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” assignment to your R Pubs account and submit the link to Moodle. Points will be deducted for uploading the improper format.

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)
View(anscombe)
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)
## Loading required package: timeDate
## Loading required package: timeSeries
#summriz each column (min, median, mean, max, quntile)
summary(data)
##        x1             x2             x3             x4    
##  Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8  
##  1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8  
##  Median : 9.0   Median : 9.0   Median : 9.0   Median : 8  
##  Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9  
##  3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8  
##  Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19  
##        y1               y2              y3              y4        
##  Min.   : 4.260   Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
##  1st Qu.: 6.315   1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
##  Median : 7.580   Median :8.140   Median : 7.11   Median : 7.040  
##  Mean   : 7.501   Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
##  3rd Qu.: 8.570   3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
##  Max.   :10.840   Max.   :9.260   Max.   :12.74   Max.   :12.500
#The correlation between each pair of variabes
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
  1. Create scatter plots for each \(x, y\) pair of data.
x1<-data$x1
x2<-data$x2
x3<-data$x3
x4<-data$x4

y1<-data$y1
y2<-data$y2
y3<-data$y3
y4<-data$y4

#scatter plot for x1 and y1
plot(x1,y1, main="Scatter plots for x1 and y1",
     xlab="x1", ylab="y1")

#scatter plot for x2 and y2
plot(x2,y2, main="Scatter plots for x2 and y2",
     xlab="x2", ylab="y2")

#scatter plot for x3 and y3
plot(x3,y3, main="Scatter plots for x3 and y3",
     xlab="x3", ylab="y3")

#scatter plot for x4 and y4 
plot(x4,y4, main="Scatter plots for x4 and y4",
     xlab="x4", ylab="y4")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
#List 4 plots 2 by 2.  
par(mfrow=c(2,2))
#scatter plot for x1 and y1
plot(x1,y1, main="Scatterplot for x1 and y1",
     xlab="x1", ylab="y1",
     pch=19)
#scatter plot for x2 and y2
plot(x2,y2, main="Scatterplot for x2 and y2",
     xlab="x2", ylab="y2",
     pch=19)
#scatter plot for x3 and y3
plot(x3,y3, main="Scatterplot for x3 and y3",
     xlab="x3", ylab="y3",
     pch=19)
#scatter plot for x4 and y4 
plot(x4,y4, main="Scatterplot for x4 and y4",
     xlab="x4", ylab="y4",
     pch=19)

  1. Now fit a linear model to each data set using the lm() function.
#Build Linear model for each set
linearMod_1<-lm(y1~x1, data=data)
print(linearMod_1)
## 
## Call:
## lm(formula = y1 ~ x1, data = data)
## 
## Coefficients:
## (Intercept)           x1  
##      3.0001       0.5001
linearMod_2<-lm(y2~x2, data=data)
print(linearMod_2)
## 
## Call:
## lm(formula = y2 ~ x2, data = data)
## 
## Coefficients:
## (Intercept)           x2  
##       3.001        0.500
linearMod_3<-lm(y3~x3, data=data)
print(linearMod_3)
## 
## Call:
## lm(formula = y3 ~ x3, data = data)
## 
## Coefficients:
## (Intercept)           x3  
##      3.0025       0.4997
linearMod_4<-lm(y4~x4, data=data)
print(linearMod_4)
## 
## 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))

# Add regression line

#scatter plot for x1 and y1
plot(x1,y1, main="Scatterplot for x1 and y1",
     xlab="x1", ylab="y1",
     pch=19)
abline(linearMod_1)
#scatter plot for x2 and y2
plot(x2,y2, main="Scatterplot for x2 and y2",
     xlab="x2", ylab="y2",
     pch=19)
abline(linearMod_2)
#scatter plot for x3 and y3
plot(x3,y3, main="Scatterplot for x3 and y3",
     xlab="x3", ylab="y3",
     pch=19)
abline(linearMod_3)
#scatter plot for x4 and y4 
plot(x4,y4, main="Scatterplot for x4 and y4",
     xlab="x4", ylab="y4",
     pch=19)
abline(linearMod_4)

  1. Now compare the model fits for each model object.
#look at statisitc rsult of each model

summary(linearMod_1)

Call: lm(formula = y1 ~ x1, data = data)

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 * 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

summary(linearMod_2)

Call: lm(formula = y2 ~ x2, data = data)

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 * 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

summary(linearMod_3)

Call: lm(formula = y3 ~ x3, data = data)

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 * 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

summary(linearMod_4)

Call: lm(formula = y4 ~ x4, data = data)

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 * 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

#From the summary of ststistical result, 4 models all achieve the significant level.However, from the scattorplots, only model 1 looks the best fit the randomness and fit the regression line the best.
# model 2 show the dots with curve shape
#model 3 and 4 has outliner, also model 4 data is not random distributed
  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.

This article shows how visualized graphs can help to check regression model in diferent aspects other than statistical method. Graphics help us to perceive the broad features of the data. It also helps to see the data outside of the assumption that made from statistical calculation. This artical uses the example of scatterplot to check regression model to show how graphics can help the ststistic analysis.