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” 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.

data<-anscombe

2. 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 3.3.3
## Loading required package: timeDate
## Loading required package: timeSeries
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

3. 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")

4. 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)

5. 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)

6. 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)

7. 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

8. 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 the fit1 is less linear than fit 2.

It is also seen that the fit 3 and fit 4 are very vunerable to outliers, as shown in the visualization.