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 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")
show(anscombe)
##    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(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
#correlation
sapply(1:4, function(x) cor(anscombe[, x], anscombe[, x+4]))
## [1] 0.8164205 0.8162365 0.8162867 0.8165214
#variance
sapply(5:8, function(x) var(anscombe[, x]))
## [1] 4.127269 4.127629 4.122620 4.123249
  1. Create scatter plots for each \(x, y\) pair of data.
attach(anscombe)
plot(x1,y1, main="Scatterplot First pair", xlab = "x1", ylab = "y1", pch=19)

plot(x2,y2, main="Scatterplot Second pair", xlab = "x2", ylab = "y2", pch=19)

plot(x3,y3, main="Scatterplot Third pair", xlab = "x3", ylab = "y3", pch=19)

plot(x4,y4, main="Scatterplot Forth pair", xlab = "x4", ylab = "y4", pch=19)

  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 First pair", xlab = "x1", ylab = "y1", pch=19)
plot(x2,y2, main="Scatterplot Second pair", xlab = "x2", ylab = "y2", pch=19)
plot(x3,y3, main="Scatterplot Third pair", xlab = "x3", ylab = "y3", pch=19)
plot(x4,y4, main="Scatterplot Forth pair", xlab = "x4", ylab = "y4", pch=19)

  1. Now fit a linear model to each data set using the lm() function.
lm(formula = x1 ~ y1, data = anscombe)
## 
## Call:
## lm(formula = x1 ~ y1, data = anscombe)
## 
## Coefficients:
## (Intercept)           y1  
##     -0.9975       1.3328
lm(formula = x2 ~ y2, data = anscombe)
## 
## Call:
## lm(formula = x2 ~ y2, data = anscombe)
## 
## Coefficients:
## (Intercept)           y2  
##     -0.9948       1.3325
lm(formula = x3 ~ y3, data = anscombe)
## 
## Call:
## lm(formula = x3 ~ y3, data = anscombe)
## 
## Coefficients:
## (Intercept)           y3  
##      -1.000        1.333
lm(formula = x4 ~ y4, data = anscombe)
## 
## Call:
## lm(formula = x4 ~ y4, data = anscombe)
## 
## Coefficients:
## (Intercept)           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!)
mod1 <- lm (x1 ~ y1)
plot(x1,y1, main="Scatterplot First pair", xlab = "x1", ylab = "y1", pch=19)
mod1 <- lm (x1 ~ y1)
abline(mod1)

mod2 <- lm (x2 ~ y2)
plot(x2,y2, main="Scatterplot First pair", xlab = "x2", ylab = "y2", pch=19)
abline(mod2)

mod3 <- lm (x3 ~ y3)
plot(x3,y3, main="Scatterplot First pair", xlab = "x3", ylab = "y3", pch=19)
abline(mod3)

mod4 <- lm (x4 ~ y4)
plot(x4,y4, main="Scatterplot First pair", xlab = "x4", ylab = "y4", pch=19)
abline(mod4)

  1. Now compare the model fits for each model object.
anova(mod1, test="Chisq")

Analysis of Variance Table

Response: x1 Df Sum Sq Mean Sq F value Pr(>F)
y1 1 73.32 73.320 17.99 0.00217 ** Residuals 9 36.68 4.076
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(mod2, test="Chisq")

Analysis of Variance Table

Response: x2 Df Sum Sq Mean Sq F value Pr(>F)
y2 1 73.287 73.287 17.966 0.002179 ** Residuals 9 36.713 4.079
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(mod3, test="Chisq")

Analysis of Variance Table

Response: x3 Df Sum Sq Mean Sq F value Pr(>F)
y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(mod4, test="Chisq")

Analysis of Variance Table

Response: x4 Df Sum Sq Mean Sq F value Pr(>F)
y4 1 73.338 73.338 18.003 0.002165 ** Residuals 9 36.662 4.074
— 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 includes four different data sets that clearly represents the importance of data visualization given that the data sets are identical, but the visualizations are completely different