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 upload your document to rpubs.com and share the link 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.
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)
## Loading required package: timeDate
## Loading required package: timeSeries
## 
## 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
means <- colMeans(data)
vars <- colVars(data)
cor1 <- cor(data$x1,data$y1)
cor2 <- cor(data$x2,data$y2)
cor3 <- cor(data$x3,data$y3)
cor4 <- cor(data$x4,data$y4)
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1,data$y1, main="Scatterplot of x1,y1", xlab="x1", ylab="y1")

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

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

plot(data$x4,data$y4, main="Scatterplot of x4,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, main="Scatterplot of x1,y1", xlab="x1", ylab="y1", pch=19)
plot(data$x2,data$y2, main="Scatterplot of x2,y2", xlab="x2", ylab="y2", pch=19)
plot(data$x3,data$y3, main="Scatterplot of x3,y3", xlab="x3", ylab="y3", pch=19)
plot(data$x4,data$y4, main="Scatterplot of x4,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)
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!)
par(mfrow=c(2,2))
plot(data$x1,data$y1, main="Scatterplot of x1,y1", xlab="x1", ylab="y1", pch=19)
abline(lm1)
plot(data$x2,data$y2, main="Scatterplot of x2,y2", xlab="x2", ylab="y2", pch=19)
abline(lm2)
plot(data$x3,data$y3, main="Scatterplot of x3,y3", xlab="x3", ylab="y3", pch=19)
abline(lm3)
plot(data$x4,data$y4, main="Scatterplot of x4,y4", xlab="x4", ylab="y4", pch=19)
abline(lm4)

  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.

There are four pairs of data of Anscombe’s Quartet. From the data visualization results, we can see they are quite different. In addition, linear model doesn’t fit all data pairs. Only x3,y3 fit the linear model well. The data visualization help us understand the dataset clear and directly.