knitr::opts_chunk$set(echo = TRUE)
library(ggplot2)
library(dplyr)
## Warning: package 'dplyr' was built under R version 3.4.4
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(datasets)
library(fBasics)
## Loading required package: timeDate
## Loading required package: timeSeries
library(ggpubr)
## Warning: package 'ggpubr' was built under R version 3.4.4
## Loading required package: magrittr

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” assignmenet on Moodle.

Questions

Question 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.

data = datasets::anscombe

Question 2

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!)

# load required library
require("fBasics")

# mean of each column
colMeans(data)
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
#variance of each column
colSds(data)
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 3.316625 3.316625 3.316625 3.316625 2.031568 2.031657 2.030424 2.030579
#correlation
cor(data$x1,data$y1)
## [1] 0.8164205
cor(data$x2,data$y2)
## [1] 0.8162365
cor(data$x3,data$y3)
## [1] 0.8162867
fBasics::correlationTest(data$x1,data$y1,method = "spearman")
## 
## Title:
##  Spearman's rho Correlation Test
## 
## Test Results:
##   SAMPLE ESTIMATES:
##     rho: 0.8182
##   STATISTIC:
##     S: 40
##   P VALUE:
##     Alternative Two-Sided: 0.003734 
##     Alternative      Less: 0.9984 
##     Alternative   Greater: 0.001867 
## 
## Description:
##  Tue Oct  2 18:30:33 2018
fBasics::correlationTest(data$x2,data$y2,method = "spearman")
## 
## Title:
##  Spearman's rho Correlation Test
## 
## Test Results:
##   SAMPLE ESTIMATES:
##     rho: 0.6909
##   STATISTIC:
##     S: 68
##   P VALUE:
##     Alternative Two-Sided: 0.02306 
##     Alternative      Less: 0.9896 
##     Alternative   Greater: 0.01153 
## 
## Description:
##  Tue Oct  2 18:30:33 2018
fBasics::correlationTest(data$x3,data$y3,method = "spearman")
## 
## Title:
##  Spearman's rho Correlation Test
## 
## Test Results:
##   SAMPLE ESTIMATES:
##     rho: 0.9909
##   STATISTIC:
##     S: 2
##   P VALUE:
##     Alternative Two-Sided: < 2.2e-16 
##     Alternative      Less: 1 
##     Alternative   Greater: < 2.2e-16 
## 
## Description:
##  Tue Oct  2 18:30:34 2018

Question 3

Create scatter plots for each \(x, y\) pair of data.

# for x1, y1
ggplot(data = data, aes(x1, y1)) + 
    geom_point(colour = '#7C9EB2') + 
    labs(title = "Scatterplot of x1, y1") + 
    theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))

# for x2, y2
ggplot(data = data, aes(x2, y2)) + 
    geom_point(colour = '#52528C') + 
    labs(title = "Scatterplot of x2, y2") + 
    theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))

# for x3, y3
ggplot(data = data, aes(x3, y3)) + 
    geom_point(colour = 'royal blue') + 
    labs(title = "Scatterplot of x3, y3") + 
    theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))

# for x4, y4
ggplot(data = data, aes(x4, y4)) + 
    geom_point(colour = '#14BDEB') + 
    labs(title = "Scatterplot of x4, y4") + 
    theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))

Question 4

Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic

# solid circles has shape number of 19
# for x1, y1
sc1 = ggplot(data = data, aes(x1, y1)) + 
      geom_point(colour = '#7C9EB2', shape = 19) + 
      labs(title = "Scatterplot of x1, y1") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))
# for x2, y2
sc2 = ggplot(data = data, aes(x2, y2)) + 
      geom_point(colour = '#52528C', shape = 19) + 
      labs(title = "Scatterplot of x2, y2") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))
# for x3, y3
sc3 = ggplot(data = data, aes(x3, y3)) + 
      geom_point(colour = 'royal blue', shape = 19) + 
      labs(title = "Scatterplot of x3, y3") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))
# for x4, y4
sc4 = ggplot(data = data, aes(x4, y4)) + 
      geom_point(colour = '#14BDEB', shape = 19) + 
      labs(title = "Scatterplot of x4, y4") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))

# create panel of 4
ggarrange(sc1, sc2, sc3, sc4, nrow = 2, ncol = 2)

Question5

Now fit a linear model to each data set using the lm() function.

# linear model for y1 ~ x1
fit1 = lm(y1 ~ x1, data = data)

# linear model for y2 ~ x2
fit2 = lm(y2 ~ x2, data = data)

# linear model for y3 ~ x3
fit3 = lm(y3 ~ x3, data = data)

# linear model for y4 ~ x4
fit4 = lm(y4 ~ x4, data = data)

Question6

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!)

# create scatterplot with regression lines
# for x1, y1 
scr1 = ggplot(data = data, aes(x1, y1)) + 
      geom_point(colour = '#7C9EB2', shape = 19) + 
      geom_smooth(method=lm) +
      labs(title = "Scatterplot of x1, y1") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))
# for x2, y2
scr2 = ggplot(data = data, aes(x2, y2)) + 
      geom_point(colour = '#52528C', shape = 19) + 
      geom_smooth(method=lm) +
      labs(title = "Scatterplot of x2, y2") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))
# for x3, y3
scr3 = ggplot(data = data, aes(x3, y3)) + 
      geom_point(colour = 'royal blue', shape = 19) + 
      geom_smooth(method=lm) +
      labs(title = "Scatterplot of x3, y3") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))
# for x4, y4
scr4 = ggplot(data = data, aes(x4, y4)) + 
      geom_point(colour = '#14BDEB', shape = 19) + 
      geom_smooth(method=lm) +
      labs(title = "Scatterplot of x4, y4") + 
      theme(plot.title = element_text(size=22,face = "bold",colour = "black", hjust = 0.5))

# create panel of 4
ggarrange(scr1, scr2, scr3, scr4, nrow = 2, ncol = 2)

Question7

Now compare the model fits for each model object.

summary(fit1)

Call: lm(formula = y1 ~ x1, data = data)

Residuals: Min 1Q Median 3Q Max -1.92127 -0.45577 -0.04136 0.70941 1.83882

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.0001 1.1247 2.667 0.02573 * x1 0.5001 0.1179 4.241 0.00217 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 1.237 on 9 degrees of freedom Multiple R-squared: 0.6665, Adjusted R-squared: 0.6295 F-statistic: 17.99 on 1 and 9 DF, p-value: 0.00217

summary(fit2)

Call: lm(formula = y2 ~ x2, data = data)

Residuals: Min 1Q Median 3Q Max -1.9009 -0.7609 0.1291 0.9491 1.2691

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.001 1.125 2.667 0.02576 * x2 0.500 0.118 4.239 0.00218 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 1.237 on 9 degrees of freedom Multiple R-squared: 0.6662, Adjusted R-squared: 0.6292 F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002179

summary(fit3)

Call: lm(formula = y3 ~ x3, data = data)

Residuals: Min 1Q Median 3Q Max -1.1586 -0.6146 -0.2303 0.1540 3.2411

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.0025 1.1245 2.670 0.02562 * x3 0.4997 0.1179 4.239 0.00218 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 1.236 on 9 degrees of freedom Multiple R-squared: 0.6663, Adjusted R-squared: 0.6292 F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002176

summary(fit4)

Call: lm(formula = y4 ~ x4, data = data)

Residuals: Min 1Q Median 3Q Max -1.751 -0.831 0.000 0.809 1.839

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.0017 1.1239 2.671 0.02559 * x4 0.4999 0.1178 4.243 0.00216 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 1.236 on 9 degrees of freedom Multiple R-squared: 0.6667, Adjusted R-squared: 0.6297 F-statistic: 18 on 1 and 9 DF, p-value: 0.002165

Question 8

In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.

Anscombe’s Quartet is a great example of the limitation of statistical summary. Statistical summary such as mean, median, standard deviation gives us very high level information and overall picture of a distribution. And often times, we believe together with the additional information of correlation, linear regression, we would have a good view of the data. However, we might neglect the non-linear pattern in the data pairs.

In Anscombe’s Quartet, it gives us 4 pairs of data with all the same/almost same statistical summary and linear model formula and result. They seems to be pretty similar in this case. However, when we check the scatterplot, we can directly see the difference, and identify the different patterns between different pairs. The value of data visualization, as it shows, can help people easily, directly detect patterns and understand better of the relationships.