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, upload a link to your file hosted on rpubs.com

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!)
library(fBasics)
## Loading required package: timeDate
## Loading required package: timeSeries
basicStats(data)
##                    x1        x2        x3        x4        y1        y2
## nobs        11.000000 11.000000 11.000000 11.000000 11.000000 11.000000
## NAs          0.000000  0.000000  0.000000  0.000000  0.000000  0.000000
## Minimum      4.000000  4.000000  4.000000  8.000000  4.260000  3.100000
## Maximum     14.000000 14.000000 14.000000 19.000000 10.840000  9.260000
## 1. Quartile  6.500000  6.500000  6.500000  8.000000  6.315000  6.695000
## 3. Quartile 11.500000 11.500000 11.500000  8.000000  8.570000  8.950000
## Mean         9.000000  9.000000  9.000000  9.000000  7.500909  7.500909
## Median       9.000000  9.000000  9.000000  8.000000  7.580000  8.140000
## Sum         99.000000 99.000000 99.000000 99.000000 82.510000 82.510000
## SE Mean      1.000000  1.000000  1.000000  1.000000  0.612541  0.612568
## LCL Mean     6.771861  6.771861  6.771861  6.771861  6.136083  6.136024
## UCL Mean    11.228139 11.228139 11.228139 11.228139  8.865735  8.865795
## Variance    11.000000 11.000000 11.000000 11.000000  4.127269  4.127629
## Stdev        3.316625  3.316625  3.316625  3.316625  2.031568  2.031657
## Skewness     0.000000  0.000000  0.000000  2.466911 -0.048374 -0.978693
## Kurtosis    -1.528926 -1.528926 -1.528926  4.520661 -1.199123 -0.514319
##                    y3        y4
## nobs        11.000000 11.000000
## NAs          0.000000  0.000000
## Minimum      5.390000  5.250000
## Maximum     12.740000 12.500000
## 1. Quartile  6.250000  6.170000
## 3. Quartile  7.980000  8.190000
## Mean         7.500000  7.500909
## Median       7.110000  7.040000
## Sum         82.500000 82.510000
## SE Mean      0.612196  0.612242
## LCL Mean     6.135943  6.136748
## UCL Mean     8.864057  8.865070
## Variance     4.122620  4.123249
## Stdev        2.030424  2.030579
## Skewness     1.380120  1.120774
## Kurtosis     1.240044  0.628751
x <- data[1:4]
y <- data[5:8]
cor(x,y)
##            y1         y2         y3         y4
## x1  0.8164205  0.8162365  0.8162867 -0.3140467
## x2  0.8164205  0.8162365  0.8162867 -0.3140467
## x3  0.8164205  0.8162365  0.8162867 -0.3140467
## x4 -0.5290927 -0.7184365 -0.3446610  0.8165214
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1, data$y1)

plot(data$x2, data$y2)

plot(data$x3, data$y3)

plot(data$x4, data$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, pch =18)
plot(data$x2, data$y2, pch =18)
plot(data$x3, data$y3, pch =18)
plot(data$x4, data$y4, pch =18)

  1. Now fit a linear model to each data set using the lm() function.
l1 <- lm(data$y1~data$x1)
l2 <- lm(data$y2~data$x2)
l3 <- lm(data$y3~data$x3)
l4 <- 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, pch =19)
abline(l1)
plot(data$x2, data$y2, pch =19)
abline(l2)
plot(data$x3, data$y3, pch =19)
abline(l3)
plot(data$x4, data$y4, pch =19)
abline(l4)

  1. Now compare the model fits for each model object.
anova(l1)

Analysis of Variance Table

Response: data\(y1 Df Sum Sq Mean Sq F value Pr(>F) data\)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(l2)

Analysis of Variance Table

Response: data\(y2 Df Sum Sq Mean Sq F value Pr(>F) data\)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(l3)

Analysis of Variance Table

Response: data\(y3 Df Sum Sq Mean Sq F value Pr(>F) data\)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(l4)

Analysis of Variance Table

Response: data\(y4 Df Sum Sq Mean Sq F value Pr(>F) data\)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. This lesson is a great example that although the data might look similar but when we use visualization tools we see how different they are. Hence it establishes the importance of visualization techniques.We also learn different techniques of data visualization. We also see the linear fit sets well with the third set only.