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

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

colMeans(data)

##       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(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

cor(data)

##            x1         x2         x3         x4         y1         y2
## x1  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x2  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x3  1.0000000  1.0000000  1.0000000 -0.5000000  0.8164205  0.8162365
## x4 -0.5000000 -0.5000000 -0.5000000  1.0000000 -0.5290927 -0.7184365
## y1  0.8164205  0.8164205  0.8164205 -0.5290927  1.0000000  0.7500054
## y2  0.8162365  0.8162365  0.8162365 -0.7184365  0.7500054  1.0000000
## y3  0.8162867  0.8162867  0.8162867 -0.3446610  0.4687167  0.5879193
## y4 -0.3140467 -0.3140467 -0.3140467  0.8165214 -0.4891162 -0.4780949
##            y3         y4
## x1  0.8162867 -0.3140467
## x2  0.8162867 -0.3140467
## x3  0.8162867 -0.3140467
## x4 -0.3446610  0.8165214
## y1  0.4687167 -0.4891162
## y2  0.5879193 -0.4780949
## y3  1.0000000 -0.1554718
## y4 -0.1554718  1.0000000

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)

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=20)
plot(data$x2, data$y2, pch=20)
plot(data$x3, data$y3, pch=20)
plot(data$x4, data$y4, pch=20)

#another method of doing this!
pairs(data)

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

reg1 <- lm(data$y1~data$x1, data=data)
reg2 <- lm(data$y2~data$x2, data=data)
reg3 <- lm(data$y3~data$x3, data=data)
reg4 <- lm(data$y4~data$x4, data=data)

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=20)
abline(reg1)
plot(data$x2, data$y2, pch=20)
abline(reg2)
plot(data$x3, data$y3, pch=20)
abline(reg3)
plot(data$x4, data$y4, pch=20)
abline(reg4)

Now compare the model fits for each model object.

library(fit.models)
x <- fit.models(reg1, reg2, reg3, reg4)
summary(x)

Calls: reg1: lm(formula = data\(y1 ~ data\)x1, data = data) reg2: lm(formula = data\(y2 ~ data\)x2, data = data) reg3: lm(formula = data\(y3 ~ data\)x3, data = data) reg4: lm(formula = data\(y4 ~ data\)x4, data = data)

Residual Statistics: Min 1Q Median 3Q Max reg1: -1.921 -0.4558 -4.136e-02 0.7094 1.839 reg2: -1.901 -0.7609 1.291e-01 0.9491 1.269 reg3: -1.159 -0.6146 -2.303e-01 0.1540 3.241 reg4: -1.751 -0.8310 1.110e-16 0.8090 1.839

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept): reg1: 3.0001 1.1247 2.667 0.02573 * reg2: 3.0009 1.1253 2.667 0.02576 * reg3: 3.0025 1.1245 2.670 0.02562 * reg4: 3.0017 1.1239 2.671 0.02559 *

data$x1: reg1:   0.5001     0.1179   4.241  0.00217 **
         reg2:                                        
         reg3:                                        
         reg4:                                        
                                                      
data$x2: reg1:                                        
         reg2:   0.5000     0.1180   4.239  0.00218 **
         reg3:                                        
         reg4:                                        
                                                      
data$x3: reg1:                                        
         reg2:                                        
         reg3:   0.4997     0.1179   4.239  0.00218 **
         reg4:                                        
                                                      
data$x4: reg1:                                        
         reg2:                                        
         reg3:                                        
         reg4:   0.4999     0.1178   4.243  0.00216 **

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

Residual Scale Estimates: reg1: 1.237 on 9 degrees of freedom reg2: 1.237 on 9 degrees of freedom reg3: 1.236 on 9 degrees of freedom reg4: 1.236 on 9 degrees of freedom

Multiple R-squared: reg1: 0.6665 reg2: 0.6662 reg3: 0.6663 reg4: 0.6667

plot(x)

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

The quartet is used to illustrate the importance of looking at a set of data graphically before starting to analyze according to a particular type of relationship, and the inadequacy of basic statistic properties for describing realistic datasets.

ANLY 512 - Problem Set 2

Anscombe’s quartet

Rohit Mishra

2017-07-22

Objectives

Questions