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
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
  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
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
cor(data$x1, data$y1)
## [1] 0.8164205
cor(data$x2, data$y2)
## [1] 0.8162365
cor(data$x3, data$y3)
## [1] 0.8162867
cor(data$x4, data$y4)
## [1] 0.8165214
  1. Create scatter plots for each \(x, y\) pair of data.
plot_1<-plot(data$x1, data$y1, pch = 19, main='Plot 1',xlab = "x1", ylab = "y1", cex=0.9)

plot_2<-plot(data$x2, data$y2, pch = 19, main='Plot 2',xlab = "x2", ylab = "y2", cex=0.9)

plot_3<-plot(data$x3, data$y3, pch = 19, main='Plot 3',xlab = "x3", ylab = "y3", cex=0.9)

plot_4<-plot(data$x4, data$y4, pch = 19, main='Plot 4',xlab = "x4", ylab = "y4", cex=0.9)

  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, pch = 16, main='Plot 1',xlab = "x1", ylab = "y1", cex=0.9)
plot(data$x2, data$y2, pch = 16, main='Plot 2',xlab = "x2", ylab = "y2", cex=0.9)
plot(data$x3, data$y3, pch = 16, main='Plot 3',xlab = "x3", ylab = "y3", cex=0.9)
plot(data$x4, data$y4, pch = 16, main='Plot 4',xlab = "x4", ylab = "y4", cex=0.9)

  1. Now fit a linear model to each data set using the lm() function.
plot_1<-plot(data$x1, data$y1, pch = 19, main='Plot 1',xlab = "x1", ylab = "y1", cex=0.9)
abline(lm(data$y1 ~ data$x1),col="darkblue")

plot_2<-plot(data$x2, data$y2, pch = 19, main='Plot 2',xlab = "x2", ylab = "y2", cex=0.9)
abline(lm(data$y2 ~ data$x2),col="darkblue")

plot_3<-plot(data$x3, data$y3, pch = 19, main='Plot 3',xlab = "x3", ylab = "y3", cex=0.9)
abline(lm(data$y3 ~ data$x3),col="darkblue")

plot_4<-plot(data$x4, data$y4, pch = 19, main='Plot 4',xlab = "x4", ylab = "y4", cex=0.9)
abline(lm(data$y4 ~ data$x4),col="darkblue")

  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, pch = 16, main='Plot 1',xlab = "x1", ylab = "y1", cex=0.9)
abline(lm(data$y1 ~ data$x1),col="darkred")
plot(data$x2, data$y2, pch = 16, main='Plot 2',xlab = "x2", ylab = "y2", cex=0.9)
abline(lm(data$y2 ~ data$x2),col="darkred")
plot(data$x3, data$y3, pch = 16, main='Plot 3',xlab = "x3", ylab = "y3", cex=0.9)
abline(lm(data$y3 ~ data$x3),col="darkred")
plot(data$x4, data$y4, pch = 16, main='Plot 4',xlab = "x4", ylab = "y4", cex=0.9)
abline(lm(data$y4 ~ data$x4),col="darkred")

  1. Now compare the model fits for each model object.
model1<-lm(data$y1 ~ data$x1)
anova(model1)

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

model2<-lm(data$y2 ~ data$x2)
anova(model2)

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

model3<-lm(data$y3 ~ data$x3)
anova(model3)

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

model4<-lm(data$y4 ~ data$x4)
anova(model4)

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.

Anscombe’s quarter contains 4 different data sets which are having identically descriptive statistics but when we plot them they appear to be very different have very different distributions. This was contracted to explain the importance of data visualization. In 1973 these data sets were used to show how data seem to be identical but how it appears when we actually plot it in a graph. This actually explains that how important is to plot the data to see how data is distributed and what are the outliers and how the data points are correlated. We can clearly say that summarizing the data always might not help in all the cases we need to plot the data to have a clear picture.