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 post your assignment on Rpubs and upload a link to it to the “Problem Set 2” assignmenet on Moodle.

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.
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 3.3.3
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.3.3
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.3.3
## 
## Rmetrics Package fBasics
## Analysing Markets and calculating Basic Statistics
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
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
colVars(data)
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249
correlationTest(data$x1,data$y1,title = "Pearson's Correlation Test between x1 and y1", description = NULL)
## 
## Title:
##  Pearson's Correlation Test between x1 and y1
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8164
##   STATISTIC:
##     t: 4.2415
##   P VALUE:
##     Alternative Two-Sided: 0.00217 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001085 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4244, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5113, 1
## 
## Description:
##  Sun Jul 23 22:05:30 2017
correlationTest(data$x2,data$y2,title = "Pearson's Correlation Test between x2 and y2", description = NULL)
## 
## Title:
##  Pearson's Correlation Test between x2 and y2
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8162
##   STATISTIC:
##     t: 4.2386
##   P VALUE:
##     Alternative Two-Sided: 0.002179 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001089 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4239, 0.9506
##          Less: -1, 0.9387
##       Greater: 0.5109, 1
## 
## Description:
##  Sun Jul 23 22:05:30 2017
correlationTest(data$x3,data$y3,title = "Pearson's Correlation Test between x3 and y3",description = NULL)
## 
## Title:
##  Pearson's Correlation Test between x3 and y3
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8163
##   STATISTIC:
##     t: 4.2394
##   P VALUE:
##     Alternative Two-Sided: 0.002176 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001088 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4241, 0.9507
##          Less: -1, 0.9387
##       Greater: 0.511, 1
## 
## Description:
##  Sun Jul 23 22:05:30 2017
correlationTest(data$x4,data$y4,title = "Pearson's Correlation Test between x4 and y4",description = NULL)
## 
## Title:
##  Pearson's Correlation Test between x4 and y4
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8165
##   STATISTIC:
##     t: 4.243
##   P VALUE:
##     Alternative Two-Sided: 0.002165 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001082 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4246, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5115, 1
## 
## Description:
##  Sun Jul 23 22:05:30 2017
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1,data$y1,xlab="x1", ylab="y1")

plot(data$x2,data$y2,xlab="x2", ylab="y2")

plot(data$x3,data$y3,xlab="x3", ylab="y3")

plot(data$x4,data$y4,xlab="x4", ylab="y4")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
par(mfrow=c(2,2))
plot(data$x1,data$y1,xlab="x1", ylab="y1",pch=19)
plot(data$x2,data$y2,xlab="x2", ylab="y2",pch=19)
plot(data$x3,data$y3,xlab="x3", ylab="y3",pch=19)
plot(data$x4,data$y4,xlab="x4", ylab="y4",pch=19)

  1. Now fit a linear model to each data set using the lm() function.
lm1 = lm(data$y1~data$x1)
lm1
## 
## Call:
## lm(formula = data$y1 ~ data$x1)
## 
## Coefficients:
## (Intercept)      data$x1  
##      3.0001       0.5001
lm2 = lm(data$y2~data$x2)
lm2
## 
## Call:
## lm(formula = data$y2 ~ data$x2)
## 
## Coefficients:
## (Intercept)      data$x2  
##       3.001        0.500
lm3 = lm(data$y3~data$x3)
lm3
## 
## Call:
## lm(formula = data$y3 ~ data$x3)
## 
## Coefficients:
## (Intercept)      data$x3  
##      3.0025       0.4997
lm4 = lm(data$y4~data$x4)
lm4
## 
## Call:
## lm(formula = data$y4 ~ data$x4)
## 
## Coefficients:
## (Intercept)      data$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!)
par(mfrow=c(2,2))
plot(data$x1,data$y1,xlab="x1", ylab="y1",pch=19)
abline(lm1,col="blue")
plot(data$x2,data$y2,xlab="x2", ylab="y2",pch=19)
abline(lm2,col="blue")
plot(data$x3,data$y3,xlab="x3", ylab="y3",pch=19)
abline(lm3,col="blue")
plot(data$x4,data$y4,xlab="x4", ylab="y4",pch=19)
abline(lm4,col="blue")

  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 summary statistics of each (x,y) pair of Anscombe’s Quartet dataset are close to identical regarding to their means, variances, and correlations. When we plot the data, however, it appears that they are dramatically different from each other. It can be dangerous if we draw conclusions only based upon summary statistics. Visualization helps us quickly tell the general pattern of a dataset and see the details which cannot be told by summary statistics.