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” assignmenet on Moodle.

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.
#View(anscombe)
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 <- anscombe
View(data)
  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
mean(data$x1)
## [1] 9
var(data$x1)
## [1] 11
mean(data$x2)
## [1] 9
var(data$x2)
## [1] 11
mean(data$x3)
## [1] 9
var(data$x3)
## [1] 11
mean(data$x4)
## [1] 9
var(data$x4)
## [1] 11
mean(data$y1)
## [1] 7.500909
var(data$y1)
## [1] 4.127269
mean(data$y2)
## [1] 7.500909
var(data$y2)
## [1] 4.127629
mean(data$y3)
## [1] 7.5
var(data$y3)
## [1] 4.12262
mean(data$y4)
## [1] 7.500909
var(data$y4)
## [1] 4.123249
correlationTest(data$x1, data$y1)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8164
##   STATISTIC:
##     t: 4.2415
##   P VALUE:
##     Alternative Two-Sided: 0.00217 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001085 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4244, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5113, 1
## 
## Description:
##  Wed Jan 23 17:40:37 2019
correlationTest(data$x2, data$y2)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8162
##   STATISTIC:
##     t: 4.2386
##   P VALUE:
##     Alternative Two-Sided: 0.002179 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001089 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4239, 0.9506
##          Less: -1, 0.9387
##       Greater: 0.5109, 1
## 
## Description:
##  Wed Jan 23 17:40:37 2019
correlationTest(data$x3, data$y3)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8163
##   STATISTIC:
##     t: 4.2394
##   P VALUE:
##     Alternative Two-Sided: 0.002176 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001088 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4241, 0.9507
##          Less: -1, 0.9387
##       Greater: 0.511, 1
## 
## Description:
##  Wed Jan 23 17:40:37 2019
correlationTest(data$x4, data$y4)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8165
##   STATISTIC:
##     t: 4.243
##   P VALUE:
##     Alternative Two-Sided: 0.002165 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001082 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4246, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5115, 1
## 
## Description:
##  Wed Jan 23 17:40:37 2019
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1, data$y1,main = "ScaterPlot of x1,y1",xlab = "Values of x1", ylab = "Values of y1",type='p',col = 'red')

plot(data$x2, data$y2,main = "ScaterPlot of x2,y2", xlab ="Values of x2", ylab ="Values of y2",type ='p',col ='blue')

plot(data$x3,data$y3,main = "ScaterPlot of x3,y3",xlab ="Values of x3", ylab = "Values of y3",type ='p',col = 'black')

plot(data$x4, data$y4,main = "ScaterPlot of x4,y4",xlab = "Values of x4", ylab = "Values of y4",type ='p',col='brown')

  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,main = "ScaterPlot of x1,y1",xlab = "Values of x1", ylab = "Values of y1",type='p',pch =19,col = 'red')
plot(data$x2, data$y2,main = "ScaterPlot of x2,y2", xlab ="Values of x2", ylab ="Values of y2",type ='p',pch =19,col ='blue')
plot(data$x3,data$y3,main = "ScaterPlot of x3,y3",xlab ="Values of x3",ylab = "Values of y3",type ='p',pch =19,col = 'brown')
plot(data$x4, data$y4,main = "ScaterPlot of x4,y4",xlab = "Values of x4", ylab ="Values of y4",type ='p',pch =19,col='black')

  1. Now fit a linear model to each data set using the lm() function.
fit1 <- lm(data$y1 ~ data$x1)
summary(fit1)
## 
## 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$x1       0.5001     0.1179   4.241  0.00217 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.237 on 9 degrees of freedom
## Multiple R-squared:  0.6665, Adjusted R-squared:  0.6295 
## F-statistic: 17.99 on 1 and 9 DF,  p-value: 0.00217
fit2 <- lm(data$y2 ~ data$x2)
summary(fit2)
## 
## Call:
## lm(formula = data$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
fit3 <- lm(data$y3 ~ data$x3)
summary(fit3)
## 
## 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$x3       0.4997     0.1179   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.236 on 9 degrees of freedom
## Multiple R-squared:  0.6663, Adjusted R-squared:  0.6292 
## F-statistic: 17.97 on 1 and 9 DF,  p-value: 0.002176
fit4 <- lm(data$y4 ~ data$x4)
summary(fit4)
## 
## Call:
## lm(formula = data$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. 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, main = "Scater Plot of x1,y1", xlab = "Values of x1", ylab = "Values of y1",pch = 20)
abline(fit1, col = "red")

plot(data$x2, data$y2, main = "Scater Plot of x2,y2", xlab = "Values of x2", ylab = "Values of y2", pch = 20)
abline(fit2, col = "blue")

plot(data$x3, data$y3, main = "Scater Plot of x3,y3", xlab = "Values of x3", ylab = "Values of y3", pch = 20)
abline(fit3, col = "brown")

plot(data$x4, data$y4, main = "Scater Plot of x4,y4", xlab = "Values of x4", ylab = "Values of y4", pch = 20)
abline(fit4, col = "green")

  1. Now compare the model fits for each model object.
anova(fit1)

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(fit2)

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(fit3)

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(fit4)

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.

Summary:

Anscombe’s Quartet consists of four datasets, with the data of similar kind. Each data set consists of 11 observations values for x and y. The datasets appears to be identical. But, when we plot the 4 datasets on x, y plane we find the datasets are not similar.Overall, statistics is not able to show comparison, instead it’s important to visulaize the data to get a better picture.This proves the importance of data visualization in analyzing huge datasets.