Xiao Huang_ANLY512-51-2017-Fall-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 = 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)

## 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, na.rm = FALSE, dims = 1 )

##       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, na.rm = FALSE)

##        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, y = NULL)

##            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.

pairs(data, main="Paired Scatter Plot")

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)
lm1

## 
## Call:
## lm(formula = data$y1 ~ data$x1)
## 
## Coefficients:
## (Intercept)      data$x1  
##      3.0001       0.5001

lm2 = lm(data$y2 ~ data$x2)
lm2

## 
## Call:
## lm(formula = data$y2 ~ data$x2)
## 
## Coefficients:
## (Intercept)      data$x2  
##       3.001        0.500

lm3 = lm(data$y3 ~ data$x3)
lm3

## 
## Call:
## lm(formula = data$y3 ~ data$x3)
## 
## Coefficients:
## (Intercept)      data$x3  
##      3.0025       0.4997

lm4 = lm(data$y4 ~ data$x4)
lm4

## 
## Call:
## lm(formula = data$y4 ~ data$x4)
## 
## Coefficients:
## (Intercept)      data$x4  
##      3.0017       0.4999

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, col="red")

plot(data$x2, data$y2, pch=19)
abline(lm2, col="red")

plot(data$x3, data$y3, pch=19)
abline(lm3, col="red")

plot(data$x4, data$y4, pch=19)
abline(lm4, col="red")

Now compare the model fits for each model object.

summary(lm1)

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

anova(lm1)

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

summary(lm2)

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

anova(lm2)

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

summary(lm3)

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

anova(lm3)

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

summary(lm4)

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

anova(lm4)

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

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

From Anscombe’s Quartet, I think too many statistic numbers cannot simplely and directly tell us conclusions. Instead, it’s every simple and easy for us to visualize the data which can give us a clear picture of what’s going on.
Data visualization is the presentation of data in a pictorial or graphical format. It helps us to focus on the information that is relevant and important, representing the data so that we can immediately understand what’s going on. Or we have to do as many researches as we can to identify all the calculated numbers’ meanings. After that, we may be able to fully understand the data. So Data Visualization is really helpful and save us a lot of time.

Xiao Huang_ANLY512-51-2017-Fall-Problem Set 2

Anscombe’s quartet

Xiao Huang

2017-09-10

Objectives

Questions