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.
anscombe
##    x1 x2 x3 x4    y1   y2    y3    y4
## 1  10 10 10  8  8.04 9.14  7.46  6.58
## 2   8  8  8  8  6.95 8.14  6.77  5.76
## 3  13 13 13  8  7.58 8.74 12.74  7.71
## 4   9  9  9  8  8.81 8.77  7.11  8.84
## 5  11 11 11  8  8.33 9.26  7.81  8.47
## 6  14 14 14  8  9.96 8.10  8.84  7.04
## 7   6  6  6  8  7.24 6.13  6.08  5.25
## 8   4  4  4 19  4.26 3.10  5.39 12.50
## 9  12 12 12  8 10.84 9.13  8.15  5.56
## 10  7  7  7  8  4.82 7.26  6.42  7.91
## 11  5  5  5  8  5.68 4.74  5.73  6.89
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(fBasics)
## Loading required package: timeDate
## Loading required package: timeSeries
## 
## 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
colAvgs(anscombe) ##Mean
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
colVars(anscombe) ##Variance
##        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(anscombe$x1, anscombe$y1, method = c("pearson", "kendall", "spearman"),
title = "Correlation of x1 and y1") ##correlation
## 
## Title:
##  Correlation of x1 and y1
## 
## 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 Nov 15 08:00:09 2017
correlationTest(anscombe$x2, anscombe$y2, method = c("pearson", "kendall", "spearman"),
title = "Correlation of x2 and y2") ##correlation
## 
## Title:
##  Correlation of x2 and y2
## 
## 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 Nov 15 08:00:09 2017
correlationTest(anscombe$x3, anscombe$y3, method = c("pearson", "kendall", "spearman"),
title = "Correlation of x3 and y3") ##correlation
## 
## Title:
##  Correlation of x3 and y3
## 
## 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 Nov 15 08:00:09 2017
correlationTest(anscombe$x4, anscombe$y4, method = c("pearson", "kendall", "spearman"),
title = "Correlation of x4 and y4") ##correlation
## 
## Title:
##  Correlation of x4 and y4
## 
## 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 Nov 15 08:00:09 2017
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1, data$y1, xlab="x1", ylab="y1", main="Scatter plot of x1 and y1")

plot(data$x2, data$y2, xlab="x2", ylab="y2", main="Scatter plot of x2 and y2")

plot(data$x3, data$y3, xlab="x3", ylab="y3", main="Scatter plot of x3 and y3")

plot(data$x4, data$y4, xlab="x4", ylab="y4", main="Scatter plot of x4 and y4")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
op <- par(mfcol = c(2, 2)) # 2 x 2 pictures on one plot
plot(data$x1, data$y1, xlab="x1", ylab="y1", main="Scatter plot of x1 and y1",  pch=19, mfg=c(1, 1))
plot(data$x2, data$y2, xlab="x2", ylab="y2", main="Scatter plot of x2 and y2", pch=19, mfg=c(1, 2))
plot(data$x3, data$y3, xlab="x3", ylab="y3", main="Scatter plot of x3 and y3", pch=19, mfg=c(2, 1))
plot(data$x4, data$y4, xlab="x4", ylab="y4", main="Scatter plot of x4 and y4", pch=19, mfg=c(2, 2))

par(op); #Restore graphics parameters
  1. Now fit a linear model to each data set using the lm() function.
linear1<-lm(data$y1~data$x1)
summary(linear1)
## 
## 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

Model 1: y1=3.0001+0.5001*x1

linear2<-lm(data$y2~data$x2)
summary(linear2)
## 
## 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

Model 2: y21=3.0001+0.500*x2

linear3<-lm(data$y3~data$x3)
summary(linear3)
## 
## 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

Model 3: y3=3.0025+0.4997*x3

linear4<-lm(data$y4~data$x4)
summary(linear4)
## 
## 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

Model 4: y4=3.0017+0.4999*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!)
op <- par(mfcol = c(2, 2)) # 2 x 2 pictures on one plot
plot(data$x1, data$y1, xlab="x1", ylab="y1", main="Scatter plot of x1 and y1",  pch=19, mfg=c(1, 1))
abline(lm(data$y1~data$x1))
plot(data$x2, data$y2, xlab="x2", ylab="y2", main="Scatter plot of x2 and y2", pch=19, mfg=c(1, 2))
abline(lm(data$y2~data$x2))
plot(data$x3, data$y3, xlab="x3", ylab="y3", main="Scatter plot of x3 and y3", pch=19, mfg=c(2, 1))
abline(lm(data$y3~data$x3))
plot(data$x4, data$y4, xlab="x4", ylab="y4", main="Scatter plot of x4 and y4", pch=19, mfg=c(2, 2))
abline(lm(data$y4~data$x4))

par(op); #Restore graphics parameters
  1. Now compare the model fits for each model object.
anova(linear1, test="Chisq")
## 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(linear2, test="Chisq")
## 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(linear3, test="Chisq")
## 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(linear4, test="Chisq")
## 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

Model fitting is an optimization algorithm. Each of the methods above optimizes a likelihood function to find the “best fitting”" model.

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

Anscombe’s Quartet contains four datasets. Though the simple statistical values of four datasets are identical, graphs of four data sets are completely different.This proves the importance of data visualization in form of graphical datasets.