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” assignment to your R Pubs account and submit the link to Moodle. Points will be deducted for uploading the improper format.

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)
data<-anscombe
head(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
  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
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         x2         x3         x4         y1         y2
## x1  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x2  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x3  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x4 -0.5000000 -0.5000000 -0.5000000  1.0000000 -0.5290927 -0.7184365
## y1  0.8164205  0.8164205  0.8164205 -0.5290927  1.0000000  0.7500054
## y2  0.8162365  0.8162365  0.8162365 -0.7184365  0.7500054  1.0000000
## y3  0.8162867  0.8162867  0.8162867 -0.3446610  0.4687167  0.5879193
## y4 -0.3140467 -0.3140467 -0.3140467  0.8165214 -0.4891162 -0.4780949
##            y3         y4
## x1  0.8162867 -0.3140467
## x2  0.8162867 -0.3140467
## x3  0.8162867 -0.3140467
## x4 -0.3446610  0.8165214
## y1  0.4687167 -0.4891162
## y2  0.5879193 -0.4780949
## y3  1.0000000 -0.1554718
## y4 -0.1554718  1.0000000
  1. Create scatter plots for each \(x, y\) pair of data.
layout(matrix(1:4, nrow = 2, ncol = 2, byrow = TRUE))
plot(data$x1,data$y1,xlab="x1",ylab="y1",main="Scatter Plots for x1 & y1",col="blue")
plot(data$x2,data$y2,xlab="x2",ylab="y2",main="Scatter Plots for x2 & y2",col="green")
plot(data$x3,data$y3,xlab="x3",ylab="y3",main="Scatter Plots for x3 & y3",col="orange")
plot(data$x4,data$y4,xlab="x4",ylab="y4",main="Scatter Plots for x4 & y4",col="red")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
layout(matrix(1:4, nrow = 2, ncol = 2, byrow = TRUE))
plot(data$x1,data$y1,xlab="x1",ylab="y1",main="Scatter Plots for x1 & y1",col="blue",pch=19)
plot(data$x2,data$y2,xlab="x2",ylab="y2",main="Scatter Plots for x2 & y2",col="green",pch=19)
plot(data$x3,data$y3,xlab="x3",ylab="y3",main="Scatter Plots for x3 & y3",col="orange",pch=19)
plot(data$x4,data$y4,xlab="x4",ylab="y4",main="Scatter Plots for x4 & y4",col="red",pch=19)

  1. Now fit a linear model to each data set using the lm() function.
lm1<-lm(data$y1~data$x1)
lm2<-lm(data$y2~data$x2)
lm3<-lm(data$y3~data$x3)
lm4<-lm(data$y4~data$x4)
  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!)
layout(matrix(1:4, nrow = 2, ncol = 2, byrow = TRUE))
plot(data$x1,data$y1,xlab="x1",ylab="y1",main="Scatter Plots for x1 & y1 with regression line",col="blue",pch=19)
abline(lm1, col="blue")
plot(data$x2,data$y2,xlab="x2",ylab="y2",main="Scatter Plots for x2 & y2 with regression line",col="green",pch=19)
abline(lm2, col="green")
plot(data$x3,data$y3,xlab="x3",ylab="y3",main="Scatter Plots for x3 & y3 with regression line",col="orange",pch=19)
abline(lm3, col="orange")
plot(data$x4,data$y4,xlab="x4",ylab="y4",main="Scatter Plots for x4 & y4 with regression line",col="red",pch=19)
abline(lm4, col="red")

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

Call: lm(formula = data\(y1 ~ data\)x1)

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 * data$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(lm2)

Call: lm(formula = data\(y2 ~ data\)x2)

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 * data$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(lm3)

Call: lm(formula = data\(y3 ~ data\)x3)

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 * data$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(lm4)

Call: lm(formula = data\(y4 ~ data\)x4)

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 * data$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

A: By comparing th r square for each linear model, the model fit rank is lm4>lm1>lm3>lm2.

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

The data variables in Anscombe appear same mean and vairance for each variables and relatively same correlations / linear relations between each pair. Only when we plotting these variables pairs can we see the true relationship between each pair. Data visulization is valuble when approaching the analysis of dataset.