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” assignmenet on Moodle.

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.
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!)
mean(data$x1)
## [1] 9
var(data$x1)
## [1] 11
mean(data$x2)
## [1] 9
var(data$x2)
## [1] 11
mean(data$x3)
## [1] 9
var(data$x3)
## [1] 11
mean(data$x4)
## [1] 9
var(data$x4)
## [1] 11
mean(data$y1)
## [1] 7.500909
var(data$y1)
## [1] 4.127269
mean(data$y2)
## [1] 7.500909
var(data$y2)
## [1] 4.127629
mean(data$y3)
## [1] 7.5
var(data$y3)
## [1] 4.12262
mean(data$y4)
## [1] 7.500909
var(data$y4)
## [1] 4.123249
library("fBasics")
## Warning: package 'fBasics' was built under R version 3.4.2
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.4.2
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.4.2
correlationTest(data$x1,data$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:
##  Wed Nov 29 17:16:21 2017
correlationTest(data$x2,data$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:
##  Wed Nov 29 17:16:21 2017
correlationTest(data$x3,data$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:
##  Wed Nov 29 17:16:21 2017
correlationTest(data$x4,data$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:
##  Wed Nov 29 17:16:21 2017
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data$x1, data$y1, main = "Scater Plot - x1 and y1", xlab = "x1", 
     ylab = "y1")

plot(data$x2, data$y2, main = "Scater Plot - x2 and y2", xlab = "x2",
     ylab = "y2")

plot(data$x3, data$y3, main = "Scater Plot - x3 and y3", xlab = "x3",
     ylab = "y3")

plot(data$x4, data$y4, main = "Scater Plot - 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
par(mfrow= c(2,2))
plot(data$x1, data$y1, main = "Scater Plot - x1 and y1", xlab = "x1", 
     ylab = "y1", pch = 20)
plot(data$x2, data$y2, main = "Scater Plot - x2 and y2", xlab = "x2",
     ylab = "y2", pch = 20)
plot(data$x3, data$y3, main = "Scater Plot - x3 and y3", xlab = "x3",
     ylab = "y3", pch = 20)
plot(data$x4, data$y4, main = "Scater Plot - x4 and y4", xlab = "x4",
     ylab = "y4", pch = 20)

  1. Now fit a linear model to each data set using the lm() function.
fit_1 <- lm(y1 ~ x1, data = data)
summary(fit_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
fit_2 <- lm(y2 ~ x2, data = data)
summary(fit_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
fit_3 <- lm(y3 ~ x3, data = data)
summary(fit_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
fit_4 <- lm(y4 ~ x4, data = data)
summary(fit_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
  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(data$x1, data$y1, main = "Scater Plot - x1 and y1", xlab = "x1", 
     ylab = "y1", pch = 20)
abline(fit_1, col = "blue")
plot(data$x2, data$y2, main = "Scater Plot - x2 and y2", xlab = "x2", 
     ylab = "y2", pch = 20)
abline(fit_2, col = "blue")
plot(data$x3, data$y3, main = "Scater Plot - x3 and y3", xlab = "x3", 
     ylab = "y3", pch = 20)
abline(fit_3, col = "blue")
plot(data$x4, data$y4, main = "Scater Plot - x4 and y4", xlab = "x4", 
     ylab = "y4", pch = 20)
abline(fit_4, col = "blue")

  1. Now compare the model fits for each model object.
anova(fit_1)

Analysis of Variance Table

Response: y1 Df Sum Sq Mean Sq F value Pr(>F)
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(fit_2)

Analysis of Variance Table

Response: y2 Df Sum Sq Mean Sq F value Pr(>F)
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(fit_3)

Analysis of Variance Table

Response: y3 Df Sum Sq Mean Sq F value Pr(>F)
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(fit_4)

Analysis of Variance Table

Response: y4 Df Sum Sq Mean Sq F value Pr(>F)
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.

Through the analysis of Anscombe’s Quarter we found that for the four different data sets, the mean and variance of x and y was the same. Not only that, the four data set pairs (e.g.: x1 and y1) had identical correlation. However, when all these data set pairs are visualized through a scatter plot, all of them tell vastly different story from each other as to how the x and y variables are related for each data set pair. This shows that numerical analysis does not provide a complete and accurate picture of the data all the time.

Addditionally, after performing regressions analysis for these pairs, it can be concluded that all of them have similar coefficients and model r-square values. Again, this quantitative analysis fails to show any difference among different pairs. In fact, even the model fit calculations show no numeric difference of square error or independet variable signficance value among pairs. However, when the regression fit line is plotted on top of the scatter plots of pairs, it clear shows that the fit lines display a completely different fit for each of them.

This entire exerices demonstrates that numerical analysis does not always provide an accurate interpretation of the relationship between variables and data visualization is a great tool to overcome these limitations. Data visualization also helps in supporting those data driven decisions and providing a proper context and narrative to the entire analysis.