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.
#Install the anscombe data from library(datasets) and assign as request:
library(datasets)
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!)
library(fBasics)
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.4.3
## Loading required package: timeSeries
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
#mean and variance for each column
mean(x1)
## [1] 9
var(x1)
## [1] 11
mean(x2) 
## [1] 9
var(x2)
## [1] 11
mean(x3) 
## [1] 9
var(x3)
## [1] 11
mean(x4) 
## [1] 9
var(x4)
## [1] 11
mean(y1)
## [1] 7.500909
var(y1)
## [1] 4.127269
mean(y2)
## [1] 7.500909
var(y2)
## [1] 4.127629
mean(y3)
## [1] 7.5
var(y3)
## [1] 4.12262
mean(y4)
## [1] 7.500909
var(y4)
## [1] 4.123249
#correlation between x and y
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:
##  Wed May  2 22:21:27 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:
##  Wed May  2 22:21:27 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:
##  Wed May  2 22:21:27 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:
##  Wed May  2 22:21:27 2018
  1. Create scatter plots for each \(x, y\) pair of data.
#Create scatter plots:
plot(x1,y1, main="Scatterplot for x1 and y1")

plot(x2,y2, main="Scatterplot for x2 and y2")

plot(x3,y3, main="Scatterplot for x3 and y4")

plot(x4,y4, main="Scatterplot for x4 and y4")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
#Reformat the plots:
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.
#Using liinear regression model:
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!)
# add regression line (y~x)
par(mfrow=c(2,2))
plot(x1,y1, main="Scatterplot between x1,y1",pch=19) 
abline(fit1, col="red") 
plot(x2,y2, main="Scatterplot between x2,y2",pch=19) 
abline(fit2, col="red")
plot(x3,y3, main="Scatterplot between x3,y3",pch=19) 
abline(fit3, col="red") 
plot(x4,y4, main="Scatterplot between x4,y4",pch=19) 
abline(fit4, col="red")

  1. Now compare the model fits for each model object.
#Use ANOVA to compare each model
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 comprises four datasets that have relatively similar simple descriptive statistics, yet appear very different when graphed. Each dataset consists of eleven (x,y) points. Though the simple statistical values of four datasets are identical, graphs of four data sets are completely different. This proves that it is important to utilize data visualization for statistics in form of graphical datasets so people are able to see the data’s trend in different spectrum.