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.

data = anscombe
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

2. 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!)

require("fBasics")
## Loading required package: fBasics
## Warning: package 'fBasics' was built under R version 3.4.2
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.4.3
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.4.2
basic.stats <- colStats(data,
                        FUN = function(x){c(mean(x), var(x))})

rownames(basic.stats) <- c("Mean", "Variance")

basic.stats      
##          x1 x2 x3 x4       y1       y2      y3       y4
## Mean      9  9  9  9 7.500909 7.500909 7.50000 7.500909
## Variance 11 11 11 11 4.127269 4.127629 4.12262 4.123249
cor <- rbind(cor(data$x1, data$y1),
             cor(data$x2, data$y2),
             cor(data$x3, data$y3),
             cor(data$x4, data$y4))

colnames(cor) <- c("Correlation")
rownames(cor) <- c("(x1, y1)", "(x2, y2)", "(x3, y3)", "(x4, y4)")

cor
##          Correlation
## (x1, y1)   0.8164205
## (x2, y2)   0.8162365
## (x3, y3)   0.8162867
## (x4, y4)   0.8165214

3. Create scatter plots for each \(x, y\) pair of data.

plot(data$x1, data$y1)

plot(data$x2, data$y2)

plot(data$x3, data$y3)

plot(data$x4, data$y4)

4. 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, pch = 16, xlab="x", ylab="y", main= "1") 
plot(data$x2, data$y2, pch = 16,xlab="x", ylab="y", main= "2") 
plot(data$x3, data$y3, pch = 16,xlab="x", ylab="y", main= "3") 
plot(data$x4, data$y4, pch = 16,xlab="x", ylab="y", main= "4")

5. Now fit a linear model to each data set using the lm() function.

m1 = lm(y1~x1, data = data)
m2 = lm(y2~x2, data = data)
m3 = lm(y3~x3, data = data)
m4 = lm(y4~x4, data = data)

6. 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,pch = 16, xlab="x", ylab="y", main= "1")
abline(m1)
plot(data$x2, data$y2,pch = 16, xlab="x", ylab="y", main= "2") 
abline(m2)
plot(data$x3, data$y3,pch = 16, xlab="x", ylab="y", main= "3") 
abline(m3)
plot(data$x4, data$y4,pch = 16, xlab="x", ylab="y", main= "4") 
abline(m4)

7. Now compare the model fits for each model object.

summary(m1)$r.squared

[1] 0.6665425

summary(m2)$r.squared

[1] 0.666242

summary(m3)$r.squared

[1] 0.666324

summary(m4)$r.squared

[1] 0.6667073

R-squared for four models are close to 0.67, which means the four models explains about 67% of the variability of the response data around its mean.

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

summary(m1)
## 
## 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(m2)
## 
## 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(m3)
## 
## 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(m4)
## 
## 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

It demonstrate the value of data visualizations: the four datasets have almost the same statistical chareacteriscs: same mean of x, y, same variance of x,y, even the co-efficients, intercept and R-squared of the regression models are almost the same. If we don’t visualize the data points and just look at these statistical characteristics, we wouldn’t be able to know that the data are distributed so differently. Especially, if we didn’t visualize dataset #4, we wouldn’t see there might be an outlier that had a big impact on the regression model, and there probably should be a warning that one observation has played a critical role.