ANLY 512 - Problem Set 2
Anscombe’s quartet
Xizi Tong 2018-04-20
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
anscombedata that is part of thelibrary(datasets)inR. And assign that data to a new object calleddata.
library(datasets)
data("anscombe")
View(anscombe)
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')## Warning: package 'fBasics' was built under R version 3.4.4## Loading required package: timeDate## Warning: package 'timeDate' was built under R version 3.4.3## Loading required package: timeSeries## Warning: package 'timeSeries' was built under R version 3.4.4basicStats(data)## x1 x2 x3 x4 y1 y2
## nobs 11.000000 11.000000 11.000000 11.000000 11.000000 11.000000
## NAs 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## Minimum 4.000000 4.000000 4.000000 8.000000 4.260000 3.100000
## Maximum 14.000000 14.000000 14.000000 19.000000 10.840000 9.260000
## 1. Quartile 6.500000 6.500000 6.500000 8.000000 6.315000 6.695000
## 3. Quartile 11.500000 11.500000 11.500000 8.000000 8.570000 8.950000
## Mean 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909
## Median 9.000000 9.000000 9.000000 8.000000 7.580000 8.140000
## Sum 99.000000 99.000000 99.000000 99.000000 82.510000 82.510000
## SE Mean 1.000000 1.000000 1.000000 1.000000 0.612541 0.612568
## LCL Mean 6.771861 6.771861 6.771861 6.771861 6.136083 6.136024
## UCL Mean 11.228139 11.228139 11.228139 11.228139 8.865735 8.865795
## Variance 11.000000 11.000000 11.000000 11.000000 4.127269 4.127629
## Stdev 3.316625 3.316625 3.316625 3.316625 2.031568 2.031657
## Skewness 0.000000 0.000000 0.000000 2.466911 -0.048374 -0.978693
## Kurtosis -1.528926 -1.528926 -1.528926 4.520661 -1.199123 -0.514319
## y3 y4
## nobs 11.000000 11.000000
## NAs 0.000000 0.000000
## Minimum 5.390000 5.250000
## Maximum 12.740000 12.500000
## 1. Quartile 6.250000 6.170000
## 3. Quartile 7.980000 8.190000
## Mean 7.500000 7.500909
## Median 7.110000 7.040000
## Sum 82.500000 82.510000
## SE Mean 0.612196 0.612242
## LCL Mean 6.135943 6.136748
## UCL Mean 8.864057 8.865070
## Variance 4.122620 4.123249
## Stdev 2.030424 2.030579
## Skewness 1.380120 1.120774
## Kurtosis 1.240044 0.628751correlationTest(data$x1,data$y1,method = 'pearson',title = 'x1 and y1 correlation')##
## Title:
## x1 and y1 correlation
##
## 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:
## Mon Apr 30 14:45:22 2018correlationTest(data$x2,data$y2,method = 'pearson',title = 'x2 and y2 correlation')##
## Title:
## x2 and y2 correlation
##
## 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:
## Mon Apr 30 14:45:22 2018correlationTest(data$x3,data$y3,method = 'pearson',title = 'x3 and y3 correlation')##
## Title:
## x3 and y3 correlation
##
## 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:
## Mon Apr 30 14:45:22 2018correlationTest(data$x4,data$y4,method = 'pearson',title = 'x4 and y4 correlation')##
## Title:
## x4 and y4 correlation
##
## 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:
## Mon Apr 30 14:45:22 2018- Create scatter plots for each \(x, y\) pair of data.
library(ggplot2)## Warning: package 'ggplot2' was built under R version 3.4.3pl1=ggplot(data,aes(x=x1,y=y1))+geom_point(color='red')
pl1
pl2=ggplot(data,aes(x=x2,y=y2))+geom_point(color='red')
pl2
pl3=ggplot(data,aes(x=x3,y=y3))+geom_point(color='red')
pl3
pl4=ggplot(data,aes(x=x4,y=y4))+geom_point(color='red')
pl4
- Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
library('gridExtra')## Warning: package 'gridExtra' was built under R version 3.4.4grid.arrange(pl2,pl2,pl3,pl4, top="ScatterPlot")
- Now fit a linear model to each data set using the
lm()function.
pl10=plot(data$x1,data$y1)
abline(lm(y1~x1,data=data))
pl20=plot(data$x2,data$y2)
abline(lm(y2~x2,data=data))
pl30=plot(data$x3,data$y3)
abline(lm(y3~x3,data=data))
pl40=plot(data$x4,data$y4)
abline(lm(y4~x4,data=data))
- 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))
pl10=plot(data$x1,data$y1)
abline(lm(y1~x1,data=data))
pl20=plot(data$x2,data$y2)
abline(lm(y2~x2,data=data))
pl30=plot(data$x3,data$y3)
abline(lm(y3~x3,data=data))
pl40=plot(data$x4,data$y4)
abline(lm(y4~x4,data=data))
- Now compare the model fits for each model object.
summary(lm(y1~x1,data))$r.squared[1] 0.6665425
summary(lm(y2~x2,data))$r.squared[1] 0.666242
summary(lm(y3~x3,data))$r.squared[1] 0.666324
summary(lm(y4~x4,data))$r.squared[1] 0.6667073
- In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.