existe relación lineal entre las variables.
Prueba una solución factorial con 1 factor
datos.fa1<-factanal(datos,factors=1)
datos.fa1##
## Call:
## factanal(x = datos, factors = 1)
##
## Uniquenesses:
## perimeter area smoothness compactness concavity
## 0.276 0.323 0.696 0.305 0.147
## concave points symmetry
## 0.005 0.785
##
## Loadings:
## Factor1
## perimeter 0.851
## area 0.823
## smoothness 0.551
## compactness 0.833
## concavity 0.923
## concave points 0.998
## symmetry 0.464
##
## Factor1
## SS loadings 4.463
## Proportion Var 0.638
##
## Test of the hypothesis that 1 factor is sufficient.
## The chi square statistic is 2449.98 on 14 degrees of freedom.
## The p-value is 0Prueba una solución factorial con 2 factor
datos.fa2<-factanal(datos,factors=2)
datos.fa2##
## Call:
## factanal(x = datos, factors = 2)
##
## Uniquenesses:
## perimeter area smoothness compactness concavity
## 0.018 0.005 0.458 0.110 0.096
## concave points symmetry
## 0.034 0.566
##
## Loadings:
## Factor1 Factor2
## perimeter 0.974 0.182
## area 0.990 0.119
## smoothness 0.731
## compactness 0.403 0.853
## concavity 0.604 0.735
## concave points 0.756 0.628
## symmetry 0.654
##
## Factor1 Factor2
## SS loadings 3.041 2.671
## Proportion Var 0.434 0.382
## Cumulative Var 0.434 0.816
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 238.03 on 8 degrees of freedom.
## The p-value is 5.91e-47- La solución con dos factores resulta la más apropiada, por lo tanto trabajaremos con dos factores.
# Diagrama de arbol para interpretar el análisis de factores
modelo<-fa(matriz_correlaciones,rotate = "varimax",nfactors = 2,fm="minres")
fa.diagram(modelo)
# Fracción de la varianza total de la variable explicada por el factor.
apply(datos.fa2$loadings^2,1,sum)## perimeter area smoothness compactness concavity
## 0.9822539 0.9950088 0.5421682 0.8895946 0.9040730
## concave points symmetry
## 0.9658948 0.43336821-apply(datos.fa2$loadings^2,1,sum)## perimeter area smoothness compactness concavity
## 0.017746149 0.004991218 0.457831833 0.110405437 0.095926956
## concave points symmetry
## 0.034105180 0.566631804Lambda<-datos.fa2$loadings
Psi<-diag(datos.fa2$uniquenesses)
S<-datos.fa2$correlation
Sigma<-Lambda%*%t(Lambda)+Psi
round(S-Sigma,6)## perimeter area smoothness compactness concavity
## perimeter 0.000000 0.000026 -0.012306 0.008905 -0.005729
## area 0.000026 -0.000009 0.002451 -0.001886 0.001008
## smoothness -0.012306 0.002451 0.000015 0.000070 -0.068575
## compactness 0.008905 -0.001886 0.000070 0.000004 0.013356
## concavity -0.005729 0.001008 -0.068575 0.013356 -0.000003
## concave points 0.000074 0.000004 0.027609 -0.009080 0.003714
## symmetry -0.006248 0.002430 0.073081 0.015666 -0.023489
## concave points symmetry
## perimeter 0.000074 -0.006248
## area 0.000004 0.002430
## smoothness 0.027609 0.073081
## compactness -0.009080 0.015666
## concavity 0.003714 -0.023489
## concave points 0.000000 -0.002799
## symmetry -0.002799 0.000184La matriz residual muestra números cercanos a 0, lo cual es una indicación que nuestro modelo factorial es una buena representación.
La unicidad corresponde a la proporción de variabilidad, que no puede explicarse mediante una combinación lineal de los factores y las cargas son la contribución de cada variable original al factor.
Como las variables tienen una baja unicidad, por lo tanto los factores dan cuenta de su varianza.
La variable perimeter tiene una carga alta, por lo tanto dicha variable esta bien explicada por el factor1.
La variable area tiene una carga alta, por lo tanto dicha variable esta bien explicada por el factor1.
La variable concave points_mean tiene una carga alta, por lo tanto dicha variable esta bien explicada por el factor1.
La variable compactness tiene una carga alta, por lo tanto dicha variable esta bien explicada por el factor2.
La variable smoothness tiene una carga alta, por lo tanto dicha variable estabien explicada por el factor2.
La variable concavity tiene una carga alta, por lo tanto dicha variable estabien explicada por el factor2.
La variable symmetry tiene una carga alta, por lo tanto dicha variable estabien explicada por el factor2.
Se observa que la mayoria de las variables tienen valores altos para la comunalidad.
Se observa que la mayoria de las variables tienen valores bajo de unicidad
Podemos concluir que nuestro modelo de factor es apropiado ya que posee valores bajos para la unicidad y valores altos para la comunalidad.
El Factor1 esta mas relacionado con las variables perimeter, area y concave points, así que este factor describe las dimensiones que compoenen el núcleo de las celulas cancerigenas.
El Factor2 esta mas relacionado con las variables smoothness, compactness, concavity, symmetry y concave points, así que este factor describe las irregularidades de la forma del núcleo de las celulas cancerigenas.
La variables smoothness no tienen relación con el Factor1.
Observamos que el Factor1 y el Factor2 estan explicando en total el 81,6% de la variación de los dtaos.
Prueba de Hipotesis
H0: El número de factores en el modelo, en nuestro caso dos factores no es suficiente para capturar la dimensionalidad completa del conjunto de datos.
H1: El número de factores en el modelo, en nuestro caso dos factores, es suficiente para capturar la dimensionalidad completa del conjunto de datos.
como el p-value < 5.91e-47 < 0.05, rechazamos la hipotesis nula, por lo tanto, el factor1 y el factor2 es suficiente para capturar la dimensionalidad del conjunto de datos.
Verificar si el número de factores es el apropiado.
REGLA DE Kaiser
Para determinar que efectivamente dos es el número de Factores adecuados, usaremos la regla de Kaiser la cual establece: calcular los valores propios de la matriz de correlación y conservar aquellos factores cuyos valores propios (eigenvalues) son mayores a la unidad.
# Determinar el número de Factores.
ev<-eigen(matriz_correlaciones) # Obtención de loa autovalores.
ev## eigen() decomposition
## $values
## [1] 4.649747861 1.452602451 0.448388433 0.315225609 0.088708521 0.035346390
## [7] 0.009980735
##
## $vectors
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.3833763 0.44141012 0.069008751 0.25734309 0.23776895 -0.09734920
## [2,] -0.3689628 0.47146114 0.074613476 0.32868883 0.06500827 -0.27943272
## [3,] -0.2855983 -0.50206353 -0.628839424 0.49138106 -0.05324454 -0.15910908
## [4,] -0.4127042 -0.23127602 -0.089299016 -0.51369668 0.69955757 -0.02441467
## [5,] -0.4358866 -0.01545665 -0.008864521 -0.49027288 -0.62054082 -0.41701006
## [6,] -0.4533638 0.08655670 -0.076010918 -0.01972101 -0.24794623 0.84396877
## [7,] -0.2635566 -0.51922457 0.761840565 0.28220942 -0.02181364 -0.01865564
## [,7]
## [1,] 0.72192876
## [2,] -0.66759163
## [3,] 0.03634291
## [4,] -0.11979649
## [5,] 0.10173297
## [6,] -0.08391664
## [7,] 0.00875683- Se observa que son dos los eigenvalues que están por encima de la unidad, por lo tanto efectivamente dos es el número adecuado de factores.
scree(matriz_correlaciones)## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect. Try a
## different factor score estimation method.## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected. Examine the results carefully
- En la gráfica se observa un punto de inflexión en la curva a apartir del factor dos.
Cálculo de las puntaciones Factoriales
Se realiza el cálculo de las puntuaciones para observar los individuos mas extremos.
Puntuaciones de los individuos en sus Factores.
scores <-factanal(datos, factors = 2, rotation = "none", scores = "regression")$scores
scores## Factor1 Factor2
## [1,] 1.2999879918 2.898779293
## [2,] 1.6978514807 -1.904311064
## [3,] 1.6349806817 0.649366872
## [4,] -0.3938940728 4.030544941
## [5,] 1.7960571662 -0.335802412
## [6,] -0.3021486754 2.046287636
## [7,] 1.0534096817 -0.626711040
## [8,] -0.1141611734 1.042307676
## [9,] -0.1296666025 2.450596416
## [10,] -0.2362588499 2.801687129
## [11,] 0.2979460676 -1.331359383
## [12,] 0.3958904110 0.125162412
## [13,] 1.4914137176 1.187877578
## [14,] 0.3594922541 -0.365630667
## [15,] 0.0232139354 2.237794204
## [16,] 0.1528992043 1.276858919
## [17,] 0.0724450832 -0.336618764
## [18,] 0.6068858920 1.663525427
## [19,] 1.6310918667 -0.836874843
## [20,] -0.2295002963 0.027713431
## [21,] -0.3643360548 0.160105999
## [22,] -1.1082792879 0.224040817
## [23,] 0.3790560352 2.093649364
## [24,] 1.9414998555 -1.561262863
## [25,] 0.7783323718 0.645323769
## [26,] 1.0131728038 2.537919086
## [27,] 0.1642628897 1.602956342
## [28,] 1.1993406024 -0.621623896
## [29,] 0.3790604989 1.269765710
## [30,] 0.8586940739 -0.274212701
## [31,] 1.3790242475 1.264932568
## [32,] -0.4991271728 1.334680193
## [33,] 0.8907584501 1.634303477
## [34,] 1.4067041296 -0.327571879
## [35,] 0.5249547214 0.633933633
## [36,] 0.6237763164 -0.075974905
## [37,] 0.0036367986 0.414913019
## [38,] -0.4280601115 -0.749710533
## [39,] 0.0275571048 -1.166209314
## [40,] -0.1921087041 0.599997587
## [41,] -0.3321027110 -0.838547610
## [42,] -0.6908450843 1.352400940
## [43,] 1.3833418211 0.823858661
## [44,] -0.2067818822 0.926634375
## [45,] -0.2911585110 0.403164285
## [46,] 1.2723315847 0.577068657
## [47,] -1.3638776878 0.002407781
## [48,] -0.2077847952 1.228500632
## [49,] -0.5815260715 0.078222795
## [50,] -0.2865498384 -0.381277788
## [51,] -0.7228199181 -0.741481609
## [52,] -0.3203654966 -1.086317819
## [53,] -0.6885926340 -0.675941500
## [54,] 1.1591571379 0.652541648
## [55,] 0.0955882294 -0.899082867
## [56,] -0.7347951375 -0.250231177
## [57,] 1.3746472653 -0.399696527
## [58,] 0.1360190234 1.071304519
## [59,] -0.4873620763 -1.279706503
## [60,] -1.2931216392 -0.016486001
## [61,] -1.0123478902 0.247050423
## [62,] -1.2722608026 0.478755109
## [63,] 0.1827349785 1.891595208
## [64,] -1.1234346459 0.522934983
## [65,] -0.3194825785 1.172708366
## [66,] 0.1976749850 1.262595584
## [67,] -1.1307380546 0.180781046
## [68,] -0.7819687164 -0.397381401
## [69,] -0.9883288676 2.427538728
## [70,] -0.4710918468 -0.481481718
## [71,] 1.2714050653 -0.925853436
## [72,] -1.1118430169 1.258968764
## [73,] 0.8506314957 0.615485444
## [74,] -0.1442335370 0.343026480
## [75,] -0.5606744947 -0.404574030
## [76,] 0.4562250187 -0.319966161
## [77,] -0.1833941487 0.807896185
## [78,] 1.1285779336 1.134461630
## [79,] 2.0357465767 3.281758863
## [80,] -0.4450894139 -0.267637671
## [81,] -0.7264813411 0.149245747
## [82,] -0.2341237666 1.300906915
## [83,] 3.5202321957 0.786672239
## [84,] 1.5359415344 1.277974111
## [85,] -0.6366236576 -0.232139191
## [86,] 1.1751409039 -0.331586063
## [87,] 0.0163493758 0.156740211
## [88,] 1.1801454012 -0.360328720
## [89,] -0.5127273822 0.168536564
## [90,] 0.0926666018 0.777056426
## [91,] -0.0359606306 -0.789881473
## [92,] 0.2783564135 0.267348556
## [93,] -0.3576447782 -0.879158517
## [94,] -0.3146352811 -0.423994093
## [95,] 0.3051910797 1.236382347
## [96,] 1.6503817503 -0.752016757
## [97,] -0.5953928444 -0.136238223
## [98,] -1.0970043628 -0.334365331
## [99,] -0.6858283695 0.015825306
## [100,] 0.0232316038 0.299302076
## [101,] -0.1926595495 -0.042443095
## [102,] -1.5521001510 0.342286893
## [103,] -0.6268674196 -0.737943966
## [104,] -0.9863183435 0.668287240
## [105,] -0.9393378657 0.078229906
## [106,] -0.1045473875 2.421428672
## [107,] -0.6531007544 0.501737390
## [108,] -0.5826004733 -0.547837273
## [109,] 2.6754472974 2.590858246
## [110,] -0.7673308900 -0.255175183
## [111,] -1.0550346825 0.258769614
## [112,] -0.3869207011 0.908209344
## [113,] 0.1758863953 2.076056890
## [114,] -0.8623620741 0.852230829
## [115,] -1.2224501380 0.530915024
## [116,] -0.6482114991 -0.258074475
## [117,] -1.1339193651 0.848335102
## [118,] 0.2545837561 1.518155521
## [119,] 0.5410913215 1.538047236
## [120,] 0.8252487058 -1.189577663
## [121,] -0.7336678686 -0.134686983
## [122,] 1.1844177822 -0.278679905
## [123,] 3.3321909170 2.323526289
## [124,] 0.0148681778 0.310220778
## [125,] -0.3153088115 -0.524372992
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## [522,] 3.2576305876 -0.517242367
## [523,] -0.8473101960 -0.774417411
## [524,] -0.2291492235 -0.075135452
## [525,] -1.0392012097 0.192306548
## [526,] -1.2703349991 0.331142302
## [527,] -0.2941420888 -0.079769845
## [528,] -0.5623638660 -0.397613517
## [529,] -0.0873571816 0.679281908
## [530,] -0.5993026006 0.016316623
## [531,] -0.6141689092 0.456348729
## [532,] -0.6899070335 0.035152852
## [533,] -0.2977503515 -0.848552793
## [534,] 1.7746662083 -0.556625496
## [535,] -0.8082837569 0.291208519
## [536,] 1.9241479117 0.523181463
## [537,] 0.0187597030 0.653724338
## [538,] -0.6186867243 1.074765995
## [539,] -1.4627474507 -0.165170991
## [540,] -1.3869911597 1.007632624
## [541,] -0.6963318855 0.351054411
## [542,] 0.0206240224 -0.055481114
## [543,] -0.0231807319 -0.853840239
## [544,] -0.3600810029 -0.496879481
## [545,] -0.2346685832 -0.509117325
## [546,] -0.2913281547 -0.720079432
## [547,] -1.0150824283 -0.415685356
## [548,] -0.9547719264 0.190064851
## [549,] -1.1155974173 -0.277985537
## [550,] -0.9011260507 -0.407059934
## [551,] -0.9413047230 -0.875054103
## [552,] -0.7946874157 0.116371404
## [553,] -0.5035989883 -0.968746135
## [554,] -1.1584212379 0.017006944
## [555,] -0.4422775041 -0.571015787
## [556,] -0.9440384837 0.273334334
## [557,] -1.0254978502 -0.115832819
## [558,] -1.1887564392 -0.485435707
## [559,] 0.0248105542 -0.119883007
## [560,] -0.6526193497 0.596722078
## [561,] -0.1395103492 -0.110982164
## [562,] -0.8785107211 -1.123336374
## [563,] 0.4158601643 2.058378037
## [564,] 2.1339086183 1.463320125
## [565,] 2.3032883319 -0.146526268
## [566,] 1.6491806987 -0.770461437
## [567,] 0.5417678250 -0.669547448
## [568,] 1.9910994665 2.372715525
## [569,] -1.4657573224 -0.423154664En la puntución de la fila 1 se puede observar que los pesos de cada uno de los factores los definen porque son mayores a la unidad.
En la puntación de la fila 4 se observa que el peso 4.030544941 define al factor dos, ya que es mayor a la unidad.
En la puntación de la fila 5 se observa que el peso 1.7960571662 define al factor uno, ya que es mayor a la unidad.
En la puntución de la fila 13 se puede observar que los pesos de cada uno de los factores los definen porque son mayores a la unidad.
En la puntución de la fila 79 se puede observar que los pesos de cada uno de los factores los definen porque son mayores a la unidad.
pairs(scores) 
scatterplot3d(scores, angle=35, col.grid="lightblue", main="Grafica de las
puntuaciones", pch=20)
- En el gráfico podemos observar que la mayoria de las puntuaciones se encuentran en el rango de -1 a 1.
CONCLUSIÓN
Mediante la técnica del análisis factorial podemos identificar como se puede diagnosticar que una celula sea cancerigena de acuerdo a las caracteristicas descritas anteriormente.
REFERENCIAS
http://www3.udg.edu/dghha/cat/secciogeografia/prac/models/factorial(5).htm#:~:text=Uno%20de%20los%20m%C3%A1s%20conocidos,los%20programas%20estad%C3%ADsticos%20por%20defecto.
Practical Guide To Clauter Analysis in R Unsupervised Machine Learning by Alboukadel kassambara, Edition 1