ANLY 512 - Problem Set 2

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.

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]

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

install.packages("fBasics", repos="http://cran.rstudio.com/")

## Installing package into 'C:/Users/sunny.duggal/Documents/R/win-library/3.5'
## (as 'lib' is unspecified)

## package 'fBasics' successfully unpacked and MD5 sums checked
## 
## The downloaded binary packages are in
##  C:\Users\sunny.duggal\AppData\Local\Temp\Rtmpwjix1U\downloaded_packages

library(fBasics)

## Warning: package 'fBasics' was built under R version 3.5.2

## Loading required package: timeDate

## Warning: package 'timeDate' was built under R version 3.5.2

## Loading required package: timeSeries

## Warning: package 'timeSeries' was built under R version 3.5.2

correlationTest(x1,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 Dec 31 05:34:46 2018

correlationTest(x2,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 Dec 31 05:34:46 2018

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

library(ggplot2)
plot(x1,y1, main = "Scatter plot between x1 & y1")

plot(x2,y2,main = "Scatter plot between x2 & y2")

plot(x3,y3, main = "Scatter plot between x3 & y3")

plot(x4,y4, main = "Scatter plot between x4 & 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))
plot(x1,y1, main = "Scatter plot between x1 & y1", pch = 19)
plot(x2,y2,main = "Scatter plot between x2 & y2", pch = 19)
plot(x3,y3, main = "Scatter plot between x3 & y3", pch = 19)
plot(x4,y4, main = "Scatter plot between x4 & y4", pch = 19)

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

Lm1 <- lm( x1~y1)
summary(Lm1)

## 
## Call:
## lm(formula = x1 ~ 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   
## 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

Lm2 <- lm(x2~y2)
summary(Lm2)

## 
## Call:
## lm(formula = x2 ~ 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   
## 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

Lm3 <- lm(x3~y3)
summary(Lm3)

## 
## Call:
## lm(formula = x3 ~ 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   
## 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

Lm4 <- lm(x4~y4)
summary(Lm4)

## 
## Call:
## lm(formula = x4 ~ 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   
## 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))
plot(Lm1)

plot(Lm2)

plot(Lm3)

plot(Lm4)

Now compare the model fits for each model object.

anova(Lm1, test ="Chisq")

Analysis of Variance Table

Response: x1 Df Sum Sq Mean Sq F value Pr(>F)
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(Lm2, test ="Chisq")

Analysis of Variance Table

Response: x2 Df Sum Sq Mean Sq F value Pr(>F)
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(Lm3, test ="Chisq")

Analysis of Variance Table

Response: x3 Df Sum Sq Mean Sq F value Pr(>F)
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(Lm4, test ="Chisq")

Analysis of Variance Table

Response: x4 Df Sum Sq Mean Sq F value Pr(>F)
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.

There are four pairs of data of Anscombe’s Quartet. From the data visualization results, we can see they are quite different. In addition, linear model doesn’t fit all data pairs. Only x3,y3 fit the linear model well. The data visualization help us understand the dataset clear and directly.

ANLY 512 - Problem Set 2

Anscombe’s quartet

Your Name

2018-12-31

Objectives

Questions

There are four pairs of data of Anscombe’s Quartet. From the data visualization results, we can see they are quite different. In addition, linear model doesn’t fit all data pairs. Only x3,y3 fit the linear model well. The data visualization help us understand the dataset clear and directly.