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 enter your code or text response in the code chunk that completes/answers the activity or question requested. To submit this homework you will create the document in Rstudio, using the knitr package (button included in Rstudio) and then submit the document to your Rpubs account. Once uploaded you will submit the link to that document on Canvas. Please make sure that this link is hyper linked and that I can see the visualization and the code required to create it. Each question is worth 5 points.

Questions

  1. Anscombe’s 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.
#Load the required library
library(datasets)

#Assign 'anscombe' data to a new object and name it 'data'
data = anscombe

#Check the data set
head(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
  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 dplyr package!)
#Load the required library
library(resample)
## Registered S3 method overwritten by 'resample':
##   method         from  
##   print.resample modelr
#Data set summary
summary(data)
##        x1             x2             x3             x4           y1        
##  Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8   Min.   : 4.260  
##  1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8   1st Qu.: 6.315  
##  Median : 9.0   Median : 9.0   Median : 9.0   Median : 8   Median : 7.580  
##  Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9   Mean   : 7.501  
##  3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8   3rd Qu.: 8.570  
##  Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19   Max.   :10.840  
##        y2              y3              y4        
##  Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
##  1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
##  Median :8.140   Median : 7.11   Median : 7.040  
##  Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
##  3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
##  Max.   :9.260   Max.   :12.74   Max.   :12.500
#Mean calculation
colMeans(data, na.rm = FALSE)
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
#Variance calculation
colVars(data, na.rm = FALSE)
##        x1        x2        x3        x4        y1        y2        y3        y4 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620  4.123249
#Correlation b/w each pair
cor(data[ , 1:4], data[ , 5:8])
##            y1         y2         y3         y4
## x1  0.8164205  0.8162365  0.8162867 -0.3140467
## x2  0.8164205  0.8162365  0.8162867 -0.3140467
## x3  0.8164205  0.8162365  0.8162867 -0.3140467
## x4 -0.5290927 -0.7184365 -0.3446610  0.8165214
  1. Using ggplot, create scatter plots for each \(x, y\) pair of data (maybe use ‘facet_grid’ or ‘facet_wrap’).
#Load required library
library(ggplot2)

#Create multiple plots in one view
par(mfrow = c(2, 2))

#Pair 1
plot(data$x1, data$y1, main = "Scatterplot for Pair 1", xlab = "x1", ylab = "y1")

#Pair 2
plot(data$x2, data$y2, main = "Scatterplot for Pair 2", xlab = "x2", ylab = "y2")

#Pair 3
plot(data$x3, data$y3, main = "Scatterplot for Pair 3", xlab = "x3", ylab = "y3")

#Pair 4
plot(data$x4, data$y4, main = "Scatterplot for Pair 4", xlab = "x4", ylab = "y4")

  1. Now change the symbols on the scatter plots to solid blue circles.
#Create multiple plots in one view
par(mfrow = c(2, 2))

#Pair 1
plot(data$x1, data$y1, main = "Scatterplot for Pair 1", xlab = "x1", ylab = "y1", pch = 18, col = "blue")

#Pair 2
plot(data$x2, data$y2, main = "Scatterplot for Pair 2", xlab = "x2", ylab = "y2", pch = 18, col = "blue")

#Pair 3
plot(data$x3, data$y3, main = "Scatterplot for Pair 3", xlab = "x3", ylab = "y3", pch = 18, col = "blue")

#Pair 4
plot(data$x4, data$y4, main = "Scatterplot for Pair 4", xlab = "x4", ylab = "y4", pch = 18, col = "blue")

  1. Now fit a linear model to each data set using the lm() function.
lm_1 = lm(data$y1~data$x1)
lm_2 = lm(data$y2~data$x2)
lm_3 = lm(data$y3~data$x3)
lm_4 = 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!)
#Create multiple plots in one view
par(mfrow = c(2, 2))

#Pair 1
plot(data$x1, data$y1, main = "Scatterplot for Pair 1", xlab = "x1", ylab = "y1", pch = 18, col = "blue",abline(lm_1, col = "red"))

#Pair 2
plot(data$x2, data$y2, main = "Scatterplot for Pair 2", xlab = "x2", ylab = "y2", pch = 18, col = "blue",abline(lm_2, col = "red"))

#Pair 3
plot(data$x3, data$y3, main = "Scatterplot for Pair 3", xlab = "x3", ylab = "y3", pch = 18, col = "blue",abline(lm_3, col = "red"))

#Pair 4
plot(data$x4, data$y4, main = "Scatterplot for Pair 4", xlab = "x4", ylab = "y4", pch = 18, col = "blue", abline(lm_4, col = "red"))

  1. Now compare the model fits for each model object.
#Model fit for pair 1
anova(lm_1)

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

#Model fit for pair 2
anova(lm_2)

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

#Model fit for pair 3
anova(lm_3)

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

#Model fit for pair 4
anova(lm_4)

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

  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.
#Based on the ANOVA analysis model outcome, we can say that a data set summary statistics could be misleading if the analysis is solely relying on it. The underlying structure of the data set could vary vastly and could be identified by through visual analysis, even though the statistical parameters are same for different data sets. Thus data visualization is an integral part in data analysis, and the visuals could be used to communicate the data. Data visuals could be further used to make informed decisions.