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.
library(datasets)
library(tidyverse)
## Warning: package 'tidyverse' was built under R version 4.2.2
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.3.6      ✔ purrr   1.0.1 
## ✔ tibble  3.1.8      ✔ dplyr   1.0.10
## ✔ tidyr   1.2.1      ✔ stringr 1.4.1 
## ✔ readr   2.1.3      ✔ forcats 0.5.2
## Warning: package 'purrr' was built under R version 4.2.2
## Warning: package 'dplyr' was built under R version 4.2.2
## Warning: package 'forcats' was built under R version 4.2.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
library(reshape)
## Warning: package 'reshape' was built under R version 4.2.2
## 
## Attaching package: 'reshape'
## 
## The following object is masked from 'package:dplyr':
## 
##     rename
## 
## The following objects are masked from 'package:tidyr':
## 
##     expand, smiths
#install.packages("gridExtra")
library(gridExtra)
## Warning: package 'gridExtra' was built under R version 4.2.2
## 
## Attaching package: 'gridExtra'
## 
## The following object is masked from 'package:dplyr':
## 
##     combine
data = datasets::anscombe
head(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
summary(data)
##        x1             x2             x3             x4           y1        
##  Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8   Min.   : 4.260  
##  1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8   1st Qu.: 6.315  
##  Median : 9.0   Median : 9.0   Median : 9.0   Median : 8   Median : 7.580  
##  Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9   Mean   : 7.501  
##  3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8   3rd Qu.: 8.570  
##  Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19   Max.   :10.840  
##        y2              y3              y4        
##  Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
##  1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
##  Median :8.140   Median : 7.11   Median : 7.040  
##  Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
##  3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
##  Max.   :9.260   Max.   :12.74   Max.   :12.500
  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’).
plot1 = ggplot(data, aes(x=x1, y=y1)) + 
  geom_point() + 
  labs(title="X/Y Pair 1")

plot2 = ggplot(data, aes(x=x2, y=y2)) + 
  geom_point() + 
  labs(title="X/Y Pair 2")

plot3 = ggplot(data, aes(x=x3, y=y3)) + 
  geom_point() + 
  labs(title="X/Y Pair 3")


plot4 = ggplot(data, aes(x=x4, y=y4)) + 
  geom_point() + 
  labs(title="X/Y Pair 4")

grid.arrange(plot1, plot2, plot3, plot4, nrow = 2, ncol = 2)

  1. Now change the symbols on the scatter plots to solid blue circles.
plot1blue = ggplot(data, aes(x=x1, y=y1)) + 
  geom_point(color = "blue") + 
  labs(title="X/Y Pair 1")

plot2blue = ggplot(data, aes(x=x2, y=y2)) + 
  geom_point(color = "blue") + 
  labs(title="X/Y Pair 2")

plot3blue = ggplot(data, aes(x=x3, y=y3)) + 
  geom_point(color = "blue") + 
  labs(title="X/Y Pair 3")


plot4blue = ggplot(data, aes(x=x4, y=y4)) + 
geom_point(color = "blue") + 
  labs(title="X/Y Pair 4")

grid.arrange(plot1blue, plot2blue, plot3blue, plot4blue, nrow = 2, ncol = 2)

  1. Now fit a linear model to each data set using the lm() function.
lm1 = lm(data$x1~data$y1)
lm2 = lm(data$x2~data$y2)
lm3 = lm(data$x3~data$y3)
lm4 = lm(data$x4~data$y4)
  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!)
plot1blue = ggplot(data, aes(x=x1, y=y1)) + 
  geom_point(color = "blue") + 
  labs(title="X/Y Pair 1")+ 
  geom_smooth(method="lm",color = "green", se=FALSE)

plot2blue = ggplot(data, aes(x=x2, y=y2)) + 
  geom_point(color = "blue") + 
  labs(title="X/Y Pair 2")+ 
  geom_smooth(method="lm",color = "green", se=FALSE)

plot3blue = ggplot(data, aes(x=x3, y=y3)) + 
  geom_point(color = "blue") + 
  labs(title="X/Y Pair 3")+ 
  geom_smooth(method="lm",color = "green", se=FALSE)


plot4blue = ggplot(data, aes(x=x4, y=y4)) + 
geom_point(color = "blue") + 
  labs(title="X/Y Pair 4")+ 
  geom_smooth(method="lm",color = "green", se=FALSE)

grid.arrange(plot1blue, plot2blue, plot3blue, plot4blue, 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\(x1 Df Sum Sq Mean Sq F value Pr(>F) data\)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(lm2)

Analysis of Variance Table

Response: data\(x2 Df Sum Sq Mean Sq F value Pr(>F) data\)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(lm3)

Analysis of Variance Table

Response: data\(x3 Df Sum Sq Mean Sq F value Pr(>F) data\)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(lm4)

Analysis of Variance Table

Response: data\(x4 Df Sum Sq Mean Sq F value Pr(>F) data\)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.

From the data & Anscombre we ran above, what we can identify is that all of the four data sets/models that we ran are very similar. If you look at the linear regression in question 6, we can notice that all four of the data sets have similar fits. Additionally, this is confirmed when we ran specific statistics such as standard deviation and mean, with all foru data sets having similar points as can be seen in question 2, with just slight variation between data sets.Thus from the above we can see the importance of data visualization when trying to understand data and analyze data.