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. Finally, upon completion name your final output .html file as: YourName_ANLY512-Section-Year-Semester.html and upload it to the “Problem Set 2” assignmenet on Moodle.

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.
View(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!)
summary(anscombe)
##        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
library(fBasics)
## Loading required package: timeDate
## Loading required package: timeSeries
var(data$x1)
## [1] 11
var(data$x2)
## [1] 11
var(data$x3)
## [1] 11
var(data$x4)
## [1] 11
var(data$y1)
## [1] 4.127269
var(data$y2)
## [1] 4.127629
var(data$y3)
## [1] 4.12262
var(data$y4)
## [1] 4.123249

load fBasic

library(fBasics)

find correlation

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:
##  Tue Sep 18 22:02:25 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:
##  Tue Sep 18 22:02:25 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:
##  Tue Sep 18 22:02:25 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:
##  Tue Sep 18 22:02:25 2018
  1. Create scatter plots for each \(x, y\) pair of data.
x1<-data[,1]
x2<-data[,2]
x3<-data[,3]
x4<-data[,4]
y1<-data[,5]
y2<-data[,6]
y3<-data[,7]
y4<-data[,8]
plot(x1, y1, main="Scatterplot between x1,y1") 

plot(x2, y2, main="Scatterplot between x2,y2") 

plot(x3, y3, main="Scatterplot between x3,y3") 

plot(x4, y4, main="Scatterplot between x4,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(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!)
library(ggplot2)
par(mfrow=c(2,2))
plot(x1,y1, main="Scatterplot between x1,y1",pch=19) 
abline(fit1, col="blue") 
plot(x2,y2, main="Scatterplot between x2,y2",pch=19) 
abline(fit2, col="green") 
plot(x3,y3, main="Scatterplot between x3,y3",pch=19) 
abline(fit3, col="orange") 
plot(x4,y4, main="Scatterplot between x4,y4",pch=19) 
abline(fit4, col="red")

  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.

There are four related datasets in Anscombe’s Quatet. First, we summarize the datasets by calculate the mean, variance and the correlation. Then we plot them in columns pairs by scatter plot. In addition, we dig in the analysis comparison through statistical analysis (by using anova analysis) to show the details of each set of data and plot in data with regression line.

From the lesson of Anscombe’s Quartet, we can learn that summary statistics can be used to learn the data from the surface, and it sometimes fail to tell us the full or true story. While data visualization can help us get a clearer picture of the dataset. It can show some information ignored by the summary statistics.