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 Rpubs site and submit the link to the hosted file via 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.
library(datasets)
library(fBasics)
## Warning: package 'fBasics' was built under R version 3.2.5
## Loading required package: timeDate
## Loading required package: timeSeries
data <- anscombe
  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!)
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(data$x1,data$y1,main="Scatter Plot of x1 & y1")

plot(data$x2,data$y2,main="Scatter Plot of x2 & y2")

plot(data$x3,data$y3,main="Scatter Plot of x3 & y3")

plot(data$x4,data$y4,main="Scatter Plot of x4 & 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(data$x1,data$y1,main="Scatter Plot of x1 & y1",pch=19)
plot(data$x2,data$y2,main="Scatter Plot of x2 & y2",pch=19)
plot(data$x3,data$y3,main="Scatter Plot of x3 & y3",pch=19)
plot(data$x4,data$y4,main="Scatter Plot of x4 & y4",pch=19)

  1. Now fit a linear model to each data set using the lm() function.
summary(lm(y1~x1,data=data))
## 
## Call:
## lm(formula = y1 ~ x1, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.92127 -0.45577 -0.04136  0.70941  1.83882 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   3.0001     1.1247   2.667  0.02573 * 
## x1            0.5001     0.1179   4.241  0.00217 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.237 on 9 degrees of freedom
## Multiple R-squared:  0.6665, Adjusted R-squared:  0.6295 
## F-statistic: 17.99 on 1 and 9 DF,  p-value: 0.00217
summary(lm(y2~x2,data=data))
## 
## Call:
## lm(formula = y2 ~ x2, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9009 -0.7609  0.1291  0.9491  1.2691 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)    3.001      1.125   2.667  0.02576 * 
## x2             0.500      0.118   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.237 on 9 degrees of freedom
## Multiple R-squared:  0.6662, Adjusted R-squared:  0.6292 
## F-statistic: 17.97 on 1 and 9 DF,  p-value: 0.002179
summary(lm(y3~x3,data=data))
## 
## Call:
## lm(formula = y3 ~ x3, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.1586 -0.6146 -0.2303  0.1540  3.2411 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   3.0025     1.1245   2.670  0.02562 * 
## x3            0.4997     0.1179   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.236 on 9 degrees of freedom
## Multiple R-squared:  0.6663, Adjusted R-squared:  0.6292 
## F-statistic: 17.97 on 1 and 9 DF,  p-value: 0.002176
summary(lm(y4~x4,data=data))
## 
## Call:
## lm(formula = y4 ~ x4, data = data)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -1.751 -0.831  0.000  0.809  1.839 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   3.0017     1.1239   2.671  0.02559 * 
## x4            0.4999     0.1178   4.243  0.00216 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.236 on 9 degrees of freedom
## Multiple R-squared:  0.6667, Adjusted R-squared:  0.6297 
## F-statistic:    18 on 1 and 9 DF,  p-value: 0.002165
  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!)
Regression1<-lm(y1~x1,data=data)
Regression2<-lm(y2~x2,data=data)
Regression3<-lm(y3~x3,data=data)
Regression4<-lm(y4~x4,data=data)

par(mfrow=c(2,2))
plot(data$x1,data$y1,main="Regression x1 and y1",pch=19)
abline(Regression1)
plot(data$x2,data$y2,main="Regression x2 and y2",pch=19)
abline(Regression2)
plot(data$x3,data$y3,main="Regression x3 and y3",pch=19)
abline(Regression3)
plot(data$x4,data$y4,main="Regression x4 and y4",pch=19)
abline(Regression4)

  1. Now compare the model fits for each model object.
CompareTable<-data.frame(c(summary(Regression1)$adj.r.squared,summary(Regression2)$adj.r.squared,summary(Regression3)$adj.r.squared,summary(Regression4)$adj.r.squared))
rownames(CompareTable)<-c("x1&y1","x2&y2","x3&y3","x4&y4")
colnames(CompareTable)<-c("Adjusted R-square")
CompareTable
  Adjusted R-square

x1&y1 0.6294916 x2&y2 0.6291578 x3&y3 0.6292489 x4&y4 0.6296747

anova(Regression1)
## 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(Regression2)
## 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(Regression3)
## 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(Regression4)
## 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

Results from above models are quite close.

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

The lesson of Anscombe’s Quartet discusses that eventough summary statistics allow us to have a general idea about the data, it could be misleading for certain situation when only depend on stat summary. Anscombe is a typical example to show the necessary for data visualization comparing with only using data summary. Above four datasets show extremely similar pattern statistically while they actually tell quite different story when we visualize the data.