Xiao’ou Zhu’s Anscombe’s quartet Analysis

Questions

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

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

colStats(anscombe,FUN = mean)

##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909

colStats(anscombe, FUN= var)

##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249

x1<-data[,1]
x2<-data[,2]
x3<-data[,3]
x4<-data[,4]
y1<-data[,5]
y2<-data[,6]
y3<-data[,7]
y4<-data[,8]

cor(x1,y1)

## [1] 0.8164205

cor(x2,y2)

## [1] 0.8162365

cor(x3,y3)

## [1] 0.8162867

cor(x4,y4)

## [1] 0.8165214

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

plot(x1,y1, main="Scatterplot for X1 and Y1")

plot(x2,y2, main="Scatterplot for X2 and Y2")

plot(x3,y3, main="Scatterplot for X3 and Y3")

plot(x4,y4, main="Scatterplot for X4 and Y4")

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(x1,y1,pch=16)
plot(x2,y2,pch=16)
plot(x3,y3,pch=16)
plot(x4,y4,pch=16)

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

fit1 <- lm(y1~x1)
fit2 <- lm(y2~x2)
fit3 <- lm(y3~x3)
fit4 <- lm(y4~x4)

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(x1,y1,pch=16)
abline(fit1)
plot(x2,y2,pch=16)
abline(fit2)
plot(x3,y3,pch=16)
abline(fit3)
plot(x4,y4,pch=16)
abline(fit4)

Now compare the model fits for each model object.

anova(fit1)

Analysis of Variance Table

Response: y1 Df Sum Sq Mean Sq F value Pr(>F)
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(fit2)

Analysis of Variance Table

Response: y2 Df Sum Sq Mean Sq F value Pr(>F)
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(fit3)

Analysis of Variance Table

Response: y3 Df Sum Sq Mean Sq F value Pr(>F)
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(fit4)

Analysis of Variance Table

Response: y4 Df Sum Sq Mean Sq F value Pr(>F)
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

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

The summary statistics have shown similar responses for all four fit models. In particular, fit 1 and fit 2 both demonstrate positive correlation. However, the data visualization has shown that fit is less linear than fit 2.

Also, fit 3 and fit 4 are very vunerable to outliers, as shown in the visualization.

Xiao’ou Zhu’s Anscombe’s quartet Analysis

Xiao’ou Zhu

May 30th, 2017

Objectives

Questions