Everitt B.S. The Cambridge Dictionary of Statistics

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Kuiper’s test: A test that a circular random variable has an

angular uniform distribution

. Given a

set of observed values, 

;

; ...;

the test statistic is

¼ D

þ D



where

¼ maxðj=n  x

Þ; j ¼ 1; ...; n



¼ max



j 1



; j ¼ 1; ...; n

and x

¼ 

=2p. V

ﬃﬃﬃ

has an approximate standard normal distribution under the hypo-

thesis of an angular uniform distribution. See also Watson’s test. [MV2 Chapter 14.]

K u llback^ Lei bl er in f o rmati o n: A function, I,deﬁned for two probability distributions, f(x)

and g(x) and given by

Iðf : gÞ¼

1

f ðxÞlog

f ðxÞ

gðxÞ



Essentially an asymmetric distance function for the two distributions.

Jeffreys’s distance

measure is a symmetric combination of the Kullback–Leibler information given by

Iðf ; gÞþIðg; f Þ

See also Bhattacharya ’s distance and Hellinger distance. [MV2 Chapter 11.]

Kurtosis: The extent to which the peak of a unimodal probability distribution or frequency distri-

bution departs from the shape of a normal distribution, by either being more pointed

(leptokurtic)orﬂatter (platykurtic). Usually measured for a probability distribution as



=

where 

is the fourth central

moment

of the distribution, and 

is its variance.

(Corresponding functions of the sample moments are used for frequency distributions.)

For a normal distribution this index takes the value three and often the index is redeﬁned

as the value above minus three so that the normal distribution would have a value zero.

(Other distributions with zero kurtosis are called mesokurtic.) For a distribution which is

leptokurtic the index is positive and for a platykurtic curve it is negative. See Fig. 82.

See also skewness. [KA1 Chapter 3.]

Kurtosis procedure: See many-outlier detection procedures.

Kwiatkowski-Phillips-Schmidt-Shin test (KPSS test): A test of the hypothesis that

time series

stationary

around a deterministic trend. [Journal of Economics, 1992, 54,

159–178.]

Fig. 82 Curves with differing degrees

of kurtosis: A, mesokurtic; B, platykurtic;

C, leptokurtic.

239

L’Abbe

plot: A plot often used in the

meta-analysis

clinical trials

where the outcome is a binary

response. The event risk (number of events/number of patients in a group) in the treatment

groups are plotted against the risk for the controls of each trial. If the trials are fairly

homogeneous the points will form a ‘cloud’ close to a line, the gradient of which will

correspond to the pooled treatment effect. Large deviations or scatter would indicate

possible heterogeneity. [Annals of Internal Medicine, 1987, 107, 224–33.]

Labour force survey: A survey carried out in the UK on a quarterly basis since the spring of 1992.

It covers 60 000 households and provides labour force and other details for about 120 000

people aged 16 and over. The survey covers not only unemployment, but employment, self-

employment, hours of work, redundancy, education and training.

Lackof memory property: A property possessed by a random variable Y, namely that

PrfY  x þ y jY  yg¼PrfY  xg

Variables having an

exponential distribution

have this property, as do those following a

geometric distribution

.[A Primer on Statistical Distributions, 2003, N. Balakrishnan and

V. B. Nevzorov, Wiley, New York.]

Ladder of re-expression: See one-bend transformation.

Lagging indicators: Part of a collection of economic

time series

designed to provide information

about broad swings in measures of aggregate economic activity known as business cycles.

Used primarily to conﬁrm recent turning points in such cycles. Such indicators change after

the overall economy has changed and examples include labour costs, business spending and

the unemployment rate.

Lagrange multipliers: A method of evaluating maxima or minima of a function of possibly several

variables, subject to one or more linear constraints. [Optimization, Theory and Applications,

1979, S. S. Rao, Wiley Eastern, New Delhi.]

Lagrange multiplier test: Synonym for score test.

Lancaster,HenryOliver(1913^2001): Born in Kempsey, New South Wales, Lancaster intended

to train as an actuary, but in 1931 enrolled as a medical student at the University of Sydney,

qualifying in 1937. Encountering Yule’s Introduction to the Theory of Statistics, he became

increasingly interested in statistical problems, particularly the analysis of 2 × 2 × 2

tables. After World War II this interest led him to obtain a post as Lecturer in Statistics at

the Sydney School of Public Health and Tropical Medicine. In 1948 Lancaster left Australia

to study at the London School of Hygiene and Tropical Medicine under

Bradford Hill

returning to Sydney in 1950 to study trends in Australian mortality. Later he undertook more

theoretical work that led to his 1969 book, The Chi-squared Distribution. In 1966 he was

made President of the Statistical Society of Australia. Lancaster died in Sydney on 2

December 2001.

240

Lancast ermod els: A means of representing the

joint distribution

of a set of variables in terms of the

marginal distributions

, assuming all interactions higher than a certain order vanish. Such

models provide a way to capture dependencies between variables without making the

sometimes unrealisitic assumptions of total independence on the one hand, yet having a

model that does not require an unrealistic number of observations to provide precise

parameter estimates. [Statistical Pattern Recognition, 1999, A. Webb, Arnold, London.]

Landmark an alysis: A term applied to a form of analysis occasionally applied to data consisting of

survival times in which a test is used to assess whether ‘treatment’ predicts subsequent

survival time among subjects who survive to a ‘landmark’ time (for example, 6 months post-

randomization) and who have, at this time, a common prophylaxis status and history of all

other covariates. [Statistics in Medicine, 1996, 15, 2797–812.]

Lan dma r k-based shape an alys i s: An approach to quantifying the shape and shape differences

of widely-spaced landmarks (coordinate points) that allows visualization of these differ-

ences between groups. [Statistical Shape Analysis, 1998, I. Dryden and K. V. Mardia, Wiley,

New York.]

Landmark registrat i on: A method for aligning the proﬁles in a set of

repeated measures data

, for

example

human growth curves

, by identifying the timing of salient features such as peaks,

troughs, or inﬂection points. Curves are then aligned by transforming individual time so that

landmark events become synchronized. [The Annals of Statistics, 1992, 16,82–112.]

Lapl a ce ap pr ox im a tion: An approach to approximating the integral of a function ∫ f(x)dx by ﬁtting

a multivariate normal density at the maximum

x of f(x) and computing the volume under

the density. The covariance matrix of the ﬁtted

multivariate normal distribution

is deter-

mined by the

Hessian matrix

of log f(x) at the maximum

x. The term is also used

for approximating the

posterior distribution

with a multivariate normal centered at the

maximum a posteriori estimate. [Bayesian Data Analysis, 2004, A.Gelman, J. B. Carlin,

H. S. Stern and D. B. Rubin, Chapman and Hall/CRC, Boca Raton.]

Lapl a ce distrib utio n: The probability distribution, f (x), given by

f ðxÞ¼



1

exp 

jx j







0; 1

Can be derived as the distribution of the difference of two independent random variables

each having an identical

exponential distribution

. Examples of the distribution are shown in

Fig. 83. The mean, variance,

skewness

and

kurtosis

of the distribution are as follows:

mean ¼ 

variance ¼ 2

skewness ¼ 0

kurtosis ¼ 6

Also known as the double exponential distribution. [STD Chapter 22.]

Laplace, Pie rre-Si mon , M arq u is de ( 1749^1 827): Born in Beaumont-en-Auge near to Caen

where he ﬁrst studied, Laplace made immense contributions to physics, particularly astron-

omy, and mathematics. Became a professor at the École Militaire in 1771 and entered the

Académie des Sciences in 1773. Most noted for his ﬁve-volume work Traité de Mécanique

Céleste he also did considerable work in probability including independently discovering

Bayes’ theorem

, ten years after Bayes’ essay on the topic. Much of this work was published

in Théorie Analytique des Probabilités. Laplace died on 5 March 1827, in Paris.

241

Large samplemethods: Methods for evaluating the performance of estimators and statistics as the

sample size tends to inﬁnity. See also asymptotic distribution .

Lasagna’slaw: Once a

clinical trial

has started the number of suitable patients dwindles to a

tenth of what was calculated before the trial began. [British Medical Journal, 2001, 322,

1457–62.]

Lasso: A penalized

least squares regression

method that can be used for variable selection on high-

dimensional data. For a

multiple linear regression

model of the form

¼ x

β þ 

The lasso estimator of β is the value that minimizes

i¼1

ðy

 x

βÞ

þ l

j¼1

jβ

where q is the number of variables and λ is the penalty parameter. The estimator minimizes

the usual sum of squared errors but has a bound on the sum of the absolute values of these

coefﬁcents. When the bound is large enough the constraint has no effect and the solution is

simply that of multiple regression. When however the bound is smaller the parameter

estimates are ‘shrunken’ versions of the least squares estimates. [Journal of the Royal

Statistical Society, Series B, 1996, 58, 267–288.]

Last observation carried forward: A method for replacing the observations of patients who

drop out of a

clinical trial

carried out over a period of time. It consists of substituting for each

missing value

the subject’s last available assessment of the same type. Although widely

applied, particularly in the pharmaceutical industry, its usefulness is very limited since it

makes very unlikely assumptions about the data, for example, that the (unobserved) post

drop-out response remains frozen at the last observed value. See also imputation.[Statistics

in Medicine, 1992, 11, 2043–62.]

f(x)

–20 –10 0 10 20

0.0 0.1 0.2 0.3 0.4 0.5

alpha = 0, beta = 1

alpha = 0, beta = 2

alpha = 0, beta = 3

Fig. 83 Laplace

distributions for α =0;

β =1,2,3.

242

Latent c lass analys i s: A method of assessing whether a set of observations involving q categorical

variables, in particular , binary variables, consists of a number of dif ferent groups or classes within

which the variables are independent. Essentially a

ﬁnite mixture model

in which the component

distributions are the product of q Bernoulli distributions, one for each of the binary variables in

the data. Parameters in such models can be estimated by

maximum likelihood estimation

via the

EM algorithm

. Can be considered as either an analogue of

factor analysis

for categorical

variables, or a model of

cluster analysis

for such data. See also grade of membership model.

[An Introduction to Latent Variable Models, 1984, B. S. Everitt, Chapman and Hall, London.]

Latent c lass i dentif i ab ility d i s pl a y: A graphical diagnostic for recognizing weakly identiﬁed

models in applications of

latent class analysis

.[Biometrics, 2000, 56, 1055–67.]

Latent period: The time interval between the initiation time of a disease process and the time of the

ﬁrst occurrence of a speciﬁcally deﬁned manifestation of the disease. An example is the

period between exposure to a tumorigenic dose of radioactivity and the appearance of

tumours. [The Mathematical Theory of Infectious Diseases and its Applications, 1975,

N. T. J. Bailey, Arnold, London.]

Latent root d istributions: Probability distributions for the latent roots of a square matrix whose

elements are random variables having a

joint distribution

. Those of primary importance arise

in multivariate analyses based on the assumption of

multivariate normal distributions

. See

also Bartlett’s test for eigenvalues.[An Introduction to Multivariate Statistics, 1979, M. S.

Srivastava and C. G. Khatri, North Holland, New York.]

Latent roots: Synonym for eigenvalues.

Latent var iabl e: Avariable that cannot be measured directly, but is assumed to be related to a number

of observable or manifest variables. Examples include racial prejudice and social class. See

also indicator variable.[An Introduction to Latent Variable Models, 1984, B. S. Everitt,

Chapman and Hall/CRC Press, London.]

Latent vectors: Synonym for eigenvectors.

Latin hypercube sampling (LHS): A stratiﬁed random sampling technique in which a sample of

size N from multiple (continuous) variables is drawn such that for each individual variable

the sample is (marginally) maximally stratiﬁed, where a sample is maximally stratiﬁed when

the number of strata equals the sample size N and when the probability of falling in each of

the strata equals N

−1

. An example is shown in Fig. 84, involving two independent uniform

[0,1] variables, the number of categories per variable equals the sample size (6), each row or

each column contains one element and the width of rows and columns is 1/6. [Reliability

Engineering and System Safety, 2003, 81,23–69.]

Latin sq uar e: An experimental design aimed at removing from the experimental error the variation

from two extraneous sources so that a more sensitive test of the treatment effect can be

achieved. The rows and columns of the square represent the levels of the two extraneous

factors. The treatments are represented by Roman letters arranged so that no letter appears

more than once in each row and column. The following is an example of a 4 × 4 Latin square

ABCD

BCDA

CDAB

DABC

See also Graeco-Latin square.[Handbook of Experimental Methods for Process

Improvement, 1997, D. Drain, Chapman and Hall, London.]

243

Lattice designs: A class of

incomplete block designs

introduced to increase the precision of treat-

ment comparisons particularly in agricultural crop trials. [The Design of Experiments, 1988,

R. Mead, Cambridge University Press, Cambridge.]

Lattice distribution: A class of probability distributions to which most distributions for discrete

random variables used in statistics belong. In such distributions the intervals between values

of any one random variable for which there are non-zero probabilities are all integral

multiples of one quantity. Points with these coordinates thus form a lattice. By an approx-

imate linear transformation it can be arranged that all variables take values which are

integers. [Univariate Discrete Distributions, 2005, N. I. Johnson, A. W. Kemp and

S. Kotz, Wiley, New York.]

Lattice square: An

incomplete block design

for ν =(s − 1)s treatments in b = rs

blocks

, each

containing s − 1 units. In addition the blocks can be arranged into r complete replications.

[Biometrics, 1977, 33, 410–13.]

Law of large numbers: A ‘law’ that attempts to formalize the intuitive notion of probability

which assumes that if in n identical trials an event A occurs n

times, and if n is very

large, then n

/n should be near the probability of A. The formalization involves translating

‘identical trials’ as

Bernoulli trials

with probability p of a success. The law then states that as

n increases, the probability that the average number of successes deviates from p by more

then any preassigned value  where 

0 is arbitrarily small but ﬁxed, tends to zero.

[An Introduction to Probability Theory, Volume 1, 3rd edition, 1968, W. Feller, Wiley,

New York.]

Law o flikelih ood: Within the framework of a statistical model, a particular set of data supports one

statistical hypothesis better than another if the

likelihood

of the ﬁrst hypothesis, on the data,

exceeds the likelihood of the second hypothesis. [Likelihood, 1992, A. W. F. Edwards,

Cambridge University Press, Cambridge.]

Law of primacy: A ‘law’ relevant to work in market research which says that an individual for

whom, at the moment of choice, n brands are tied for ﬁrst place in brand strength chooses

each of these n brands with probability 1/n.[Marketing Research: State of the Art

Perspectives, edited by C. Chakrapani, 2000, American Marketing Association, Chicago.]

Law of truly large numbers: The law that says that, with a large enough sample, any outrageous

thing is likely to happen. See also coincidences.

LD50: Abbreviation for lethal dose 50.

Fig. 84 Latin hypercube sampling.

244

LDUtest: A test for the rank of a matrix A using an estimate of A based on a sample of observations.

[Journal of the American Statistical Association, 1996, 91, 1301–9.]

Leadt im e: An indicator of the effectiveness of a screening test for chronic diseases given by the length

of time the diagnosis is advanced by screening. [International Journal of Epidemiology,

1982, 11, 261–7.]

Lead timeb ias: A term used particularly with respect to cancer studies for the bias that arises when the

time from early detection to the time when the cancer would have been symptomatic is added

to the

survival time

of each case. [Journal of the National Cancer Institute, 1968, 41,665–81.]

Leaps-and-bounds algorithm: An algorithm used to ﬁnd the optimal solution in problems that

may have a very large number of possible solutions. Begins by splitting the possible

solutions into a number of exclusive subsets and limits the number of subsets that need to

be examined in searching for the optimal solution by a number of different strategies. Often

used in

all subsets regression

to restrict the number of models that have to be examined.

[ARA Chapter 7.]

Least absolute deviation reg ression: An alternative to

least squares estimation

for determin-

ing the parameters in

multiple regression

. The criterion minimized to ﬁnd the estimators of

the regression coefﬁcients is S, given by;

S ¼

i¼1

j y

 β



j¼1

where y

and x

; j ¼ 1; ...; q are the response variable and explanatory variable values for

individual i. The estimators, b

; b

; ...; b

of the regression coefﬁcients, β

; β

; ...; β

are

such that the median(b

Þ¼β

(median unbiased) and are

maximum likelihood estimators

when the errors have a

Laplace distribution

. This type of regression has greater power than

that based on least squares for asymmetric error distributions and heavy tailed, symmetric

error distributions; it also has greater resistance to the inﬂuence of a few outlying values of

the dependent variable. [Transformation and Weighting in Regression, 1988, R. J. Carroll

and D. Ruppert, Chapman and Hall/CRC Press, London.]

Least significant difference test: An approach to comparing a set of means that controls the

familywise error rate

at some particular level, say α. The hypothesis of the equality of the

means is tested ﬁrst by an α-level

F-test

. If this test is not signiﬁcant, then the procedure

terminates without making detailed inferences on pairwise differences; otherwise each

pairwise difference is tested by an α-level,

Student’s t-test

Least squares cross-validation: A method of cross validation in which models are assessed by

calculating the sum of squares of differences between the observed values of a sub-set of the

data and the relevant predicted values calculated from ﬁtting a model to the remainder of the

data. [ Biometrika, 1984, 71, 353–60.]

Least sq ua r es esti m ation: A method used for estimating parameters, particularly in regression

analysis, by minimizing the difference between the observed response and the value predicted

by the model. For example, if the expected value of a response variable y is of the form

E ð yÞ¼α þ βx

where x is an explanatory variable, then least squares estimators of the parameters α and β

may be obtained from n pairs of sample values ðx

; y

Þ; ðx

; y

Þ; ...; ðx

; y

Þ by minimizing

S given by

245

S ¼

i¼1

ðy

 α  βx

to give

α ¼



y 



β ¼

i¼1

ðx





xÞðy





yÞ

i¼1

ðx





xÞ

Often referred to as ordinary least squares to differentiate this simple version of the

technique from more involved versions such as,

weighted least squares

and

iteratively

reweighted least squares

. [ARA Chapter 1.]

Leave-one-out estimator: See error-rate estimation.

LeCam, Lucien (1924^2000): Born in Felletin, France, LeCam studied at the University of Paris

and received a Licence en Sciences (university diploma) in 1945. His career began with

work on power and hydraulic systems at the Electricité de France, but in 1950 he went to

Berkeley at the invitation of

Jerzy Neyman

. LeCam received a Ph.D. from the university in

1952. He was made Assistant Professor of Mathematics in 1953 and in 1955 joined the new

Department of Statistics at Berkeley. He was departmental chairman from 1961 to 1965.

LeCam was a brilliant mathematician and made important contributions to the asymptotic

theory of statistics, much of the work being published in his book Asymptotic Methods in

Statistical Decision Theory, published in 1986.

Left-censored: See censored observations.

Lehmann, Erich Leo (1917^2009): Lehmann obtained his MA and PhD from the University of

California, Berkeley where he worked from 1942. During the Second World War he worked

as an analyst for the US Air Force. In the early 1950s Lehmann worked at Columbia,

Princeton and Stanford universities, before returning to Berkeley. Erich Lehmann made

numerous important contributions to mathematical statistics, particularly to estimation

theory and the theory of hypothesis testing. He served as editor of The Annals of

Mathematical Statistics, was president of the Institute of Mathematical Statistics, and an

elected member of the American Academy of Arts and Sciences and the US National

Academy of Science. Erich Lehmann died on September 12th 2009.

Length -b iased data: Data that arise when the probability that an item is sampled is proportional to

its length. A prime example of this situation occurs in

renewal theory

where inter-event time

data are of this type if they are obtained by sampling lifetimes in progress at a randomly

chosen point in time. [Biometrika, 1996, 83, 343–54.]

Length-biased sampling: The bias that arises in a sampling scheme based on patient visits, when

some individuals are more likely to be selected than others simply because they make more

frequent visits. In a

screening study

for cancer, for example, the sample of cases detected is

likely to contain an excess of slow-growing cancers compared to the sample diagnosed

positive because of their symptoms. [Australian Journal of Statistics, 1981, 23,91–4.]

Lenth’smethod: An objective method for deciding which effects are active in the analysis of

unreplicated experiments, when the model is saturated and hence there are no degrees of

freedom for estimating the error variance. [Technometrics, 1989, 31, 469–473.]

Lepagetest: A

distribution free test

for either location or dispersion. The test statistic is related to that

used in the

Mann–Whitney test

and the

Ansari–Bradley test

. [NSM Chapter 5.]

246

Leptokurtic curve: See kurtosis.

Les lie matr ix mo del: A model often applied in demographic and animal population studies in

which the vector of the number of individuals of each age at time t, N

, is related to the initial

number of individuals of each age, N

, by the equation

¼ M

where M is what is known as the population projection matrix given by

M ¼

 B

v1

00 00

0 P

0  00



000 P

v1

where B

equals the number of females born to females of age x in one unit of time that

survive to the next unit of time, P

equals the proportion of females of age x at time t that

survive to time t + 1 and v is the greatest age attained. See also population growth model.

[Bulletin of Mathematical Biology, 1995, 57, 381–99.]

Lethal dose 50: The administered dose of a compound that causes death to 50% of the animals

during a speciﬁed period, in an experiment involving toxic material. [Modelling Binary

Data, 2nd edition, 2003, D. Collett, Chapman and Hall/CRC Press, London.]

Levene test: A test used for detecting heterogeneity of variance, which consists of an

analysis of

variance

applied to

residuals

calculated as the differences between observations and group

means. [ Journal of the American Statistical Association, 1974, 69, 364–7.]

Leverage poi nts: A term used in regression analysis for those observations that have an extreme

value on one or more explanatory variables. The effect of such points is to force the ﬁtted

model close to the observed value of the response leading to a small residual. See also hat

matrix, inﬂuence statistics and Cook’s distance. [ARA Chapter 10.]

vy concentrationfunction: A function Q(x; λ) of a random variable X deﬁned by the equality

QðX ; lÞ¼

sup

Prðx  X  x þ lÞ

for every l  0; QðX ; lÞ is a non-decreasing function of λ satisfying the inequalities

0  QðX ; lÞ1 for every l  0. A measure of the variability of the random variable that

is used to investigate convergence problems for sums of independent random variables.

[Concentration Functions, 1973, W. Hengarner and R. Theodorescu, Academic Press, New

York.]

vy distribution: A probability distribution, f(x), given by

f ðxÞ¼ð2pÞ



exp½

x



; 0  x

None of the moments of the distribution exist. An example of such a distribution is given in

Fig. 85. [KA1 Chapter 4.]

vy, Paul-Pierre (1886^1971): Born in Paris, Lévy was educated at the Ecole Polytechnique

between 1904 and 1906. His main work was in the calculus of probability where he

introduced the

characteristic function

, and in

Markov chains

martingales

and

game theory

Lévy died on 15 December 1971 in Paris.

247

vy process: A

stochastic process

that has independent and stationary increments and a stochasti-

cally continuous sample path. Such processes provide a natural generalization of the sum of

independent and identically distributed random variables. [Journal of Financial Economics,

2004, 71,113–141.]

Lexian d ist ri butio ns:

Finite mixture distributions

having component

binomial distributions

with

common n.[Univariate Discrete Distributions, 2005, N. I. Johnson, A. W. Kemp and

S. Kotz, Wiley, New York.]

Lexicostatistics: A term sometimes used for investigations of the evolution times of languages,

involving counting the number of cognate words shared by each pair of present-day

languages and using these data to reconstruct the ancestry of the family. [Statistique

Textuelle, 1994, L. Lebart and A. Salem, Dunod, Paris.]

Lexisdiagram: A diagram for displaying the simultaneous effects of two time scales (usually age and

calendar time) on a rate. For example, mortality rates from cancer of the cervix depend upon

age, as a result of the age-dependence of the

incidence

, and upon calendar time as a result of

changes in treatment, population screening and so on. The main feature of such a diagram is

a series of rectangular regions corresponding to a combination of two time bands, one from

each scale. Rates for these combinations of bands can be estimated by allocating failures to

the rectangles in which they occur and dividing the total observation time for each subject

between rectangles according to how long the subjects spend in each. An example of such a

diagram is given in Fig. 86.[Statistics in Medicine, 1987, 6, 449–68.]

Lexis,Wilhelm (1837^1914): Lexis’s early studies were in science and mathematics. He graduated

from the University of Bonn in 1859 with a thesis on analytic mechanics and a degree in

mathematics. In 1861 he went to Paris to study social science and his most important

statistical work consisted of several articles published between 1876 and 1880 on population

and

vital statistics

f(x)

0246810

0.0 0.1 0.2 0.3 0.4

Fig. 85 An example of a Lévy

distribution.

248