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, upload a link to your file hosted on rpubs.com

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 <- datasets::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
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
cor(data)
##            x1         x2         x3         x4         y1         y2
## x1  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x2  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x3  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x4 -0.5000000 -0.5000000 -0.5000000  1.0000000 -0.5290927 -0.7184365
## y1  0.8164205  0.8164205  0.8164205 -0.5290927  1.0000000  0.7500054
## y2  0.8162365  0.8162365  0.8162365 -0.7184365  0.7500054  1.0000000
## y3  0.8162867  0.8162867  0.8162867 -0.3446610  0.4687167  0.5879193
## y4 -0.3140467 -0.3140467 -0.3140467  0.8165214 -0.4891162 -0.4780949
##            y3         y4
## x1  0.8162867 -0.3140467
## x2  0.8162867 -0.3140467
## x3  0.8162867 -0.3140467
## x4 -0.3446610  0.8165214
## y1  0.4687167 -0.4891162
## y2  0.5879193 -0.4780949
## y3  1.0000000 -0.1554718
## y4 -0.1554718  1.0000000
mean(data$x1)
## [1] 9
mean(data$x2)
## [1] 9
mean(data$x3)
## [1] 9
mean(data$x4)
## [1] 9
mean(data$x4)
## [1] 9
mean(data$y1)
## [1] 7.500909
mean(data$y2)
## [1] 7.500909
mean(data$y3)
## [1] 7.5
mean(data$y4)
## [1] 7.500909
var(data$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$y4)
## [1] 4.123249
cor(data$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
  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)

  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=16)
plot(data$x2, data$y2, pch=16)
plot(data$x3, data$y3, pch=16)
plot(data$x4, data$y4, pch=16)

  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 = "Scater Plot - 1", pch = 20)
abline(lm1, col = "green")

plot(data$x2, data$y2, main = "Scater Plot - 2", pch = 20)
abline(lm2, col = "green")

plot(data$x3, data$y3, main = "Scater Plot - 3", pch = 20)
abline(lm3, col = "green")

plot(data$x4, data$y4, main = "Scater Plot - 4", pch = 20)
abline(lm4, col = "green")

  1. Now compare the model fits for each model object.
summary(lm1)$r.squared

[1] 0.6665425

summary(lm2)$r.squared

[1] 0.666242

summary(lm3)$r.squared

[1] 0.666324

summary(lm4)$r.squared

[1] 0.6667073

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

The biggest learning from this exercise using Anscombe’s Quartet is that without the visual representation this data could’ve been completely misleading. The mean, variance, correlation, even the fit of regression based on r-square value is almost exact for all x,y pairs. Only with the visual plotting do we see that in reality the x,y pairs are vastly different in nature. x1, y1 pair is linear. x2,y2 pair is hyperbolic. x3, y3 is linear with an outlier. x4,y4 is almost vertical with a large outlier. These plots showcase the power of visualiztaion and the context it offers when reading and trying to understand data.