El archivo Credit Approval Decisions.csv contiene información acerca de la aceptación o rechazo de un crédito. Se dispone también de información acerca de otras variables adicionales. El objetivo es analizar la variable Decision en función del resto de variables.
d = read.csv("datos/Credit Approval Decisions.csv")
str(d)## 'data.frame': 50 obs. of 6 variables:
## $ Homeowner : chr "Y" "Y" "Y" "N" ...
## $ Credit_Score : int 725 573 677 625 527 795 733 620 591 660 ...
## $ Years_Credit_History : int 20 9 11 15 12 22 7 5 17 24 ...
## $ Revolving_Balance : num 11320 7200 20000 12800 5700 ...
## $ Revolving_Utilization: num 0.25 0.7 0.55 0.65 0.75 0.12 0.2 0.62 0.5 0.35 ...
## $ Decision : chr "Approve" "Reject" "Approve" "Reject" ...Convertimos las variables cualitativas a factor:
d$Homeowner = factor(d$Homeowner)
d$Decision = factor(d$Decision)
str(d)## 'data.frame': 50 obs. of 6 variables:
## $ Homeowner : Factor w/ 2 levels "N","Y": 2 2 2 1 1 2 1 1 2 2 ...
## $ Credit_Score : int 725 573 677 625 527 795 733 620 591 660 ...
## $ Years_Credit_History : int 20 9 11 15 12 22 7 5 17 24 ...
## $ Revolving_Balance : num 11320 7200 20000 12800 5700 ...
## $ Revolving_Utilization: num 0.25 0.7 0.55 0.65 0.75 0.12 0.2 0.62 0.5 0.35 ...
## $ Decision : Factor w/ 2 levels "Approve","Reject": 1 2 1 2 2 1 1 2 2 1 ...Se crean variables auxiliares con valores cero - uno. Por ejemplo
HomeownerY = ifelse(d$Homeowner == "Y", 1, 0)El modelo que se estima es:
\[ P(Decision_i = 1) = \frac{exp(\beta_0 + \beta_1 HomeownerY_i)}{1 + exp(\beta_0 + \beta_1 HomeownerY_i)} \]
Al final estamos trabajando con dos modelos:
\[ P(Decision_i = 1) = \frac{exp(\beta_0)}{1 + exp(\beta_0)} \]
\[ P(Decision_i = 1) = \frac{exp(\beta_0+\beta_1)}{1 + exp(\beta_0+\beta_1)} \]
Por tanto, si \(\beta_1 = 0\), entonces P(Decision = 1 | Homeowner = Y) = P(Decision = 1 | Homeowner = N). Si \(\beta_1 > 0\), entonces:
\[ \beta_0 < \beta_0 + \beta_1 \\ -\beta_0 > -\beta_0 - \beta_1 \\ exp(-\beta_0) > exp(-\beta_0 - \beta_1) \\ 1 + exp(-\beta_0) > 1 + exp(-\beta_0 - \beta_1) \\ \frac{1}{1 + exp(-\beta_0)} < \frac{1}{1 + exp(-\beta_0 - \beta_1)} \]
Multiplicando numerador y denominador por el mismo número la desigualdad no cambia:
\[ \frac{exp(\beta_0)}{1 + exp(\beta_0)} < \frac{exp(\beta_0 + \beta_1)}{1 + exp(\beta_0 + \beta_1)} \\ P(Decision = 1 | Homeowner = N) < P(Decision = 1 | Homeowner = Y) \]
Estimamos el modelo. Recordemos que la variable respuesta toma valores 0 y 1. Por tanto si definimos:
Decision1 = ifelse(d$Decision == "Approve", 1, 0)el modelo que estamos estimando es P(Y = 1) = P(Decision1 = 1) = P(Decision = “Approve”).
m1 = glm(Decision1 ~ HomeownerY, family = binomial)
summary(m1)##
## Call:
## glm(formula = Decision1 ~ HomeownerY, family = binomial)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.3514 0.7400 -3.177 0.00149 **
## HomeownerY 3.6041 0.8729 4.129 3.64e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 68.994 on 49 degrees of freedom
## Residual deviance: 42.194 on 48 degrees of freedom
## AIC: 46.194
##
## Number of Fisher Scoring iterations: 5Como \(\beta_1 > 0\) quiere decir que P(Decision = Approve|Homeowner = N) < P(Decision = Approve|Homeowner = Y).
HomeownerN = ifelse(d$Homeowner == "N", 1, 0)m2 = glm(Decision1 ~ HomeownerN, family = binomial)
summary(m2)##
## Call:
## glm(formula = Decision1 ~ HomeownerN, family = binomial)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.2528 0.4629 2.706 0.0068 **
## HomeownerN -3.6041 0.8729 -4.129 3.64e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 68.994 on 49 degrees of freedom
## Residual deviance: 42.194 on 48 degrees of freedom
## AIC: 46.194
##
## Number of Fisher Scoring iterations: 5El modelo que se estima ahora es:
\[ P(Decision_i = 1) = \frac{exp(\beta_0^* + \beta_1^* HomeownerY_i)}{1 + exp(\beta_0^* + \beta_1^* HomeownerY_i)} \]
Que equivale a dos modelos:
\[ P(Decision_i = 1) = \frac{exp(\beta_0^*)}{1 + exp(\beta_0^*)} \]
\[ P(Decision_i = 1) = \frac{exp(\beta_0^*+\beta_1^*)}{1 + exp(\beta_0^*+\beta_1^*)} \]
La probabilidad de que la Decision = 1 cuando el cliente es propietario de una casa tiene que ser igual en ambos modelos:
\[ P(Decision_i = 1 | HomeownerY = 1 ) = P(Decision_i = 1 | HomeownerN = 0 ) \]
\[ \frac{exp(\beta_0 + \beta_1)}{1 + exp(\beta_0 + \beta_1)} = \frac{exp(\beta_0^*)}{1 + exp(\beta_0^*)} \]
Por tanto
\[ \beta_0 + \beta_1 = \beta_0^* \Rightarrow -2.3514 + 3.6041 = 1.2528 \]
Por otro lado, la probabilidad de que la Decision = 1 cuando el cliente NO es propietario de una casa tiene que ser igual en ambos modelos:
\[ P(Decision_i = 1 | HomeownerY = 0 ) = P(Decision_i = 1 | HomeownerN = 1 ) \]
\[ \frac{exp(\beta_0)}{1 + exp(\beta_0)} = \frac{exp(\beta_0^*+\beta_1^*)}{1 + exp(\beta_0^*+\beta_1^*)} \]
Es decir
\[ \beta_0 = \beta_0^* + \beta_1^* \Rightarrow -2.3514 = 1.2528 - 3.6041 \]
Es importante utilizar bien los niveles de las variables:
levels(d$Decision)## [1] "Approve" "Reject"levels(d$Homeowner)## [1] "N" "Y"Por tanto, si estimamos el modelo:
m3 = glm(Decision ~ Homeowner, data = d, family = binomial)
summary(m3)##
## Call:
## glm(formula = Decision ~ Homeowner, family = binomial, data = d)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.3514 0.7400 3.177 0.00149 **
## HomeownerY -3.6041 0.8729 -4.129 3.64e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 68.994 on 49 degrees of freedom
## Residual deviance: 42.194 on 48 degrees of freedom
## AIC: 46.194
##
## Number of Fisher Scoring iterations: 5En el fondo estamos estimando P(Decision = Reject|Homeowner = “Y”), ya que
Si queremos estimar el modelo m1 anterior tenemos que hacer:
d$Decision = relevel(d$Decision, ref = "Reject")
m4 = glm(Decision ~ Homeowner, data = d, family = binomial)
summary(m4)##
## Call:
## glm(formula = Decision ~ Homeowner, family = binomial, data = d)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.3514 0.7400 -3.177 0.00149 **
## HomeownerY 3.6041 0.8729 4.129 3.64e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 68.994 on 49 degrees of freedom
## Residual deviance: 42.194 on 48 degrees of freedom
## AIC: 46.194
##
## Number of Fisher Scoring iterations: 5El funcionamiento es idéntico que el modelo de regresión lineal:
d$Decision = relevel(d$Decision, ref = "Approve")
m4 = glm(Decision ~ Homeowner + Credit_Score + Years_Credit_History, data = d, family = binomial)
summary(m4)##
## Call:
## glm(formula = Decision ~ Homeowner + Credit_Score + Years_Credit_History,
## family = binomial, data = d)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 38.05066 14.85133 2.562 0.0104 *
## HomeownerY -3.13441 1.73873 -1.803 0.0714 .
## Credit_Score -0.04946 0.01933 -2.559 0.0105 *
## Years_Credit_History -0.31716 0.21846 -1.452 0.1466
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 68.994 on 49 degrees of freedom
## Residual deviance: 15.208 on 46 degrees of freedom
## AIC: 23.208
##
## Number of Fisher Scoring iterations: 8Igual que en regresión lineal, se tienen que crear tantas variables auxiliares como niveles del factor menos una.