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.
str(anscombe)
## 'data.frame':    11 obs. of  8 variables:
##  $ x1: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x2: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x3: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x4: num  8 8 8 8 8 8 8 19 8 8 ...
##  $ y1: num  8.04 6.95 7.58 8.81 8.33 ...
##  $ y2: num  9.14 8.14 8.74 8.77 9.26 8.1 6.13 3.1 9.13 7.26 ...
##  $ y3: num  7.46 6.77 12.74 7.11 7.81 ...
##  $ y4: num  6.58 5.76 7.71 8.84 8.47 7.04 5.25 12.5 5.56 7.91 ...
  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!)
data <- data("anscombe")
x1 <- anscombe[,1]
x2 <- anscombe[,2]
x3 <- anscombe[,3]
x4 <- anscombe[,4]
y1 <- anscombe[,5]
y2 <- anscombe[,6]
y3 <- anscombe[,7]
y4 <- anscombe[,8]
mean(x1)
## [1] 9
var(x1)
## [1] 11
mean(x2)
## [1] 9
var(x2)
## [1] 11
mean(x3)
## [1] 9
var(x3)
## [1] 11
mean(x4)
## [1] 9
var(x4)
## [1] 11
mean(y1)
## [1] 7.500909
var(y1)
## [1] 4.127269
mean(y2)
## [1] 7.500909
var(y2)
## [1] 4.127629
mean(y3)
## [1] 7.5
var(y3)
## [1] 4.12262
mean(y4)
## [1] 7.500909
var(y4)
## [1] 4.123249
library(fBasics)

correlationTest(x1,y1)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8164
##   STATISTIC:
##     t: 4.2415
##   P VALUE:
##     Alternative Two-Sided: 0.00217 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001085 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4244, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5113, 1
## 
## Description:
##  Sun Mar 12 16:34:38 2023
correlationTest(x2,y2)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8162
##   STATISTIC:
##     t: 4.2386
##   P VALUE:
##     Alternative Two-Sided: 0.002179 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001089 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4239, 0.9506
##          Less: -1, 0.9387
##       Greater: 0.5109, 1
## 
## Description:
##  Sun Mar 12 16:34:38 2023
correlationTest(x3,y3)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8163
##   STATISTIC:
##     t: 4.2394
##   P VALUE:
##     Alternative Two-Sided: 0.002176 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001088 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4241, 0.9507
##          Less: -1, 0.9387
##       Greater: 0.511, 1
## 
## Description:
##  Sun Mar 12 16:34:38 2023
correlationTest(x4,y4)
## 
## Title:
##  Pearson's Correlation Test
## 
## Test Results:
##   PARAMETER:
##     Degrees of Freedom: 9
##   SAMPLE ESTIMATES:
##     Correlation: 0.8165
##   STATISTIC:
##     t: 4.243
##   P VALUE:
##     Alternative Two-Sided: 0.002165 
##     Alternative      Less: 0.9989 
##     Alternative   Greater: 0.001082 
##   CONFIDENCE INTERVAL:
##     Two-Sided: 0.4246, 0.9507
##          Less: -1, 0.9388
##       Greater: 0.5115, 1
## 
## Description:
##  Sun Mar 12 16:34:38 2023
  1. Using ggplot, create scatter plots for each \(x, y\) pair of data (maybe use ‘facet_grid’ or ‘facet_wrap’).
library(ggplot2)
plot(x1,y1, main = "Scatter Plot: x1 & y1")

plot(x2,y2,main = "Scatter Plot: x2 & y2")

plot(x3,y3, main = "Scatter Plot: x3 & y3")

plot(x4,y4, main = "Scatter Plot: x4 & y4")

  1. Now change the symbols on the scatter plots to solid blue circles.
par(mfrow = c(2,2))
plot(x1,y1, main = "Scatter Plot: x1 & y1", pch = 19)
plot(x2,y2,main = "Scatter Plot: x2 & y2", pch = 19)
plot(x3,y3, main = "Scatter Plot: x3 & y3", pch = 19)
plot(x4,y4, main = "Scatter Plot: x4 & y4", pch = 19)

  1. Now fit a linear model to each data set using the lm() function.
Lm1 <- lm( x1~y1)
summary(Lm1)
## 
## Call:
## lm(formula = x1 ~ y1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6522 -1.5117 -0.2657  1.2341  3.8946 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9975     2.4344  -0.410  0.69156   
## y1            1.3328     0.3142   4.241  0.00217 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.019 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
Lm2 <- lm(x2~y2)
summary(Lm2) 
## 
## Call:
## lm(formula = x2 ~ y2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8516 -1.4315 -0.3440  0.8467  4.2017 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9948     2.4354  -0.408  0.69246   
## y2            1.3325     0.3144   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.02 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
Lm3 <- lm(x3~y3)
summary(Lm3)
## 
## Call:
## lm(formula = x3 ~ y3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9869 -1.3733 -0.0266  1.3200  3.2133 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0003     2.4362  -0.411  0.69097   
## y3            1.3334     0.3145   4.239  0.00218 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.019 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
Lm4 <- lm(x4~y4)
summary(Lm4)
## 
## Call:
## lm(formula = x4 ~ y4)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7859 -1.4122 -0.1853  1.4551  3.3329 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0036     2.4349  -0.412  0.68985   
## y4            1.3337     0.3143   4.243  0.00216 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.018 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
  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))
plot(Lm1)

plot(Lm2)

plot(Lm3)

plot(Lm4)

  1. Now compare the model fits for each model object.
anova(Lm1, test ="Chisq")

Analysis of Variance Table

Response: x1 Df Sum Sq Mean Sq F value Pr(>F)
y1 1 73.32 73.320 17.99 0.00217 ** Residuals 9 36.68 4.076
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

anova(Lm2, test ="Chisq")

Analysis of Variance Table

Response: x2 Df Sum Sq Mean Sq F value Pr(>F)
y2 1 73.287 73.287 17.966 0.002179 ** Residuals 9 36.713 4.079
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

anova(Lm3, test ="Chisq")

Analysis of Variance Table

Response: x3 Df Sum Sq Mean Sq F value Pr(>F)
y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ’ ’ 1

anova(Lm4, test ="Chisq")

Analysis of Variance Table

Response: x4 Df Sum Sq Mean Sq F value Pr(>F)
y4 1 73.338 73.338 18.003 0.002165 ** Residuals 9 36.662 4.074
— 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 is a set if 4 datasets that consists of nearly identical statistical properties but visualized differently. It demonstrated the importance of data visualization and disadvantage of relying solely on numerical summaries.

The lesson of Anscombe’s Quartet is that statistical measures can be misleading if used in isolation, and that visualizing the data is crucial for exploring and analyzing data.