Introducción

Este trabajo exploratorio es un primer ejercicio de acercamiento al uso de R y Rstudio en el análisis de datos. La base de datos elegida atiende a las indicaciones del ejercicio y buscamos poner en práctica los conocimientos adquiridos en la asignatura.

El análisis de la base de datos Housing pretende dará luz sobre los principales factores que incrementan el valor de una vivienda.

Objetivos

Inspeccionar y preparar la base de datos para su análisis Analizar las variables de interés para determinar su influencia en el aumento del precio de las viviendas

Análisis de estadística descriptiva

La base de datos Housing contiene 545 observaciones y consta de 13 variables que se mencionan a continuación:

  1. Price: The price of the house.
  2. Area: The total area of the house in square feet.
  3. Bedrooms: The number of bedrooms in the house.
  4. Bathrooms: The number of bathrooms in the house.
  5. Stories: The number of stories in the house.
  6. Mainroad: Whether the house is connected to the main road (Yes/No).
  7. Guestroom: Whether the house has a guest room (Yes/No).
  8. Basement: Whether the house has a basement (Yes/No).
  9. Hot water heating: Whether the house has a hot water heating system (Yes/No).
  10. Airconditioning: Whether the house has an air conditioning system (Yes/No).
  11. Parking: The number of parking spaces available within the house.
  12. Prefarea: Whether the house is located in a preferred area (Yes/No).
  13. Furnishing status: The furnishing status of the house (Fully Furnished, Semi-Furnished, Unfurnished)

Exploración de la base de datos

Preparación de RStudio: llamando librerías

library(dplyr)
library(DescTools)
library(ggplot2)
library(moments)
library(corrplot)
library(car)

Carga de la base de datos

housing = read.csv("Housing.csv", header = TRUE, sep=",")

Inspección

head(housing, 8)
##      price  area bedrooms bathrooms stories mainroad guestroom basement
## 1 13300000  7420        4         2       3      yes        no       no
## 2 12250000  8960        4         4       4      yes        no       no
## 3 12250000  9960        3         2       2      yes        no      yes
## 4 12215000  7500        4         2       2      yes        no      yes
## 5 11410000  7420        4         1       2      yes       yes      yes
## 6 10850000  7500        3         3       1      yes        no      yes
## 7 10150000  8580        4         3       4      yes        no       no
## 8 10150000 16200        5         3       2      yes        no       no
##   hotwaterheating airconditioning parking prefarea furnishingstatus
## 1              no             yes       2      yes        furnished
## 2              no             yes       3       no        furnished
## 3              no              no       2      yes   semi-furnished
## 4              no             yes       3      yes        furnished
## 5              no             yes       2       no        furnished
## 6              no             yes       2      yes   semi-furnished
## 7              no             yes       2      yes   semi-furnished
## 8              no              no       0       no      unfurnished
tail(housing, 10) 
##       price area bedrooms bathrooms stories mainroad guestroom basement
## 536 2100000 3360        2         1       1      yes        no       no
## 537 1960000 3420        5         1       2       no        no       no
## 538 1890000 1700        3         1       2      yes        no       no
## 539 1890000 3649        2         1       1      yes        no       no
## 540 1855000 2990        2         1       1       no        no       no
## 541 1820000 3000        2         1       1      yes        no      yes
## 542 1767150 2400        3         1       1       no        no       no
## 543 1750000 3620        2         1       1      yes        no       no
## 544 1750000 2910        3         1       1       no        no       no
## 545 1750000 3850        3         1       2      yes        no       no
##     hotwaterheating airconditioning parking prefarea furnishingstatus
## 536              no              no       1       no      unfurnished
## 537              no              no       0       no      unfurnished
## 538              no              no       0       no      unfurnished
## 539              no              no       0       no      unfurnished
## 540              no              no       1       no      unfurnished
## 541              no              no       2       no      unfurnished
## 542              no              no       0       no   semi-furnished
## 543              no              no       0       no      unfurnished
## 544              no              no       0       no        furnished
## 545              no              no       0       no      unfurnished
dim(housing)
## [1] 545  13
class(housing)
## [1] "data.frame"
str(housing)
## 'data.frame':    545 obs. of  13 variables:
##  $ price           : int  13300000 12250000 12250000 12215000 11410000 10850000 10150000 10150000 9870000 9800000 ...
##  $ area            : int  7420 8960 9960 7500 7420 7500 8580 16200 8100 5750 ...
##  $ bedrooms        : int  4 4 3 4 4 3 4 5 4 3 ...
##  $ bathrooms       : int  2 4 2 2 1 3 3 3 1 2 ...
##  $ stories         : int  3 4 2 2 2 1 4 2 2 4 ...
##  $ mainroad        : chr  "yes" "yes" "yes" "yes" ...
##  $ guestroom       : chr  "no" "no" "no" "no" ...
##  $ basement        : chr  "no" "no" "yes" "yes" ...
##  $ hotwaterheating : chr  "no" "no" "no" "no" ...
##  $ airconditioning : chr  "yes" "yes" "no" "yes" ...
##  $ parking         : int  2 3 2 3 2 2 2 0 2 1 ...
##  $ prefarea        : chr  "yes" "no" "yes" "yes" ...
##  $ furnishingstatus: chr  "furnished" "furnished" "semi-furnished" "furnished" ...
sum(complete.cases(housing))
## [1] 545
summary(housing)
##      price               area          bedrooms       bathrooms    
##  Min.   : 1750000   Min.   : 1650   Min.   :1.000   Min.   :1.000  
##  1st Qu.: 3430000   1st Qu.: 3600   1st Qu.:2.000   1st Qu.:1.000  
##  Median : 4340000   Median : 4600   Median :3.000   Median :1.000  
##  Mean   : 4766729   Mean   : 5151   Mean   :2.965   Mean   :1.286  
##  3rd Qu.: 5740000   3rd Qu.: 6360   3rd Qu.:3.000   3rd Qu.:2.000  
##  Max.   :13300000   Max.   :16200   Max.   :6.000   Max.   :4.000  
##     stories        mainroad          guestroom           basement        
##  Min.   :1.000   Length:545         Length:545         Length:545        
##  1st Qu.:1.000   Class :character   Class :character   Class :character  
##  Median :2.000   Mode  :character   Mode  :character   Mode  :character  
##  Mean   :1.806                                                           
##  3rd Qu.:2.000                                                           
##  Max.   :4.000                                                           
##  hotwaterheating    airconditioning       parking         prefarea        
##  Length:545         Length:545         Min.   :0.0000   Length:545        
##  Class :character   Class :character   1st Qu.:0.0000   Class :character  
##  Mode  :character   Mode  :character   Median :0.0000   Mode  :character  
##                                        Mean   :0.6936                     
##                                        3rd Qu.:1.0000                     
##                                        Max.   :3.0000                     
##  furnishingstatus  
##  Length:545        
##  Class :character  
##  Mode  :character  
##                    
##                    
##

Convirtiendo a factores

Las variables se transformaran a factores para un mejor manejo. Las variables bathrooms, stories y parking se convertiran a as.numeric.

housing$bedrooms <- factor(housing$bedrooms)
housing$bathrooms <- as.numeric(housing$bathrooms)
housing$stories <- as.numeric(housing$stories)
housing$mainroad <- factor(housing$mainroad)
housing$guestroom <- factor(housing$guestroom)
housing$basement <- factor(housing$basement)
housing$hotwaterheating <- factor(housing$hotwaterheating)
housing$airconditioning <- factor(housing$airconditioning)
housing$parking <- as.numeric(housing$parking)
housing$prefarea <- factor(housing$prefarea)
housing$furnishingstatus <- factor(housing$furnishingstatus)

Comprobamos la estructura:

str(housing)
## 'data.frame':    545 obs. of  13 variables:
##  $ price           : int  13300000 12250000 12250000 12215000 11410000 10850000 10150000 10150000 9870000 9800000 ...
##  $ area            : int  7420 8960 9960 7500 7420 7500 8580 16200 8100 5750 ...
##  $ bedrooms        : Factor w/ 6 levels "1","2","3","4",..: 4 4 3 4 4 3 4 5 4 3 ...
##  $ bathrooms       : num  2 4 2 2 1 3 3 3 1 2 ...
##  $ stories         : num  3 4 2 2 2 1 4 2 2 4 ...
##  $ mainroad        : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
##  $ guestroom       : Factor w/ 2 levels "no","yes": 1 1 1 1 2 1 1 1 2 2 ...
##  $ basement        : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 1 1 2 1 ...
##  $ hotwaterheating : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
##  $ airconditioning : Factor w/ 2 levels "no","yes": 2 2 1 2 2 2 2 1 2 2 ...
##  $ parking         : num  2 3 2 3 2 2 2 0 2 1 ...
##  $ prefarea        : Factor w/ 2 levels "no","yes": 2 1 2 2 1 2 2 1 2 2 ...
##  $ furnishingstatus: Factor w/ 3 levels "furnished","semi-furnished",..: 1 1 2 1 1 2 2 3 1 3 ...
summary(housing)
##      price               area       bedrooms   bathrooms        stories     
##  Min.   : 1750000   Min.   : 1650   1:  2    Min.   :1.000   Min.   :1.000  
##  1st Qu.: 3430000   1st Qu.: 3600   2:136    1st Qu.:1.000   1st Qu.:1.000  
##  Median : 4340000   Median : 4600   3:300    Median :1.000   Median :2.000  
##  Mean   : 4766729   Mean   : 5151   4: 95    Mean   :1.286   Mean   :1.806  
##  3rd Qu.: 5740000   3rd Qu.: 6360   5: 10    3rd Qu.:2.000   3rd Qu.:2.000  
##  Max.   :13300000   Max.   :16200   6:  2    Max.   :4.000   Max.   :4.000  
##  mainroad  guestroom basement  hotwaterheating airconditioning    parking      
##  no : 77   no :448   no :354   no :520         no :373         Min.   :0.0000  
##  yes:468   yes: 97   yes:191   yes: 25         yes:172         1st Qu.:0.0000  
##                                                                Median :0.0000  
##                                                                Mean   :0.6936  
##                                                                3rd Qu.:1.0000  
##                                                                Max.   :3.0000  
##  prefarea        furnishingstatus
##  no :417   furnished     :140    
##  yes:128   semi-furnished:227    
##            unfurnished   :178    
##                                  
##                                  
##

Análisis de la variable precio

str(housing$price)
##  int [1:545] 13300000 12250000 12250000 12215000 11410000 10850000 10150000 10150000 9870000 9800000 ...
class(housing$price)
## [1] "integer"
summary(housing$price)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##  1750000  3430000  4340000  4766729  5740000 13300000

El análisis muestra que es una variable cuantitativa continua.

Medidas de tendencia central de la variable “Precio”

mean(housing$price) 
## [1] 4766729
median(housing$price) 
## [1] 4340000

Medidas de dispersión

var(housing$price) 
## [1] 3.498544e+12
sd(housing$price) 
## [1] 1870440
IQR(housing$price) 
## [1] 2310000
range(housing$price)
## [1]  1750000 13300000

Medidas de posición

quantile(housing$price)
##       0%      25%      50%      75%     100% 
##  1750000  3430000  4340000  5740000 13300000

Esto nos indica que el 50% de nuestros datos se encuentra entre 3,430,000 y 5,740,000 por el máximo de precio es probable tengamos valores atípicos y un sesgo a la derecha.

Para confirmar esto aplicamos medidas de forma.

Medidas de forma

skewness(housing$price) 
## [1] 1.2089
kurtosis(housing$price) 
## [1] 4.931205

Tabla de frecuencias

Preparamos los elementos requeridos para armar una tabla de frecuencias.

k <- nclass.Sturges(housing$price) # Número de clases
ac <- (max(housing$price)-min(housing$price))/k # Calculamos el ancho
bins <- seq(min(housing$price), max(housing$price), by = ac) # Secuencia del valor mínimo al máximo
precio.clases <- cut(housing$price, breaks = bins, include.lowest=TRUE, dig.lab = 8)

Creamos ahora la tabla de frecuencias:

dist.freq <- table(precio.clases)
dist.freq
## precio.clases
##   [1750000,2800000]   (2800000,3850000]   (3850000,4900000]   (4900000,5950000] 
##                  53                 152                 140                  83 
##   (5950000,7000000]   (7000000,8050000]   (8050000,9100000]  (9100000,10150000] 
##                  53                  28                  21                   9 
## (10150000,11200000] (11200000,12250000] (12250000,13300000] 
##                   1                   4                   1
transform(dist.freq, 
          rel.freq=prop.table(Freq),
          cum.freq=cumsum(Freq),
          cum.rel.freq=cumsum(prop.table(Freq)))
##          precio.clases Freq    rel.freq cum.freq cum.rel.freq
## 1    [1750000,2800000]   53 0.097247706       53   0.09724771
## 2    (2800000,3850000]  152 0.278899083      205   0.37614679
## 3    (3850000,4900000]  140 0.256880734      345   0.63302752
## 4    (4900000,5950000]   83 0.152293578      428   0.78532110
## 5    (5950000,7000000]   53 0.097247706      481   0.88256881
## 6    (7000000,8050000]   28 0.051376147      509   0.93394495
## 7    (8050000,9100000]   21 0.038532110      530   0.97247706
## 8   (9100000,10150000]    9 0.016513761      539   0.98899083
## 9  (10150000,11200000]    1 0.001834862      540   0.99082569
## 10 (11200000,12250000]    4 0.007339450      544   0.99816514
## 11 (12250000,13300000]    1 0.001834862      545   1.00000000

Gráficas: histograma y boxplot

Generamos ahora histograma y gráfica boxplot.

hist(housing$price, main = "Histograma Precio", xlab="Precio",col = "darkblue")

boxplot(housing$price, horizontal = FALSE,col = "white",border=4)
points (mean(housing$price), col = 3, pch = 19)

Primeros hallazgos

Los precios de la vivienda fluctuan entre $1,750,000 y $13,300,000. El 50% de las viviendas en venta se encuentran en el rango de $3,430,000 a $5,740,000 Se indentificaron valores atípicos que podrían estar afectando el modelo de análisis. Estos valores serán tratados para mejorar el modelo. La distribución de la variable muestra un sesgo a la derecha

Análisis de variables categóricas

Llevaremos a cabo un análisis de las variables categóricas:

Mainroad

table1::table1(~mainroad, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$mainroad,xlab=“mainroad”)

Overall
(N=545)
mainroad
no 77 (14.1%)
yes 468 (85.9%)

Guestroom

table1::table1(~guestroom, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$guestroom,xlab=“guestroom”)

Overall
(N=545)
guestroom
no 448 (82.2%)
yes 97 (17.8%)

Basement

table1::table1(~basement, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$basement,xlab=“basement”)

Overall
(N=545)
basement
no 354 (65.0%)
yes 191 (35.0%)

Airconditioning

table1::table1(~airconditioning, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$basement,xlab=“airconditioning”)

Overall
(N=545)
airconditioning
no 373 (68.4%)
yes 172 (31.6%)

Hot water heating

table1::table1(~hotwaterheating, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$basement,xlab=“hotwaterheating”)

Overall
(N=545)
hotwaterheating
no 520 (95.4%)
yes 25 (4.6%)

Preferred area

table1::table1(~prefarea, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$basement,xlab=“prefarea”)

Overall
(N=545)
prefarea
no 417 (76.5%)
yes 128 (23.5%)

Furnishing house

table1::table1(~furnishingstatus, data = housing, na.rm = TRUE, digits = 1, format.number = TRUE) plot(housing$furnishingstatus,xlab=“furnishing”)

Overall
(N=545)
furnishingstatus
furnished 140 (25.7%)
semi-furnished 227 (41.7%)
unfurnished 178 (32.7%)

Análisis de cada variable vs precio (price)

  • boxplot(price~mainroad,data = housing)
  • boxplot(price~guestroom,data = housing)
  • boxplot(price~basement,data = housing)
  • boxplot(price~airconditioning,data = housing)
  • boxplot(price~hotwaterheating,data = housing)
  • boxplot(price~prefarea,data = housing)
  • boxplot(price~furnishingstatus,data = housing)

Del análisis se concluye que la inclusión de los servicios y comodidades influye en un precio de venta más elevado.

Para confirmar esto llevamos a cabo la ejecución de t.test para 3 variables teniendo en cuenta:

  • H0: M1=M2, por lo tanto son independientes
  • H1: M1=!M2, por lo tanto son dependientes
t.test(price~mainroad,data = housing)
## 
##  Welch Two Sample t-test
## 
## data:  price by mainroad
## t = -11.853, df = 210.68, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group no and group yes is not equal to 0
## 95 percent confidence interval:
##  -1857781 -1327964
## sample estimates:
##  mean in group no mean in group yes 
##           3398905           4991777

2.2e-16, p-value < 0.05 de significancia, por lo tanto, se rechaza H0, las variables son dependientes

t.test(price~guestroom,data = housing)
## 
##  Welch Two Sample t-test
## 
## data:  price by guestroom
## t = -6.3847, df = 146.28, p-value = 2.144e-09
## alternative hypothesis: true difference in means between group no and group yes is not equal to 0
## 95 percent confidence interval:
##  -1634767.3  -861935.3
## sample estimates:
##  mean in group no mean in group yes 
##           4544546           5792897

2.144e-09, p-value < 0.05 de significancia, por lo tanto, se rechaza H0, las variables son dependientes

t.test(price~prefarea,data = housing)
## 
##  Welch Two Sample t-test
## 
## data:  price by prefarea
## t = -7.4923, df = 187.47, p-value = 2.592e-12
## alternative hypothesis: true difference in means between group no and group yes is not equal to 0
## 95 percent confidence interval:
##  -1836514 -1070980
## sample estimates:
##  mean in group no mean in group yes 
##           4425299           5879046

2.592e-12, p-value < 0.05 de significancia, por lo tanto, se rechaza H0, las variables son dependientes

Análisis de variables discretas

str(housing)
## 'data.frame':    545 obs. of  13 variables:
##  $ price           : int  13300000 12250000 12250000 12215000 11410000 10850000 10150000 10150000 9870000 9800000 ...
##  $ area            : int  7420 8960 9960 7500 7420 7500 8580 16200 8100 5750 ...
##  $ bedrooms        : Factor w/ 6 levels "1","2","3","4",..: 4 4 3 4 4 3 4 5 4 3 ...
##  $ bathrooms       : num  2 4 2 2 1 3 3 3 1 2 ...
##  $ stories         : num  3 4 2 2 2 1 4 2 2 4 ...
##  $ mainroad        : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
##  $ guestroom       : Factor w/ 2 levels "no","yes": 1 1 1 1 2 1 1 1 2 2 ...
##  $ basement        : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 1 1 2 1 ...
##  $ hotwaterheating : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
##  $ airconditioning : Factor w/ 2 levels "no","yes": 2 2 1 2 2 2 2 1 2 2 ...
##  $ parking         : num  2 3 2 3 2 2 2 0 2 1 ...
##  $ prefarea        : Factor w/ 2 levels "no","yes": 2 1 2 2 1 2 2 1 2 2 ...
##  $ furnishingstatus: Factor w/ 3 levels "furnished","semi-furnished",..: 1 1 2 1 1 2 2 3 1 3 ...
freq_bedrooms <- table(housing$bedrooms)
freq_bedrooms
## 
##   1   2   3   4   5   6 
##   2 136 300  95  10   2
transform(freq_bedrooms, 
          rel.freq=prop.table(Freq),
          cum.freq=cumsum(Freq),
          cum.rel.freq=cumsum(prop.table(Freq)))
##   Var1 Freq    rel.freq cum.freq cum.rel.freq
## 1    1    2 0.003669725        2  0.003669725
## 2    2  136 0.249541284      138  0.253211009
## 3    3  300 0.550458716      438  0.803669725
## 4    4   95 0.174311927      533  0.977981651
## 5    5   10 0.018348624      543  0.996330275
## 6    6    2 0.003669725      545  1.000000000
barplot(freq_bedrooms, xlab = "Bedrooms", col = "#10454F", main = "Bedrooms")

freq_bathrooms <- table(housing$bathrooms)
freq_bathrooms
## 
##   1   2   3   4 
## 401 133  10   1
transform(freq_bathrooms, 
          rel.freq=prop.table(Freq),
          cum.freq=cumsum(Freq),
          cum.rel.freq=cumsum(prop.table(Freq)))
##   Var1 Freq    rel.freq cum.freq cum.rel.freq
## 1    1  401 0.735779817      401    0.7357798
## 2    2  133 0.244036697      534    0.9798165
## 3    3   10 0.018348624      544    0.9981651
## 4    4    1 0.001834862      545    1.0000000
barplot(freq_bathrooms, xlab = "Bathrooms", col = "#10454F", main = "Bathrooms")

freq_stories <- table(housing$stories)
freq_stories
## 
##   1   2   3   4 
## 227 238  39  41
transform(freq_stories, 
          rel.freq=prop.table(Freq),
          cum.freq=cumsum(Freq),
          cum.rel.freq=cumsum(prop.table(Freq)))
##   Var1 Freq   rel.freq cum.freq cum.rel.freq
## 1    1  227 0.41651376      227    0.4165138
## 2    2  238 0.43669725      465    0.8532110
## 3    3   39 0.07155963      504    0.9247706
## 4    4   41 0.07522936      545    1.0000000
barplot(freq_stories, xlab = "Stories", col = "#10454F", main = "Bathrooms")

freq_parking <- table(housing$parking)
freq_parking
## 
##   0   1   2   3 
## 299 126 108  12
transform(freq_parking, 
          rel.freq=prop.table(Freq),
          cum.freq=cumsum(Freq),
          cum.rel.freq=cumsum(prop.table(Freq)))
##   Var1 Freq   rel.freq cum.freq cum.rel.freq
## 1    0  299 0.54862385      299    0.5486239
## 2    1  126 0.23119266      425    0.7798165
## 3    2  108 0.19816514      533    0.9779817
## 4    3   12 0.02201835      545    1.0000000
barplot(freq_parking, xlab = "Parking", col = "#10454F", main = "Parking")

Analizando su relación con el precio

  • boxplot(housing$price~housing$bedrooms)
  • boxplot(housing$price~housing$bathrooms)
  • boxplot(housing$price~housing$parking)
  • boxplot(housing$price~housing$stories)

Se observa una variación en el precio a medida que aumentan el número de servicios y habitaciones disponibles en la vivienda. Persisten también los casos atípicos.

Relación de correlación

Objetivo: Identificar relaciones de correlación entre al menos dos variables y plantear un problema de regresión.

Selección de variables y matriz de correlación

housing.select<-select(housing,price,area)
round(cor(housing.select),4)
##       price  area
## price 1.000 0.536
## area  0.536 1.000

Graficando correlación

pairs(~ price+area, 
      data=housing, gap=.4, cex_labels=1.5)

corrplot(cor(housing.select))

La correlación observada es positiva.

Modelo de regresión lineal múltiple

attach(housing)

model1 <- lm(price ~ area+bedrooms+bathrooms+stories+parking+mainroad+guestroom
                      +basement+hotwaterheating+airconditioning+prefarea+furnishingstatus)
summary(model1)
## 
## Call:
## lm(formula = price ~ area + bedrooms + bathrooms + stories + 
##     parking + mainroad + guestroom + basement + hotwaterheating + 
##     airconditioning + prefarea + furnishingstatus)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2603715  -652407   -78857   515376  5210924 
## 
## Coefficients:
##                                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                     357446.85  775559.37   0.461 0.645068    
## area                               244.65      24.48   9.992  < 2e-16 ***
## bedrooms2                      -114030.93  766523.74  -0.149 0.881797    
## bedrooms3                        44379.14  767625.38   0.058 0.953919    
## bedrooms4                        95121.74  776261.47   0.123 0.902519    
## bedrooms5                       225794.30  840612.20   0.269 0.788337    
## bedrooms6                       725852.04 1075980.80   0.675 0.500228    
## bathrooms                       995395.42  104634.07   9.513  < 2e-16 ***
## stories                         447940.57   66210.11   6.765 3.54e-11 ***
## parking                         277396.50   58975.46   4.704 3.27e-06 ***
## mainroadyes                     420192.91  143761.60   2.923 0.003617 ** 
## guestroomyes                    302160.05  132199.82   2.286 0.022672 *  
## basementyes                     348044.60  111709.52   3.116 0.001936 ** 
## hotwaterheatingyes              855162.11  225019.54   3.800 0.000161 ***
## airconditioningyes              865512.39  108940.97   7.945 1.19e-14 ***
## prefareayes                     646125.01  117161.83   5.515 5.48e-08 ***
## furnishingstatussemi-furnished  -42608.78  117741.15  -0.362 0.717583    
## furnishingstatusunfurnished    -412029.93  126841.54  -3.248 0.001235 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1071000 on 527 degrees of freedom
## Multiple R-squared:  0.6822, Adjusted R-squared:  0.6719 
## F-statistic: 66.54 on 17 and 527 DF,  p-value: < 2.2e-16
anova(model1)
## Analysis of Variance Table
## 
## Response: price
##                   Df     Sum Sq    Mean Sq  F value    Pr(>F)    
## area               1 5.4678e+14 5.4678e+14 476.3753 < 2.2e-16 ***
## bedrooms           5 1.8368e+14 3.6736e+13  32.0062 < 2.2e-16 ***
## bathrooms          1 2.2437e+14 2.2437e+14 195.4817 < 2.2e-16 ***
## stories            1 7.4811e+13 7.4811e+13  65.1784 4.671e-15 ***
## parking            1 4.6539e+13 4.6539e+13  40.5467 4.183e-10 ***
## mainroad           1 2.4823e+13 2.4823e+13  21.6268 4.192e-06 ***
## guestroom          1 3.3227e+13 3.3227e+13  28.9489 1.120e-07 ***
## basement           1 2.6272e+13 2.6272e+13  22.8895 2.231e-06 ***
## hotwaterheating    1 7.2006e+12 7.2006e+12   6.2734  0.012557 *  
## airconditioning    1 7.8517e+13 7.8517e+13  68.4068 1.092e-15 ***
## prefarea           1 3.5577e+13 3.5577e+13  30.9964 4.123e-08 ***
## furnishingstatus   2 1.6523e+13 8.2616e+12   7.1979  0.000824 ***
## Residuals        527 6.0489e+14 1.1478e+12                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

El R2 nos indica qie el % de la variación del precio que es explicado por el modelo, es decir, el 68.89% de la variación de precio es explicada por el modelo.

Adicional nos indica que a un nivel de confianza del 95% la variable bedrooms no resulta significativa para el modelo’

Ejecutamos nuevamente el modelo sin variable bedrooms:

model2 <- update(model1,~.-bedrooms)
summary(model2)
## 
## Call:
## lm(formula = price ~ area + bathrooms + stories + parking + mainroad + 
##     guestroom + basement + hotwaterheating + airconditioning + 
##     prefarea + furnishingstatus)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2640857  -681531   -67385   495051  5210247 
## 
## Coefficients:
##                                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                     268518.21  222746.65   1.205 0.228551    
## area                               247.17      24.25  10.194  < 2e-16 ***
## bathrooms                      1024077.94  100906.12  10.149  < 2e-16 ***
## stories                         486503.39   60160.58   8.087 4.16e-15 ***
## parking                         283577.52   58464.89   4.850 1.62e-06 ***
## mainroadyes                     393735.49  141352.60   2.785 0.005535 ** 
## guestroomyes                    295836.20  131862.32   2.244 0.025274 *  
## basementyes                     374040.80  109394.16   3.419 0.000676 ***
## hotwaterheatingyes              861783.03  223431.02   3.857 0.000129 ***
## airconditioningyes              863270.43  108501.89   7.956 1.07e-14 ***
## prefareayes                     655385.53  115819.75   5.659 2.50e-08 ***
## furnishingstatussemi-furnished  -44100.39  116729.66  -0.378 0.705730    
## furnishingstatusunfurnished    -417991.06  126315.89  -3.309 0.000999 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1069000 on 532 degrees of freedom
## Multiple R-squared:  0.6803, Adjusted R-squared:  0.6731 
## F-statistic: 94.34 on 12 and 532 DF,  p-value: < 2.2e-16
anova(model2)
## Analysis of Variance Table
## 
## Response: price
##                   Df     Sum Sq    Mean Sq F value    Pr(>F)    
## area               1 5.4678e+14 5.4678e+14 478.079 < 2.2e-16 ***
## bathrooms          1 3.3838e+14 3.3838e+14 295.861 < 2.2e-16 ***
## stories            1 1.2531e+14 1.2531e+14 109.564 < 2.2e-16 ***
## parking            1 5.2007e+13 5.2007e+13  45.473 4.043e-11 ***
## mainroad           1 2.1928e+13 2.1928e+13  19.173 1.438e-05 ***
## guestroom          1 3.6165e+13 3.6165e+13  31.621 3.030e-08 ***
## basement           1 3.3160e+13 3.3160e+13  28.994 1.091e-07 ***
## hotwaterheating    1 7.2019e+12 7.2019e+12   6.297 0.0123900 *  
## airconditioning    1 7.9306e+13 7.9306e+13  69.341 7.048e-16 ***
## prefarea           1 3.7419e+13 3.7419e+13  32.717 1.780e-08 ***
## furnishingstatus   2 1.7110e+13 8.5549e+12   7.480 0.0006256 ***
## Residuals        532 6.0845e+14 1.1437e+12                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

El modelo no muestra una mejora significativa en el R2, nos quedamos con model1.

Validación de colinealidad

vif(model1)
##                      GVIF Df GVIF^(1/(2*Df))
## area             1.337976  1        1.156709
## bedrooms         1.641134  5        1.050786
## bathrooms        1.310094  1        1.144593
## stories          1.563568  1        1.250427
## parking          1.223704  1        1.106211
## mainroad         1.190594  1        1.091143
## guestroom        1.214095  1        1.101860
## basement         1.348833  1        1.161393
## hotwaterheating  1.052265  1        1.025800
## airconditioning  1.217197  1        1.103267
## prefarea         1.171277  1        1.082255
## furnishingstatus 1.132983  2        1.031706

No se identifica colinealidad entre variables.

Mejora del modelo: outliers y puntos de apalancamiento

Eliminando outliers para mejorar el modelo:

residuos=rstudent(model1)
summary(residuos)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -2.489156 -0.621834 -0.074590  0.000397  0.483670  5.047234
plot(residuos)

Eliminando residuos superiores a 3

outliers = which(residuos>2.5)
outliers
##  1  3  4  5 14 15 16 21 28 
##  1  3  4  5 14 15 16 21 28
housing.outliers= housing[-outliers,]
nrow((housing.outliers))
## [1] 536

Model3

model3 <- lm(price ~ area+bathrooms+stories+mainroad+guestroom
              +basement+hotwaterheating+airconditioning+parking+prefarea+furnishingstatus,
              data = housing.outliers)
summary(model3)
## 
## Call:
## lm(formula = price ~ area + bathrooms + stories + mainroad + 
##     guestroom + basement + hotwaterheating + airconditioning + 
##     parking + prefarea + furnishingstatus, data = housing.outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2552754  -609397   -31035   518760  2751079 
## 
## Coefficients:
##                                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                     370300.80  194333.73   1.905 0.057265 .  
## area                               232.38      21.27  10.927  < 2e-16 ***
## bathrooms                      1007014.90   88516.23  11.377  < 2e-16 ***
## stories                         466204.87   52381.33   8.900  < 2e-16 ***
## mainroadyes                     403684.31  122773.50   3.288 0.001077 ** 
## guestroomyes                    389659.78  115363.42   3.378 0.000785 ***
## basementyes                     298016.71   96220.46   3.097 0.002058 ** 
## hotwaterheatingyes              664822.11  201168.27   3.305 0.001016 ** 
## airconditioningyes              890033.41   94696.16   9.399  < 2e-16 ***
## parking                         198370.31   51521.16   3.850 0.000133 ***
## prefareayes                     603498.11  101808.02   5.928 5.58e-09 ***
## furnishingstatussemi-furnished  -16248.06  102482.08  -0.159 0.874088    
## furnishingstatusunfurnished    -344189.54  110107.64  -3.126 0.001871 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 928000 on 523 degrees of freedom
## Multiple R-squared:  0.7153, Adjusted R-squared:  0.7088 
## F-statistic: 109.5 on 12 and 523 DF,  p-value: < 2.2e-16

El nuevo modelo presenta una mejora de 3 puntos. Ahora explica el 72.8% de la variación.

Se eliminaron solo 9 registros del dataset, lo que representa solo el 1.6% de los registros.

Eliminando puntos de apalancamiento y creando nuevo modelo

apalancamiento=hatvalues(model3)
plot(apalancamiento)

puntosapa=which(apalancamiento>.08)
puntosapa
##  8 67 
##  4 58
housing.outliers.heat = housing.outliers[-puntosapa,] 
nrow(housing.outliers.heat)
## [1] 534
model4 <- lm(price ~ area+bathrooms+stories+parking+mainroad+guestroom
              +basement+hotwaterheating+airconditioning+prefarea,
              data = housing.outliers.heat)
summary(model4)
## 
## Call:
## lm(formula = price ~ area + bathrooms + stories + parking + mainroad + 
##     guestroom + basement + hotwaterheating + airconditioning + 
##     prefarea, data = housing.outliers.heat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2724277  -566034   -55210   535405  2844063 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        1.948e+05  1.705e+05   1.142 0.253958    
## area               2.219e+02  2.261e+01   9.814  < 2e-16 ***
## bathrooms          1.003e+06  9.034e+04  11.099  < 2e-16 ***
## stories            4.789e+05  5.286e+04   9.061  < 2e-16 ***
## parking            2.318e+05  5.230e+04   4.433 1.13e-05 ***
## mainroadyes        4.432e+05  1.235e+05   3.588 0.000364 ***
## guestroomyes       4.180e+05  1.167e+05   3.582 0.000373 ***
## basementyes        3.279e+05  9.720e+04   3.373 0.000799 ***
## hotwaterheatingyes 6.843e+05  2.072e+05   3.302 0.001025 ** 
## airconditioningyes 9.175e+05  9.558e+04   9.600  < 2e-16 ***
## prefareayes        6.234e+05  1.031e+05   6.046 2.82e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 936900 on 523 degrees of freedom
## Multiple R-squared:  0.7032, Adjusted R-squared:  0.6975 
## F-statistic: 123.9 on 10 and 523 DF,  p-value: < 2.2e-16

El modelo no representa mejora. Se mantiene el model3.

Supuestos de la regresión lineal

La regresión es un modelo estadístico paramétrico, lo que significa que involucra la existencia de parámetros asociados a una distribución específica.

  1. Se asume que los errores siguen una distribución normal; es decir, se espera que todas las distribuciones de errores se ajusten a una distribución normal.

  2. En cuanto a la endogeneidad, se establece que el error, que representa las variables no modeladas de manera explícita, no debe estar correlacionado con las variables explicativas.

Estimación de residuos estandarizados

StanRes2 <- rstandard(model3)

qqnorm(StanRes2) 
qqline(StanRes2)

Gráficamente se concluye que los residuos se distribuyen como una normal. Los puntos forman una línea de 45 grados y solo se observan variaciones en los extremos.

Prueba KS para residuos

ks.test(StanRes2,"pnorm")
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  StanRes2
## D = 0.046186, p-value = 0.203
## alternative hypothesis: two-sided
hist(StanRes2)

H0: la variable se distribuya como una normal H1: la variable no se distribuya como una normal

Dado que el p-value es mayor a 0.5 de significancia, entonces se acepta H0 y distribuye como normal.

Nuestra recta de regresión sería: y=370300.80+232.38 área + 1007014.90 bathrooms + 466204.87 stories + 403684.31 mainroadyes + 389659.78 guestroomsyes + 298016.71 basementsyes + 664822.11 hotwaterheatingyes + 890033.41 airconditioningyes + 198370.31 parking + 603498.11 prefareayes -16248.06 furnishingstatussemi-furnished -344189.54 furnishingstatusunfurnished

Predicciones

Una vez que hemos realizado las validaciones, podemos usar el modelo para hacer predicciones.

data <- data.frame(
  area= c(7510,5760,3500,8520),
  bathrooms= c(2,2,2,1),
  stories= c(2,4,2,1),
  mainroad=c("yes","yes","yes","yes"),
  guestroom=c("no","yes","no","no"),
  basement=c("yes","no","no","no"),
  hotwaterheating=c("no","no","yes","no"),
  airconditioning=c("yes","yes","no","yes"),
  parking=c(3,1,2,2),
  prefarea=c("yes","yes","no","no"),
  furnishingstatus=c("furnished","furnished","furnished","furnished")
)

predict(model3,newdata= data, interval= "confidence", level=0.95)
##       fit     lwr     upr
## 1 7852236 7526055 8178417
## 2 8072888 7734623 8411154
## 3 5595307 5144288 6046326
## 4 5513832 5234351 5793313

Conclusiones

El modelo identifica la influencia significativa de las variables analizadas en el precio de las viviendas.

Estos resultados pueden ser valiosos para entender las características específicas que contribuyen al aumento del valor de una propiedad y podrían ser considerados por agentes inmobiliarios, compradores o vendedores para tomar decisiones informadas sobre transacciones inmobiliarias ya que las predicciones se apoyan en los resultados del model3, mismo que explica el 72.8% de la variación del precio.

Es importante tener en cuenta que estas conclusiones se derivan del análisis específico de la base de datos proporcionada y deben considerarse en el contexto de ese conjunto de datos particular.