library(datasets)
  1. Anscombes quartet is a set of 4 \(x,y\) data sets that were published by Francis Anscombe in a 1973 paper Graphs in statistical analysis. Firstly, for this loading the anscombe data that is part of the library(datasets) in R and assigning that data to a new object called data.
data=anscombe
summary(data)
##        x1             x2             x3             x4           y1        
##  Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8   Min.   : 4.260  
##  1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8   1st Qu.: 6.315  
##  Median : 9.0   Median : 9.0   Median : 9.0   Median : 8   Median : 7.580  
##  Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9   Mean   : 7.501  
##  3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8   3rd Qu.: 8.570  
##  Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19   Max.   :10.840  
##        y2              y3              y4        
##  Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
##  1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
##  Median :8.140   Median : 7.11   Median : 7.040  
##  Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
##  3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
##  Max.   :9.260   Max.   :12.74   Max.   :12.500
  1. Summarizing the data by calculating the mean, variance, for each column and the correlation between each pair (eg. x1 and y1, x2 and y2, etc)
library(fBasics)
mean(data$x1)
## [1] 9
var(data$x1)
## [1] 11
mean(data$x2)
## [1] 9
var(data$x2)
## [1] 11
mean(data$x3)
## [1] 9
var(data$x3)
## [1] 11
mean(data$x4)
## [1] 9
var(data$x4)
## [1] 11
mean(data$y1)
## [1] 7.500909
var(data$y1)
## [1] 4.127269
mean(data$y2)
## [1] 7.500909
var(data$y2)
## [1] 4.127629
mean(data$y3)
## [1] 7.5
var(data$y3)
## [1] 4.12262
mean(data$y4)
## [1] 7.500909
var(data$y4)
## [1] 4.123249
correlationTest(data$x1,data$y1)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8164
##   STATISTIC:
##     t: 4.2415
##   P VALUE:
##     Alternative Two-Sided: 0.00217 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001085 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4244, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5113, 1
## 
## Description:
##  Mon Nov 23 21:15:08 2020
correlationTest(data$x2,data$y2)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8162
##   STATISTIC:
##     t: 4.2386
##   P VALUE:
##     Alternative Two-Sided: 0.002179 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001089 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4239, 0.9506
##          Less: -1, 0.9387
##       Greater: 0.5109, 1
## 
## Description:
##  Mon Nov 23 21:15:08 2020
correlationTest(data$x3,data$y3)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8163
##   STATISTIC:
##     t: 4.2394
##   P VALUE:
##     Alternative Two-Sided: 0.002176 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001088 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4241, 0.9507
##          Less: -1, 0.9387
##       Greater: 0.511, 1
## 
## Description:
##  Mon Nov 23 21:15:08 2020
correlationTest(data$x4,data$y4)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8165
##   STATISTIC:
##     t: 4.243
##   P VALUE:
##     Alternative Two-Sided: 0.002165 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001082 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4246, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5115, 1
## 
## Description:
##  Mon Nov 23 21:15:08 2020
  1. Creating scatter plots for each \(x, y\) pair of data.
library("ggplot2")
ggplot(data = data, mapping = aes(x = x1,y=y1)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x1 and y1") +
    xlab("x1")+
    ylab("y1")+
  theme(plot.title = element_text(hjust = 0.5)) 

ggplot(data = data, mapping = aes(x = x2,y=y2)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x2 and y2") +
    xlab("x2")+
    ylab("y2")+
  theme(plot.title = element_text(hjust = 0.5)) 

ggplot(data = data, mapping = aes(x = x3,y=y3)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x3 and y3") +
    xlab("x3")+
    ylab("y3")+
  theme(plot.title = element_text(hjust = 0.5)) 

ggplot(data = data, mapping = aes(x = x4,y=y4)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x4 and y4") +
    xlab("x4")+
    ylab("y4")+
  theme(plot.title = element_text(hjust = 0.5))

  1. Now changing the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic.
library("ggplot2")
library(gridExtra)
p1=ggplot(data = data, mapping = aes(x = x1,y=y1)) +
    geom_point()+
  ggtitle("Relationship between x1 and y1") +
    xlab("x1")+
    ylab("y1")+
  theme(plot.title = element_text(hjust = 0.5)) 
p2=ggplot(data = data, mapping = aes(x = x2,y=y2)) +
    geom_point()+
  ggtitle("Relationship between x2 and y2") +
    xlab("x2")+
    ylab("y2")+
  theme(plot.title = element_text(hjust = 0.5)) 
p3=ggplot(data = data, mapping = aes(x = x3,y=y3)) +
    geom_point()+
  ggtitle("Relationship between x3 and y3") +
    xlab("x3")+
    ylab("y3")+
  theme(plot.title = element_text(hjust = 0.5)) 
p4=ggplot(data = data, mapping = aes(x = x4,y=y4)) +
    geom_point()+
  ggtitle("Relationship between x4 and y4") +
    xlab("x4")+
    ylab("y4")+
  theme(plot.title = element_text(hjust = 0.5)) 
grid.arrange(p1,p2,p3,p4, nrow = 2)

  1. Now fitting a linear model to each data set using the lm() function.
fit1=lm(x1~y1,data = data)
summary(fit1)
## 
## Call:
## lm(formula = x1 ~ y1, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6522 -1.5117 -0.2657  1.2341  3.8946 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9975     2.4344  -0.410  0.69156   
## y1            1.3328     0.3142   4.241  0.00217 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.019 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
fit2=lm(x2~y2,data = data)
summary(fit2)
## 
## Call:
## lm(formula = x2 ~ y2, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8516 -1.4315 -0.3440  0.8467  4.2017 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9948     2.4354  -0.408  0.69246   
## y2            1.3325     0.3144   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.02 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
fit3=lm(x3~y3,data = data)
summary(fit3)
## 
## Call:
## lm(formula = x3 ~ y3, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9869 -1.3733 -0.0266  1.3200  3.2133 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0003     2.4362  -0.411  0.69097   
## y3            1.3334     0.3145   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.019 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
fit4=lm(x4~y4,data = data)
summary(fit4)
## 
## Call:
## lm(formula = x4 ~ y4, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7859 -1.4122 -0.1853  1.4551  3.3329 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0036     2.4349  -0.412  0.68985   
## y4            1.3337     0.3143   4.243  0.00216 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.018 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
  1. Now combining the last two tasks by Creating a four panel scatter plot matrix that has both the data points and the regression lines.
library("ggplot2")
library(gridExtra)
p1=ggplot(data = data, mapping = aes(x = x1,y=y1)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x1 and y1") +
    xlab("x1")+
    ylab("y1")+
    geom_smooth(method = 'lm', color = 'black')+
  theme(plot.title = element_text(hjust = 0.5)) 
p2=ggplot(data = data, mapping = aes(x = x2,y=y2)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x2 and y2") +
    xlab("x2")+
    ylab("y2")+
    geom_smooth(method = 'lm', color = 'black')+
  theme(plot.title = element_text(hjust = 0.5)) 
p3=ggplot(data = data, mapping = aes(x = x3,y=y3)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x3 and y3") +
    xlab("x3")+
    ylab("y3")+
    geom_smooth(method = 'lm', color = 'black')+
  theme(plot.title = element_text(hjust = 0.5)) 
p4=ggplot(data = data, mapping = aes(x = x4,y=y4)) +
    geom_point()+
  ggtitle("(Scatterplot)Relationship between x4 and y4") +
    xlab("x4")+
    ylab("y4")+
    geom_smooth(method = 'lm', color = 'black')+
  theme(plot.title = element_text(hjust = 0.5)) 
grid.arrange(p1,p2,p3,p4, nrow = 2)
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'

  1. Now comparing the model fits for each model object.
a1=anova(fit1)
a1

Analysis of Variance Table

Response: x1 Df Sum Sq Mean Sq F value Pr(>F)
y1 1 73.32 73.320 17.99 0.00217 ** Residuals 9 36.68 4.076
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

plot(a1)

plot(fit1)

a2=anova(fit2)
a2

Analysis of Variance Table

Response: x2 Df Sum Sq Mean Sq F value Pr(>F)
y2 1 73.287 73.287 17.966 0.002179 ** Residuals 9 36.713 4.079
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

plot(a2)

plot(fit2)

a3=anova(fit3)
a3

Analysis of Variance Table

Response: x3 Df Sum Sq Mean Sq F value Pr(>F)
y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

plot(a3)

plot(fit3)

a4=anova(fit4)
a4

Analysis of Variance Table

Response: x4 Df Sum Sq Mean Sq F value Pr(>F)
y4 1 73.338 73.338 18.003 0.002165 ** Residuals 9 36.662 4.074
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

plot(a4)

plot(fit4)

  1. summarizing the lesson of Anscombe’s Quartet and what it says about the value of data visualization:

The Anscombe’s Quartet is a masterpiece dataset from the Greatest Statistician “Anscombe” which has four datasets resembling descriptive statistics but different distributions which points out the importance of Data Visualization. In this Problem set, we can clearly see the importance and essence of Data Visualization to ourselves from the Anscombe’s Quartlet. For clearly understanding of the data check out (https://en.wikipedia.org/wiki/Anscombe's_quartet). From above Scatterplots and the model Fits alongside its plots, we can clearly see that the datapoints of each of the datasets are scattered differently from each other.