Answers

  1. For this first question load the anscombe data that is part of the library(datasets) in R. And assign that data to a new object called data.
data<-anscombe
x1<-data[,1]
x2<-data[,2]
x3<-data[,3]
x4<-data[,4]
y1<-data[,5]
y2<-data[,6]
y3<-data[,7]
y4<-data[,8]
  1. Summarise the data by calculating the mean, variance, for each column and the correlation between each pair (eg. x1 and y1, x2 and y2, etc) (Hint: use the fBasics() package!)
summary(data)
##        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  
##        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
var(x1)
## [1] 11
var(y1)
## [1] 4.127269
var(x2)
## [1] 11
var(y2)
## [1] 4.127629
var(x3)
## [1] 11
var(y3)
## [1] 4.12262
var(x4)
## [1] 11
var(y4)
## [1] 4.123249
library(fBasics)
## Warning: package 'fBasics' was built under R version 3.4.2
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.4.2
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.4.2
correlationTest(x1,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:
##  Sun Apr  8 16:33:17 2018
correlationTest(x2,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:
##  Sun Apr  8 16:33:17 2018
correlationTest(x3,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:
##  Sun Apr  8 16:33:18 2018
correlationTest(x4,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:
##  Sun Apr  8 16:33:18 2018
  1. Create scatter plots for each \(x, y\) pair of data.
plot(x1, y1, main="Scatterplot between x1,y1", pch=8, col="red")

plot(x2, y2, main="Scatterplot between x2,y2", pch=8, col="green")

plot(x3, y3, main="Scatterplot between x3,y3", pch=8, col="orange") 

plot(x4, y4, main="Scatterplot between x4,y4", pch=8, col="blue")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
par(mfrow=c(2,2))
plot(x1,y1, main="Scatterplot between x1,y1",pch=19) 
plot(x2,y2, main="Scatterplot between x2,y2",pch=19) 
plot(x3,y3, main="Scatterplot between x3,y3",pch=19) 
plot(x4,y4, main="Scatterplot between x4,y4",pch=19)

  1. Now fit a linear model to each data set using the lm() function.
fit1<-lm(y1~x1)
fit2<-lm(y2~x2)
fit3<-lm(y3~x3)
fit4<-lm(y4~x4)
  1. Now combine the last two tasks. Create a four panel scatter plot matrix that has both the data points and the regression lines. (hint: the model objects will carry over chunks!)
par(mfrow=c(2,2))
plot(x1,y1, main="Scatterplot between x1,y1",pch=19) 
abline(fit1, col="red") # regression line (y~x)
plot(x2,y2, main="Scatterplot between x2,y2",pch=19) 
abline(fit2, col="green") # regression line (y~x)
plot(x3,y3, main="Scatterplot between x3,y3",pch=19) 
abline(fit3, col="orange") # regression line (y~x)
plot(x4,y4, main="Scatterplot between x4,y4",pch=19) 
abline(fit4, col="blue") # regression line (y~x)

  1. Now compare the model fits for each model object.
anova(fit1)

Analysis of Variance Table

Response: y1 Df Sum Sq Mean Sq F value Pr(>F)
x1 1 27.510 27.5100 17.99 0.00217 ** Residuals 9 13.763 1.5292
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(fit2)

Analysis of Variance Table

Response: y2 Df Sum Sq Mean Sq F value Pr(>F)
x2 1 27.500 27.5000 17.966 0.002179 ** Residuals 9 13.776 1.5307
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(fit3)

Analysis of Variance Table

Response: y3 Df Sum Sq Mean Sq F value Pr(>F)
x3 1 27.470 27.4700 17.972 0.002176 ** Residuals 9 13.756 1.5285
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(fit4)

Analysis of Variance Table

Response: y4 Df Sum Sq Mean Sq F value Pr(>F)
x4 1 27.490 27.4900 18.003 0.002165 ** Residuals 9 13.742 1.5269
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.

Anscombe’s Quartet is a classic example of why data visualization is imperative. The data contains four datasets and through the simple statistical values of four datasets the results look similar. However, the graphs of four data sets are completely different. On running the visualization functions. There are so many ways we can ways to visualise the data for the concerned audience. We ran scatter plots and fit regression lines to see the relationships between variables. Thus data visualization gives us exploratory tools to better understand the data, which may seem simila at first but is different in many ways as proved in our analysis.