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
- 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
- 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!)
summary(data)
x1 x2 x3 x4
Min. : 4.0 Min. : 4.0 Min. : 4.0 Min. : 8
1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 8
Median : 9.0 Median : 9.0 Median : 9.0 Median : 8
Mean : 9.0 Mean : 9.0 Mean : 9.0 Mean : 9
3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.: 8
Max. :14.0 Max. :14.0 Max. :14.0 Max. :19
y1 y2 y3 y4
Min. : 4.260 Min. :3.100 Min. : 5.39 Min. : 5.250
1st Qu.: 6.315 1st Qu.:6.695 1st Qu.: 6.25 1st Qu.: 6.170
Median : 7.580 Median :8.140 Median : 7.11 Median : 7.040
Mean : 7.501 Mean :7.501 Mean : 7.50 Mean : 7.501
3rd Qu.: 8.570 3rd Qu.:8.950 3rd Qu.: 7.98 3rd Qu.: 8.190
Max. :10.840 Max. :9.260 Max. :12.74 Max. :12.500
- Create scatter plots for each \(x, y\) pair of data.
attach(anscombe) plot( x1, y1, main=“Scatter Plot of x1 and y1 data”, xlab=“x1”, ylab=“y1”) plot( x2, y2, main=“Scatter Plot of x2 and y2 data”, xlab=“x2”, ylab=“y2”) plot( x3, y3, main=“Scatter Plot of x3 and y3 data”, xlab=“x3”, ylab=“y3”) plot( x4, y4, main=“Scatter Plot of x4 and y4 data”, xlab=“x4”, ylab=“y4”)
4. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
attach(anscombe)
## The following objects are masked from anscombe (pos = 3):
##
## x1, x2, x3, x4, y1, y2, y3, y4
par(mfrow = c(2,2))
plot( x1, y1, main="Scatter Plot of x1 and y1 data", xlab="x1", ylab="y1", pch=16)
plot( x2, y2, main="Scatter Plot of x2 and y2 data", xlab="x2", ylab="y2", pch=16)
plot( x3, y3, main="Scatter Plot of x3 and y3 data", xlab="x3", ylab="y3", pch=16)
plot( x4, y4, main="Scatter Plot of x4 and y4 data", xlab="x4", ylab="y4", pch=16)
- Now fit a linear model to each data set using the
lm() function.
lm1 <- lm( x1~y1, data) summary(lm1)
Call:
lm(formula = x1 ~ y1, data = data)
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, data) summary(lm2)
Call:
lm(formula = x2 ~ y2, data = data)
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, data) summary(lm3)
Call:
lm(formula = x3 ~ y3, data = data)
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, data) summary(lm4)
Call:
lm(formula = x4 ~ y4, data = data)
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
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(lm1)
plot(lm2)
plot(lm3)
plot(lm4)
- Now compare the model fits for each model object. anova(lm1, test=“Chisq”)
anova(lm2, test=“Chisq”)
anova(lm3, test=“Chisq”)
anova(lm4, test=“Chisq”)
- In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.
Anscombe’s quartet helps us understand that although there can be different datasets with significantly identical statistical characteristics they can appear to be totally different when graphed. This is a significant characteristic and the important feature emphasizing the need for a good visualization model. With the current processing and data storage capabilities of cheap machines that are accessible, there are large amounts of datasets with similar statistical measures that are being generated. Anscombe’s quarter explains the importance of good visualization models to understand these datasets accurately.