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)
str(anscombe)
## 'data.frame':    11 obs. of  8 variables:
##  $ x1: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x2: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x3: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x4: num  8 8 8 8 8 8 8 19 8 8 ...
##  $ y1: num  8.04 6.95 7.58 8.81 8.33 ...
##  $ y2: num  9.14 8.14 8.74 8.77 9.26 8.1 6.13 3.1 9.13 7.26 ...
##  $ y3: num  7.46 6.77 12.74 7.11 7.81 ...
##  $ y4: num  6.58 5.76 7.71 8.84 8.47 7.04 5.25 12.5 5.56 7.91 ...
data <- data("anscombe")
x1 <- anscombe[,1]
x2 <- anscombe[,2]
x3 <- anscombe[,3]
x4 <- anscombe[,4]
y1 <- anscombe[,5]
y2 <- anscombe[,6]
y3 <- anscombe[,7]
y4 <- anscombe[,8]
  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(x1)
## [1] 9
var(x1)
## [1] 11
mean(x2)
## [1] 9
var(x2)
## [1] 11
mean(x3)
## [1] 9
var(x3)
## [1] 11
mean(x4)
## [1] 9
var(x4)
## [1] 11
mean(y1)
## [1] 7.500909
var(y1)
## [1] 4.127269
mean(y2)
## [1] 7.500909
var(y2)
## [1] 4.127629
mean(y3)
## [1] 7.5
var(y3)
## [1] 4.12262
mean(y4)
## [1] 7.500909
var(y4)
## [1] 4.123249
cor(x1,y1)
## [1] 0.8164205
cor(x2,y2)
## [1] 0.8162365
cor(x3,y3)
## [1] 0.8162867
cor(x4,y4)
## [1] 0.8165214
  1. Create scatter plots for each \(x, y\) pair of data.
library(ggplot2)
plot(x1,y1, main = "Scatter plot - Pair1 (x1 & y1)")

plot(x2,y2,main = "Scatter plot - Pair2 (x2 & y2)")

plot(x3,y3,main = "Scatter plot - Pair3 (x3 & y3)")

plot(x4,y4,main = "Scatter plot - Pair4 (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(x1,y1, main = "Scatter plot - Pair1 (x1 & y1)", pch = 20)
plot(x2,y2, main = "Scatter plot - Pair2 (x2 & y2)", pch = 20)
plot(x3,y3, main = "Scatter plot - Pair3 (x3 & y3)", pch = 20)
plot(x4,y4, main = "Scatter plot - Pair4 (x4 & y4)", pch = 20)

  1. Now fit a linear model to each data set using the lm() function.
LM1 <- lm(y1~x1)
summary(LM1)
## 
## Call:
## lm(formula = y1 ~ 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 * 
## 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
LM2 <- lm(y2~x2)
summary(LM2)
## 
## Call:
## lm(formula = y2 ~ 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 * 
## 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
LM3 <- lm(y3~x3)
summary(LM3)
## 
## Call:
## lm(formula = y3 ~ 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 * 
## 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
LM4 <- lm(y4~x4)
summary(LM4)
## 
## Call:
## lm(formula = y4 ~ 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 * 
## 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!)
par(mfrow = c(2,2))
plot(x1,y1, main = "Scatter plot - Pair1 (x1 & y1)", pch = 20)
abline(LM1, col="red")

plot(x2,y2, main = "Scatter plot - Pair2 (x2 & y2)", pch = 20)
abline(LM2, col="red")

plot(x3,y3, main = "Scatter plot - Pair3 (x3 & y3)", pch = 20)
abline(LM3, col="red")

plot(x4,y4, main = "Scatter plot - Pair4 (x4 & y4)", pch = 20)
abline(LM4, col="red")

  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.

Anscombe’s Quartet has four datasets that appear to be identical with respect to their summary statistics. However, after plotting the data, the graphs of all the four datasets are very different. Pair 1 has weak linear relationship. Pair 2 doesn’t have any linear relationship. Pair 3 has a much stronger linear relationship than pair1 except for one outlier. Pair4 shows almost the constant values of x with one outlier.
From Anscombe’s Quartet we can conclude that, there is a need to add data visualization to our statistical analysis as data visualization gives us a much better picture about our data.