ANLY 512 - Problem Set 2

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.

library(datasets)
data <- anscombe
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

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 3.5.2

## 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

means <- colMeans(data)
variances <- colVars(data)
correlation1 <- correlationTest(data$x1, data$y1)
correlation2 <- correlationTest(data$x2, data$y2)
correlation3 <- correlationTest(data$x3, data$y3)
correlation4 <- correlationTest(data$x4, data$y4)

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

par(mfrow=c(1,1))
plot(data$x1, data$y1)

plot(data$x2, data$y2)

plot(data$x3, data$y3)

plot(data$x4, data$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(data$x1, data$y1, pch = 19)
plot(data$x2, data$y2, pch = 19)
plot(data$x3, data$y3, pch = 19)
plot(data$x4, data$y4, pch = 19)

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

lm1 <- lm(data$y1~data$x1)
lm2 <- lm(data$y2~data$x2)
lm3 <- lm(data$y3~data$x3)
lm4 <- lm(data$y4~data$x4)

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 = 19)
abline(lm1)
plot(data$x2, data$y2, pch = 19)
abline(lm2)
plot(data$x3, data$y3, pch = 19)
abline(lm3)
plot(data$x4, data$y4, pch = 19)
abline(lm4)

Now compare the model fits for each model object.

library(fit.models)

## Warning: package 'fit.models' was built under R version 3.5.2

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

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

Data visualizaiton provides us a more direct and solict visual when we need to observie the relationship between independent variables and dependent variables. For instance, in question 6, we could figure out which pair of data has stronger linear relationship than the other one. A clear visualization is more applicable and understandable than piles of data in business field.

ANLY 512 - Problem Set 2

Anscombe’s quartet

Meng Zhang

2019-01-01

Objectives

Questions