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 enter your code or text response in the code chunk that completes/answers the activity or question requested. To submit this homework you will create the document in Rstudio, using the knitr package (button included in Rstudio) and then submit the document to your Rpubs account. Once uploaded you will submit the link to that document on Canvas. Please make sure that this link is hyper linked and that I can see the visualization and the code required to create it. Each question is worth 5 points.

Questions

  1. Anscombe’s 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
  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 dplyr package!)
library(dplyr)
sapply(data, mean)
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
sapply(data, var)
##        x1        x2        x3        x4        y1        y2        y3        y4 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620  4.123249
cor(data[,1:4],data[,5:8])
##            y1         y2         y3         y4
## x1  0.8164205  0.8162365  0.8162867 -0.3140467
## x2  0.8164205  0.8162365  0.8162867 -0.3140467
## x3  0.8164205  0.8162365  0.8162867 -0.3140467
## x4 -0.5290927 -0.7184365 -0.3446610  0.8165214
  1. Using ggplot, create scatter plots for each \(x, y\) pair of data (maybe use ‘facet_grid’ or ‘facet_wrap’).
library(gridExtra)
## 
## Attaching package: 'gridExtra'
## The following object is masked from 'package:dplyr':
## 
##     combine
p1 <- ggplot(data=anscombe, aes(x=x1, y=y1)) + 
  geom_point() + 
  labs(title="Pair 1")
  
p2 <- ggplot(data=anscombe, aes(x=x2, y=y2)) + 
  geom_point() + 
  labs(title="Pair 2")

p3 <-ggplot(data=anscombe, aes(x=x3, y=y3)) + 
  geom_point() + 
  labs(title="Pair 3")

p4 <- ggplot(data=anscombe, mapping=aes(x=x4, y=y4)) + 
  geom_point() + 
  labs(title="Pair 4")

grid.arrange(p1, p2, p3, p4, nrow = 2, ncol = 2)

  1. Now change the symbols on the scatter plots to solid blue circles.
p1 <- ggplot(data=anscombe, aes(x=x1, y=y1)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 1")
  
p2 <- ggplot(data=anscombe, aes(x=x2, y=y2)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 2")

p3 <-ggplot(data=anscombe, aes(x=x3, y=y3)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 3")

p4 <- ggplot(data=anscombe, mapping=aes(x=x4, y=y4)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 4")

grid.arrange(p1, p2, p3, p4, nrow = 2, ncol = 2)

  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!)
p1 <- ggplot(data=anscombe, aes(x=x1, y=y1)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 1") +
  geom_smooth(method="lm", color = "red",se=FALSE)
  
p2 <- ggplot(data=anscombe, aes(x=x2, y=y2)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 2") +
  geom_smooth(method="lm", color = "red",se=FALSE)

p3 <-ggplot(data=anscombe, aes(x=x3, y=y3)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 3") +
  geom_smooth(method="lm", color = "red",se=FALSE)

p4 <- ggplot(data=anscombe, mapping=aes(x=x4, y=y4)) + 
  geom_point(color = "blue") + 
  labs(title="Pair 4") +
  geom_smooth(method="lm", color = "red",se=FALSE)

grid.arrange(p1, p2, p3, p4, nrow = 2, ncol = 2)
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'

  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.

Anscombes quartet teaches us the importance of visualizing data instead of relying only on summary statistics when exploring data.