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.
library(datasets)
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")
## Warning: package 'fBasics' was built under R version 3.4.3
## Loading required package: timeDate
## Loading required package: timeSeries
basicStats(data)
##                    x1        x2        x3        x4        y1        y2
## nobs        11.000000 11.000000 11.000000 11.000000 11.000000 11.000000
## NAs          0.000000  0.000000  0.000000  0.000000  0.000000  0.000000
## Minimum      4.000000  4.000000  4.000000  8.000000  4.260000  3.100000
## Maximum     14.000000 14.000000 14.000000 19.000000 10.840000  9.260000
## 1. Quartile  6.500000  6.500000  6.500000  8.000000  6.315000  6.695000
## 3. Quartile 11.500000 11.500000 11.500000  8.000000  8.570000  8.950000
## Mean         9.000000  9.000000  9.000000  9.000000  7.500909  7.500909
## Median       9.000000  9.000000  9.000000  8.000000  7.580000  8.140000
## Sum         99.000000 99.000000 99.000000 99.000000 82.510000 82.510000
## SE Mean      1.000000  1.000000  1.000000  1.000000  0.612541  0.612568
## LCL Mean     6.771861  6.771861  6.771861  6.771861  6.136083  6.136024
## UCL Mean    11.228139 11.228139 11.228139 11.228139  8.865735  8.865795
## Variance    11.000000 11.000000 11.000000 11.000000  4.127269  4.127629
## Stdev        3.316625  3.316625  3.316625  3.316625  2.031568  2.031657
## Skewness     0.000000  0.000000  0.000000  2.466911 -0.048374 -0.978693
## Kurtosis    -1.528926 -1.528926 -1.528926  4.520661 -1.199123 -0.514319
##                    y3        y4
## nobs        11.000000 11.000000
## NAs          0.000000  0.000000
## Minimum      5.390000  5.250000
## Maximum     12.740000 12.500000
## 1. Quartile  6.250000  6.170000
## 3. Quartile  7.980000  8.190000
## Mean         7.500000  7.500909
## Median       7.110000  7.040000
## Sum         82.500000 82.510000
## SE Mean      0.612196  0.612242
## LCL Mean     6.135943  6.136748
## UCL Mean     8.864057  8.865070
## Variance     4.122620  4.123249
## Stdev        2.030424  2.030579
## Skewness     1.380120  1.120774
## Kurtosis     1.240044  0.628751
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:
##  Thu Feb 22 00:19:32 2018
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:
##  Thu Feb 22 00:19:32 2018
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:
##  Thu Feb 22 00:19:32 2018
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:
##  Thu Feb 22 00:19:32 2018
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1,data$y1)

plot(data$x2,data$y2)

plot(data$x3,data$y3)

plot(data$x4,data$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))
plot1<-plot(data$x1,data$y1, pch=16)
plot2<- plot(data$x2,data$y2,pch=16)
plot3<- plot(data$x3,data$y3, pch=16)
plot4<- plot(data$x4,data$y4, pch=16)

  1. Now fit a linear model to each data set using the lm() function.
model1<-lm(y1~x1, data=data)
model1
## 
## Call:
## lm(formula = y1 ~ x1, data = data)
## 
## Coefficients:
## (Intercept)           x1  
##      3.0001       0.5001
model2<-lm(y2~x2, data=data)
model2
## 
## Call:
## lm(formula = y2 ~ x2, data = data)
## 
## Coefficients:
## (Intercept)           x2  
##       3.001        0.500
model3<-lm(y3~x3, data=data)
model3
## 
## Call:
## lm(formula = y3 ~ x3, data = data)
## 
## Coefficients:
## (Intercept)           x3  
##      3.0025       0.4997
model4<-lm(y4~x4, data=data)
model4
## 
## Call:
## lm(formula = y4 ~ x4, data = data)
## 
## Coefficients:
## (Intercept)           x4  
##      3.0017       0.4999
  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))
plot1<-plot(data$x1,data$y1, pch=16)
abline(lm(y1~x1,data=data), col="red")
plot2<- plot(data$x2,data$y2,pch=16)
abline(lm(y2~x2,data=data), col="red")
plot3<- plot(data$x3,data$y3, pch=16)
abline(lm(y3~x3,data=data), col="red")
plot4<- plot(data$x4,data$y4, pch=16)
abline(lm(y4~x4,data=data), col="red")

  1. Now compare the model fits for each model object.
library(fit.models)
## Warning: package 'fit.models' was built under R version 3.4.3
fit <- fit.models(lm(data$y1~data$x1), lm(data$y2~data$x2), lm(data$y3~data$x3), lm(data$y4~data$x4))
summary(fit)

Calls: lm(data\(y1 ~ data\)x1): lm(formula = data\(y1 ~ data\)x1) lm(data\(y2 ~ data\)x2): lm(formula = data\(y2 ~ data\)x2) lm(data\(y3 ~ data\)x3): lm(formula = data\(y3 ~ data\)x3) lm(data\(y4 ~ data\)x4): lm(formula = data\(y4 ~ data\)x4)

Residual Statistics: Min 1Q Median 3Q Max lm(data\(y1 ~ data\)x1): -1.921 -0.4558 -4.136e-02 0.7094 1.839 lm(data\(y2 ~ data\)x2): -1.901 -0.7609 1.291e-01 0.9491 1.269 lm(data\(y3 ~ data\)x3): -1.159 -0.6146 -2.303e-01 0.1540 3.241 lm(data\(y4 ~ data\)x4): -1.751 -0.8310 1.110e-16 0.8090 1.839

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept): lm(data\(y1 ~ data\)x1): 3.0001 1.1247 2.667 0.02573 lm(data\(y2 ~ data\)x2): 3.0009 1.1253 2.667 0.02576 lm(data\(y3 ~ data\)x3): 3.0025 1.1245 2.670 0.02562 lm(data\(y4 ~ data\)x4): 3.0017 1.1239 2.671 0.02559

data$x1: lm(data$y1 ~ data$x1):   0.5001     0.1179   4.241  0.00217
         lm(data$y2 ~ data$x2):                                     
         lm(data$y3 ~ data$x3):                                     
         lm(data$y4 ~ data$x4):                                     
                                                                    
data$x2: lm(data$y1 ~ data$x1):                                     
         lm(data$y2 ~ data$x2):   0.5000     0.1180   4.239  0.00218
         lm(data$y3 ~ data$x3):                                     
         lm(data$y4 ~ data$x4):                                     
                                                                    
data$x3: lm(data$y1 ~ data$x1):                                     
         lm(data$y2 ~ data$x2):                                     
         lm(data$y3 ~ data$x3):   0.4997     0.1179   4.239  0.00218
         lm(data$y4 ~ data$x4):                                     
                                                                    
data$x4: lm(data$y1 ~ data$x1):                                     
         lm(data$y2 ~ data$x2):                                     
         lm(data$y3 ~ data$x3):                                     
         lm(data$y4 ~ data$x4):   0.4999     0.1178   4.243  0.00216

(Intercept): lm(data\(y1 ~ data\)x1): * lm(data\(y2 ~ data\)x2): * lm(data\(y3 ~ data\)x3): * lm(data\(y4 ~ data\)x4): *

data$x1: lm(data$y1 ~ data$x1): **
         lm(data$y2 ~ data$x2):   
         lm(data$y3 ~ data$x3):   
         lm(data$y4 ~ data$x4):   
                                  
data$x2: lm(data$y1 ~ data$x1):   
         lm(data$y2 ~ data$x2): **
         lm(data$y3 ~ data$x3):   
         lm(data$y4 ~ data$x4):   
                                  
data$x3: lm(data$y1 ~ data$x1):   
         lm(data$y2 ~ data$x2):   
         lm(data$y3 ~ data$x3): **
         lm(data$y4 ~ data$x4):   
                                  
data$x4: lm(data$y1 ~ data$x1):   
         lm(data$y2 ~ data$x2):   
         lm(data$y3 ~ data$x3):   
         lm(data$y4 ~ data$x4): **

Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual Scale Estimates: lm(data\(y1 ~ data\)x1): 1.237 on 9 degrees of freedom lm(data\(y2 ~ data\)x2): 1.237 on 9 degrees of freedom lm(data\(y3 ~ data\)x3): 1.236 on 9 degrees of freedom lm(data\(y4 ~ data\)x4): 1.236 on 9 degrees of freedom

Multiple R-squared: lm(data\(y1 ~ data\)x1): 0.6665 lm(data\(y2 ~ data\)x2): 0.6662 lm(data\(y3 ~ data\)x3): 0.6663 lm(data\(y4 ~ data\)x4): 0.6667

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

Summary: the Anscombe’s Quartet is the perfect example to demonstrate the value of data visualization. From pure statstical analysis, we could see that the four sets of data display the exact same characteristics. Wrong assuption could’ve easily been made about the data, and wrong analysis methods could’ve been applied which would in turn lead to he wrong conclusion. It was only through data visualization that we could see the data sets are actually wildly different - for example, not all four of them demonstrated linear relationship as we would’ve assumed from just the statistcal analysis. Data visualization gives us a more direct understanding of the data, and when combined with the correct statistical analysis, will provide us with a more whole some understanding of the data/ situation we are trying to analyze.

install.packages(sys.date)