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)
data<-anscombe
  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(psych)
describe(data)
##    vars  n mean   sd median trimmed  mad  min   max range  skew kurtosis
## x1    1 11  9.0 3.32   9.00    9.00 4.45 4.00 14.00 10.00  0.00    -1.53
## x2    2 11  9.0 3.32   9.00    9.00 4.45 4.00 14.00 10.00  0.00    -1.53
## x3    3 11  9.0 3.32   9.00    9.00 4.45 4.00 14.00 10.00  0.00    -1.53
## x4    4 11  9.0 3.32   8.00    8.00 0.00 8.00 19.00 11.00  2.47     4.52
## y1    5 11  7.5 2.03   7.58    7.49 1.82 4.26 10.84  6.58 -0.05    -1.20
## y2    6 11  7.5 2.03   8.14    7.79 1.47 3.10  9.26  6.16 -0.98    -0.51
## y3    7 11  7.5 2.03   7.11    7.15 1.53 5.39 12.74  7.35  1.38     1.24
## y4    8 11  7.5 2.03   7.04    7.20 1.90 5.25 12.50  7.25  1.12     0.63
##      se
## x1 1.00
## x2 1.00
## x3 1.00
## x4 1.00
## y1 0.61
## y2 0.61
## y3 0.61
## y4 0.61
library(fBasics)
## Warning: package 'fBasics' was built under R version 3.5.3
## Loading required package: timeDate
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.5.3
## 
## Attaching package: 'timeSeries'
## The following object is masked from 'package:psych':
## 
##     outlier
## 
## Attaching package: 'fBasics'
## The following object is masked from 'package:psych':
## 
##     tr
colVars(data)
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  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:
##  Sun Jun 02 20:16:08 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:
##  Sun Jun 02 20:16:08 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:
##  Sun Jun 02 20:16:08 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:
##  Sun Jun 02 20:16:08 2019
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1, data$y1, main = "x1,y1")

plot(data$x2, data$y2, main = "x2,y2")

plot(data$x3, data$y3, main = "x3,y3")

plot(data$x4, data$y4, main = "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(data$x1, data$y1, main = "x1,y1", pch = 20,bg="black")
plot(data$x2, data$y2, main = "x2,y2", pch = 20,bg="black")
plot(data$x3, data$y3, main = "x3,y3", pch = 20,bg="black")
plot(data$x4, data$y4, main = "x4,y4", pch = 20,bg="black")

  1. Now fit a linear model to each data set using the lm() function.
m1<-lm(data$y1~data$x1)
m2<-lm(data$y2~data$x2)
m3<-lm(data$y3~data$x3)
m4<-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, main = "x1,y1", pch = 20,bg="black")
abline(m1,col="green")
plot(data$x2, data$y2, main = "x2,y2", pch = 20,bg="black")
abline(m2,col="green")
plot(data$x3, data$y3, main = "x3,y3", pch = 20,bg="black")
abline(m3,col="green")
plot(data$x4, data$y4, main = "x4,y4", pch = 20,bg="black")
abline(m4,col="green")

  1. Now compare the model fits for each model object.
summary(m1)

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

summary(m2)

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

summary(m3)

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

summary(m4)

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

This lesson tells us data visualization is very important. We can’t analyze data without ploting the data first. All 4 dependant variables Y share the same mean values and variance and same as all 4 independant variables X. All 4 linear models share the same adjusted r^2. No one can imagine the linear models look that different on the scatter plots by just looking at the model summaries and the analysis on the variables. That’s why data visualization is very important.