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.
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
  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!)
library(dplyr)
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
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
apply(data,2,var)
##        x1        x2        x3        x4        y1        y2        y3        y4 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620  4.123249
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’).
library(ggplot2)
data(anscombe)

data <- data.frame(x = c(data$x1, data$x2, data$x3, data$x4),
                 y = c(data$y1, data$y2, data$y3, data$y4),
                 dataset = factor(rep(1:4, each = 11)))

ggplot(data, aes(x = x, y = y)) +
  geom_point() +
  facet_wrap(~ dataset, nrow = 2)

  1. Now change the symbols on the scatter plots to solid blue circles.
data <- data.frame(x = c(anscombe$x1, anscombe$x2, anscombe$x3, anscombe$x4),
                 y = c(anscombe$y1, anscombe$y2, anscombe$y3, anscombe$y4),
                 dataset = factor(rep(1:4, each = 11)))

ggplot(data, aes(x = x, y = y)) +
  geom_point(shape = 19, color = "blue", size = 3) +
  facet_wrap(~ dataset, nrow = 2)

  1. Now fit a linear model to each data set using the lm() function.
lm1 = lm(anscombe$y1~anscombe$x1)
lm2 = lm(anscombe$y2~anscombe$x2)
lm3 = lm(anscombe$y3~anscombe$x3)
lm4 = lm(anscombe$y4~anscombe$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!)
ggplot(data, aes(x=x, y=y)) +
  geom_point(pch=19, col="blue")+
  geom_smooth(method = "lm", se=FALSE)+
  facet_wrap(~dataset)
## `geom_smooth()` using formula 'y ~ x'

  1. Now compare the model fits for each model object.
anova(lm1)

Analysis of Variance Table

Response: anscombe\(y1 Df Sum Sq Mean Sq F value Pr(>F) anscombe\)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

anova(lm2)
## Analysis of Variance Table
## 
## Response: anscombe$y2
##             Df Sum Sq Mean Sq F value   Pr(>F)   
## anscombe$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
anova(lm3)
## Analysis of Variance Table
## 
## Response: anscombe$y3
##             Df Sum Sq Mean Sq F value   Pr(>F)   
## anscombe$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
anova(lm4)
## Analysis of Variance Table
## 
## Response: anscombe$y4
##             Df Sum Sq Mean Sq F value   Pr(>F)   
## anscombe$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.
# Anscombe's Quartet teaches us that data alone cannot tell the full story, but when you visualize it, you find patterns within the data that help interpret the data better. Even nearly identical data can have peculiarities that can be deceptive and misleading, but plotting the same data in scatter plots for example can show us how different the data sets appear.