Objectives

The objectives of this problem set is to orient you to a number of activities in R. And to conduct a thoughtful exercise in appreciating the importance of data visualization. For each question create a code chunk or text response that completes/answers the activity or question requested.

Questions

  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.
data("anscombe")
data<-anscombe
  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!)
library('fBasics')
## Warning: package 'fBasics' was built under R version 3.4.4
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.4.4
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.4.4
colMeans(data)
##       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(data)
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  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 May 21 21:46:21 2018
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 May 21 21:46:21 2018
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 May 21 21:46:21 2018
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 May 21 21:46:21 2018

The correlation between x1 and y1, x2 and y2, x3 and y3, and x4 and y4 is 0.8164, 0.8162, 0.8163, 0.8165. There is a high correlation for all the pairs.

  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1, data$y1, main = "Scatter Plot: x1 vs y1",xlab="x1",ylab="y1")

plot(data$x2, data$y2, main = "Scatter Plot: x2 vs y2",xlab="x2",ylab="y2")

plot(data$x3, data$y3, main = "Scatter Plot: x3 vs y3",xlab="x3",ylab="y3")

plot(data$x4, data$y4, main = "Scatter Plot: x4 vs y4",xlab="x4",ylab="y4")

  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(data$x1, data$y1, main = "Scatter Plot: x1 vs y1",xlab="x1",ylab="y1",pch=20)
plot(data$x2, data$y2, main = "Scatter Plot: x2 vs y2",xlab="x2",ylab="y2",pch=20)
plot(data$x3, data$y3, main = "Scatter Plot: x3 vs y3",xlab="x3",ylab="y3",pch=20)
plot(data$x4, data$y4, main = "Scatter Plot: x4 vs y4",xlab="x4",ylab="y4",pch=20)

  1. Now fit a linear model to each data set using the lm() function.
lm1<-lm(data$y1~data$x1)
lm2<-lm(data$y2~data$x2)
lm3<-lm(data$y3~data$x3)
lm4<-lm(data$y4~data$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(data$x1, data$y1, main = "Scatter Plot: x1 vs y1",xlab="x1",ylab="y1",pch=20)
abline(lm1)
plot(data$x2, data$y2, main = "Scatter Plot: x2 vs y2",xlab="x2",ylab="y2",pch=20)
abline(lm2)
plot(data$x3, data$y3, main = "Scatter Plot: x3 vs y3",xlab="x3",ylab="y3",pch=20)
abline(lm3)
plot(data$x4, data$y4, main = "Scatter Plot: x4 vs y4",xlab="x4",ylab="y4",pch=20)
abline(lm4)

  1. Now compare the model fits for each model object.
qqnorm(lm1$residuals)
qqline(lm1$residuals)

shapiro.test(lm1$residuals)
Shapiro-Wilk normality test

data: lm1$residuals W = 0.94211, p-value = 0.5456

qqnorm(lm2$residuals)
qqline(lm2$residuals)

shapiro.test(lm2$residuals)
Shapiro-Wilk normality test

data: lm2$residuals W = 0.87615, p-value = 0.09295

qqnorm(lm3$residuals)
qqline(lm3$residuals)

shapiro.test(lm3$residuals)
Shapiro-Wilk normality test

data: lm3$residuals W = 0.74073, p-value = 0.001574

qqnorm(lm4$residuals)
qqline(lm4$residuals)

shapiro.test(lm4$residuals)
Shapiro-Wilk normality test

data: lm4$residuals W = 0.96067, p-value = 0.78

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

Anscombe’s Quartet tell us that data visualization is very important to interpret data results. Statistics and graphs need to be looked at simultaneously. The correlation and regression statistics show that the relationships each pair shares is the same. But visual graphs show a completely different picture. Hence it is essential to interpre data using both statistics and graphs.