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 <- datasets::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!)
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$x1,data$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, xlab="x1", ylab="y1")

plot(data$x2, data$y2, xlab="x2", ylab="y2")

plot(data$x3, data$y3, xlab="x3", ylab="y3")

plot(data$x4, data$y4, 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(data$x1, data$y1, xlab="x1", ylab="y1", pch=19)
plot(data$x2, data$y2, xlab="x2", ylab="y2", pch=19)
plot(data$x3, data$y3, xlab="x3", ylab="y3", pch=19)
plot(data$x4, data$y4, xlab="x4", ylab="y4", 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!)
par(mfrow = c(2, 2))
plot(data$x1, data$y1, xlab="x1", ylab="y1", pch=19); abline(lm1)
plot(data$x2, data$y2, xlab="x2", ylab="y2", pch=19); abline(lm2)
plot(data$x3, data$y3, xlab="x3", ylab="y3", pch=19); abline(lm3)
plot(data$x4, data$y4, xlab="x4", ylab="y4", pch=19); abline(lm4)

  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

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

The lesson I learned from Anscombe’s Quartet is that we shouldn’t reach any conclusion about data without visualizing it. The 4 datasets of Anscombe’s Quartet have very similar statistics in terms of mean, variance and correlation. When trying to fit each of them with a linear model, we see similar F-statistics as well. All of above is suggesting that these 4 datasets are very similar to each other. However, after visualizing the data sets, we realize they are fundamentally different. Some of them are very good fits except outliers, while in other cases, the relationship between variables requires a more complicated model than linear ones to fit.