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

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.

data <- anscombe

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!)

colMeans(data[,1:4])

## x1 x2 x3 x4 
##  9  9  9  9

colMeans(data[,5:8])

##       y1       y2       y3       y4 
## 7.500909 7.500909 7.500000 7.500909

var(data$x1)

## [1] 11

var(data$x2)

## [1] 11

var(data$x3)

## [1] 11

var(data$x4)

## [1] 11

var(data$y1)

## [1] 4.127269

var(data$y2)

## [1] 4.127629

var(data$y3)

## [1] 4.12262

var(data$y4)

## [1] 4.123249

library(fBasics)

## Warning: package 'fBasics' was built under R version 3.3.3

## Loading required package: timeDate

## Warning: package 'timeDate' was built under R version 3.3.3

## Loading required package: timeSeries

## Warning: package 'timeSeries' was built under R version 3.3.3

##

## Rmetrics Package fBasics

## Analysing Markets and calculating Basic Statistics

## Copyright (C) 2005-2014 Rmetrics Association Zurich

## Educational Software for Financial Engineering and Computational Science

## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.

## https://www.rmetrics.org --- Mail to: info@rmetrics.org

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:
##  Mon Nov 13 23:14:02 2017

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:
##  Mon Nov 13 23:14:02 2017

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:
##  Mon Nov 13 23:14:02 2017

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:
##  Mon Nov 13 23:14:02 2017

Create scatter plots for each \(x, y\) pair of data.

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 3.3.3

ggplot(data, aes(x=x1, y=y1)) + 
  geom_point(shape=1) + 
  ggtitle("Scatter_Plot between x1 and y1")

ggplot(data, aes(x=x2, y=y2)) + 
  geom_point(shape=1) + 
  ggtitle("Scatter_Plot between x2 and y2")

ggplot(data, aes(x=x3, y=y3)) + 
  geom_point(shape=1) + 
  ggtitle("Scatter_Plot between x3 and y3")

ggplot(data, aes(x=x4, y=y4)) + 
  geom_point(shape=1) + 
  ggtitle("Scatter_Plot between x4 and y4")

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))
ggplot(data, aes(x=x1, y=y1)) + 
  geom_point(shape=20) + 
  ggtitle("Scatter_Plot between x1 and y1")

ggplot(data, aes(x=x2, y=y2)) + 
  geom_point(shape=20) + 
  ggtitle("Scatter_Plot between x2 and y2")

ggplot(data, aes(x=x3, y=y3)) + 
  geom_point(shape=20) + 
  ggtitle("Scatter_Plot between x3 and y3")

ggplot(data, aes(x=x4, y=y4)) + 
  geom_point(shape=20) + 
  ggtitle("Scatter_Plot between x4 and y4")

Now fit a linear model to each data set using the lm() function.

linear_model_1 <- lm(data$x1~data$y1)
summary(linear_model_1)

## 
## Call:
## lm(formula = data$x1 ~ data$y1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6522 -1.5117 -0.2657  1.2341  3.8946 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9975     2.4344  -0.410  0.69156   
## data$y1       1.3328     0.3142   4.241  0.00217 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.019 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

linear_model_2 <- lm(data$x2~data$y2)
summary(linear_model_2)

## 
## Call:
## lm(formula = data$x2 ~ data$y2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8516 -1.4315 -0.3440  0.8467  4.2017 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9948     2.4354  -0.408  0.69246   
## data$y2       1.3325     0.3144   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.02 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

linear_model_3 <- lm(data$x3~data$y3)
summary(linear_model_3)

## 
## Call:
## lm(formula = data$x3 ~ data$y3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9869 -1.3733 -0.0266  1.3200  3.2133 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0003     2.4362  -0.411  0.69097   
## data$y3       1.3334     0.3145   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.019 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

linear_model_4 <- lm(data$x4~data$y4)
summary(linear_model_4)

## 
## Call:
## lm(formula = data$x4 ~ data$y4)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7859 -1.4122 -0.1853  1.4551  3.3329 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0036     2.4349  -0.412  0.68985   
## data$y4       1.3337     0.3143   4.243  0.00216 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.018 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

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))
ggplot(data, aes(x=x1, y=y1)) + 
  geom_point(shape=20) + 
  geom_smooth(method = lm)+
  ggtitle("Scatter_Plot between x1 and y1")

ggplot(data, aes(x=x2, y=y2)) + 
  geom_point(shape=20) + 
  geom_smooth(method = lm)+
  ggtitle("Scatter_Plot between x2 and y2")

ggplot(data, aes(x=x3, y=y3)) + 
  geom_point(shape=20) + 
  geom_smooth(method = lm)+
  ggtitle("Scatter_Plot between x3 and y3")

ggplot(data, aes(x=x4, y=y4)) + 
  geom_point(shape=20) + 
  geom_smooth(method = lm)+
  ggtitle("Scatter_Plot between x4 and y4")

Now compare the model fits for each model object.

anova(linear_model_1)

Analysis of Variance Table

Response: data\(x1 Df Sum Sq Mean Sq F value Pr(>F) data\)y1 1 73.32 73.320 17.99 0.00217 ** Residuals 9 36.68 4.076
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(linear_model_2)

Analysis of Variance Table

Response: data\(x2 Df Sum Sq Mean Sq F value Pr(>F) data\)y2 1 73.287 73.287 17.966 0.002179 ** Residuals 9 36.713 4.079
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(linear_model_3)

Analysis of Variance Table

Response: data\(x3 Df Sum Sq Mean Sq F value Pr(>F) data\)y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

anova(linear_model_4)

Analysis of Variance Table

Response: data\(x4 Df Sum Sq Mean Sq F value Pr(>F) data\)y4 1 73.338 73.338 18.003 0.002165 ** Residuals 9 36.662 4.074
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

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

In this dataset the mean and sample variance of x are 9 and 11 respectively. For y, the mean and sample variance are 7.5 and 4.125(+/-0.003). The correlation between x and y is 0.816. The four datasets have nearly identical simple descriptive statistics but looks different when plotted. In the first scatter plot, the variables have simple linear relationship. In the second scatter plot, the relationship between variables looks logarithmic.The third scatter plot shows linear relationship with poor goodness of fit. In the fourth scatter plot relationship between variables is difficult to define. Hence Anscombe’s Quartet proves that similar looking datasets may be statistically different when properly studied.

ANLY 512 - Problem Set 2

Anscombe’s quartet

Vivek Verma

2017-11-13

Objectives

Questions