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.
#The quartet consists of four datasets, each containing eleven (x, y) pairs. When these datasets are analyzed using summary statistics, they appear to be nearly identical, with the same mean, variance, and correlation coefficient. However, when plotted graphically, each dataset exhibits a unique pattern, demonstrating the importance of visualizing data.

#The lesson of Anscombe's Quartet is that data visualization is a critical tool in data analysis, as it allows us to uncover patterns and relationships that might be missed by purely numerical analysis. In addition, visualization can help us to identify outliers, assess the validity of statistical assumptions, and communicate findings to others more effectively.

#By using data visualization, we can gain a deeper understanding of our data and make better-informed decisions. In short, Anscombe's Quartet reminds us that summary statistics alone cannot fully capture the complexity of data, and that data visualization is an essential tool in any data analysis toolkit.