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 Rpubs site and submit the link to the hosted file via 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)
data("anscombe")     # load data
data <- anscombe     # assign data to a new object
data
##    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
  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
## Warning: package 'timeDate' was built under R version 3.4.3
## Loading required package: timeSeries
means <- colMeans(data)    # compute mean value in each column
means
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
variances <- colVars(data)     # compute variance value in each column
variances
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249
#Performs correlation test on each pair
corr_x1y1 <- correlationTest(data$x1, data$y1)
corr_x2y2 <- correlationTest(data$x2, data$y2)
corr_x3y3 <- correlationTest(data$x3, data$y3)
corr_x4y4 <- correlationTest(data$x4, data$y4)

# see correlation results
corr_x1y1
## 
## 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:
##  Tue Apr  3 19:25:03 2018
corr_x2y2
## 
## 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:
##  Tue Apr  3 19:25:03 2018
corr_x3y3
## 
## 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:
##  Tue Apr  3 19:25:03 2018
corr_x4y4
## 
## 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:
##  Tue Apr  3 19:25:03 2018
  1. Create scatter plots for each \(x, y\) pair of data.
par(mfrow=c(2,2))
plot(data$x1, data$y1, title("x1y1"))
plot(data$x2, data$y2, title("x2y2"))
plot(data$x3, data$y3, title("x3y3"))
plot(data$x4, data$y4, title("x4y4"))

  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, pch = 16, title("x1y1"))
plot(data$x2, data$y2, pch = 16, title("x2y2"))
plot(data$x3, data$y3, pch = 16, title("x3y3"))
plot(data$x4, data$y4, pch = 16, title("x4y4"))

  1. Now fit a linear model to each data set using the lm() function.
lm_x1y1 <- lm(data$y1~data$x1)
lm_x2y2 <- lm(data$y2~data$x2)
lm_x3y3 <- lm(data$y3~data$x3)
lm_x4y4 <- 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, pch= 16, title("x1y1"))
abline(lm_x1y1)
abline(lm_x1y1)
plot(data$x2, data$y2, pch= 16, title("x2y2"))
abline(lm_x2y2)
plot(data$x3, data$y3, pch= 16, title("x3y3"))
abline(lm_x3y3)
plot(data$x4, data$y4, pch= 16, title("x4y4"))
abline(lm_x4y4)

  1. Now compare the model fits for each model object.
library(fit.models)
modelfits <- fit.models(lm(data$y1~data$x1), 
                        lm(data$y2~data$x2),
                        lm(data$y3~data$x3),
                        lm(data$y4~data$x4))
summary(modelfits)

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.

Data visualization helps us understand the hiden true structure and reveal the unknown information of dataset that is hardly found by statistical values such as mean and variance. Let alone learning the data with our naked eyes…