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 Rpubs site and submit the link to the hosted file via 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.
library(datasets)
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
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
colVars(data)
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249
attach(data)
correlationTest(x1,y1, method = "pearson")
## 
## 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:
##  Mon May 21 22:16:18 2018
correlationTest(x2,y2, method = "pearson")
## 
## 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:
##  Mon May 21 22:16:18 2018
correlationTest(x3,y3, method = "pearson")
## 
## 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:
##  Mon May 21 22:16:18 2018
correlationTest(x4,y4, method = "pearson")
## 
## 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:
##  Mon May 21 22:16:18 2018
  1. Create scatter plots for each \(x, y\) pair of data.
plot(x1,y1, main="Relationship between x1 and y1")

plot(x2,y2, main="Relationship between x2 and y2")

plot(x3,y3, main="Relationship between x3 and y3")

plot(x4,y4, main="Relationship between x4 and 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(x1,y1, pch=19, main="Relationship between x1 and y1")
plot(x2,y2, pch=19, main="Relationship between x2 and y2")
plot(x3,y3, pch=19, main="Relationship between x3 and y3")
plot(x4,y4, pch=19, main="Relationship between x4 and y4")

  1. Now fit a linear model to each data set using the lm() function.
fit1 <- lm(y1 ~ x1, data=data)
fit2 <- lm(y2 ~ x2, data=data)
fit3 <- lm(y3 ~ x3, data=data)
fit4 <- lm(y4 ~ x4, data=data)
  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(x1,y1, pch=19, main="Relationship between x1 and y1")
abline(fit1, pch=2, col="red")
plot(x2,y2, pch=19, main="Relationship between x2 and y2")
abline(fit2, pch=2, col="red")
plot(x3,y3, pch=19, main="Relationship between x3 and y3")
abline(fit3, pch=2, col="red")
plot(x4,y4, pch=19, main="Relationship between x4 and y4")
abline(fit4, pch=2, col="red")

  1. Now compare the model fits for each model object.
summary(fit1)

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(fit2)

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(fit3)

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(fit4)

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 the four linear regression models, we can see that x variables are all significant. The coefficients of x are also similar. The adjusted R-squared are close to each other.

  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.

The example Anscombe’s Quartet shows indicates why data visualization is important. The descriptive statistics show the means and standard deviations of the x variables are very similar (same for y variables). The correlation coefficients of the four pairs of x and y variables are almost the same. Besides, from the summary of the four linear regression models, we can see that x variables are all significant. The coefficients of x are also similar. The adjusted R-squared are close to each other. All the statistics reveal the four models are almost the same. However, the graphs show the relationship between x and y are very different in fact. Thus, data visualization can tell us something that calculation cannot provide. It demonstrates the importance of graphing data before analyzing it.