Examen N°03

Los datos presentados a continuación contienen información de los casos de personas que padecen cáncer en los 14 distritos de la provincia “Mundo Feliz” (5p).:

Casos: Cantidad de personas con cáncer por distrito.
Edad: Rango de edad de las personas que padecen cáncer en el distrito.
Área: Lugar donde ha residido el último año, urbano (1) o rural (0)

#Cargar los datos
datos <- read.table("CASOSCANCER.txt", header = T, sep = "\t")
attach(datos)
head(datos,5) #Observar los 5 datos de la base de datos

##   Casos  Edad Area
## 1    11 30-40    0
## 2    15 40-50    1
## 3    21 40-50    1
## 4    25 30-40    0
## 5    24 40-50    1

Para no tener mayor inconveniente con las variables categoricas o “CHR”, vamos hacer las transformaciones siguientes para edad.

De 20 a 30 (1)
De 30 a 40 (2)
De 40 a 50 (3)
De 50 a 60 (4)
De 60 a 70 (5)

datos$Edad <- ifelse(datos$Edad=="20-30",1,
                      ifelse(datos$Edad=="30-40",2,
                             ifelse(datos$Edad=="40-50",3,
                                    ifelse(datos$Edad=="50-60",4,5))))
head(datos,5) #Observar los 5 datos de la base de datos

##   Casos Edad Area
## 1    11    2    0
## 2    15    3    1
## 3    21    3    1
## 4    25    2    0
## 5    24    3    1

De acuerdo a los datos, Se pide:

Regresión poisson clásico

Estimar el modelo de regresión poisson clásico e interpretar los parámetros.

# Ajusta el modelo de regresión de Poisson
modelo_poisson <- glm(Casos ~   Edad +  Area, data = datos, family = "poisson")

# Obtén un resumen de los resultados del modelo
summary(modelo_poisson)

## 
## Call:
## glm(formula = Casos ~ Edad + Area, family = "poisson", data = datos)
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  2.29234    0.18497  12.393  < 2e-16 ***
## Edad         0.14860    0.05113   2.906  0.00366 ** 
## Area         0.10768    0.14238   0.756  0.44947    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 29.477  on 13  degrees of freedom
## Residual deviance: 20.512  on 11  degrees of freedom
## AIC: 90.098
## 
## Number of Fisher Scoring iterations: 4

Nota. Los coeficiente del modelo de regresión poisson clasico, nos expresan los siguiente: * B(Edad)=0.1486 : Nos da ha entender que, si la edad aumenta, y el área donde a recidido los ultimos años se mantiene constantes, Cantidad de personas con cáncer por distrito aumentaría. * B(Área)=0.10768 : Nos da ha entender que, el área donde ha residido el último año sea urbano y la edad se mantienen constantes, los casos de covid por día aumentaría.

Evaluar los supuestos del modelo.

#Evaluación de supuestos
# Verifica la bondad de ajuste del modelo
#Bondad de ajuste del modelo
plot(modelo_poisson, which = 1)

Nota. Se observa que los datos no se ajustan a la recta, lo que nos da ha entender que el modelo es no es bueno.

library(AER)

## Loading required package: car

## Loading required package: carData

## Loading required package: lmtest

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## Loading required package: sandwich

## Loading required package: survival

#Examina la sobredispersión o subdispersión:
dispersion_test <- dispersiontest(modelo_poisson)
dispersion_test

## 
##  Overdispersion test
## 
## data:  modelo_poisson
## z = 0.87546, p-value = 0.1907
## alternative hypothesis: true dispersion is greater than 1
## sample estimates:
## dispersion 
##   1.570365

Nota. De acuerdo a la dispersión de los datos, se observa que, es no significativo, p-value mayor al 5%, dando a entender que la verdadera dispersión es menor que 1.

Regresión poisson bayesiano

Considerando los datos del anterior ejercicio:

Estimar el modelo de regresión bayesiano.

#BAYESIANO
# install.packages("brms")
# Carga el paquete brms
library(brms)

## Loading required package: Rcpp

## Loading 'brms' package (version 2.20.3). Useful instructions
## can be found by typing help('brms'). A more detailed introduction
## to the package is available through vignette('brms_overview').

## 
## Attaching package: 'brms'

## The following object is masked from 'package:survival':
## 
##     kidney

## The following object is masked from 'package:stats':
## 
##     ar

# Utiliza la función brm() para ajustar el modelo
model <- brm(Casos ~ Edad + Area, family = poisson(), data = datos)

## Compiling Stan program...

## Start sampling

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# Resumen del modelo
summary(model)

##  Family: poisson 
##   Links: mu = log 
## Formula: Casos ~ Edad + Area 
##    Data: datos (Number of observations: 14) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept     2.28      0.18     1.91     2.63 1.00     3242     2682
## Edad          0.15      0.05     0.05     0.25 1.00     3499     2821
## Area          0.11      0.15    -0.17     0.41 1.00     3377     2274
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Nota. Se observa que los coeficiente del modelo de regresión poisson bayesiano.

Entonces: el modelo de regresión bayesiano seria: ln(y)=alfa + beta1(x1) + beta2(x2)

ln(y) = 2.28 + 0.15x1 + 0.11x2

A MODO PRÁCTICA

Regresión poisson bayesiano

Rojas Rojas Henry

2023-09-24

Examen N°03

Regresión poisson clásico

Regresión poisson bayesiano