Everitt B.S. The Cambridge Dictionary of Statistics

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H aplotype analysis: The analysis of

haplotype

frequencies in one or more populations, with the

aim of establishing associations between two or more alleles, between a haplotype and a

phenotypic trait, or the genetic relationship between populations. [Statistics in Human

Genetics, 1998, P. Sham, Arnold, London.]

Hardy ^ Weinberg law: The law stating that both gene frequencies and genotype frequencies will

remain constant from generation to generation in an inﬁnitely large interbreeding population

in which mating is at random and there is no selection, migration or mutation. In a situation

where a single pair of alleles (A and a) is considered, the frequencies of germ cells carrying A

and a are deﬁned as p and q, respectively. At equilibrium the frequencies of the genotype

classes are p

(AA), 2pq(Aa) and q

(aa). [Statistics in Medicine, 1986, 5, 281–8.]

Harmonic analysis: A method of determining the period of the cyclical term S

in a

time series

of the form

¼ S

þ 

where 

represents the random ﬂuctuations of the series about S

. The cyclical term is

represented as a sum of sine and cosine terms so that X

becomes

ðA

cos ω

t þ B

sin ω

tÞþ

For certain series the periodicity of the cyclical component can be speciﬁed very accu-

rately, as, for example, in the case of economic or geophysical series which contain a strict

12-month cycle. In such cases, the coefﬁcients {A

}, {B

} can be estimated by regression

techniques. For many series, however, there may be several periodic terms present with

unknown periods and so not only the coefﬁcients {A

}, {B

} have to be estimated but

also the unknown frequencies {ω

}. The so-called

hidden periodicities

can often be

determined by examination of the

periodogram

which is a plot of I(ω) against ω where

IðωÞ¼

i¼1

cosðωtÞ

i¼1

sinðωtÞ

;

and ω ¼ 2pp=N ; p ¼ 1; 2; ...; ½N=2; N is the length of the series. Large ordinates on this

plot indicate the presence of a cyclical component at a particular frequency. As an

example of the application of this procedure Fig. 70 shows the sunspot series and the

Fig. 70 Sunspot activity and its periodogram (Reproduced from

Applications of Time Series

Analysis in Astronomy and Meteorology

by permission of the publisher, Chapman and Hall/CRC

Press London.)

199

periodogram of this series based on 281 observations. It is clear that the ordinate at

ω ¼ 2p 28=281, corresponding to a period of approximately 10 years, is appreciably

larger than the other ordinates. If several peaks are observed in the periodogram it

cannot be concluded that each of these corresponds to a genuine periodic component in

since it is possible that peaks may occur due simply to random ﬂuctuations in the noise

term 

. Various procedures for formally assessing periodic ordinates are available of

which the most commonly used are

Schuster’s test

Walker’s test

and

Fisher’s g statistic

See also spectral analysis, fast Fourier transform and window estimates. [TMS

Chapter 7.]

Harmonic mean: The reciprocal of the arithmetic mean of the reciprocals of a set of observations

; x

; ...; x

. Speciﬁcally obtained from the equation

i¼1

Used in some methods of analysing

non-orthogonal designs

. The harmonic mean is either

smaller than or equal to the arithmetic mean and the

geometric mean

. [ARA Chapter 16.]

H a rr i ngto n and Fl em i ng G

tests: A class of linear rank test for comparing two

interval

censored samples

. Useful in testing for a difference between two or more

survival curves

or for a single curve against a known alternative. [

Biometrika

, 1982, 69, 553–66.]

H arris and Stevens forecast ing: A method of making short term forecasts in a

time series

that

is subject to abrupt changes in pattern and transient effects. Examples of such series are

those arising from measuring the concentration of certain biochemicals in biological

organisms, or the concentration of plasma growth hormone. The changes are modelled

by adding a random perturbation vector having zero mean to a linearly updated parameter

vector. [

Bayesian Forecasting and Dynamic Models

, 2nd edition, 1999, M. West and

J. Harrison, Springer Verlag, New York.]

Harris walk: A

random walk

on the set of nonnegative integers, for which the matrix of

transition

probabilities

consists of zeros except for the elements;

0;1

¼ 1

j;jþ1

¼ a

1; 1  j

j;j1

¼ 1  a

Har tley, Herman Otto (191 2^1980): Born in Berlin, Hartley obtained a Ph.D. degree in

mathematics from the University of Berlin in 1933. In 1934 he emigrated to England

where he worked with

Egon Pearson

in producing the

Biometrika Tables for Statisticians

In 1953 Hartley moved to Iowa State College in the USA and from 1963 to 1977

established and ran the Institute of Statistics at Texas A&M University. Contributed to

data processing,

analysis of variance

, sampling theory and

sample surveys

. Hartley was

awarded the S. S. Wilks medal in 1973. He died on 30 December 1980 in Durham, USA.

Har tley’s test: A simple test of the equality of variances of the populations corresponding to the

groups in a

one way design

. The test statistic (if each group has the same number of

observations) is the ratio of the largest (s

largest) to the smallest (s

smallest) within

group variance, i.e.

F ¼

largest

smallest

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Critical values are available in many statistical tables. The test is sensitive to departures

from normality. See also Bartlett’s test and Box’s test.[Biometrika, 1950, 37, 308–12.]

Hat matrix : A matrix, H, arising in

multiple regression

, which is used to obtain the predicted values

of the response variable corresponding to each observed value via the equation

y ¼ Hy

The matrix H puts the ‘hats’ on the elements of y; it is a symmetric matrix and is also

idempotent

. Given explicitly in terms of

design matrix

, X as

H ¼ X ð X

XÞ

1

The diagonal elements of H are often useful diagnostically in assessing the results from

the analysis. See also Cook’s distance.[

Regression Analysis

, Volume 2, 1993, Ed.

M. S. Lewis-Beck, Sage Publications, London.]

Haugh’s test: A test for the independence of two

time series

which is based on the sum of

ﬁnitely many squares of residual cross-correlations. [

Canadian Journal of Statistics

, 1997,

25, 233–56.]

Hausd o rff di m ens io n: Synonym for fractal dimension.

Hausman misspecification test: A test for model misspeciﬁcation that in its standard version

considers one estimator

β which is consistent under both correct and incorrect model

speciﬁcation and another estimator

β which is consistent and efﬁcient under correct

speciﬁcation but inconsistent under misspeciﬁcation. The null hypothesis is that both

and

β are consistent. The test statistic used is

β 

βÞ

½covð

βÞcovð

βÞ

1

β 

βÞ

which asymptotically has a chi-square null distribution with degrees of freedom equal to

the dimension of β. A common application is in testing multilevel models with random

effects versus ﬁxed-effects models for clustered data. [Econometrica, 1978, 46, 1251–

1271.]

Hausman-Taylor estimator: An

instrumental variable estimator

for linear models for

panel

studies

where some

time-dependent covariates

and some time-invariant covariates are

endogenous in the sense of being correlated with the random intercept. [Econometrica,

1981, 49, 1377–1398.]

Hawthorne effect: A term used for the effect that might be produced in an experiment simply

from the awareness by the subjects that they are participating in some form of scientiﬁc

investigation. The name comes from a study of industrial efﬁciency at the Hawthorne Plant

in Chicago in the 1920s. [

Annals of Emergency Medicine

, 1995, 26, 590–4.]

Hazard function: The risk that an individual experiences an event (death, improvement etc.) in a

small time interval, given that the individual has survived up to the beginning of the interval.

It is a measure of how likely an individual is to experience an event as a function of time.

Usually denoted h(t), the function can be expressed in terms of the probability distribution

of the

survival times

f (t) and the

survival function

S(t), as h(t )=f (t)/S(t). The hazard function

may remain constant, increase, decrease or take on some more complex shape. The function

can be estimated as the proportion of individuals experiencing an event in an interval per unit

time, given that they have survived to the beginning of the interval, that is

hðtÞ¼

number of individuals experiencing an event in interval beginning at t

ðnumber of individuals surviving at tÞðinterval widthÞ

201

Care is needed in the interpretation of the hazard function because of both selection effects

due to variation between individuals and variation within each individual over time. For

example, individuals with a high risk are more prone to experience an event early, and

those remaining at risk will tend to be a selected group with a lower risk. This will result

in the hazard rate being ‘pulled down’ to an increasing extent as time passes. See also

survival function, bathtub curve and frailty models. [SMR Chapter 13.]

Hazard plotting: Based on the

hazard function

of a distribution, this procedure provides estimates

of distribution parameters, the proportion of units failing by a given time, percentiles of

the distribution, the behaviour of the failure rate of the units as a function of time, and

conditional failure probabilities for units at any time. [ Technometrics, 2000, 42,12–25.]

Hazard regression: A procedure for modelling the

hazard function

that does not depend on the

assumptions made in

Cox’s proportional hazards model

, namely that the log-hazard function is

an additive function of both time and the vector of covariates. In this approach,

spline functions

are often used to model the log-hazard function. [Statistics in Medicine, 1996, 15,1757–70.]

Head-banging smoother: A procedure for smoothing

spatial data

. The basic algorithm proceeds

as follows:

for each point or area whose value y

is to be smoothed, determine the N nearest

neighbours to location x

from among these N nearest neighbours, deﬁne a set of points around the point area,

such that the ‘triple’ (pair plus target point at x

) are roughly collinear. Let NTRIP be

the maximum number of such triplets

let (a

) denote the (higher, lower) of the two values in the kth pair and let

A = median{a

}, B= median{b

}

the smoothed value corresponding to y

median{A, y

, B}. [IEEE Transactions

on Geosciences and Remote Sensing, 1991, 29, 369–78.]

Heady , James Austin (191 7^20 04): Born in Kuling, China, Heady studied mathematics at

Merton College, Oxford, United Kingdom from where he graduated in 1939. In 1946 he

was appointed to start a new Department of Statistics at St. Bartholomew’s Hospital in

London, and then in 1949 became a statistician in the Social Medicine Unit of the Medical

Research Council where he remained until 1975. During this period Heady produced

a long series of publications concerned with the social and biological factors in infant

mortality. Later he became interested in the methodology of dietary surveys. Heady died

in London on November 4th, 2004.

Healthylifeexpectancy(HLAE):The average number of years that a newborn child can

expect to live in good health. The measure is useful in assessing a health system’s effective-

ness in reducing the burden of disease. See also disability adjusted life years (DALYs).

[Demographic Research, 2009, 20, 467–494.]

Healthy worker effect: The phenomenon whereby employed individuals tend to have lower

mortality rates than those unemployed. The effect, which can pose a serious problem in

the interpretation of industrial

cohort studies

, has two main components:

selection at recruitment to exclude the chronically sick resulting in low

standardized

mortality rates

among recent recruits to an industry,

a secondary selection process by which workers who become unﬁt during employ-

ment tend to leave, again leading to lower standardized mortality ratios among long-

serving employees.

[Statistics in Medicine, 1986, 5,61–72.]

202

H eckman selecti on model: A model designed to obtain correct inference for regression models

based on nonrandom or “selected” samples. The Heckman selection model contains two

equations, a linear regression model of primary interest

¼ x

β þ"

and a probit selection equation



¼ z

γ þ

determining that unit i is sampled if S



> 0 and not sampled if S



 0. The error terms "

and



are assumed to be bivariate normal with zero means, Varð"

Þ¼

and Varð

Þ¼1, and

correlation  .If ¼ 0, valid inference regarding the parameters of interest β can be

based on the substantive equation alone. However, there is “informative selection” if

 6¼ 0. Valid inference must in this case be based on joint estimation of both equations

and estimators solely based on the substantive equation would be inconsistent. See also

Selection model.[Econometrica, 1979, 47, 153–161.]

Hellinger distance: A measure of distance between two probability distributions, f (x) and g(x)

given by

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ð1 Þ

where

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

f ðxÞgðxÞ

See also Bhattacharya ’s distance. [MV2 Chapter 14.]

Hello-goodbye effect: A phenomenon originally described in psychotherapy research, but one

which may arise whenever a subject is assessed on two occasions, with some interven-

tion between the visits. Before an intervention a person may present himself/herself in

as bad a light as possible, thereby hoping to qualify for treatment, and impressing staff

with the seriousness of his/her problems. At the end of the study the person may want

to ‘please’ the staff with his/her improvement, and so may minimize any problems. The

result is to make it appear that there has been some improvement when none has

occurred, or to magnify the effects that did occur. [Journal of Clinical Psychology,

2000, 56, 853–9.]

Helmert contrast: A

contrast

often used in the

analysis of variance

, in which each level of a factor

is tested against the average of the remaining levels. So, for example, if three groups

are involved of which the ﬁrst is a control, and the other two treatment groups, the ﬁrst

contrast tests the control group against the average of the two treatments and the second

tests whether the two treatments differ. [KA1 Chapter 11.]

Helsinki declaration: A set of principles to guide clinicians on the ethics of clinical trials and other

clinical research. See also Nuremberg code.[Biostatistics: A Methodology for the Health

Sciences, 2nd edition, 2004, E. Van Belle, L. D. Fisher, P. J. Heagerty and T. S. Lumley,

Wiley, New York.]

Herbicide bioassay: A procedure for establishing a

dose-response curve

in the development of

new herbicides. As no objective death criteria can be given for plants, a graded response

such as biomass or height reduction must be considered. [Herbicide Bioassays, 1993, ed.

J. C. Streibig and P. Kudsk, CRC, Boca Raton, Florida.]

Hess, Irene (1910^2009): Born in Muhlenberg County, Kansas, Hess received a bachelor’s

degree in mathematics from Indiana University in 1931. After then studying statistics at

the University of Michigan, she joined the US. Bureau of Labour Statistics in 1941 and

then joined the US Census Bureau where she implemented the new ideas of

probability

203

sampling

. Later Hess became director of the Institute of Social Research at the University

of Michigan and was responsible for teaching several generations of sampling statisti-

cians how to translate theory into practice. She died on the 5th July, 2009, in Ann Arbor,

USA.

Herfindahl index: An index of industry concentration given by

H ¼

i¼1

½s

=S

where S is the combined size of all ﬁrms in an industry (scaled in terms of employees,

sales, etc.) and s

is the size of the ith ﬁrm and there are n ﬁrms ranked from largest to

smallest. Concentration increases with the value of the index. [Statistical Science, 1994, 9,

94–108.]

Hermite functions: Functions 

ðxÞ given by



ðxÞ¼ð2

j!p



expð

ÞH

ðxÞ

where H

ðxÞ is the Hermite polynomial deﬁned by

ðxÞ¼ð1Þ

expðx

Þd

fexpðx

Þg=dx

[KA1 Chapter 6.]

H e rm ite po lyno m ia ls: See Hermite functions.

Herzberg’sformula: See shrinkage formulae.

Hessian matrix: The matrix of second-order partial derivatives of a function f of n independent

variables x

, x

, ..., x

with respect to those variables. The element in the ith row and the

jth column of the matrix is therefore ∂

y/∂x

∂x

. In statistics the matrix most often occurs

for the

log-likelihood

as a function of a set of parameters. The inverse of the matrix then

gives standard errors and covariances of the

maximum likelihood estimates

of the

parameters. The Hessian is essentially the observed information matrix in contrast to the

Fisher information matrix

where expectations are taken. [MVA Chapter 6.]

Heterogeneous: A term used in statistics to indicate the inequality of some quantity of interest

(usually a variance) in a number of different groups, populations, etc. See also homo-

geneous. [MV1 Chapter 3.]

Heteroscedasticity : A random variable is heteroscedastic if its variance depends on the values

of another variable. The opposite case is called homoscedastic.

Hettmansperger^McKean test: A

distribution free method

for assessing whether speciﬁc

subsets of the regression parameters in a

multiple regression

are zero. [NSM Chapter 9.]

Heuristic computer program: A computer program which attempts to use the same sort of

selectivity in searching for solutions that human beings use.

Heywood cases: Solutions obtained when using

factor analysis

in which one or more of the

variances of the

speciﬁc variates

become negative. [MV2 Chapter 12.]

H idden Ma rko v models: An extension of

ﬁnite mixture models

which provides a ﬂexible

class of models exhibiting dependence and a possibly large degree of variability. [Hidden

Markov and Other Models for Discrete Valued Time Series , 1997, I. L. MacDonald and

W. Zucchini, Chapman and Hall/CRC Press, London.]

204

Hidden periodicities: See harmonic analysis.

Hidden time effects: Effects that arise in data sets that may simply be a result of collecting the

observations over a period of time. See also cusum. [SMR Chapter 7.]

H ierarchical clusteri ng: See agglomerative hierarchical clustering.

Hierarchical de sign: Synonym for nested design.

H ierarchical li kelihood: A somewhat controversial approach to estimating the parameters of

generalized linear mixed models

that treats random effects as parameters. [Generalized

Linear Models with Random Effects: Uniﬁed Analysis via H-Likelihood, 2006, Y. Lee,

J. A. Nelder and Y. Pawitan, Chapman and Hall/CRC, Boca Raton.]

Hierarchical models: A series of models for a set of observations where each model results from

adding, deleting or constraining parameters from other models in the series. Alternatively,

synonym for

multilevel model

Higgins’slaw: A ‘law’ that states that the prevalance of any condition is inversely proportional

to the number of experts whose agreement is required to establish its presence.

High breakdown methods : Methods that are designed to be resistant to even multiple severe

outliers

. Such methods are an extreme example of

robust statistics

.[Computational

Statistics, 1996, 11, 137–46.]

High-dimensional data: A term used for data sets which are characterised by a very large

number of variables and a much more modest number of observations. In the 21

century

such data sets are collected in many areas, for example, text/web

data mining

and

bioinformatics

. The task of extracting meaningful statistical and biological information

from such data sets present many challenges for which a number of recent methodological

developments, for example,

sure screening methods, lasso

, and

Dantzig selector

, may be

helpful. [Nature Reviews Cancer, 2008, 8,37–49.]

H i gher criticism: A

multiple-comparison test

concept arising from the situation where there are

many independent tests of signiﬁcance and interest lies in rejecting the joint null hypothesis.

A fraction of the observed signiﬁcances at a given α-level is compared to the expected

fraction under the joint null and the difference between the two quantities is standardized

and this standardized value then maximised over a range of signiﬁcance levels to give the

higher criticism statistic. This statistic is effective in assessing whether n normal means are

all zero against the alternative that a small fraction is nonzero. [Annals of Statistics, 2004,

32, 962–994.]

Highest post er ior density regi on: See Bayesian conﬁdence interval.

High-throughput data: Data produced by modern biological assays and experiments which

supply thousands of measurements per sample and where the sheer amount of data increases

the need for better models to enhance inference. See also high-dimensional data.

[Bioinformatics, 2007, 23, 906–909.]

Hill, Austin Bradford (1897^1991 ): Born in Hampstead, London, Hill served as a pilot in World

War I. Contracting tuberculosis prevented him taking a medical qualiﬁcation, as he would

have liked, so instead he studied for a London University degree in economics by corre-

spondence. His interest in medicine drew him to work with the Industrial Fatigue Research

Board, a body associated with the Medical Research Council. He improved his knowledge

of statistics at the same time by attending

Karl Pearson

’s lectures at University College. In

205

1933 Hill became Reader in Epidemiology and Vital Statistics at the recently formed

London School of Hygiene and Tropical Medicine (LSHTM). He was an extremely suc-

cessful lecturer and a series of papers on Principles of Medical Statistics published in

the Lancet in 1937 were almost immediately reprinted in book form. The resulting text

remained in print until its ninth edition in 1971. In 1947 Hill became Professor of

Medical Statistics at the LSHTM and Honorary Director of the MRC Statistical Research

Unit. He had strong inﬂuence in the MRC in particular in their setting up of a series of

randomized controlled clinical trials, the ﬁrst involving the use of streptomycin in pulmo-

nory tuberculosis. Hill’s other main achievment was his work with Sir Richard Doll

on a case-control study of smoking and lung cancer. Hill received the CBE in 1951, was

elected Fellow of the Royal Society in 1954 and was knighted in 1961. He received the

Royal Statistical Society’s Guy medal in gold in 1953. Hill died on 18 April, 1991, in

Cumbria, UK.

Hill-climbing algorithm: An algorithm used in those techniques of

cluster analysis

which seek

to ﬁnd a partition of n individuals into g clusters by optimizing some numerical index of

clustering. Since it is impossible to consider every partition of the n individuals into g

groups (because of the enormous number of partitions), the algorithm begins with some

given initial partition and considers individuals in turn for moving into other clusters,

making the move if it causes an improvement in the value of the clustering index. The

process is continued until no move of a single individual causes an improvement. See

also K-means cluster analysis.[Cluster Analysis, 4th edition, 2001, B. S. Everitt,

S. Landau and M. Leese, Arnold, London.]

Hinge: A more exotic (but less desirable) term for

quartile

Histogram: A graphical representation of a set of observations in which class frequencies are

represented by the areas of rectangles centred on the class interval. If the latter are all

equal, the heights of the rectangles are also proportional to the observed frequencies.

A histogram of heights of elderly women is shown in Fig. 71. [SMR Chapter 2.]

Fig. 71 A histogram of heights of elderly women.

206

Historical controls: A group of patients treated in the past with a standard therapy, used as the

control group for evaluating a new treatment on current patients. Although used fairly

frequently in medical investigations, the approach is not to be recommended since possible

biases, due to other factors that may have changed over time, can never be satisfactorily

eliminated. See also literature controls. [SMR Chapter 15.]

Historical prospective studies: A ‘prospective study’ in which the cohort to be investigated

and its subsequent disease history are identiﬁed from past records, for example, from

information of an individual’s work history. [American Journal of Individual Medicine,

1983, 4, 651–7.]

Hit rate: A term occasionally used for the proportion of correct classiﬁcations in a

discriminant analysis

H-likelihood: See hierarchical likelihood.

HLM: Software for the analysis of

multilevel models

. See also MLwiN. [Scientiﬁc Software

International Inc., 7383 N. Lincoln Ave., Suite 100, Lincolnwood, IL 60712, USA.]

Hodges^Lehmann estimator: An estimator for the location difference of two uncensored data

samples, y

; y

; ...; y

and y

þ1

; ...; y

; n ¼ n

þ n

, given by

¼ medianfðy

 y

Þ : i ¼ 1; ...; n

; j ¼ n

þ 1; ...; ng

[Nonparametric Statistical Methods, 2nd edition, 1999, M. Hollander and D. A. Wolfe,

Wiley, New York.]

Hoeffding test: A

distribution free method

for testing for the independence of two random

variables X and Y, that is able to detect a broader class of alternatives to independence

than is possibly by using sample correlation coefﬁcients. [NSM Chapter 8.]

Hoeffding’sinequality:A result from probability theory that gives an upper bound on the

probability for the sum of random variables to deviate from its expected value. [Journal of

the American Statistical Association, 1963, 58,13–30.]

Hoeffding, Wassily (191 4^1991 ): Born in Mustamaki, Finland, Hoeffding began his university

education studying economics but quickly switched to mathematics eventually earning

a Ph.D. degree from Berlin University in 1940 with a dissertation on nonparametric

measures of association and correlation. He emigrated to the USA in 1946 settling in

Chapel Hill, North Carolina. Hoeffding made signiﬁcant contributions to

sequential

analysis

, statistical

decision theory

and

central limit theorems

. He died on 28 February

1991 in Chapel Hill.

Hogbe n, Lan celot (1895^1975): Born in Southsea, Hampshire, Hogben studied at Cambridge.

He was a man of remarkable intellect and a great communicator who made important

and original contributions in both theoretical and applied science. Hogben held academic

appointments in zoology in England, Scotland, Canada and South Africa before becoming

Professor of Zoology at Birmingham from 1941 to 1947, and then Professor of Medical

Statistics at the same university from 1947 to 1961. During his career he held ﬁve

university chairs. Best remembered for his popular book, Mathematics for the Millions.

Hogben died on 22 August 1975.

Holdout estimate: A method of estimating the

misclassiﬁcation rate

in a

discriminant analysis

The data is split into two mutually exclusive sets and the classiﬁcation rule derived from

one and its performance evaluated on the other. The method makes inefﬁcient use of the

data (using only part of them to construct the classiﬁcation rule) and gives a pessimistically

207

biased estimate of the derived rule’s misclassiﬁcation rate. [ Pattern Recognition,

A Statistical Approach, 1982, P. A. Devijver and J. Kittler, Prentice Hall Inc., London.]

H oldo ver effect: Synonym for carryover effect.

Holgate, Philip (1934^1993): Born in Chesterﬁeld, UK, Holgate attended Newton Abbot

Grammar School and then entered Exeter University in 1952 to read mathematics. He

began his career as a school teacher, teaching mathematics and physics at Burgess Hill

School, Hampstead. Holgate’s career as a statistician began in 1961 when he joined the

Scientiﬁc Civil Service at Rothamsted. In 1967 he took up a lecturer post at Birkbeck

College, London, progressing to professor in 1970. Holgate’s statistical work was primarily

in the area of stochastic processes in biology, and he also made seminal contributions to

non-associative algebras.

Hollander test: A

distribution free method

for testing for bivariate symmetry. The null hypothesis

tested is the two random variables X and Y are exchangeable. [NSM Chapter 3.]

Homogeneous: A term that is used in statistics to indicate the equality of some quantity of interest

(most often a variance), in a number of different groups, populations, etc. See also hetero-

geneous. [SMR Chapter 7.]

Horseshoe shape: See seriation.

Horvitz^Thompson estimator: An unbiased estimator of the total size of a population when

sampling is with or without replacement from a ﬁnite population and sampling unit i has

probability p

of being included in the sample. The estimator does not depend on the number

of times a unit may be selected, since each unit is utilized only once in the formula

^τ ¼



i¼1

where  is the effective sample size (the number of distinct units in the sample) and y

is the

number of individuals, animals or species observed in the ith sampling unit. If the proba-

bility that both unit i and unit j are included in the sample is p

and all these joint

inclusion

probabilities

are greater than zero then an unbiased estimator of the variance of ^τ is

varð^τÞ¼



i¼1





þ 2



i¼1





See also Hansen–Hurwitz estimator.[Experimental and Ecological Statistics, 2003, 10,

115–127.]

Hosmer-Lemeshow test: A goodness-of-ﬁt test used in

logistic regression

, particularly when

there are continuous covariates. Units are divided into

deciles

based on predicted proba-

bilities and a 2 × 10 contingency table of expected and observed frequencies for the

deciles obtained. The Hosmer-Lemeshow test statistic is the Pearson chi-square statistic

for this contingency table and has an approximate

chi-squared distribution

with 8 degrees

of freedom if the model is correct. The test has limited power. [Applied Logistic

Regression, 2nd edn, 2000, D. W. Hosmer and S. Lemeshow, Wiley, New York.].

Hospital controls: See control group.

Hot deck: A method widely used in surveys for imputing missing values. In its simplest form the

method involves sampling with replacement m values from the sample respondents A

an item y, where m is the number of non-respondents to the item and r is the number of

respondents. The sampled values are used in place of the missing values. In practice, the

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