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 post your assignment on Rpubs and upload a link to 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.
library(datasets)
data=anscombe
head(data,10)
##    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
  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
library(corrplot)
## Warning: package 'corrplot' was built under R version 3.3.3
corrplot(cor(data),method="circle")

  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1,data$y1)

plot(data$x2,data$y2)

plot(data$x3,data$y3)

plot(data$x4,data$y4)

pairs(data)

  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,pch=19)
plot(data$x2,data$y2,pch=19)
plot(data$x3,data$y3,pch=19)
plot(data$x4,data$y4,pch=19)

  1. Now fit a linear model to each data set using the lm() function.
lm(data$x1 ~ data$y1)
## 
## Call:
## lm(formula = data$x1 ~ data$y1)
## 
## Coefficients:
## (Intercept)      data$y1  
##     -0.9975       1.3328
lm(data$x2 ~ data$y2)
## 
## Call:
## lm(formula = data$x2 ~ data$y2)
## 
## Coefficients:
## (Intercept)      data$y2  
##     -0.9948       1.3325
lm(data$x3 ~ data$y3)
## 
## Call:
## lm(formula = data$x3 ~ data$y3)
## 
## Coefficients:
## (Intercept)      data$y3  
##      -1.000        1.333
lm(data$x4 ~ data$y4)
## 
## Call:
## lm(formula = data$x4 ~ data$y4)
## 
## Coefficients:
## (Intercept)      data$y4  
##      -1.004        1.334
  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,pch=19)+abline(lm(data$x1 ~ data$y1))
## numeric(0)
plot(data$x2,data$y2,pch=19)+abline(lm(data$x2 ~ data$y2))
## numeric(0)
plot(data$x3,data$y3,pch=19)+abline(lm(data$x3 ~ data$y3))
## numeric(0)
plot(data$x4,data$y4,pch=19)+abline(lm(data$x4 ~ data$y4))

## numeric(0)
  1. Now compare the model fits for each model object.
library(fit.models)
## Warning: package 'fit.models' was built under R version 3.3.3
fit.models(lm(data$x1 ~ data$y1),lm(data$x2 ~ data$y2),lm(data$x3 ~ data$y3),lm(data$x4 ~ data$y4))

Calls: lm(data\(x1 ~ data\)y1): lm(formula = data\(x1 ~ data\)y1) lm(data\(x2 ~ data\)y2): lm(formula = data\(x2 ~ data\)y2) lm(data\(x3 ~ data\)y3): lm(formula = data\(x3 ~ data\)y3) lm(data\(x4 ~ data\)y4): lm(formula = data\(x4 ~ data\)y4)

Coefficients: (Intercept) data\(y1 data\)y2 data\(y3 data\)y4 lm(data\(x1 ~ data\)y1) -0.9975 1.3328
lm(data\(x2 ~ data\)y2) -0.9948 1.3325
lm(data\(x3 ~ data\)y3) -1.0003 1.3334
lm(data\(x4 ~ data\)y4) -1.0036 1.334

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

All four sets are very similar when we eyeball the data or even when we use numberical summary methods. But interestingly, the data sets vary considerably when graphed. It proves that data visualization can expose hidden stories and pattern within the data.