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A.12 The Box–Cox Method for Transformations

A.12 The Box–Cox Method for Transformations

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A.12 

313

the box–cox method for transformations

Many statistical packages will have routines that will provide a graph of
RSS(λ) versus λ, or of (n/2) log(RSS(λ)) versus λ as shown in Figure 8.7, for
the highway accident data. Equation (A.40) shows that the confidence interval
for λ includes all values of λ for which the log-likelihood is within 1.92 units
of the maximum value of the log-likelihood, or between the two vertical lines
in the figure.
A.12.2  Multivariate Case
Although the material in this section uses more mathematical statistics than
most of this book, it is included because the details of computing the multivariate extension of Box–Cox transformations are not published elsewhere. The
basic idea was proposed by Velilla (1993).
Suppose X is a set of p variables we wish to transform and define

ψ M ( X , l ) = (ψ M ( X 1, λ1 ), … , ψ M ( X k , λ k ))
We have used the modified power transformations (8.5) for each element of
X, but the same general idea can be applied using other transformations such
as the Yeo–Johnson family introduced in Section 8.4. In analogy to the univariate case, we assume that for some λ, we will have

ψ M ( X , l ) ~ N( m , V)
where V is an unknown positive definite symmetric matrix that needs to be
estimated. If xi is the observed value of X for the ith observation, then the
likelihood function is given by
n

L( m , V, l|X ) =

1

∏ (2π V )

1/2

i =1

1
× exp  − (ψ M (x i, l ) − m )′ V −1 (ψ M (x i, l ) − m )
 2



(A.41)

where |V| is the determinant.3 After rearranging terms, the log-likelihood is
given by
n
n
log(L( m , V, l|X )) = − log(2π ) − log ( V )
2
2


3

1
2

n



V −1 (ψ M (x i, l ) − m )(ψ M (x i, l ) − m )′

i =1

The determinant is defined in any linear algebra textbook.


(A.42)

314

appendix

If we fix λ, then (A.42) is the standard log-likelihood for the multivariate
normal distribution. The values of V and μ that maximize (A.42) are the
sample mean and sample covariance matrix, the latter with divisor n rather
than n − 1,
1
m(l ) =
n
V( l ) =

1
n

n

∑ψ

M

(x i , l )

i =1
n

∑ (ψ

M

(x i, l ) − m ( l ))(ψ M (x i, l ) − m ( l ))′

i =1

Substituting these estimates into (A.42) gives the profile log-likelihood
for λ,


n
n
n
log(L( m ( l ), V( l ), l|X )) = − log(2π ) − log ( V( l ) ) −
2
2
2

(A.43)

This equation will be maximized by minimizing the determinant of V(λ) over
values of λ. This is a numerical problem for which there is no closed-form
solution, but it can be solved using a general-purpose function minimizer.
Standard theory for maximum likelihood estimates can provide tests concerning λ and standard errors for the elements of λ. To test the hypothesis that
λ = λ0 against a general alternative, compute
G 2 = 2  log(L(m (lˆ ), V(lˆ ), lˆ )) − log(L(m (l0 ), V(l0 ), l0 ))
and compare G2 with a chi-squared distribution with p df. The standard error
of lˆ is obtained from the inverse of the expected information matrix evaluated
at lˆ . The expected information for lˆ is just the matrix of second derivatives
of (A.43) with respect to λ evaluated at lˆ . Many optimization routines, such
as optim in R, will return the matrix of estimated second derivatives if
requested; all that is required is inverting this matrix, and then the square roots
of the diagonal elements are the estimated standard errors.

A.13  CASE DELETION IN LINEAR REGRESSION
Suppose X is the n × p′ matrix of regressors with linearly independent columns.
We use the subscript “(i)” to mean “without case i,” so that X(i) is an (n − 1) × p′
matrix. We can compute (X (′i ) X (i ) )−1 from the remarkable formula


(X (′i ) X ( i ) )−1 = (X ′X)−1 +

(X ′X)−1 x i x ′i (X ′X)−1

1 − hii

(A.44)

A.13 

315

case deletion in linear regression

where hii = x i′(X ′X)−1 x i is the ith leverage value, a diagonal value from the hat
matrix. This formula was used by Gauss (1821); a history of it and many variations is given by Henderson and Searle (1981). It can be applied to give all the
results that one would want relating multiple linear regression with and
without the ith case. For example,


(X ′X) x i eˆi

bˆ (i ) = bˆ −
1 − hii
−1

(A.45)

Writing ri = eˆi /σˆ 1 − hii , the estimate of variance is
−1



 n − p′ − 1 
σˆ (2i ) = σˆ 2 
 n − p′ − ri2 

(A.46)

and the studentized residual ti is


 n − p′ − 1 
ti = ri 
 n − p′ − ri2 

1/ 2



(A.47)

The diagnostic statistics examined in this book were first thought to be
practical because of simple formulas used to obtain various statistics when
cases are deleted that avoided recomputing estimates. Advances in computing
in the last 30 years have made the computational burden of recomputing
without a case much less onerous, and so diagnostic methods equivalent to
those discussed here can be applied to problems other than linear regression
where the updating formulas are not available.

References
In a few instances, the URL given for an article refers to the website http://
dx.doi.org/, used to resolve a digital object identifier (DOI) and send you to the
correct website. This may lead you to a page requesting payment before viewing
an article. Many research libraries subscribe to journals and may use a different
method to resolve a DOI so you can get to articles for free. Ask your librarian, or see
http://doi.org.
An on-line version of this bibliography with clickable links is available on the
website for this book, http://z.umn.edu/alr4ed.
Agresti, A. (2007). An Introduction to Categorical Data Analysis. 2nd ed. Wiley,
Hoboken, NJ.
Agresti, A. (2013). Categorical Data Analysis. 3rd ed. Wiley, Hoboken, NJ.
Allison, P. D. (2001). Missing Data. Quantitative Applications in the Social Sciences.
Sage, Thousand Oaks, CA.
Allison, T. and Cicchetti, D. V. (1976). Sleep in mammals: Ecological and constitutional correlates. Science, 194, 732–734. URL: http://www.jstor.org/stable/1743947.
Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27,
17–21. URL: http://www.jstor.org/stable/2682899.
Atkinson, A. C. (1985). Plots, Transformations and Regression: An Introduction to
Graphical Methods of Diagnostic Regression Analysis. Clarendon Press, Oxford.
Baes, C. and Kellogg, H. (1953). Effects of dissolved sulphur on the surface tension
of liquid copper. Journal of Metals, 5, 643–648.
Barnett, V. and Lewis, T. (1994). Outliers in Statistical Data. 3rd ed. Wiley, Hoboken,
NJ.
Bates, D. and Watts, D. (1988). Nonlinear Regression Analysis and Its Applications.
Wiley, Hoboken, NJ.
Beckman, R. J. and Cook, R. D. (1983). Outliers. Technometrics, 25, 119–149. URL:
http://www.jstor.org/stable/1268541.
Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A
practical and powerful approach to multiple testing. Journal of the Royal Statistical
Society. Series B (Methodological), 57, 289–300. URL: http://www.jstor.org/stable/
2346101.
Applied Linear Regression, Fourth Edition. Sanford Weisberg.
© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

317

318

references

Bertalanffy, L. (1938). A quantitative theory of organic growth (inquiries on
growth laws II). Human Biology, 10, 181–213. URL: http://www.jstor.org/stable/
41447359.
Berzuini, C., Dawid, P., and Bernardinelli, L. (eds.) (2012). Causality: Statistical Perspectives and Applications. Wiley, Hoboken, NJ.
Bleske-Rechek, A. and Fritsch, A. (2011). Student consensus on ratemyprofessors.
com. Practical Assessment, Research & Evaluation, 16. (Online; last accessed—
August 1, 2013), URL: http://pareonline.net/getvn.asp?v=16&n=18.
Blom, G. (1958). Statistical Estimates and Transformed Beta Variables. Wiley, New
York.
Bowman, A. W. and Azzalini, A. (1997). Applied Smoothing Techniques for Data
Analysis: The Kernel Approach with S-Plus Illustrations. Oxford University Press,
Oxford.
Box, G., Jenkins, G., and Reinsel, G. (2008). Time Series Analysis: Forecasting and
Control. 4th ed. Wiley, Hoboken, NJ.
Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations. Journal of the Royal
Statistical Society. Series B (Methodological), 26, 211–252. URL: http://www.jstor.org/
stable/2984418.
Bretz, F., Hothorn, T., Westfall, P., and Westfall, P. (2010). Multiple Comparisons
Using R. Chapman & Hall/CRC, Boca Raton, FL.
Breusch, T. S. and Pagan, A. R. (1979). A simple test for heteroscedasticity and random
coefficient variation. Econometrica, 47, 1287–1294. URL: http://www.jstor.org/
stable/1911963.
Brillinger, D. (1983). A generalized linear model with “Gaussian” regressor
variables. A Festschrift for Erich L. Lehmann in Honor of His Sixty-Fifth Birthday,
97–114.
Brown, P. (1993). Measurement, Regression, and Calibration. Oxford Scientific Publications, Oxford.
Burnside, O. C., Wilson, R. G., Weisberg, S., and Hubbard, K. G. (1996). Seed longevity
of 41 weed species buried 17 years in Eastern and Western Nebraska. Weed
Science, 44, 74–86. URL: http://www.jstor.org/stable/4045786.
Burt, C. (1966). The genetic determination of differences in intelligence: A study of
monozygotic twins reared together and apart. British Journal of Psychology, 57,
137–153. URL: http://dx.doi.org/10.1111/j.2044-8295.1966.tb01014.x.
Carpenter, J. and Kenward, M. (2012). Multiple Imputation and Its Application.
Wiley, Hoboken, NJ. (Online; last accessed August 1, 2013), URL: http://
missingdata.lshtm.ac.uk.
Casella, G. and Berger, R. (2001). Statistical Inference. Duxbury Press, Pacific Grove,
CA.
Centers for Disease Control (2013). Youth risk behavior surveillance system. (Online;
last accessed August 1, 2013), URL: http://www.cdc.gov/HealthyYouth/yrbs/
index.htm.
Chen, C.-F. (1983). Score tests for regression models. Journal of the American Statistical
Association, 78, 158–161. URL: http://www.jstor.org/stable/2287123.
Christensen, R. (2011). Plane Answers to Complex Questions: The Theory of Linear
Models. 4th ed. Springer, New York.

references

319

Clapham, A. (1934). English Romanesque Architecture after the Conquest. Clarendon
Press, Oxford.
Clark, R., Henderson, H., Hoggard, G., Ellison, R., and Young, B. (1987). The ability
of biochemical and haematological tests to predict recovery in periparturient
recumbent cows. New Zealand Veterinary Journal, 35, 126–133. URL: http://
dx.doi.org/10.1080/00480169.1987.35410.
Clausius, R. (1850). Über die bewegende Kraft der Wärme und die Gezetze welche
sich daraus für Wärmelehre selbst abeiten lassen. Annelen der Physik, 79,
500–524.
Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 74, 829–836. URL: http://
www.jstor.org/stable/2286407.
Cochran, W. (1977). Sampling Techniques. 3rd ed. Wiley, Hoboken, NJ.
Collett, D. (2003). Modelling Binary Data. 2nd ed. Chapman & Hall, Boca Raton, FL.
Colorado Climate Center (2012). Colorado climate center monthly data access.
(Online; last accessed August 1, 2013), URL: http://ccc.atmos.colostate.edu/cgi-bin/
monthlydata.pl.
Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19, 15–18. URL: http://www.jstor.org/stable/1268249.
Cook, R. D. (1986). Assessment of local influence. Journal of the Royal Statistical
Society. Series B (Methodological), 48, 133–169. URL: http://www.jstor.org/
stable/2345711.
Cook, R. D. (1998). Regression Graphics: Ideas for Studying Regressions Through
Graphics. Wiley, Hoboken, NJ.
Cook, R. D. and Prescott, P. (1981). On the accuracy of Bonferroni significance levels
for detecting outliers in linear models. Technometrics, 23, 59–63. URL: http://
www.jstor.org/stable/1267976.
Cook, R. D. and Weisberg, S. (1982). Residuals and Influence in Regression. Chapman
& Hall/CRC, Boca Raton, FL. (Online; last accessed August 1, 2013), URL: http://
conservancy.umn.edu/handle/37076.
Cook, R. D. and Weisberg, S. (1983). Diagnostics for heteroscedasticity in regression.
Biometrika, 70, 1–10. URL: http://www.jstor.org/stable/2335938.
Cook, R. D. and Weisberg, S. (1994). Transforming a response variable for linearity.
Biometrika, 81, 731–737. URL: http://www.jstor.org/stable/2337076.
Cook, R. D. and Weisberg, S. (1999a). Applied Regression Including Computing and
Graphics. Wiley, New York.
Cook, R. D. and Weisberg, S. (1999b). Graphs in statistical analysis: Is the medium
the message? The American Statistician, 53, 29–37. URL: http://www.jstor.org/
stable/2685649.
Cook, R. D. and Witmer, J. A. (1985). A note on parameter-effects curvature. Journal
of the American Statistical Association, 80, 872–878. URL: http://www.jstor.org/
stable/2288546.
Cox, D. (1958). Planning of Experiments. Wiley, Hoboken, NJ.
Cunningham, R. and Heathcote, C. (1989). Estimating a non-Gaussian regression
model with multicollinearity. Australian & New Zealand Journal of Statistics, 31,
12–17.

320

references

Dalal, S. R., Fowlkes, E. B., and Hoadley, B. (1989). Risk analysis of the space shuttle:
Pre-challenger prediction of failure. Journal of the American Statistical Association,
84, 945–957. URL: http://www.jstor.org/stable/2290069.
Daniel, C. and Wood, F. (1980). Fitting Equations to Data: Computer Analysis of Multifactor Data. Wiley, Hoboken, NJ.
Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application.
Cambridge University Press, Cambridge.
Dawson, R. (1995). The “unusual episode” data revisited. Journal of Statistics Education, 3. URL: http://www.amstat.org/publications/JSE/v3n3/datasets.dawson.html.
de Boor, C. (1978). A Practical Guide to Splines. Springer, New York.
Derrick, A. (1992). Development of the measure-correlate-predict strategy for site
assessment. In Proceedings of the 14th BWEA Conference. 259–265.
Dodson, S. (1992). Predicting crustacean zooplankton species richness. Limnology and
Oceanography, 37, 848–856. URL: http://www.jstor.org/stable/2837943.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of
Statistics, 7, 1–26. URL: http://www.jstor.org/stable/2958830.
Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman &
Hall/CRC, Boca Raton, FL.
Eicker, F. (1963). Asymptotic normality and consistency of the least squares estimators
for families of linear regressions. The Annals of Mathematical Statistics, 34, 447–
456. URL: http://www.jstor.org/stable/2238390.
Eicker, F. (1967). Limit theorems for regressions with unequal and dependent errors.
In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and
Probability, vol. 1. University of California Press, Berkeley, 59–82.
Ezekiel, M. and Fox, K. (1959). Methods of Correlation and Regression Analysis:
Linear and Curvilinear. Wiley, Hoboken, NJ.
Fair Isaac Corporation (2013). myfico. (Online; last accessed August 1, 2013), URL:
http://www.myfico.com/CreditEducation/WhatsInYourScore.aspx.
Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and
its oracle properties. Journal of the American Statistical Association, 96, 1348–1360.
URL: http://www.jstor.org/stable/3085904.
Federal Highway Administration (2001). Highway statistics 2001. (Online; last
accessed August 1, 2012), URL: http://www.fhwa.dot.gov/ohim/hs01/index.htm.
Finkelstein, M. O. (1980). The judicial reception of multiple regression studies in race
and sex discrimination cases. Columbia Law Review, 80, 737–754. URL: http://
www.jstor.org/stable/1122138.
Fisher, R. and Mackenzie, W. (1923). Studies in crop variation. II. The manurial
response of different potato varieties. The Journal of Agricultural Science, 13,
311–320. URL: http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15179/
1/32.pdf.
Fitzmaurice, G., Laird, N., and Ware, J. (2011). Applied Longitudinal Analysis. 2nd ed.
Wiley, Hoboken, NJ.
Forbes, J. D. (1857). XIV.—Further experiments and remarks on the measure­
ment of heights by the boiling point of water. Transactions of the Royal
Society of Edinburgh, 21, 235–243. URL: http://journals.cambridge.org/article
_S0080456800032075.

references

321

Fox, J. (2003). Effect displays in R for generalised linear models. Journal of Statistical
Software, 8, 1–27. URL: http://www.jstatsoft.org/v08/i15.
Fox, J. and Weisberg, S. (2011). An R Companion to Applied Regression. 2nd ed. Sage,
Thousand Oaks, CA. URL: http://z.umn.edu/carbook.
Fraley, C., Raftery, A. E., Gneiting, T., Sloughter, J., and Berrocal, V. J. (2011).
Probabilistic weather forecasting in R. R Journal, 3, 55–63. (Online; last accessed
August 1, 2013), URL: http://journal.r-project.org/archive/2011-1/RJournal_2011-1_
Fraley∼et∼al.pdf.
Freedman, D. and Lane, D. (1983). A nonstochastic interpretation of reported significance levels. Journal of Business & Economic Statistics, 1, 292–298. URL: http://
www.jstor.org/stable/1391660.
Freedman, D. A. (1983). A note on screening regression equations. The American
Statistician, 37, 152–155. URL: http://www.jstor.org/stable/2685877.
Freeman, M. and Tukey, J. (1950). Transformations related to the angular and the
square root. The Annals of Mathematical Statistics, 21, 607–611.
Furnival, G. M., Wilson, J., and Robert W. (1974). Regressions by leaps and bounds.
Technometrics, 16, 499–511. URL: http://www.jstor.org/stable/1267601.
Galton, F. (1877). Typical laws of heredity. Proceedings of the Royal Institution, 8,
282–301. (Online; last accessed August 1, 2013), URL: http://galton.org/essays/18701879/galton-1877-roy-soc-typical-laws-heredity.pdf.
Galton, F. (1886). Regression towards mediocrity in hereditary stature. The Journal of
the Anthropological Institute of Great Britain and Ireland, 15, 246–263. URL: http://
www.jstor.org/stable/2841583.
Gauss, C. (1821). Anzeige: Theoria combinationis observationum erroribus minimis
obnoxiae: Pars prior (theory of the combination of observations which leads to
the smallest errors). Göttingische gelehrte Anzeigen, 33, 321–327. Reprinted by the
Society for Industrial and Applied Mathematics, 1987, URL: http://epubs.siam.org/
doi/pdf/10.1137/1.9781611971248.fm.
Gilstein, C. Z. and Leamer, E. E. (1983). The set of weighted regression estimates.
Journal of the American Statistical Association, 78, 942–948. URL: http://
www.jstor.org/stable/2288208.
Gnanadesikan, R. (1997). Methods for Statistical Data Analysis of Multivariate Observations. 2nd ed. Wiley, Hoboken, NJ.
Goldstein, H. (2010). Multilevel Statistical Models. 4th ed. Wiley, Hoboken, NJ.
Golub, G. and Van Loan, C. (1996). Matrix Computations, vol. 3. Johns Hopkins University Press, Baltimore, MD.
Gould, S. (1966). Allometry and size in ontogeny and phylogeny. Biological Reviews,
41, 587–638. URL: http://dx.doi.org/10.1111/j.1469-185X.1966.tb01624.x.
Gould, S. J. (1973). The shape of things to come. Systematic Zoology, 22, 401–404. URL:
http://www.jstor.org/stable/2412947.
Green, P. and Silverman, B. (1994). Nonparametric Regression and Generalized Linear
Models: A Roughness Penalty Approach, vol. 58. Chapman & Hall/CRC, Boca
Raton, FL.
Greene, W. (2003). Econometric Analysis. 5th ed. Prentice Hall, Upper Saddle
River, NJ.

322

references

Haddon, M. and Haddon, M. (2010). Modelling and Quantitative Methods in Fisheries.
Chapman & Hall/CRC, Boca Raton, FL.
Hahn, A. (ed.) (1979). Development and Evolution of Brain Size. Academic Press, New
York.
Hald, A. (1960). Statistical Theory with Engineering Applications. Wiley, Hoboken, NJ.
Hall, P. and Li, K.-C. (1993). On almost linearity of low dimensional projections
from high dimensional data. The Annals of Statistics, 21, 867–889. URL: http://
www.jstor.org/stable/2242265.
Härdle, W. (1990). Applied Nonparametric Regression, vol. 26. Cambridge University
Press, Cambridge, MA.
Hart, C. W. M. (1943). The Hawthorne experiments. The Canadian Journal of Economics and Political Science/Revue Canadienne d’Economique et de Science politique,
9, 150–163. URL: http://www.jstor.org/stable/137416.
Hastie, T., Tibshirani, R., and Friedman, J. H. (2009). The Elements of Statistical Learning. 2nd ed. Springer, New York.
Hawkins, D. M. (1980). Identification of Outliers. Chapman & Hall/CRC, Boca Raton,
FL.
Hawkins, D. M., Bradu, D., and Kass, G. V. (1984). Location of several outliers in
multiple-regression data using elemental sets. Technometrics, 26, 197–208. URL:
http://www.jstor.org/stable/1267545.
Henderson, H. V. and Searle, S. R. (1981). On deriving the inverse of a sum of matrices.
SIAM Review, 23, 53–60. URL: http://www.jstor.org/stable/2029838.
Hernandez, F. and Johnson, R. A. (1980). The large-sample behavior of transformations to normality. Journal of the American Statistical Association, 75, 855–861.
URL: http://www.jstor.org/stable/2287172.
Hilbe, J. M. (2011). Negative Binomial Regression. Cambridge University Press,
Cambridge.
Hinkley, D. (1985). Transformation diagnostics for linear models. Biometrika, 72, 487–
496. URL: http://www.jstor.org/stable/2336721.
Hoaglin, D. C. and Welsch, R. E. (1978). The hat matrix in regression and
ANOVA. The American Statistician, 32, 17–22. URL: http://www.jstor.org/stable/
2683469.
Hocking, R. (1985). The Analysis of Linear Models. Brooks Cole, Monterey, CA.
Hocking, R. (2003). Methods and Applications of Linear Models: Regression and the
Analysis of Variance. Wiley, Hoboken, NJ.
Hoeting, J. A., Madigan, D., Raftery, A. E., and Volinsky, C. T. (1999). Bayesian model
averaging: A tutorial. Statistical Science, 14, 382–401. URL: http://www.jstor.org/
stable/2676803.
Hosmer, D. W., Lemeshow, S., and May, S. (2008). Applied Survival Analysis. 2nd ed.
Wiley, Hoboken, NJ.
Hosmer, D. W., Lemeshow, S., and Sturdivant, R. (2013). Applied Logistic Regression.
3rd ed. Wiley, Hoboken, NJ.
Huber, P. (1967). The behavior of maximum likelihood estimates under non-standard
conditions. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, 221–33.

references

323

Huber, P. and Ronchetti, E. M. (2009). Robust Statistics. 2nd ed. Wiley, Hoboken, NJ.
Hurvich, C. M. and Tsai, C.-L. (1990). The impact of model selection on inference in
linear regression. The American Statistician, 44, 214–217. URL: http://www.jstor.org/
stable/2685338.
Ioannidis, J. P. A. (2005). Why most published research findings are false. PLoS Med,
2, e124. URL: http://dx.doi.org/10.1371%2Fjournal.pmed.0020124.
Jevons, W. S. (1868). On the condition of the metallic currency of the United
Kingdom, with reference to the question of international coinage. Journal of
the Statistical Society of London, 31, 426–464. URL: http://www.jstor.org/stable/
2338797.
Johns, M. W. (1991). A new method for measuring daytime sleepiness: The Epworth
sleepiness scale. Sleep, 16, 540–545. URL: http://www.ncbi.nlm.nih.gov/
pubmed/1798888.
Johnson, K. (1996). Unfortunate Emigrants: Narratives of the Donner Party. Utah State
University Press, Logan, UT.
Johnson, M. P. and Raven, P. H. (1973). Species number and endemism: The Galápagos
archipelago revisited. Science, 179, 893–895. URL: http://www.jstor.org/
stable/1735348.
Joiner, B. L. (1981). Lurking variables: Some examples. The American Statistician, 35,
227–233. URL: http://www.jstor.org/stable/2683295.
Kennedy, W. and Gentle, J. (1980). Statistical Computing, vol. 33. CRC, Boca
Raton, FL.
LeBeau, M. (2004). Evaluation of the Intraspecific Effects of a 15-Inch Minimum Size
Limit on Walleye Populations in Northern Wisconsin. PhD thesis, University of
Minnesota.
Lehrer, J. (2010). The truth wears off. The New Yorker, 13. (Online; last accessed
August 1, 2012), URL: http://www.newyorker.com/reporting/2010/12/13/101213fa
_fact_lehrer?currentPage=all.
Lenth, R. V. (2006–2009). Java applets for power and sample size [computer software].
(Online; last accessed August 1, 2013), URL: http://www.stat.uiowa.edu/∼rlenth/
Power.
Lenth, R. V. (2013). lsmeans: Least-squares means. R package version 1.06-05, URL:
http://CRAN.R-project.org/package=lsmeans.
Li, K.-C. and Duan, N. (1989). Regression analysis under link violation. The Annals of
Statistics, 17, 1009–1052. URL: http://www.jstor.org/stable/2241708.
Little, R. J. A. and Rubin, D. B. (2002). Statistical Analysis with Missing Data. 2nd ed.
Wiley, Hoboken, NJ.
Loader, C. (1999). Local Regression and Likelihood. Springer, New York.
Loader, C. (2004). Smoothing: Local regression techniques. In Handbook of
Computational Statistics: Concepts and Methods. Springer, New York, 539–563.
Lohr, S. (2009). Sampling: Design and Analysis. 2nd ed. Duxbury Press, Pacific
Grove, CA.
Long, J. S. and Ervin, L. H. (2000). Using heteroscedasticity consistent standard
errors in the linear regression model. The American Statistician, 54, 217–224. URL:
http://www.jstor.org/stable/2685594.