Análisis de correlación canónica. Trabajo final AMV-CEECS 2022
MOLINA, María Angélica, MARTINEZ, Gabriela
Mayo 2022
#Librerias
library(GGally)
library(fields)
library(CCA)
library(vegan)
library(tidyverse)1 INTRODUCCIÓN
Las características radiómicas son valores cuantitativos que pueden extraerse de las imagenes médicas y que nos dan información sobre la lesión o imagen de interés (Región de interés).
En PET/CT, suelen estudiarse características radiómicas que proveen información sobre el metabolismo de las regiones de interés (ROI) y sobre su textura.
Se desea conocer si existe relación entre las características radiómicas metabólicas y las características radiómicas texturales extraídas de imagenes PET/CT 18-FDG de estadificación inicial de pacientes con carcinoma escamoso de cabeza y cuello tratados con radioterapia.
Para esto, realizamos un análsis de correlación canónica entre características radiómicas metabólicas y texurales.
Explorando esta relación entre variables metabólicas y texturales, se busca reducir el número de variables para luego realizar un modelo de regresión logística que nos permita conocer si estas variables seleccionadas estan o no relacionadas con la respuesta de los tumores al tratamiento de radioterapia.
2 CARGA DE DATOS
library(readxl)
data<-read_excel("C:/Users/Angelica Molina/Desktop/IMAGENES DOCTORADO/BASE DE DATOS (AMV).xlsx",
col_types = c("text", "text", "text",
"text", "text", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric"))
data<-data[-1,]
View(data)
summary(data)## HC Nombre Diagnostico Sexo
## Length:82 Length:82 Length:82 Length:82
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
## RTA TLG sTLG MTV
## Length:82 Min. : 19.0 Min. : 0.00 Min. : 14.00
## Class :character 1st Qu.: 132.0 1st Qu.: 2.00 1st Qu.: 46.25
## Mode :character Median : 208.0 Median : 3.00 Median : 91.50
## Mean : 375.4 Mean : 5.22 Mean :112.29
## 3rd Qu.: 432.0 3rd Qu.: 5.00 3rd Qu.:131.00
## Max. :2585.0 Max. :51.00 Max. :462.00
## sMTV SUVmax GLCM_Homogeneity[=InverseDifference]
## Min. :0.040 Min. : 2.230 Min. :0.2162
## 1st Qu.:0.590 1st Qu.: 6.482 1st Qu.:0.3580
## Median :1.120 Median :11.010 Median :0.4651
## Mean :1.637 Mean :13.716 Mean :0.4662
## 3rd Qu.:2.147 3rd Qu.:17.608 3rd Qu.:0.5828
## Max. :7.020 Max. :85.290 Max. :0.7200
## GLCM_Energy[=AngularSecondMoment] GLCM_Contrast[=Variance] GLCM_Correlation
## Min. :0.001950 Min. :1.000e+00 Min. :0.3718
## 1st Qu.:0.008628 1st Qu.:5.000e+00 1st Qu.:0.6111
## Median :0.017879 Median :2.200e+01 Median :0.7196
## Mean :0.025610 Mean :3.029e+13 Mean :0.7025
## 3rd Qu.:0.037845 3rd Qu.:7.900e+01 3rd Qu.:0.8123
## Max. :0.083536 Max. :2.484e+15 Max. :0.9330
## GLCM_Entropy_log10 GLCM_Entropy_log2[=JointEntropy] GLCM_Dissimilarity
## Min. :1.000e+00 Min. :4.000e+00 Min. :1.000e+00
## 1st Qu.:2.000e+00 1st Qu.:6.000e+00 1st Qu.:1.000e+00
## Median :2.000e+00 Median :7.000e+00 Median :3.000e+00
## Mean :2.139e+14 Mean :7.104e+13 Mean :1.324e+14
## 3rd Qu.:2.000e+00 3rd Qu.:8.000e+00 3rd Qu.:6.000e+00
## Max. :1.754e+16 Max. :5.825e+15 Max. :1.086e+16
## GLRLM_SRE GLRLM_LRE GLRLM_LGRE GLRLM_HGRE
## Min. :0.6029 Min. :1.000e+00 Min. :0.0005102 Min. :1.900e+01
## 1st Qu.:0.7941 1st Qu.:1.000e+00 1st Qu.:0.0187647 1st Qu.:5.100e+01
## Median :0.8602 Median :2.000e+00 Median :0.0307128 Median :1.340e+02
## Mean :0.8464 Mean :3.621e+14 Mean :0.0382060 Mean :9.965e+12
## 3rd Qu.:0.9138 3rd Qu.:3.000e+00 3rd Qu.:0.0511120 3rd Qu.:3.790e+02
## Max. :0.9672 Max. :2.970e+16 Max. :0.1935248 Max. :8.171e+14
## GLRLM_SRLGE GLRLM_SRHGE GLRLM_LRLGE
## Min. :0.0004776 Min. :1.300e+01 Min. :0.0007759
## 1st Qu.:0.0166527 1st Qu.:4.300e+01 1st Qu.:0.0300810
## Median :0.0228919 Median :1.280e+02 Median :0.0678693
## Mean :0.0286733 Mean :8.073e+12 Mean :0.1377269
## 3rd Qu.:0.0358861 3rd Qu.:3.680e+02 3rd Qu.:0.1555553
## Max. :0.0905675 Max. :6.620e+14 Max. :2.3704035
## GLRLM_LRHGE GLRLM_GLNU GLRLM_RLNU GLRLM_RP
## Min. :5.400e+01 Min. :7.000e+00 Min. :3.900e+01 Min. :0.4924
## 1st Qu.:1.150e+02 1st Qu.:2.100e+01 1st Qu.:1.990e+02 1st Qu.:0.7254
## Median :1.840e+02 Median :4.500e+01 Median :3.420e+02 Median :0.8041
## Mean :2.470e+14 Mean :2.228e+14 Mean :1.160e+14 Mean :0.7903
## 3rd Qu.:4.520e+02 3rd Qu.:8.400e+01 3rd Qu.:5.140e+02 3rd Qu.:0.8781
## Max. :2.025e+16 Max. :1.827e+16 Max. :9.514e+15 Max. :0.9551
## NGLDM_Coarseness NGLDM_Contrast NGLDM_Busyness
## Min. :0.0005867 Min. :0.01138 Min. :0.000e+00
## 1st Qu.:0.0064859 1st Qu.:0.04178 1st Qu.:0.000e+00
## Median :0.0094433 Median :0.07848 Median :0.000e+00
## Mean :0.0118616 Mean :0.19259 Mean :1.599e+14
## 3rd Qu.:0.0141956 3rd Qu.:0.24708 3rd Qu.:1.000e+00
## Max. :0.0643724 Max. :1.25392 Max. :1.311e+16
La base de datos cuenta con 5 variables metabólicas (columnas 6 a 10) y 21 variables texturales (columnas 11 a 21).
3 EXPLORACIÓN GRÁFICA DE LOS DATOS
3.1 Evaluación gráfica de correlacion entre cada grupo de variables.
Variables metabólicas.
metabolicas<-data[,(6:10)]
ggpairs(metabolicas, title="Caraceteristicas radiomicas metabolicas")
Variables texturales
texturales <-data[,11:21]
ggpairs(texturales, title = "Caracteristicas radiomicas texturales")
3.2 Evaluación gráfica de correlaciones entre ambos set de variables
ggduo(data,columnsX = 6:10,columnsY = 11:15,
types = list(continuous = "smooth_lm"),
title = "Correlación entre caracteristicas radiomicas metabolicas y texturales",
xlab = "Caracteristicas radiomicas metabolicas",
ylab = "Caracteristicas radiomicas texturales")
ggduo(data,columnsX = 6:10,columnsY = 16:20,
types = list(continuous = "smooth_lm"),
title = "Correlación entre caracteristicas radiomicas metabolicas y texturales",
xlab = "Caracteristicas radiomicas metabolicas",
ylab = "Caracteristicas radiomicas texturales")
ggduo(data,columnsX = 6:10,columnsY = 21:25,
types = list(continuous = "smooth_lm"),
title = "Correlación entre caracteristicas radiomicas metabolicas y texturales",
xlab = "Caracteristicas radiomicas metabolicas",
ylab = "Caracteristicas radiomicas texturales")
ggduo(data,columnsX = 6:10,columnsY = 26:31,
types = list(continuous = "smooth_lm"),
title = "Correlación entre caracteristicas radiomicas metabolicas y texturales",
xlab = "Caracteristicas radiomicas metabolicas",
ylab = "Caracteristicas radiomicas texturales")
Se observa que existe alta correlación entre varias de las características radiómicas metabólicas, especialmente entre aquellas que son la misma variable pero estandarizada y no estandarizada (ej: MTV -Metabolic Tumor Volume- y sMTV -standardize Matebolic Tumor Volume-).
Por esto, para un modelo de regresión logística, elegiriamos solo un tipo de estas variables (la estandarizada o la no estandarizada).
Dentro de las metabólicas, por relevancia clinica y por hallazgos en correlación realizada en este trabajo, seleccionaremos para nuestro modelo de regresión logística las variables SUVmax, sTLG y sMTV. Al elegir las variables estandarizadas, los rangos de valores posibles son mas acotados (más propicio para el modelo de regresión logística).
Por otro lado, dentro de las variables texturales, vemos que la correlación entre la mayoría de ellas no es tan fuerte como entre las metabólicas. Al analizar como estas variables texturales se correlacionan con las metabólicas (en ggduo), decidimos utilizar para el análisis de correlación canónica solo aquellas que muestran algun grado de correlación con las metabólicas (columna 11, 12, 14, 18, 20, 22, 28, 29,30)
Con este primer análisis gráfico, logramos reducir el número de variables explicativas (de 5 a 3 metabólicas y de 21 a 9 texturales).
Veamos si podemos mejorar nuestra selección…
4 MATRICES DE COVARIANZAS
Como el análisis de corrrelación canónica se basa en operaciones matriciales entre ambos grupos de variables, planteamos a continuación las matrices de covarianza de X (metabólicas), Y (texturales) y las matrices de covarianzas XY e YX a modo exploratorio.
La matriz de covarianzas de X (Sxx), correspondiente a características radiómicas metabólicas,es:
X<-data[,c(6, 8,10)]
cov(X)## TLG MTV SUVmax
## TLG 217250.831 36211.5423 3020.8617
## MTV 36211.542 9151.6417 171.3467
## SUVmax 3020.862 171.3467 139.2819
La matriz de covarianzas de Y (Syy), correspondientes a características radiómicas texturales, es:
Y <-data[,c(11, 12, 14, 18, 20, 22, 28, 29, 30)]
cov(Y) ## GLCM_Homogeneity[=InverseDifference]
## GLCM_Homogeneity[=InverseDifference] 0.0164118118
## GLCM_Energy[=AngularSecondMoment] 0.0021118873
## GLCM_Correlation 0.0016426409
## GLRLM_SRE -0.0096503737
## GLRLM_LGRE 0.0022409095
## GLRLM_SRLGE 0.0012949255
## GLRLM_RP -0.0118886514
## NGLDM_Coarseness 0.0001437882
## NGLDM_Contrast -0.0209677004
## GLCM_Energy[=AngularSecondMoment]
## GLCM_Homogeneity[=InverseDifference] 2.111887e-03
## GLCM_Energy[=AngularSecondMoment] 4.061477e-04
## GLCM_Correlation -6.355782e-04
## GLRLM_SRE -1.340391e-03
## GLRLM_LGRE 2.551358e-04
## GLRLM_SRLGE 1.343168e-04
## GLRLM_RP -1.636804e-03
## NGLDM_Coarseness 4.351855e-05
## NGLDM_Contrast -1.488653e-03
## GLCM_Correlation GLRLM_SRE
## GLCM_Homogeneity[=InverseDifference] 0.0016426409 -9.650374e-03
## GLCM_Energy[=AngularSecondMoment] -0.0006355782 -1.340391e-03
## GLCM_Correlation 0.0153191813 -1.432320e-03
## GLRLM_SRE -0.0014323199 6.314538e-03
## GLRLM_LGRE 0.0007698418 -1.453950e-03
## GLRLM_SRLGE 0.0003435747 -7.674145e-04
## GLRLM_RP -0.0031462782 7.907992e-03
## NGLDM_Coarseness -0.0006821395 -6.960197e-06
## NGLDM_Contrast -0.0062522088 1.005920e-02
## GLRLM_LGRE GLRLM_SRLGE GLRLM_RP
## GLCM_Homogeneity[=InverseDifference] 2.240909e-03 1.294926e-03 -0.0118886514
## GLCM_Energy[=AngularSecondMoment] 2.551358e-04 1.343168e-04 -0.0016368038
## GLCM_Correlation 7.698418e-04 3.435747e-04 -0.0031462782
## GLRLM_SRE -1.453950e-03 -7.674145e-04 0.0079079924
## GLRLM_LGRE 9.768998e-04 5.696558e-04 -0.0017200532
## GLRLM_SRLGE 5.696558e-04 3.616737e-04 -0.0008765297
## GLRLM_RP -1.720053e-03 -8.765297e-04 0.0103096481
## NGLDM_Coarseness 2.037676e-05 2.991208e-05 0.0001101216
## NGLDM_Contrast -2.899586e-03 -1.892829e-03 0.0111940982
## NGLDM_Coarseness NGLDM_Contrast
## GLCM_Homogeneity[=InverseDifference] 1.437882e-04 -0.0209677004
## GLCM_Energy[=AngularSecondMoment] 4.351855e-05 -0.0014886528
## GLCM_Correlation -6.821395e-04 -0.0062522088
## GLRLM_SRE -6.960197e-06 0.0100592011
## GLRLM_LGRE 2.037676e-05 -0.0028995858
## GLRLM_SRLGE 2.991208e-05 -0.0018928292
## GLRLM_RP 1.101216e-04 0.0111940982
## NGLDM_Coarseness 9.546452e-05 -0.0004609157
## NGLDM_Contrast -4.609157e-04 0.0629291420
La matriz de covarianzas de X e Y, Sxy es:
cov(X,Y)## GLCM_Homogeneity[=InverseDifference] GLCM_Energy[=AngularSecondMoment]
## TLG -10.8397890 -1.73396260
## MTV 1.7783926 -0.12796132
## SUVmax -0.7655905 -0.03884229
## GLCM_Correlation GLRLM_SRE GLRLM_LGRE GLRLM_SRLGE GLRLM_RP
## TLG 23.35014512 4.1176972 -0.10444942 -0.27751450 0.05338671
## MTV 7.00323359 -1.3163015 0.81326159 0.43093131 -2.51440639
## SUVmax 0.07536971 0.3366493 -0.08925777 -0.06764284 0.28919975
## NGLDM_Coarseness NGLDM_Contrast
## TLG -2.08587821 50.3167470
## MTV -0.51067215 -0.2808582
## SUVmax -0.04363688 2.3585779
Y finalmente la matriz Syx , que es la matriz traspuesta de Sxy:
cov(Y,X)## TLG MTV SUVmax
## GLCM_Homogeneity[=InverseDifference] -10.83978897 1.7783926 -0.76559050
## GLCM_Energy[=AngularSecondMoment] -1.73396260 -0.1279613 -0.03884229
## GLCM_Correlation 23.35014512 7.0032336 0.07536971
## GLRLM_SRE 4.11769721 -1.3163015 0.33664925
## GLRLM_LGRE -0.10444942 0.8132616 -0.08925777
## GLRLM_SRLGE -0.27751450 0.4309313 -0.06764284
## GLRLM_RP 0.05338671 -2.5144064 0.28919975
## NGLDM_Coarseness -2.08587821 -0.5106721 -0.04363688
## NGLDM_Contrast 50.31674702 -0.2808582 2.358577915 ANÁLISIS DE CORRELACIÓN CANÓNICA
5.1 Matriz de correlación
mat_cor <-matcor(X,Y)
mat_cor## $Xcor
## TLG MTV SUVmax
## TLG 1.0000000 0.8121136 0.5491649
## MTV 0.8121136 1.0000000 0.1517676
## SUVmax 0.5491649 0.1517676 1.0000000
##
## $Ycor
## GLCM_Homogeneity[=InverseDifference]
## GLCM_Homogeneity[=InverseDifference] 1.0000000
## GLCM_Energy[=AngularSecondMoment] 0.8179947
## GLCM_Correlation 0.1035969
## GLRLM_SRE -0.9479711
## GLRLM_LGRE 0.5596558
## GLRLM_SRLGE 0.5315059
## GLRLM_RP -0.9139710
## NGLDM_Coarseness 0.1148746
## NGLDM_Contrast -0.6524491
## GLCM_Energy[=AngularSecondMoment]
## GLCM_Homogeneity[=InverseDifference] 0.8179947
## GLCM_Energy[=AngularSecondMoment] 1.0000000
## GLCM_Correlation -0.2548058
## GLRLM_SRE -0.8369870
## GLRLM_LGRE 0.4050460
## GLRLM_SRLGE 0.3504530
## GLRLM_RP -0.7998945
## NGLDM_Coarseness 0.2210097
## NGLDM_Contrast -0.2944595
## GLCM_Correlation GLRLM_SRE GLRLM_LGRE
## GLCM_Homogeneity[=InverseDifference] 0.1035969 -0.947971062 0.55965583
## GLCM_Energy[=AngularSecondMoment] -0.2548058 -0.836987012 0.40504603
## GLCM_Correlation 1.0000000 -0.145630207 0.19900256
## GLRLM_SRE -0.1456302 1.000000000 -0.58540133
## GLRLM_LGRE 0.1990026 -0.585401333 1.00000000
## GLRLM_SRLGE 0.1459637 -0.507809412 0.95836079
## GLRLM_RP -0.2503557 0.980107101 -0.54199458
## NGLDM_Coarseness -0.5640719 -0.008964576 0.06672509
## NGLDM_Contrast -0.2013677 0.504622717 -0.36981544
## GLRLM_SRLGE GLRLM_RP NGLDM_Coarseness
## GLCM_Homogeneity[=InverseDifference] 0.5315059 -0.9139710 0.114874599
## GLCM_Energy[=AngularSecondMoment] 0.3504530 -0.7998945 0.221009737
## GLCM_Correlation 0.1459637 -0.2503557 -0.564071884
## GLRLM_SRE -0.5078094 0.9801071 -0.008964576
## GLRLM_LGRE 0.9583608 -0.5419946 0.066725090
## GLRLM_SRLGE 1.0000000 -0.4539273 0.160978257
## GLRLM_RP -0.4539273 1.0000000 0.111001726
## NGLDM_Coarseness 0.1609783 0.1110017 1.000000000
## NGLDM_Contrast -0.3967593 0.4394822 -0.188050531
## NGLDM_Contrast
## GLCM_Homogeneity[=InverseDifference] -0.6524491
## GLCM_Energy[=AngularSecondMoment] -0.2944595
## GLCM_Correlation -0.2013677
## GLRLM_SRE 0.5046227
## GLRLM_LGRE -0.3698154
## GLRLM_SRLGE -0.3967593
## GLRLM_RP 0.4394822
## NGLDM_Coarseness -0.1880505
## NGLDM_Contrast 1.0000000
##
## $XYcor
## TLG MTV SUVmax
## TLG 1.000000000 0.81211362 0.54916489
## MTV 0.812113624 1.00000000 0.15176758
## SUVmax 0.549164891 0.15176758 1.00000000
## GLCM_Homogeneity[=InverseDifference] -0.181535615 0.14511084 -0.50637344
## GLCM_Energy[=AngularSecondMoment] -0.184593758 -0.06637235 -0.16331101
## GLCM_Correlation 0.404753938 0.59146811 0.05159786
## GLRLM_SRE 0.111173914 -0.17315501 0.35897112
## GLRLM_LGRE -0.007169688 0.27199176 -0.24197676
## GLRLM_SRLGE -0.031307365 0.23686457 -0.30138112
## GLRLM_RP 0.001128056 -0.25885946 0.24133963
## NGLDM_Coarseness -0.458022943 -0.54635091 -0.37842978
## NGLDM_Contrast 0.430334265 -0.01170339 0.79666726
## GLCM_Homogeneity[=InverseDifference]
## TLG -0.1815356
## MTV 0.1451108
## SUVmax -0.5063734
## GLCM_Homogeneity[=InverseDifference] 1.0000000
## GLCM_Energy[=AngularSecondMoment] 0.8179947
## GLCM_Correlation 0.1035969
## GLRLM_SRE -0.9479711
## GLRLM_LGRE 0.5596558
## GLRLM_SRLGE 0.5315059
## GLRLM_RP -0.9139710
## NGLDM_Coarseness 0.1148746
## NGLDM_Contrast -0.6524491
## GLCM_Energy[=AngularSecondMoment]
## TLG -0.18459376
## MTV -0.06637235
## SUVmax -0.16331101
## GLCM_Homogeneity[=InverseDifference] 0.81799470
## GLCM_Energy[=AngularSecondMoment] 1.00000000
## GLCM_Correlation -0.25480577
## GLRLM_SRE -0.83698701
## GLRLM_LGRE 0.40504603
## GLRLM_SRLGE 0.35045304
## GLRLM_RP -0.79989453
## NGLDM_Coarseness 0.22100974
## NGLDM_Contrast -0.29445947
## GLCM_Correlation GLRLM_SRE GLRLM_LGRE
## TLG 0.40475394 0.111173914 -0.007169688
## MTV 0.59146811 -0.173155013 0.271991755
## SUVmax 0.05159786 0.358971123 -0.241976761
## GLCM_Homogeneity[=InverseDifference] 0.10359686 -0.947971062 0.559655831
## GLCM_Energy[=AngularSecondMoment] -0.25480577 -0.836987012 0.405046029
## GLCM_Correlation 1.00000000 -0.145630207 0.199002561
## GLRLM_SRE -0.14563021 1.000000000 -0.585401333
## GLRLM_LGRE 0.19900256 -0.585401333 1.000000000
## GLRLM_SRLGE 0.14596372 -0.507809412 0.958360789
## GLRLM_RP -0.25035566 0.980107101 -0.541994582
## NGLDM_Coarseness -0.56407188 -0.008964576 0.066725090
## NGLDM_Contrast -0.20136769 0.504622717 -0.369815441
## GLRLM_SRLGE GLRLM_RP NGLDM_Coarseness
## TLG -0.03130736 0.001128056 -0.458022943
## MTV 0.23686457 -0.258859464 -0.546350914
## SUVmax -0.30138112 0.241339628 -0.378429776
## GLCM_Homogeneity[=InverseDifference] 0.53150589 -0.913970986 0.114874599
## GLCM_Energy[=AngularSecondMoment] 0.35045304 -0.799894528 0.221009737
## GLCM_Correlation 0.14596372 -0.250355665 -0.564071884
## GLRLM_SRE -0.50780941 0.980107101 -0.008964576
## GLRLM_LGRE 0.95836079 -0.541994582 0.066725090
## GLRLM_SRLGE 1.00000000 -0.453927266 0.160978257
## GLRLM_RP -0.45392727 1.000000000 0.111001726
## NGLDM_Coarseness 0.16097826 0.111001726 1.000000000
## NGLDM_Contrast -0.39675927 0.439482197 -0.188050531
## NGLDM_Contrast
## TLG 0.43033427
## MTV -0.01170339
## SUVmax 0.79666726
## GLCM_Homogeneity[=InverseDifference] -0.65244909
## GLCM_Energy[=AngularSecondMoment] -0.29445947
## GLCM_Correlation -0.20136769
## GLRLM_SRE 0.50462272
## GLRLM_LGRE -0.36981544
## GLRLM_SRLGE -0.39675927
## GLRLM_RP 0.43948220
## NGLDM_Coarseness -0.18805053
## NGLDM_Contrast 1.00000000img.matcor(mat_cor,type = 2)
La correlacion entre las variables metabólicas es muy fuerte y positiva.
En las texturales, el tipo de correlación (fuerza y sentido) es mucho mas variable.
Esto hace que la correlación cruzada (Cross-correlation) impresione bastante heterogénea.
5.2 Cálculo de correlación canónica
cc1 <- cancor(X, Y) ### funcion del R estandar
cc2 <- cc(X, Y) ### funcion del paquete "CCA". Permite graficar y sacar coeficientes
cc3 <- cca(X, Y) ### Utilizando libreria vegan.
cc1## $cor
## [1] 0.9241150 0.7736016 0.4753204
##
## $xcoef
## [,1] [,2] [,3]
## TLG -1.501046e-05 -0.0003077731 -0.0005216941
## MTV -7.210215e-06 0.0021300538 0.0013011465
## SUVmax -9.064441e-03 0.0037468108 0.0101663798
##
## $ycoef
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] 0.5812186 0.4012756 -2.3073659
## GLCM_Energy[=AngularSecondMoment] -4.5358665 0.2276106 7.9110924
## GLCM_Correlation -0.1909948 0.2091762 -0.4563411
## GLRLM_SRE -2.9710252 1.1625631 0.1871412
## GLRLM_LGRE -0.5640464 1.5814607 3.4755369
## GLRLM_SRLGE -0.1625422 0.1269038 -3.8481438
## GLRLM_RP 2.0119598 -0.4252365 -0.8038013
## NGLDM_Coarseness 1.2910167 -7.6446096 -5.3058267
## NGLDM_Contrast -0.2176032 -0.0784971 -0.5965618
## [,4] [,5] [,6]
## GLCM_Homogeneity[=InverseDifference] -1.7097134 0.4322446 -2.4154350
## GLCM_Energy[=AngularSecondMoment] -9.3450326 -0.7206837 5.6304465
## GLCM_Correlation -0.6419778 -0.1706279 0.2959886
## GLRLM_SRE -5.2929660 1.1122888 -2.7558169
## GLRLM_LGRE -0.3578526 3.5649696 -14.3148542
## GLRLM_SRLGE 0.4218719 0.8656131 23.7070541
## GLRLM_RP 0.1073757 0.2091451 -0.1123103
## NGLDM_Coarseness 0.4507196 1.7787225 -0.8049631
## NGLDM_Contrast 0.0608614 0.1421303 -0.1703760
## [,7] [,8] [,9]
## GLCM_Homogeneity[=InverseDifference] 0.94635742 -1.3577430 1.2433975
## GLCM_Energy[=AngularSecondMoment] -5.81639895 2.1731233 3.7160272
## GLCM_Correlation -1.22892674 0.6073274 0.4331283
## GLRLM_SRE 8.68889589 2.9357723 -2.6697476
## GLRLM_LGRE 0.74453621 3.8939323 3.2176908
## GLRLM_SRLGE 0.38276074 -5.8712018 -5.9829716
## GLRLM_RP -7.05024546 -3.3030028 3.0882410
## NGLDM_Coarseness 1.75501554 13.1177561 0.8594765
## NGLDM_Contrast -0.05343354 -0.2166989 0.4194397
##
## $xcenter
## TLG MTV SUVmax
## 375.37805 112.29268 13.71598
##
## $ycenter
## GLCM_Homogeneity[=InverseDifference] GLCM_Energy[=AngularSecondMoment]
## 0.46624310 0.02561044
## GLCM_Correlation GLRLM_SRE
## 0.70249484 0.84642096
## GLRLM_LGRE GLRLM_SRLGE
## 0.03820604 0.02867332
## GLRLM_RP NGLDM_Coarseness
## 0.79035001 0.01186156
## NGLDM_Contrast
## 0.19259417cc2## $cor
## [1] 0.9241150 0.7736016 0.4753204
##
## $names
## $names$Xnames
## [1] "TLG" "MTV" "SUVmax"
##
## $names$Ynames
## [1] "GLCM_Homogeneity[=InverseDifference]"
## [2] "GLCM_Energy[=AngularSecondMoment]"
## [3] "GLCM_Correlation"
## [4] "GLRLM_SRE"
## [5] "GLRLM_LGRE"
## [6] "GLRLM_SRLGE"
## [7] "GLRLM_RP"
## [8] "NGLDM_Coarseness"
## [9] "NGLDM_Contrast"
##
## $names$ind.names
## [1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10" "11" "12" "13" "14" "15"
## [16] "16" "17" "18" "19" "20" "21" "22" "23" "24" "25" "26" "27" "28" "29" "30"
## [31] "31" "32" "33" "34" "35" "36" "37" "38" "39" "40" "41" "42" "43" "44" "45"
## [46] "46" "47" "48" "49" "50" "51" "52" "53" "54" "55" "56" "57" "58" "59" "60"
## [61] "61" "62" "63" "64" "65" "66" "67" "68" "69" "70" "71" "72" "73" "74" "75"
## [76] "76" "77" "78" "79" "80" "81" "82"
##
##
## $xcoef
## [,1] [,2] [,3]
## TLG 1.350942e-04 -0.002769958 -0.004695247
## MTV 6.489194e-05 0.019170484 0.011710319
## SUVmax 8.157997e-02 0.033721297 0.091497419
##
## $ycoef
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] -5.230967 3.6114805 -20.766293
## GLCM_Energy[=AngularSecondMoment] 40.822799 2.0484954 71.199831
## GLCM_Correlation 1.718954 1.8825857 -4.107070
## GLRLM_SRE 26.739227 10.4630677 1.684271
## GLRLM_LGRE 5.076418 14.2331461 31.279832
## GLRLM_SRLGE 1.462880 1.1421341 -34.633295
## GLRLM_RP -18.107639 -3.8271282 -7.234212
## NGLDM_Coarseness -11.619151 -68.8014861 -47.752440
## NGLDM_Contrast 1.958429 -0.7064739 -5.369056
##
## $scores
## $scores$xscores
## [,1] [,2] [,3]
## [1,] -0.57780546 -0.271665753 -0.041032278
## [2,] -0.80477521 -0.149889369 -0.180177460
## [3,] -0.89013508 -1.182948743 -0.397481204
## [4,] 0.47256501 -0.594366883 0.135613016
## [5,] -0.73396138 -0.794028326 -0.344290632
## [6,] 1.20166705 -0.065895254 0.812926368
## [7,] 0.69054749 -0.886185850 -0.571735200
## [8,] -0.87784470 -0.009710697 -0.224261395
## [9,] -0.22140044 -0.446485922 -0.193831218
## [10,] 1.06759500 -0.345341421 1.380570175
## [11,] -0.60735956 -1.062958874 -0.159795178
## [12,] -0.85868406 -0.888803945 -0.303939520
## [13,] -0.26351637 -1.048350229 -0.098858624
## [14,] -0.44515302 -0.090390295 0.112646151
## [15,] -0.96462719 -1.273360099 -0.498498674
## [16,] 0.29329231 -0.138978210 -0.344329938
## [17,] -0.28298626 0.932182461 -0.779806287
## [18,] 0.54533068 -0.530298593 0.466300540
## [19,] 0.10792300 -0.513852376 0.483616904
## [20,] 3.08470436 -0.558224668 -4.490329135
## [21,] 0.23743725 -0.660949074 0.284755578
## [22,] -0.20804236 -0.735080319 0.249371321
## [23,] 0.95065736 -0.653535012 0.337372453
## [24,] 0.56047816 -0.705202176 0.914465813
## [25,] -0.66849786 -0.217257003 -0.218301071
## [26,] 0.28310110 -0.709892987 0.219648309
## [27,] -0.30830919 -0.025985075 0.258376171
## [28,] -0.01566966 -0.720555939 0.353654653
## [29,] -0.83825685 0.394182048 -0.066308772
## [30,] 0.56794202 1.591621222 -3.727908319
## [31,] -0.22846191 -0.664201166 0.086951578
## [32,] -0.82850102 -1.098489228 -0.333174705
## [33,] 0.76422005 -0.347852173 -0.677865075
## [34,] 0.07805668 3.043941593 0.405134388
## [35,] -0.89535132 -0.778432229 -0.278141842
## [36,] -0.91081068 0.533739692 0.143123083
## [37,] -0.34491163 0.346611456 0.055649021
## [38,] -0.38281185 2.089291075 0.230976558
## [39,] 0.08761407 -0.609488033 0.417191671
## [40,] 0.21040302 -0.205113485 0.610298749
## [41,] 1.79812605 1.193990059 -4.623065039
## [42,] -0.90652918 -0.165880511 -0.228390679
## [43,] -0.62892196 -0.312494673 0.058624556
## [44,] -0.33294534 0.948671211 0.181743022
## [45,] 0.03248788 0.392844057 0.587421177
## [46,] 0.30424973 0.384340711 0.049042509
## [47,] -0.22186096 0.400300573 0.198505370
## [48,] 0.57617136 -0.007364287 -0.171453213
## [49,] 1.03353548 -0.283689133 0.136829400
## [50,] -0.50098284 0.967251935 0.441044423
## [51,] -0.33632113 -0.756464776 -0.009812488
## [52,] -0.58124853 0.602145894 0.396426831
## [53,] 0.34601252 -0.473272502 -0.238836167
## [54,] -0.92211114 -0.939816957 -0.396408319
## [55,] -0.61551232 -1.049374044 -0.139858464
## [56,] 0.20217542 -0.836074703 0.457373339
## [57,] 0.42733969 -0.647086908 0.492870634
## [58,] -0.81027630 -0.973627107 -0.344664761
## [59,] -0.48662326 -0.643751873 0.015510666
## [60,] -0.80520809 1.061911974 0.477882976
## [61,] -0.59092100 -0.723599011 -0.164474188
## [62,] 0.27806405 -0.657926003 0.412689725
## [63,] -0.54881408 -0.060405021 0.100109362
## [64,] 0.60053910 1.437997835 -0.416796418
## [65,] -0.17541795 2.784501586 -0.589578657
## [66,] -0.79390302 -0.363098300 -0.075042390
## [67,] 0.05873852 1.026240479 0.773156910
## [68,] 2.30726732 -1.252385860 0.399893029
## [69,] -0.49615509 1.192485650 0.829353439
## [70,] 0.07751313 0.302244997 0.577788313
## [71,] -0.79763823 -0.180211938 -0.208044220
## [72,] 0.44783247 -0.052527846 0.524828126
## [73,] -0.26259580 1.414591838 0.501171553
## [74,] -0.56544495 1.122686183 -0.124453180
## [75,] -0.67939012 0.672897221 0.138128325
## [76,] 0.01625316 -0.251456253 0.450707805
## [77,] 0.37162350 3.297939154 0.580357674
## [78,] 0.35292670 1.233846591 0.445119708
## [79,] -0.68585443 -0.172212313 0.084968516
## [80,] 5.92556182 -0.638206426 2.670635645
## [81,] 0.53295027 2.201212406 2.178605688
## [82,] -0.99035398 -1.144974046 -0.458486507
##
## $scores$yscores
## [,1] [,2] [,3]
## [1,] -0.455546838 0.129952064 -0.74666278
## [2,] -0.985077452 0.012426270 0.04133685
## [3,] -1.360259652 -2.058939725 1.33838292
## [4,] 0.726234547 -0.006641101 0.30492394
## [5,] -1.181572891 -0.790077705 0.17083753
## [6,] 0.887097314 -0.264598961 -0.78490731
## [7,] 1.435730845 -0.932468359 -0.33564534
## [8,] -0.696002451 -0.037908343 1.30567855
## [9,] 0.155591730 -0.399746516 0.69166691
## [10,] 0.175242195 -0.212860883 -0.51102052
## [11,] -0.611803470 -0.830459210 -0.72698515
## [12,] -1.377623024 -0.700958648 -2.05871171
## [13,] -0.209411255 -1.271318657 0.05715357
## [14,] -0.188263968 0.844448246 -1.02309309
## [15,] -0.892754692 -3.435078540 -0.90595002
## [16,] 0.544479910 -0.020614831 0.10951805
## [17,] -0.352294906 0.415842233 -1.14006972
## [18,] 0.890395407 -0.764428084 0.29179515
## [19,] 0.207467978 -0.309570421 0.73691642
## [20,] 2.545013693 0.412370085 -3.63080200
## [21,] 0.077259947 -0.953580562 0.80388915
## [22,] -0.528453518 -0.932704923 0.39115276
## [23,] 1.494891980 -0.075170343 -0.62838542
## [24,] 1.898762639 -2.195113641 -3.13144005
## [25,] -1.038570588 -0.063425627 -0.49817484
## [26,] 0.384958515 -1.155122933 1.18193822
## [27,] 0.046776838 1.081475330 -0.27760350
## [28,] -0.298239860 -1.141373289 0.44507777
## [29,] -0.947455207 0.147539053 0.59749051
## [30,] 0.623673322 1.404972919 -0.13883242
## [31,] -0.028668232 -0.638288141 0.10665905
## [32,] -1.021707501 -1.735290528 0.24322195
## [33,] 0.760331467 -0.409597197 0.06863716
## [34,] 0.061939877 1.043062893 -0.13164616
## [35,] -0.839939764 -0.464594364 0.29758768
## [36,] -0.817584699 0.792448949 0.43855273
## [37,] -0.160632109 0.314316552 0.61684520
## [38,] 0.034920498 1.279096940 -0.90673815
## [39,] 0.125252708 -0.630701711 0.99630373
## [40,] 0.100050629 0.772063252 0.08351578
## [41,] 1.675599953 0.913514286 -1.77684901
## [42,] -0.733190680 -0.187234700 1.51736071
## [43,] -1.319433134 -0.189512337 -0.23788365
## [44,] -0.387008761 0.643363892 -0.69738566
## [45,] -0.152031044 0.852435700 -0.52152451
## [46,] -0.167675022 0.506956282 -0.12752089
## [47,] 0.226878362 0.681599532 0.43247863
## [48,] 0.728534631 0.148694825 -0.17649080
## [49,] 0.516935843 0.212308262 -0.31992411
## [50,] -1.259653059 0.383114852 0.03771716
## [51,] -0.342313116 -0.932813421 0.65582923
## [52,] -0.041967123 0.839912556 0.09971798
## [53,] 0.517789411 -0.321430932 0.37302053
## [54,] -1.174621282 -1.077790835 0.30711089
## [55,] 0.166787147 0.819667350 0.02899867
## [56,] 0.054096924 -0.827234400 0.41246689
## [57,] 0.634984140 -1.046031829 0.55704240
## [58,] -0.353998736 -0.706142185 1.13910394
## [59,] -0.506625858 -0.378449604 0.26692283
## [60,] -0.825703097 1.410481907 -0.18742398
## [61,] -0.732449852 -0.829080357 -0.90806634
## [62,] 0.572926254 -0.939658718 1.23923349
## [63,] -0.457588336 0.294614360 0.20970567
## [64,] 0.459819215 1.005830432 -0.60321535
## [65,] -0.453889364 0.775613418 -0.39568596
## [66,] -0.644512556 0.900544368 -0.98442564
## [67,] -0.019648291 1.380522428 0.43103155
## [68,] 2.665785838 -1.050787442 0.98622522
## [69,] 0.007118114 2.087523550 0.55776034
## [70,] -0.023360684 0.601459847 0.89369699
## [71,] -0.973182945 -0.156179136 0.24483751
## [72,] -0.301952077 0.354693819 -0.62020802
## [73,] 0.288740397 0.955092311 0.54917455
## [74,] -0.707884729 0.810359857 -1.67622512
## [75,] -0.374970372 0.771356325 -0.19560645
## [76,] -0.671444807 0.269261671 -0.73017341
## [77,] 0.860226998 1.432773984 1.04440478
## [78,] 0.134574999 1.527372651 -0.94981047
## [79,] -0.580652381 0.313776388 -0.04321034
## [80,] 5.109545140 -0.834355886 2.28832639
## [81,] 0.146266636 2.748440698 3.31626373
## [82,] -0.745062658 -0.383965311 -0.17921375
##
## $scores$corr.X.xscores
## [,1] [,2] [,3]
## TLG 0.5967388 0.4168286 -0.6856798
## MTV 0.2034648 0.8458230 -0.4931385
## SUVmax 0.9983103 -0.0327148 0.0480240
##
## $scores$corr.Y.xscores
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] -0.49806059 0.29897838 0.013047330
## GLCM_Energy[=AngularSecondMoment] -0.16926943 0.05161043 0.153273921
## GLCM_Correlation 0.07883596 0.58267454 -0.167474769
## GLRLM_SRE 0.35153872 -0.31822854 -0.049649827
## GLRLM_LGRE -0.23173543 0.41177052 0.059097402
## GLRLM_SRLGE -0.29066722 0.35487224 0.008423556
## GLRLM_RP 0.23082310 -0.38014014 -0.031851950
## NGLDM_Coarseness -0.39658012 -0.56122782 -0.018328643
## NGLDM_Contrast 0.79404658 -0.26000947 -0.094614532
##
## $scores$corr.X.yscores
## [,1] [,2] [,3]
## TLG 0.5514552 0.32245927 -0.32591760
## MTV 0.1880249 0.65433003 -0.23439879
## SUVmax 0.9225535 -0.02530822 0.02282679
##
## $scores$corr.Y.yscores
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] -0.53895954 0.38647589 0.02744955
## GLCM_Energy[=AngularSecondMoment] -0.18316923 0.06671448 0.32246443
## GLCM_Correlation 0.08530969 0.75319714 -0.35234080
## GLRLM_SRE 0.38040582 -0.41135970 -0.10445549
## GLRLM_LGRE -0.25076471 0.53227721 0.12433172
## GLRLM_SRLGE -0.31453577 0.45872737 0.01772185
## GLRLM_RP 0.24977746 -0.49139004 -0.06701154
## NGLDM_Coarseness -0.42914586 -0.72547394 -0.03856061
## NGLDM_Contrast 0.85925084 -0.33610253 -0.19905422cc3## Call: cca(X = X, Y = Y)
##
## Inertia Proportion Rank
## Total 0.064795 1.000000
## Constrained 0.055171 0.851471 2
## Unconstrained 0.009624 0.148529 2
## Inertia is scaled Chi-square
##
## Eigenvalues for constrained axes:
## CCA1 CCA2
## 0.04548 0.00969
##
## Eigenvalues for unconstrained axes:
## CA1 CA2
## 0.007184 0.002440plot(cc3)
plt.cc(cc(X,Y),var.label = TRUE)
Para el análisis, utilizamos el modelo cc2
5.3 Correlación canónica (R2)
cc2$cor## [1] 0.9241150 0.7736016 0.4753204
[1] 0.9241150 0.7736016 0.4753204
Se observa alta correlación en las tres dimensiones, pero fundamentalente en la primera (r=0.92) y en la segunda dimensión (r=0.77).
Nuestro análisis se centrara en la primera dimensión.
5.4 Coeficientes de correlación canónica no estandarizados (raw)
cc2[3:4]## $xcoef
## [,1] [,2] [,3]
## TLG 1.350942e-04 -0.002769958 -0.004695247
## MTV 6.489194e-05 0.019170484 0.011710319
## SUVmax 8.157997e-02 0.033721297 0.091497419
##
## $ycoef
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] -5.230967 3.6114805 -20.766293
## GLCM_Energy[=AngularSecondMoment] 40.822799 2.0484954 71.199831
## GLCM_Correlation 1.718954 1.8825857 -4.107070
## GLRLM_SRE 26.739227 10.4630677 1.684271
## GLRLM_LGRE 5.076418 14.2331461 31.279832
## GLRLM_SRLGE 1.462880 1.1421341 -34.633295
## GLRLM_RP -18.107639 -3.8271282 -7.234212
## NGLDM_Coarseness -11.619151 -68.8014861 -47.752440
## NGLDM_Contrast 1.958429 -0.7064739 -5.369056
Los coeficientes canónicos sin procesar (raw) se interpretan de manera análoga a la interpretación de los coeficientes de regresión, es decir, para la variable “GLCM_Homogeneity”, un aumento de una unidad de la misma conduce a una disminución de 5.23 en la primera variable canónica del conjunto 2 cuando todas las demás variables se mantienen constantes.
5.5 Canonical Loadings
cc2 <- comput(X, Y, cc2)cc2[3:6]## $corr.X.xscores
## [,1] [,2] [,3]
## TLG 0.5967388 0.4168286 -0.6856798
## MTV 0.2034648 0.8458230 -0.4931385
## SUVmax 0.9983103 -0.0327148 0.0480240
##
## $corr.Y.xscores
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] -0.49806059 0.29897838 0.013047330
## GLCM_Energy[=AngularSecondMoment] -0.16926943 0.05161043 0.153273921
## GLCM_Correlation 0.07883596 0.58267454 -0.167474769
## GLRLM_SRE 0.35153872 -0.31822854 -0.049649827
## GLRLM_LGRE -0.23173543 0.41177052 0.059097402
## GLRLM_SRLGE -0.29066722 0.35487224 0.008423556
## GLRLM_RP 0.23082310 -0.38014014 -0.031851950
## NGLDM_Coarseness -0.39658012 -0.56122782 -0.018328643
## NGLDM_Contrast 0.79404658 -0.26000947 -0.094614532
##
## $corr.X.yscores
## [,1] [,2] [,3]
## TLG 0.5514552 0.32245927 -0.32591760
## MTV 0.1880249 0.65433003 -0.23439879
## SUVmax 0.9225535 -0.02530822 0.02282679
##
## $corr.Y.yscores
## [,1] [,2] [,3]
## GLCM_Homogeneity[=InverseDifference] -0.53895954 0.38647589 0.02744955
## GLCM_Energy[=AngularSecondMoment] -0.18316923 0.06671448 0.32246443
## GLCM_Correlation 0.08530969 0.75319714 -0.35234080
## GLRLM_SRE 0.38040582 -0.41135970 -0.10445549
## GLRLM_LGRE -0.25076471 0.53227721 0.12433172
## GLRLM_SRLGE -0.31453577 0.45872737 0.01772185
## GLRLM_RP 0.24977746 -0.49139004 -0.06701154
## NGLDM_Coarseness -0.42914586 -0.72547394 -0.03856061
## NGLDM_Contrast 0.85925084 -0.33610253 -0.19905422
Las correlaciones anteriores, entre las variables observadas y las variables canónicas, se conocen como cargas canónicas. Estas variables canónicas son en realidad un tipo de variable latente.
El número de dimensiones canónicas, también conocidas como funciones canónicas, es igual al número de variables en el conjunto más pequeño (en nuestro caso, 3 del set de metabólicas). Sin embargo, el número de dimensiones significativas puede ser incluso menor.
Para este modelo en particular hay tres dimensiones canónicas, siendo las 3 estadísticamente significativas.
5.6 Tests de dimensiones canonicas
rho <- cc1$cor
n <- dim(X)[1]
p <- length(X)
q <- length(Y)5.7 Cálculo de p-valores
CCP::p.asym(rho, n, p, q, tstat="Wilks")## Wilks' Lambda, using F-approximation (Rao's F):
## stat approx df1 df2 p.value
## 1 to 3: 0.04538339 14.304386 27 205.0784 0.000000e+00
## 2 to 3: 0.31082067 7.043913 16 142.0000 9.612311e-12
## 3 to 3: 0.77407052 3.002112 7 72.0000 8.005193e-03
Como se muestra en la salida anterior, el test de dimensiones canónicas demuestra que las tres dimensiones son significativas (F= 14.30, p menor a 0.001 en 1 to 3).
Luego, la siguiente prueba muestra que las dimensiones 2 y 3 combinadas son significativas (F= 7.04, p menor a 0.001 en 2 to 3).
Finalmente, la última prueba contrasta si la dimensión 3, por sí sola, es significativa (y lo es, con F= 3.002 y p menor a 0.001).
Por lo tanto, las tres dimensiones son significativas.
Cuando las variables en el modelo tienen desviaciones estándar muy diferentes, los coeficientes estandarizados permiten comparaciones más fáciles entre las variables. A continuación, calcularemos los coeficientes canónicos estandarizados.
5.8 Coeficientes canónicos estandarizados
u1 <- cc(X, Y)
s1 <- diag(sqrt(diag(cov(X))))
s1 %*%u1$xcoef## [,1] [,2] [,3]
## [1,] 0.062967627 -1.2910822 -2.188463
## [2,] 0.006207836 1.8339293 1.120258
## [3,] 0.962788533 0.3979712 1.079832s2 <- diag(sqrt(diag(cov(Y))))
s2 %*% u1$ycoef## [,1] [,2] [,3]
## [1,] -0.67013186 0.46266168 -2.6603406
## [2,] 0.82270620 0.04128355 1.4348978
## [3,] 0.21275605 0.23300891 -0.5083348
## [4,] 2.12480774 0.83143791 0.1338390
## [5,] 0.15866545 0.44486263 0.9776636
## [6,] 0.02782064 0.02172079 -0.6586463
## [7,] -1.83858510 -0.38859296 -0.7345361
## [8,] -0.11352601 -0.67223143 -0.4665697
## [9,] 0.49128531 -0.17722379 -1.3468643
Los coeficientes canónicos estandarizados se interpretan de manera análoga a la interpretación de los coeficientes de regresión estandarizados.
Por ejemplo, considere la variable “GLCM_Homogeneity”, un aumento de una desviación estándar dicha variable conduce a una disminución de 0.67 desviaciones estándar en el puntaje en la primera variable canónica para el conjunto 2 cuando las otras variables en el modelo se mantienen constantes.
5.9 Redundancia
can_redunds<- candisc::redundancy(candisc::cancor(X,Y))
can_redunds##
## Redundancies for the X variables & total X canonical redundancy
##
## Xcan1 Xcan2 Xcan3 total X|Y
## 0.3969 0.1776 0.0539 0.6283
##
## Redundancies for the Y variables & total Y canonical redundancy
##
## Ycan1 Ycan2 Ycan3 total Y|X
## 0.15397 0.15060 0.00756 0.31213
La redundancia de X (metabólica) dada Y (texturales) es alta (62.8%). Mientras que la redundacia de Y (texturales) dado X (metabolicas), es mucho menor (31.12%).
Esto soporta la idea que, mas allá de lo que ya se conoce en la bibliografia sobre la utilidad de las características metabólicas en explicar el comportamiento de los carcinomas escamosos de cabeza y cuello a la radioterapia, es necesario también estudiar a las texturales.
6 CONCLUSIONES
En base a los hallazgos del análisis canónico realizado, se seleccionaron las siguientes variables para nuestro modelo de regresión logística:
METABOLICAS.De 5 variables metabólicas seleccionadas inicialmente, nos quedaremos con 2. Ellas son: TLG y SUVmax. Estás características metabólicas son consideradas de primer orden. Son las más fáciles de extraer y, por lo tanto, las mas utilizadas en la práctica clínica (las más avaladas por bibliografia y conocimiento previo). Además, fueron las que mejor desempeño tuvieron en cross-loading.
TEXTURALES. De 9 variables texturales seleccionadas inicialmente, nos quedaremos con 4. Ellas son: GLCM_Homogeneity, GLRLM_SRE, NGLDM_Coarseness y NGLDM_Contrast. Estas características radiómicas texturales fueron las que mejor desempeño tuvieron en cross-loading. Excluímos las restates 5 que habiamos seleccionado graficamente al inicio.
Es decir, con el analisis de correlación canónica, pudimos reducir de 26 a solo 6 variables radiómicas explicativas para incluir en nuestro modelo logístico.