Anscombe Dataset Analysis

Import libraries

library(tidyverse)
library(gridExtra)

Load dataset

data <- anscombe
head(data, 10)

Descriptive Analysis

summary(data[,c(1,2,3,4)])

       x1             x2             x3             x4    
 Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8  
 1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8  
 Median : 9.0   Median : 9.0   Median : 9.0   Median : 8  
 Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9  
 3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8  
 Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19  

summary(data[,c(5,6,7,8)])

       y1               y2              y3              y4        
 Min.   : 4.260   Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
 1st Qu.: 6.315   1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
 Median : 7.580   Median :8.140   Median : 7.11   Median : 7.040  
 Mean   : 7.501   Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
 3rd Qu.: 8.570   3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
 Max.   :10.840   Max.   :9.260   Max.   :12.74   Max.   :12.500  

sapply(data, sd)

      x1       x2       x3       x4       y1       y2       y3       y4 
3.316625 3.316625 3.316625 3.316625 2.031568 2.031657 2.030424 2.030579 

Correlation Coefficient

cor(data$x1, data$y1)

[1] 0.8164205

cor(data$x2, data$y2)

[1] 0.8162365

cor(data$x3, data$y3)

[1] 0.8162867

cor(data$x4, data$y4)

[1] 0.8165214

Scatter Plots

p1 <- ggplot(data = data, aes(x=x1, y=y1)) + 
        geom_point(size=2)
p2 <- ggplot(data = data, aes(x=x2, y=y2)) + 
        geom_point(size=2)
p3 <- ggplot(data = data, aes(x=x3, y=y3)) + 
        geom_point(size=2)
p4 <- ggplot(data = data, aes(x=x4, y=y4)) + 
        geom_point(size=2)
grid.arrange(p1,p2,p3,p4, nrow=2, ncol=2)

Histograms

p1 <- ggplot(data = data, aes(x=x1)) + 
        geom_histogram(bins=5) 
p2 <- ggplot(data = data, aes(x=x2)) + 
        geom_histogram(bins=5) 
p3 <- ggplot(data = data, aes(x=x3)) + 
        geom_histogram(bins=5) 
p4 <- ggplot(data = data, aes(x=x4)) + 
        geom_histogram(bins=5) 
grid.arrange(p1,p2,p3,p4, nrow=2, ncol=2)

p1 <- ggplot(data = data, aes(x=y1)) + 
        geom_histogram(bins=5) 
p2 <- ggplot(data = data, aes(x=y2)) + 
        geom_histogram(bins=5) 
p3 <- ggplot(data = data, aes(x=y3)) + 
        geom_histogram(bins=5) 
p4 <- ggplot(data = data, aes(x=y4)) + 
        geom_histogram(bins=5) 
grid.arrange(p1,p2,p3,p4, nrow=2, ncol=2)

Regression Analysis

m1 <- lm(y1 ~ x1, data = data)
summary(m1)

Call:
lm(formula = y1 ~ x1, data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.92127 -0.45577 -0.04136  0.70941  1.83882 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   3.0001     1.1247   2.667  0.02573 * 
x1            0.5001     0.1179   4.241  0.00217 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.237 on 9 degrees of freedom
Multiple R-squared:  0.6665,    Adjusted R-squared:  0.6295 
F-statistic: 17.99 on 1 and 9 DF,  p-value: 0.00217

m2 <- lm(y2 ~ x2, data = data)
summary(m2)

Call:
lm(formula = y2 ~ x2, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.9009 -0.7609  0.1291  0.9491  1.2691 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    3.001      1.125   2.667  0.02576 * 
x2             0.500      0.118   4.239  0.00218 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.237 on 9 degrees of freedom
Multiple R-squared:  0.6662,    Adjusted R-squared:  0.6292 
F-statistic: 17.97 on 1 and 9 DF,  p-value: 0.002179

m3 <- lm(y3 ~ x3, data = data)
summary(m3)

Call:
lm(formula = y3 ~ x3, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.1586 -0.6146 -0.2303  0.1540  3.2411 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   3.0025     1.1245   2.670  0.02562 * 
x3            0.4997     0.1179   4.239  0.00218 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.236 on 9 degrees of freedom
Multiple R-squared:  0.6663,    Adjusted R-squared:  0.6292 
F-statistic: 17.97 on 1 and 9 DF,  p-value: 0.002176

m4 <- lm(y4 ~ x4, data = data)
summary(m4)

Call:
lm(formula = y4 ~ x4, data = data)

Residuals:
   Min     1Q Median     3Q    Max 
-1.751 -0.831  0.000  0.809  1.839 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   3.0017     1.1239   2.671  0.02559 * 
x4            0.4999     0.1178   4.243  0.00216 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.236 on 9 degrees of freedom
Multiple R-squared:  0.6667,    Adjusted R-squared:  0.6297 
F-statistic:    18 on 1 and 9 DF,  p-value: 0.002165

p1 <- ggplot(data = data, aes(x=x1, y=y1)) + 
        geom_point(size=2) + geom_smooth(method='lm', formula = y~x)
p2 <- ggplot(data = data, aes(x=x2, y=y2)) + 
        geom_point(size=2) + geom_smooth(method='lm', formula = y~x)
p3 <- ggplot(data = data, aes(x=x3, y=y3)) + 
        geom_point(size=2) + geom_smooth(method='lm', formula = y~x)
p4 <- ggplot(data = data, aes(x=x4, y=y4)) + 
        geom_point(size=2) + geom_smooth(method='lm', formula = y~x)
grid.arrange(p1,p2,p3,p4, nrow=2, ncol=2)

Residual Diagnostics

par(mfrow=c(2,2))
plot(m1,1)
plot(m1,2)

plot(m1,3)
plot(m1,4)

par(mfrow=c(2,2))
plot(m2,1)
plot(m2,2)

plot(m2,3)
plot(m2,4)

par(mfrow=c(2,2))
plot(m3,1)
plot(m3,2)

plot(m3,3)
plot(m3,4)

par(mfrow=c(2,2))
plot(m4,1)
plot(m4,2)

Warning: not plotting observations with leverage one:
  8

plot(m4,3)

Warning: not plotting observations with leverage one:
  8

plot(m4,4)

Residual Normality Test

shapiro.test(m1$residuals)

    Shapiro-Wilk normality test

data:  m1$residuals
W = 0.94211, p-value = 0.5456

shapiro.test(m2$residuals)

    Shapiro-Wilk normality test

data:  m2$residuals
W = 0.87615, p-value = 0.09295

shapiro.test(m3$residuals)

    Shapiro-Wilk normality test

data:  m3$residuals
W = 0.74073, p-value = 0.001574

shapiro.test(m4$residuals)

    Shapiro-Wilk normality test

data:  m4$residuals
W = 0.96067, p-value = 0.78