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 to your R Pubs account and submit the link to Moodle. Points will be deducted for uploading the improper format.

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
  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!)
library(fBasics)
## Warning: package 'fBasics' was built under R version 3.5.3
## Loading required package: timeDate
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.5.3
fBasics::basicStats(data)
##                    x1        x2        x3        x4        y1        y2
## nobs        11.000000 11.000000 11.000000 11.000000 11.000000 11.000000
## NAs          0.000000  0.000000  0.000000  0.000000  0.000000  0.000000
## Minimum      4.000000  4.000000  4.000000  8.000000  4.260000  3.100000
## Maximum     14.000000 14.000000 14.000000 19.000000 10.840000  9.260000
## 1. Quartile  6.500000  6.500000  6.500000  8.000000  6.315000  6.695000
## 3. Quartile 11.500000 11.500000 11.500000  8.000000  8.570000  8.950000
## Mean         9.000000  9.000000  9.000000  9.000000  7.500909  7.500909
## Median       9.000000  9.000000  9.000000  8.000000  7.580000  8.140000
## Sum         99.000000 99.000000 99.000000 99.000000 82.510000 82.510000
## SE Mean      1.000000  1.000000  1.000000  1.000000  0.612541  0.612568
## LCL Mean     6.771861  6.771861  6.771861  6.771861  6.136083  6.136024
## UCL Mean    11.228139 11.228139 11.228139 11.228139  8.865735  8.865795
## Variance    11.000000 11.000000 11.000000 11.000000  4.127269  4.127629
## Stdev        3.316625  3.316625  3.316625  3.316625  2.031568  2.031657
## Skewness     0.000000  0.000000  0.000000  2.466911 -0.048374 -0.978693
## Kurtosis    -1.528926 -1.528926 -1.528926  4.520661 -1.199123 -0.514319
##                    y3        y4
## nobs        11.000000 11.000000
## NAs          0.000000  0.000000
## Minimum      5.390000  5.250000
## Maximum     12.740000 12.500000
## 1. Quartile  6.250000  6.170000
## 3. Quartile  7.980000  8.190000
## Mean         7.500000  7.500909
## Median       7.110000  7.040000
## Sum         82.500000 82.510000
## SE Mean      0.612196  0.612242
## LCL Mean     6.135943  6.136748
## UCL Mean     8.864057  8.865070
## Variance     4.122620  4.123249
## Stdev        2.030424  2.030579
## Skewness     1.380120  1.120774
## Kurtosis     1.240044  0.628751
sapply(1:4, function(x) cor(data[ , x], data[ , x+4]))
## [1] 0.8164205 0.8162365 0.8162867 0.8165214
  1. Create scatter plots for each \(x, y\) pair of data.
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.5.2
p1 <- ggplot(data) +
  geom_point(aes(x1, y1), color = "black", size = 1.5) +
  scale_x_continuous(breaks = seq(0,20,2)) +
  scale_y_continuous(breaks = seq(0,12,2)) +
  expand_limits(x = 0, y = 0) +
  labs(x = "x1", y = "y1",
       title = "Dataset 1" ) +
  theme_bw()
p1

p2 <- ggplot(data) +
  geom_point(aes(x2, y2), color = "black", size = 1.5) +
  scale_x_continuous(breaks = seq(0,20,2)) +
  scale_y_continuous(breaks = seq(0,12,2)) +
  expand_limits(x = 0, y = 0) +
  labs(x = "x2", y = "y2",
       title = "Dataset 2" ) +
  theme_bw()
p2

p3 <- ggplot(data) +
  geom_point(aes(x3, y3), color = "black", size = 1.5) +
  scale_x_continuous(breaks = seq(0,20,2)) +
  scale_y_continuous(breaks = seq(0,12,2)) +
  expand_limits(x = 0, y = 0) +
  labs(x = "x3", y = "y3",
       title = "Dataset 3" ) +
  theme_bw()
p3

p4 <- ggplot(data) +
  geom_point(aes(x4, y4), color = "black", size = 1.5) +
  scale_x_continuous(breaks = seq(0,20,2)) +
  scale_y_continuous(breaks = seq(0,12,2)) +
  expand_limits(x = 0, y = 0) +
  labs(x = "x4", y = "y4",
       title = "Dataset 4" ) +
  theme_bw()
p4

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
library(grid)
library(gridExtra)
grid.arrange(grobs = list(p1, p2, p3, p4), 
             ncol = 2, 
             top = "Anscombe's Quartet")

  1. Now fit a linear model to each data set using the lm() function.
lm1 <- lm(y1 ~ x1, data = data)
lm1
## 
## Call:
## lm(formula = y1 ~ x1, data = data)
## 
## Coefficients:
## (Intercept)           x1  
##      3.0001       0.5001
lm2 <- lm(y2 ~ x2, data = data)
lm2
## 
## Call:
## lm(formula = y2 ~ x2, data = data)
## 
## Coefficients:
## (Intercept)           x2  
##       3.001        0.500
lm3 <- lm(y3 ~ x3, data = data)
lm3
## 
## Call:
## lm(formula = y3 ~ x3, data = data)
## 
## Coefficients:
## (Intercept)           x3  
##      3.0025       0.4997
lm4 <- lm(y4 ~ x4, data = data)
lm4
## 
## Call:
## lm(formula = y4 ~ x4, data = data)
## 
## Coefficients:
## (Intercept)           x4  
##      3.0017       0.4999
  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_fitted <- p1 + geom_abline(intercept = 3.0001, slope = 0.5001, color = "red")
p2_fitted <- p2 + geom_abline(intercept = 3.001, slope = 0.500, color = "red")
p3_fitted <- p3 + geom_abline(intercept = 3.0025, slope = 0.4997, color = "red")
p4_fitted <- p4 + geom_abline(intercept = 3.0017, slope = 0.499, color = "red")

grid.arrange(grobs = list(p1_fitted, p2_fitted,
                          p3_fitted, p4_fitted), 
             ncol = 2, 
             top = "Anscombe's Quartet")

  1. Now compare the model fits for each model object.
#The dataset 1 has moderately-positive linear fit with a correlation coefficient of 0.82 which indicates that knowing the value of x will give a value of y with very less noise. In dataset 2, we can clearly see that the relationship between x an y is not linear, however the correlation coeeficent is of 0.82 again and that indicates that knwoing the value of x will help in finding the value of y with very less noise. Hence, a linear model cannot be fitted to dataset 2. In dataset 3, although the relationship is linear the model doesn't fit the data that well since only one data point falls on the regression line. Dataset 4 also doesnot have a well fitting model as can be seen from the plot above.
  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.
#From looking at the various datasets in Anscombe's Quartet, it was intersting to learn how the different datasets having the same statistical summary in terms of mean, standard deviation and correlation can look very different when plotted in a graph. Hence, indicating that correctly interpreting the data by just reporting various statistics can be errogenous if data vizualization is overlooked.