# References

Ashenfelter, Orley, David Ashmore, and Robert Lalonde. 1995. “Bordeaux Wine Vintage Quality and the Weather.” *CHANCE* 8 (4): 7–14. doi:10.1080/09332480.1995.10542468.

Bartholomew, David J, Fiona Steele, Jane Galbraith, and Irini Moustaki. 2008. *Analysis of Multivariate Social Science Data*. CRC press.

Dalal, Siddhartha R., Edward B. Fowlkes, and Bruce Hoadley. 1989. “Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure.” *Journal of the American Statistical Association* 84 (408): 945–57. doi:10.1080/01621459.1989.10478858.

Fox, John. 2005. “The R Commander: A Basic Statistics Graphical User Interface to R.” *Journal of Statistical Software* 14 (9): 1–42. http://www.jstatsoft.org/v14/i09.

Grimmett, Geoffrey, Jean-François Laslier, Friedrich Pukelsheim, Victoriano Ramirez Gonzalez, Richard Rose, Wojciech Slomczynski, Martin Zachariasen, and Karol Życzkowski. 2011. “The Allocation Between the EU Member States of the Seats in the European Parliament - Cambridge Compromise.” http://www.europarl.europa.eu/thinktank/en/document.html?reference=IPOL-AFCO_NT(2011)432760.

Hand, David J, Fergus Daly, K McConway, D Lunn, and E Ostrowski. 1994. *A Handbook of Small Data Sets*. CRC Press.

Harrison, David, and Daniel L. Rubinfeld. 1978. “Hedonic Housing Prices and the Demand for Clean Air.” *Journal of Environmental Economics and Management* 5 (1): 81–102. doi:10.1016/0095-0696(78)90006-2.

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.

James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani. 2013. *An Introduction to Statistical Learning*. Vol. 103. Springer Texts in Statistics. Springer, New York. doi:10.1007/978-1-4614-7138-7.

Moore, Gordon E. 1965. “Cramming More Components onto Integrated Circuits.” *Electronics* 38 (8): 114–17. doi:10.1109/JPROC.1998.658762.

OECD. 2012a. “Does Money Buy Strong Performance in Pisa?” *PISA in Focus*, no. 13: 1–4. doi:10.1787/5k9fhmfzc4xx-en.

———. 2012b. *PISA 2012 Results: What Students Know and Can Do (Volume I, Revised Edition, February 2014): Student Performance in Mathematics, Reading and Science*. Paris: OECD Publishing. doi:10.1787/9789264208780-en.

Peña, Daniel. 2002. *Análisis de Datos Multivariantes*. Madrid: McGraw-Hill.

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In principle, you could pick more than one explanatory variables using the

`'Control'`

or`'Shift'`

keys, but that corresponds to the*multiple linear regression*(covered in Chapter 3).↩The decision of which points are the most

*different*from the rest is done automatically by a method known as the*Mahalanobis depth*.↩The default GUI option is set to identify

`'2'`

points. However, we know after a preliminary plot that there are three very different points in the dataset, hence this particular choice.↩The outliers have a considerable impact on the regression line, as we will see later.↩

Less populated states are given more weight than its corresponding proportional share.↩

According to EuroStat and the population stated in the Cambridge Compromise report.↩

We will be able to say more about how these test are performed after Section 2.5.↩

They are unique and always exist. They can be obtained by solving \(\frac{\partial}{\partial \beta_0}\text{RSS}(\beta_0,\beta_1)=0\) and \(\frac{\partial}{\partial \beta_1}\text{RSS}(\beta_0,\beta_1)=0\).↩

The Student’s \(t\) distribution has

*heavier tails*than the normal, which means that large observations in absolute value are more likely. \(t_n\) converges to a \(\mathcal{N}(0,1)\) when \(n\) is large. For example, with \(n\) larger than \(30\), the normal is a good approximation.↩\(\chi_n^2\) is the distribution of the sum of the squares of \(n\) random variables \(\mathcal{N}(0,1)\).↩

Recall that SSR is different from RSS (Residual Sum of Squares, Section 2.3).↩

Recall that SSE and RSS (for \((\hat \beta_0,\hat \beta_1)\)) are just different names for referring to the same quantity: \(\text{SSE}=\sum_{i=1}^n\left(Y_i-\hat Y_i\right)^2=\sum_{i=1}^n\left(Y_i-\hat \beta_0-\hat \beta_1X_i\right)^2=\mathrm{RSS}\left(\hat \beta_0,\hat \beta_1\right)\).↩

The \(F_{n,m}\) distribution arises as the quotient of two independent random variables \(\chi^2_n\) and \(\chi^2_m\), \(\frac{\chi^2_n/n}{\chi^2_m/m}\).↩

If the assumptions are not satisfied (mismatch between what is assumed to happen in theory and what the data is), then the inference results may be misleading.↩

In Ashenfelter, Ashmore, and Lalonde (1995), this variable is expressed relative to the price of the 1961 vintage, regarded as the best one ever recorded. In other words, they consider

`Price - 8.4937`

as the price variable.↩If you wanted to do so, you will need the function

`I()`

for indicating that`+`

is not including predictors in the model, but is acting as a sum operator:`Price ~ I(Age + AGST + FrancePop + HarvestRain + WinterRain)`

.↩The \(R^2\) for the multiple linear regression \(Y=\beta_0+\beta_1X_1+\ldots+\beta_kX_k+\varepsilon\) is not the sum of the \(R^2\)’s for the simple linear regressions \(Y=\beta_0+\beta_jX_j+\varepsilon\), \(j=1,\ldots,k\).↩

But be aware of the changes in units for

`medv`

,`black`

,`lstat`

and`nox`

.↩Although note that the printed messages always display

`'AIC'`

even if you choose`'BIC'`

.↩The vectors are regarded as column matrices.↩

They are unique and always exist.↩

With \(m=1\), the density of a \(\mathcal{N}_{m}\) corresponds to a bell-shaped

*curve*With \(m=2\), the density is a*surface*similar to a bell.↩SSE and RSS are two names for the same quantity (that appears in different contexts): \(\text{SSE}=\sum_{i=1}^n\left(Y_i-\hat Y_i\right)^2=\sum_{i=1}^n\left(Y_i-\hat \beta_0-\hat \beta_1X_{i1}-\ldots-\hat \beta_kX_{ik}\right)^2=\mathrm{RSS}(\hat{\boldsymbol{\beta}})\).↩

More complex – included here just for clarification of the

`anova`

’s output.↩You will need to run this piece of code whenever you want to call

`simpleAnova`

, since it is not part of`R`

nor`R Commander`

.↩After the shuttle exits the atmosphere, the solid rocket boosters separate and descend to land using a parachute where they are carefully analysed.↩

Recall that the result of a bet is binary: you win or lose the bet.↩

An equivalent way of stating this assumption is \(p(\mathbf{x})=\mathrm{logistic}(\beta_0+\beta_1x_1+\ldots+\beta_kx_k)\).↩

The probability of the sample according to the saturated is \(1\) – replace \(p(\mathbf{X}_i)=Y_i\) in (4.8).↩

We are implicitly assuming that \(n>k\). Otherwise, the maximum number of PCs would be \(\min(n-1,k)\).↩

For example, if PC\(_1\) explains all the variance of \(X_1,\ldots,X_k\) or if the variables are

*uncorrelated*, in which case the PCs will be equal to the original variables.↩Alternatively, the

`'Tools' -> 'Install auxiliary software [if not already installed]'`

will redirect you to the download links for the auxiliary software.↩This is, assuming we have performed the right steps in the analysis without making any mistake.↩