2.9 Exercises and case studies
2.9.1 Data importation
Import the following datasets into R Commander
indicating the right formatting options:

iris.txt

ads.csv

auto.txt

Boston.xlsx

anscombe.RData

airquality.txt

wine.csv

worldpopulation.RData

laliga20152016.xlsx

wdicountries.txt
Create the file datasets.RData
for saving all the datasets together.
There are a good number of datasets directly available in R
:

To inspect them, go to
‘Data’ > ‘Data in packages’ > ‘List datasets in packages’
and you will get a long list of the available datasets, the package where they are stored and a small description about them. Or, inR
, simply typedata()
. 
To load a dataset, go to
‘Data’ > ‘Data in packages’ > ‘Read dataset from an attached package…’
and select the package and dataset. Or, inR
, simply typedata(nameDataset, package = “namePackage”)
. 
To get help on the dataset, go to
‘Help’ > ‘Help on active dataset (if available)’
or simply typehelp(nameDataset)
.
As you can see, this is a handy way of accessing a good number of datasets directly from R
.
Import the datasets Titanic
(datasets
package), PublicSchools
(sandwich
package) and USArrests
(datasets
package). Describe briefly the characteristics of each dataset (dimensions, variables, context).
2.9.2 Simple data management
Perform the following data management operations. Remember to select the adequate dataset as the active one:
 Load
datasets.RData
.  Establish the case names in
laliga20152016
as the variableTeam
(if they were not set).  Establish the case names in
wdicountries
as the variableCountry
.  In
laliga20152016
, create a new variable namedGoals.wrt.mean
, defined asGoals  mean(Goals)
.  In
wdicountries
, create a new variable that standardizes^{17} theGDP.growth
. Call itGDP.growth
.  Delete the variable
Species
fromiris
.  For
laliga20152016
, go to'Edit dataset'
and change thePoints
ofGetafe
to40
. To do so, click on the cell, change the content and click OK or select a new cell to save changes. Do not hit'Enter
’ or you will add a new column!.  Explore the menu options of
'Edit dataset'
for adding and removing rows/columns. Is a useful feature for simple edits.  Create a
newDatasets.RData
file saving all the modified datasets.  Restart
R Commander
and then loadnewDatasets.RData
(ignore the error'ERROR: There is more than one object in the file...'
and check that all the datasets are indeed available).
2.9.3 Computing simple linear regressions
Import the iris
dataset, either from iris.txt
or datasets.RData
. This dataset contains measurements for 150 iris flowers. The purpose of this exercise is to do the following analyses through R Commander
while inspecting and understanding the outputted code to identify what parts are changing, how and why.

Fit the regression line for
Petal.Width
(response) onPetal.Length
(predictor) and summarize it. 
Make the scatterplot of
Petal.Width
(y) vsPetal.Length
(x) with a regression line. 
Set the
‘Graph title’
to “iris dataset: petal width vs petal length”, the‘xaxis label’
to “petal length” and‘yaxis label’
to “sepal length”. 
Identify the 5 most outlier points
‘Automatically’
. 
Redo the linear regression and scatterplot excluding the points labelled as outliers (exclude them in
‘Subset expression’
with ac(…)
).  Check that the summary for the fitted line and the scatterplot displayed are coherent.

Make the matrix scatterplot for the four variables and including
‘Leastsquares lines’
. 
Set the
‘On Diagonal’
plots to‘Histograms’
and‘Boxplots’
. 
Set the
‘Graph title’
to “iris matrix scatterplot”. 
Identify the 5 most outlier points
‘Automatically’
.  Modify the code to identify 15 points.
 Compute the regression line for the plot in the fourth row and the second column and create the scatterplot for it.

Redo the scatterplot by selecting the option
‘Plot by groups…’
and then selecting‘Species’
.
The last scatterplots are an illustration of the Simpson’s paradox. The paradox surges when there are two or more welldefined groups in the data, they all have positive (negative) correlation, but taken as a whole dataset, the correlation is the opposite.
2.9.4 R
basics
Answer briefly in your own words:

What is the operator
<
doing? What does it mean, for example, thata < 1:5
?  What is the difference between a matrix and a data frame? Why is the latter useful?
 What are the differences between a vector and a matrix?

What is
c
employed for? 
Consider the expression
lm(a ~ b, data = myData)
.
What is
lm
standing for? 
What does it mean
a ~ b
? What are the roles ofa
andb
? 
Is
myData
a matrix or a data frame? 
What must be the relation between
myData
anda
andb
? 
Explain the differences with
lm(b ~ a, data = myData)
.

What is

What are the differences between running
a < 1; a
,a < 1
and1
.  What are the differences between a list and a data frame? What are their common parts?

Why is
$
employed? How can you know in which variables you can use$
? 
If you have a vector
x
, what arex^2
andx + 1
doing to its elements?
Do the following:
 Create the vectors \(x = (1.17, 0.41, 0.34, 1.11, 1.02, 0.22, 0.24, 0.27, 0.40, 1.38)\) and \(y = (3.63, 1.69, 0.27, 5.83, 2.64, 1.33, 1.22 0.62, 1.29, 0.43)\).
 Set the positions 3, 4 and 8 of \(x\) to 0. Set the positions 1, 4, 9 of \(y\) to 0.5, 0.75 and 0.3, respectively.
 Create a new vector \(z\) containing \(\log(x^2)  y^3\sqrt{\exp(x)}\).
 Create the vector \(t=(1, 4, 9, 16, 25, \ldots, 100)\).
 Access all the elements of \(t\) except the third and fifth.
 Create the matrix \(A=\begin{pmatrix}1 & 3\\0 & 2\end{pmatrix}\). Hint: use
rbind
orcbind
.  Using \(A\), what is a short way (less amount of code) of computing \(B=\begin{pmatrix}1+\sqrt{2}\sin(3) & 3+\sqrt{2}\sin(3)\\0+\sqrt{2}\sin(3) & 2+\sqrt{2}\sin(3)\end{pmatrix}\)?
 Compute
A*B
. Check that it makes sense with the results ofA[1, 1] * B[1, 1]
,A[1, 2] * B[1, 2]
,A[2, 1] * B[2, 1]
andA[2, 2] * B[2, 2]
. Why?  Create a data frame named
worldPopulation
such that: the first variable is called
Year
and contains the valuesc(1915, 1925, 1935, 1945, 1955, 1965, 1975, 1985, 1995, 2005, 2015)
.  the second variable is called
Population
and contains the valuesc(1798.0, 1952.3, 2197.3, 2366.6, 2758.3, 3322.5, 4061.4, 4852.5, 5735.1, 6519.6, 7349.5)
.
 the first variable is called
 Write
names(worldPopulation)
. Access to the two variables.  Create a new variable in
worldPopulation
calledlogPopulation
that containslog(Population)
.  Compute the standard deviation, mean and median of the variables in
worldPopulation
.  Regress
logPopulation
intoYear
. Save the result asmod
.  Compute the summary of the model and save it as
sumMod
.  Do a
str
onA
,worldPopulation
,mod
andsumMod
.  Access the \(R^2\) and \(\hat\sigma\) in
sumMod
.  Check that \(R^2\) is the same as the squared correlation between predictor and response, and also the squared correlation between response and
mod$fitted.values
.
2.9.5 Model formulation and estimation
Answer the following conceptual questions in your own words:
 What is the difference between (β_{0}, β_{1}) and \((\hat\beta_0,\hat\beta_1)\).
 Is \(\hat\beta_0\) a random variable? What about \(\hat\beta_1\)? Justify your answer.
 What function are the least squares estimates minimizing? Is important the choice of the kind distances (horizontal, vertical, perpendicular)?
 What is the justification for the use of a vertical distance in the RSS?
 Is σ^{2} affecting the R^{2} (indirectly or directly)? Why?
 What are the residuals? What is their interpretation?
 What are the fitted values? What is their interpretation?
 What is the relation of \(\hat \beta_1\) with r_{xy}?
Finally, check that the regression line goes through \((\bar X, \bar Y)\), in other words, that \(\bar Y=\hat\beta_0+\hat\beta_1\bar X\).
2.9.6 Assumptions of the linear model
The dataset moreAssumptions.RData
(download) contains the variables x1
, …, x9
and y1
, …, y9
. For each regression y1 ~ x1
, …, y9 ~ x9
describe whether the assumptions of the linear model are being satisfied or not. Justify your answer and state which assumption(s) you think are violated.
2.9.7 Nonlinear relations
Load moreAssumptions
. For the regressions y1 ~ x1
, y2 ~ x2
, y6 ~ x6
and y9 ~ x9
, identify which nonlinear transformation yields the largest \(R^2\). For that transformation, check wether the assumptions are verified.
y9 ~ x9
, try with (5  x9)^2
, abs(x9  5)
and abs(x9  5)^3
.
2.9.8 Case study: Moore’s law
Moore’s law (Moore 1965) is an empirical law that states that the power of a computer doubles approximately every two years. More precisely:
Translated into a mathematical formula, Moore’s law is \[\begin{align*} \text{transistors}\approx 2^{\text{years}/2}. \end{align*}\] Applying logarithms to both sides gives (why?) \[\begin{align*} \log(\text{transistors})\approx \frac{\log(2)}{2}\text{years}. \end{align*}\] We can write the above formula more generally \[\begin{align*} \log(\text{transistors})=\beta_0+\beta_1 \text{years}+\varepsilon, \end{align*}\]Moore’s law is the observation that the number of transistors in a dense integrated circuit [e.g. a CPU] doubles approximately every two years.
— Wikipedia article on Moore’s law
where \(\varepsilon\) is a random error. This is a linear model!
The dataset cpus.txt
(download) contains the transistor counts for the CPUs appeared in the time range 1971–2015. For this data, do the following:

Import conveniently the data and name it as
cpus
. 
Show a scatterplot of
Transistor.count
vsDate.of.introduction
with a linear regression. 
Are the assumptions verified in
Transistor.count ~ Date.of.introduction
? Which ones are which are more “problematic”? 
Create a new variable, named
Log.Transistor.count
, containing the logarithm ofTransistor.count
. 
Show a scatterplot of
Log.Transistor.count
vsDate.of.introduction
with a linear regression. 
Are the assumptions verified in
Log.Transistor.count ~ Date.of.introduction
? Which ones are which are more “problematic”? 
Regress
Log.Transistor.count ~ Date.of.introduction
.  Summarize the fit. What are the estimates \(\hat\beta_0\) and \(\hat\beta_1\)? Is \(\hat\beta_1\) close to \(\frac{\log(2)}{2}\)?
 Compute the CI for β_{1} at α = 0.05. Is \(\frac{\log(2)}{2}\) inside it? What happens at levels α = 0.10, 0.01?
 We want to forecast the average lognumber of transistors for the CPUs to be released in 2017. Compute the adequate prediction and CI.
 A new CPU design is expected for 2017. What is the range of lognumber of transistors expected for it, at a 95% level of confidence?

Compute the ANOVA table for
Log.Transistor.count ~ Date.of.introduction
. Is β_{1} significative?
The dataset gpus.txt
(download) contains the transistor counts for the GPUs appeared in the period 1997–2016. Repeat the previous analysis for this dataset.
2.9.9 Case study: Growth in a time of debt
In the aftermath of the 2007–2008 financial crisis, the paper Growth in a time of debt (Reinhart and Rogoff 2010), from Carmen M. Reinhart and Kenneth Rogoff (both at Harvard), provided an important economical support for proausterity policies. The paper claimed that for levels of external debt in excess of 90% of the GDP, the GDP growth of a country was dramatically different than for lower levels of external debt. Therefore, it concludes the existence of a magical threshold – 90% – for which the level of external debt must be kept below in order to have a growing economy. Figure 2.29, extracted from Reinhart and Rogoff (2010), illustrates the main finding.
Herndon, Ash, and Pollin (2013) replicated the analysis of Reinhart and Rogoff (2010) and found that “selective exclusion of available data, coding errors and inappropriate weighting of summary statistics lead to serious miscalculations that inaccurately represent the relationship between public debt and GDP growth among 20 advanced economies”. The authors concluded that “both mean and median GDP growth when public debt levels exceed 90% of GDP are not dramatically different from when the public debt/GDP ratios are lower”. As a consequence, Reinhart and Rogoff (2010) led to an unjustified support for the adoption of austerity policies for countries with various levels of public debt.
You can read the full story at BBC, The New York Times and The Economist. Also, the video in Figure 2.30 contains a quick summary of the story by Nobel Prize laureate Paul Krugman.
Herndon, Ash, and Pollin (2013) made the data of the study publicly available. You can download it here.
The dataset hap.txt
(download) contains data for 20 advanced economies in the time period 1946–2009, and is the data source for the papers aforementioned. The variable dRGDP
represents the real GDP growth (as a percentage) and debtgdp
represents the percentage of public debt with respect to the GDP.

Import the data and save it as
hap
. 
Set the case names of
hap
asCountry.Year
. 
Summarize
dRGDP
anddebtgdp
. What are their minimum and maximum values? 
What is the correlation between
dRGDP
anddebtgdp
? What is the standard deviation of each variable? 
Show the scatterplot of
dRGDP
vsdebtgdp
with the regression line. Is this coherent with what was stated in the video at 1:30? 
Do you see any gap on the data around 90%? Is there any substantial change for
dRGDP
around there? 
Compute the linear regression of
dRGDP
ondebtgdp
and summarize the fit.  What are the fitted coefficients? What are their standard errors? What is the R^{2}?
 Compute the ANOVA table. How many degrees of freedom are? What is the SSR? What is SSE? What is the pvalue for H_{0} : β_{1} = 0?
 Is SSR larger than SSE? Is this coherent with the resulting R^{2}?
 Are β_{0} and β_{1} significant for the regression at level α = 0.05? And at level α = 0.10, 0.01?

Compute the CIs for the coefficients. Can we conclude that the effect of
debtgdp
ondRGDP
is positive at α = 0.05? And negative?  Predict the average growth for levels of debt of 60%, 70%, 80%, 90%, 100% and 110%. Compute the 95% CIs for all of them.
 Predict the growth for the previous levels of debt. Compute also the CI for them. Is there a marked difference on the CIs for debt levels below and above 90%?
 Which assumptions of the linear model you think are satisfied? Should we trust blindly the inferential results obtained assuming that the assumptions were satisfied?
References
Moore, Gordon E. 1965. “Cramming More Components onto Integrated Circuits.” Electronics 38 (8): 114–17. doi:10.1109/JPROC.1998.658762.
Reinhart, Carmen M., and Kenneth S. Rogoff. 2010. “Growth in a Time of Debt.” Working Paper 15639. Working Paper Series. National Bureau of Economic Research. doi:10.3386/w15639.
Herndon, Thomas, Michael Ash, and Robert Pollin. 2013. “Does High Public Debt Consistently Stifle Economic Growth? A Critique of Reinhart and Rogoff.” Cambridge Journal of Economics. doi:10.1093/cje/bet075.
The standardization of \(X_1,\ldots,X_n\) is \(Z_1,\ldots,Z_n\), with \(Z_i=\frac{X_i\bar X}{s_x}\), where \(s_x\) is the standard deviation of \(X_1,\ldots,X_n\).↩