Ques-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. 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.

library(datasets)
data(anscombe)
anscombedata <- anscombe
anscombedata
##    x1 x2 x3 x4    y1   y2    y3    y4
## 1  10 10 10  8  8.04 9.14  7.46  6.58
## 2   8  8  8  8  6.95 8.14  6.77  5.76
## 3  13 13 13  8  7.58 8.74 12.74  7.71
## 4   9  9  9  8  8.81 8.77  7.11  8.84
## 5  11 11 11  8  8.33 9.26  7.81  8.47
## 6  14 14 14  8  9.96 8.10  8.84  7.04
## 7   6  6  6  8  7.24 6.13  6.08  5.25
## 8   4  4  4 19  4.26 3.10  5.39 12.50
## 9  12 12 12  8 10.84 9.13  8.15  5.56
## 10  7  7  7  8  4.82 7.26  6.42  7.91
## 11  5  5  5  8  5.68 4.74  5.73  6.89

Ques-2

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!) #install.packages(“fBasics”)

library(fBasics)
## Warning: package 'fBasics' was built under R version 3.4.3
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.4.3
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.4.3
colMeans(anscombedata)
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
colVars(anscombedata)
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249
x1<-anscombedata[,1]
x2<-anscombedata[,2]
x3<-anscombedata[,3]
x4<-anscombedata[,4]
y1<-anscombedata[,5]
y2<-anscombedata[,6]
y3<-anscombedata[,7]
y4<-anscombedata[,8]
correlationTest(x1,y1,method = c("pearson", "kendall", "spearman"))
## 
## 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:
##  Wed Jan 31 16:45:15 2018
correlationTest(x2,y2,method = c("pearson", "kendall", "spearman"))
## 
## 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:
##  Wed Jan 31 16:45:16 2018
correlationTest(x3,y3,method = c("pearson", "kendall", "spearman"))
## 
## 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:
##  Wed Jan 31 16:45:16 2018
correlationTest(x4,y4,method = c("pearson", "kendall", "spearman"))
## 
## 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:
##  Wed Jan 31 16:45:16 2018

Ques-3

Create scatter plots for each \(x, y\) pair of data.

plot(x1,y1,main="Scatterplot for data Pair-1",xlab="x1",ylab="y1")

plot(x2,y2,main="Scatterplot for data Pair-2",xlab="x2",ylab="y2")

plot(x3,y3,main="Scatterplot for data Pair-3",xlab="x3",ylab="y3")

plot(x4,y4,main="Scatterplot for data Pair-4",xlab="x4",ylab="y4")

Ques-4

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 for data Pair-1",xlab="x1",ylab="y1",pch=20)
plot(x2,y2,main="Scatterplot for data Pair-2",xlab="x2",ylab="y2",pch=20)
plot(x3,y3,main="Scatterplot for data Pair-3",xlab="x3",ylab="y3",pch=20)
plot(x4,y4,main="Scatterplot for data Pair-4",xlab="x4",ylab="y4",pch=20)

Ques-5

Now fit a linear model to each data set using the lm() function.

lm.1<-lm(y1~x1)
lm.2<-lm(y2~x2)
lm.3<-lm(y3~x3)
lm.4<-lm(y4~x4)
lm.1
## 
## Call:
## lm(formula = y1 ~ x1)
## 
## Coefficients:
## (Intercept)           x1  
##      3.0001       0.5001
lm.2
## 
## Call:
## lm(formula = y2 ~ x2)
## 
## Coefficients:
## (Intercept)           x2  
##       3.001        0.500
lm.3
## 
## Call:
## lm(formula = y3 ~ x3)
## 
## Coefficients:
## (Intercept)           x3  
##      3.0025       0.4997
lm.4
## 
## Call:
## lm(formula = y4 ~ x4)
## 
## Coefficients:
## (Intercept)           x4  
##      3.0017       0.4999

Ques-6

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 Pair 1",xlab="x1",ylab="y1",pch=20)
abline(lm(y1 ~ x1))
plot(x2,y2,main="Scatterplot Pair 2",xlab="x2",ylab="y2",pch=20)
abline(lm(y2 ~ x2))
plot(x3,y3,main="Scatterplot Pair 3",xlab="x3",ylab="y3",pch=20)
abline(lm(y3 ~ x3))
plot(x4,y4,main="Scatterplot Pair 4",xlab="x4",ylab="y4",pch=20)
abline(lm(y4 ~ x4))

Ques-7

Now compare the model fits for each model object.

anova(lm.1)

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(lm.2)

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(lm.3)

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(lm.4)

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

Ques-8

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

Ans: The outcome shown by ANOVA analysis of these datasets is quite intriguing. The initial descriptive statistics did not indicate any significant differences between the four datasets. Pearson’s Correlation test shows the correlation between all four datasets are ~0.81xx at 9 degrees of freedom. Also the linear models for all four datasets indicate a strong positive linear correlation between x,y pairs of all four datasets. Looking at the summary statistics alone one could easily misread them as identical or similar sets of data. Data visualization, however, gives us a much direct, simple and intuitive way to look at data. It might not be as accurate as the scientific methods of data analysis but is user-oriented, visually intuitive and easily comprehensible.