Objectives

The objectives of this problem set is to perform various activities in R. And to conduct a thoughtful exercise in appreciating the importance of data visualization. For each question created a code chunk or text response that completes/answers the activity or question requested.

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
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!)
## Mean of each variable

mu <- lapply(data, mean)
MU <- unlist(mu)
MU
##       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 of each variable

sigma.Sq <- lapply(data, var)
sigma.Sqr <- unlist(sigma.Sq)
sigma.Sqr
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249
## Correlation between $x, $y

cor.func <- function(x){
 cordata <- vector("numeric", 4L)
  for(i in 1:4) {
    cordata[i] <- cor(x[,i], x[,i+4])
  }
  return(cordata)
}

cor.func(data)
## [1] 0.8164205 0.8162365 0.8162867 0.8165214
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1,data$y1, xlab ="x1", ylab="y1", main = "plot of (x1,y1)")

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

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

plot(data$x4,data$y4, xlab ="x4", ylab="y4", main = "plot of (x4,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", main = "plot of (x1,y1)", pch =16, cex =1.4)
plot(data$x2,data$y2, xlab ="x2", ylab="y2", main = "plot of (x2,y2)", pch =16, cex =1.4)
plot(data$x3,data$y3, xlab ="x3", ylab="y3", main = "plot of (x3,y3)", pch =16, cex =1.4)
plot(data$x4,data$y4, xlab ="x4", ylab="y4", main = "plot of (x4,y4)", pch =16, cex =1.4)

  1. Now fit a linear model to each data set using the lm() function.
lm1 <- lm(y1 ~ x1, data)
lm2 <- lm(y2 ~ x2, data)
lm3 <- lm(y3 ~ x3, data)
lm4 <- lm(y4 ~ x4, data)
  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", main = "model fit of (x1,y1)", pch =16, cex =1.4)
abline(lm1)
plot(data$x2,data$y2, xlab ="x2", ylab="y2", main = "model fit of (x2,y2)", pch =16, cex =1.4)
abline(lm2)
plot(data$x3,data$y3, xlab ="x3", ylab="y3", main = "model fit of (x3,y3)", pch =16, cex =1.4)
abline(lm3)
plot(data$x4,data$y4, xlab ="x4", ylab="y4", main = "model fit of (x4,y4)", pch =16, cex =1.4)
abline(lm4)

  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.

At first glance datasets looked identical, summarizing the data too lead us to believe that data are identical. Only after visualizing the data it shows indeed data are not at all identical. Hence, Visualization helped in avoiding false conclusions.