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] (https://www.jstor.org/stable/2682899?seq=1#page_scan_tab_contents). 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)
data <- anscombe
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
## 7   6  6  6  8  7.24 6.13  6.08  5.25
## 8   4  4  4 19  4.26 3.10  5.39 12.50
## 9  12 12 12  8 10.84 9.13  8.15  5.56
## 10  7  7  7  8  4.82 7.26  6.42  7.91
## 11  5  5  5  8  5.68 4.74  5.73  6.89
  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

Mean

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

Variance

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

Correlation x1 with y1

correlationTest(x1,y1,method = c("pearson", "kendall", "spearman"))
## 
## Title:
##  Pearson's Correlation Test
## 
## 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:
##  Thu Apr 19 00:43:45 2018

Correlation x2 with y2

correlationTest(x2,y2,method = c("pearson", "kendall", "spearman"))
## 
## Title:
##  Pearson's Correlation Test
## 
## 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:
##  Thu Apr 19 00:43:45 2018

Correlation x3 with y3

correlationTest(x3,y3,method = c("pearson", "kendall", "spearman"))
## 
## Title:
##  Pearson's Correlation Test
## 
## 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:
##  Thu Apr 19 00:43:45 2018

Correlation x4 with y4

correlationTest(x4,y4,method = c("pearson", "kendall", "spearman"))
## 
## Title:
##  Pearson's Correlation Test
## 
## 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:
##  Thu Apr 19 00:43:45 2018
  1. Create scatter plots for each \(x, y\) pair of data.
plot(x1,y1,main="Scatterplot Pair 1",xlab="x1",ylab="y1")

plot(x2,y2,main="Scatterplot Pair 2",xlab="x2",ylab="y2")

plot(x3,y3,main="Scatterplot Pair 3",xlab="x3",ylab="y3")

plot(x4,y4,main="Scatterplot Pair 4",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(x1,y1,main="Scatterplot Pair 1",xlab="x1",ylab="y1",pch=20)
plot(x2,y2,main="Scatterplot Pair 2",xlab="x2",ylab="y2",pch=20)
plot(x3,y3,main="Scatterplot Pair 3",xlab="x3",ylab="y3",pch=20)
plot(x4,y4,main="Scatterplot Pair 4",xlab="x4",ylab="y4",pch=20)

  1. Now fit a linear model to each data set using the lm() function.
lm1<-lm(y1~x1)
lm2<-lm(y2~x2)
lm3<-lm(y3~x3)
lm4<-lm(y4~x4)
lm1
## 
## Call:
## lm(formula = y1 ~ x1)
## 
## Coefficients:
## (Intercept)           x1  
##      3.0001       0.5001
lm2
## 
## Call:
## lm(formula = y2 ~ x2)
## 
## Coefficients:
## (Intercept)           x2  
##       3.001        0.500
lm3
## 
## Call:
## lm(formula = y3 ~ x3)
## 
## Coefficients:
## (Intercept)           x3  
##      3.0025       0.4997
lm4
## 
## Call:
## lm(formula = y4 ~ x4)
## 
## Coefficients:
## (Intercept)           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(x1,y1,main="Scatterplot Pair 1",xlab="x1",ylab="y1",pch=20)
abline(lm(y1 ~ x1))
plot(x2,y2,main="Scatterplot Pair 2",xlab="x2",ylab="y2",pch=20)
abline(lm(y2 ~ x2))
plot(x3,y3,main="Scatterplot Pair 3",xlab="x3",ylab="y3",pch=20)
abline(lm(y3 ~ x3))
plot(x4,y4,main="Scatterplot Pair 4",xlab="x4",ylab="y4",pch=20)
abline(lm(y4 ~ x4))

  1. Now compare the model fits for each model object.
anova(lm1)

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(lm2)

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(lm3)

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(lm4)

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

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

This is an unexpected result, all the descriptive statistics can not show how different those groups of numbers are. If only by looking at the summary, I might think they are very similar groups of numbers. Data visualization gives us a very direct way to know those data.