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” assignment to your R Pubs account and submit the link to Moodle. Points will be deducted for uploading the improper format.

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
##    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
  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
cor(data$x1, data$y1)
## [1] 0.8164205
cor(data$x2, data$y2)
## [1] 0.8162365
cor(data$x3, data$y3)
## [1] 0.8162867
cor(data$x4, data$y4)
## [1] 0.8165214
##from the correlation values among each pair,they are close
mean(data$x1)
## [1] 9
var(data$x1)
## [1] 11
mean(data$x2)
## [1] 9
var(data$x2)
## [1] 11
mean(data$x3)
## [1] 9
var(data$x3)
## [1] 11
mean(data$x4)
## [1] 9
var(data$x4)
## [1] 11
mean(data$y1)
## [1] 7.500909
var(data$y1)
## [1] 4.127269
mean(data$y2)
## [1] 7.500909
var(data$y2)
## [1] 4.127629
mean(data$y3)
## [1] 7.5
var(data$y3)
## [1] 4.12262
mean(data$y4)
## [1] 7.500909
var(data$y4)
## [1] 4.123249
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1,data$y1, main="Scatter plot x1,y1") 

plot(data$x2,data$y2, main="Scatter plot x2,y2") 

plot(data$x3,data$y3, main="Scatter plot x3,y3") 

plot(data$x4,data$y4, main="Scatter plot 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(data$x1,data$y1, main="Scatter plot for x1,y1",pch=20) 
plot(data$x2,data$y2, main="Scatter plot for x1,y1",pch=20) 
plot(data$x3,data$y3, main="Scatter plot for x1,y1",pch=20) 
plot(data$x4,data$y4, main="Scatter plot for x1,y1",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 for x1,y1",pch=20) 
abline(lm1)
plot(data$x2,data$y2,main="Scatter plot for x2,y2",pch=20) 
abline(lm2)
plot(data$x3,data$y3,main="Scatter plot for x3,y3",pch=20) 
abline(lm3)
plot(data$x4,data$y4,main="Scatter plot for x4,y4",pch=20) 
abline(lm4)

  1. Now compare the model fits for each model object.
anova(lm1)

Analysis of Variance Table

Response: data\(y1 Df Sum Sq Mean Sq F value Pr(>F) data\)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(lm2)

Analysis of Variance Table

Response: data\(y2 Df Sum Sq Mean Sq F value Pr(>F) data\)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(lm3)

Analysis of Variance Table

Response: data\(y3 Df Sum Sq Mean Sq F value Pr(>F) data\)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(lm4)

Analysis of Variance Table

Response: data\(y4 Df Sum Sq Mean Sq F value Pr(>F) data\)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.

From anscombe quartet, if only viewing with numeric tables/lists/metrics, the data look similar. Especially from Question 2, after applying with “summary”function, you can see the summary results of X1, X2, X3 are the same. In addition, it is hard to distinguish the correlation when comparing with each pair (x1, y1). Again in Question 2, if only looking at the correlation value for each pair, you barely could tell the distinctive difference with 0.01% flation.

However, after data visualization, it is apparently can see the relationship for each pair within the same chart or graph displaying.