Everitt B.S. The Cambridge Dictionary of Statistics

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

¼ y

 αy

t1

and x



¼ x

 αx

t1

an estimate of β can be obtained from

ordinary least

squares

of y



and x



. The unknown parameter α can be calculated from the

residuals

from the

regression, leading to an iterative process in which a new set of transformed variables are

calculated and thus a new set of regression estimates and so on until convergence.

[Communications in Statistics, 1993, 22, 1315–33.]

Codominance: The relationship between the

genotype

at a locus and a

phenotype

that it in ﬂuences. If

individuals with heterozygote (for example, AB) genotype is phenotypically different from

individuals with either homozygous genotypes (AA and BB), then the genotype-phenotype

relationship is said to be codominant. [Statistics in Human Genetics, 1998, P. Sham, Arnold,

London.]

Coefficientofalienation: A name sometimes used for 1  r

, where r is the estimated value of the

correlation coefﬁcient of two random variables. See also coefﬁcient of determination.

Coefficientofconcordance: A coefﬁcient used to assess the

agreement

among m raters ranking

n individuals according to some speciﬁc characteristic. Calculated as

W ¼ 12S=½m

ðn

 nÞ

where S is the sum of squares of the differences between the total of the ranks assigned to

each individual and the value mðn þ1Þ=2. W can vary from 0 to 1 with the value 1 indicating

perfect agreement. [Quick Statistics, 1981, P. Sprent, Penguin Books, London.]

Coefficientofdetermination: The square of the correlation coefﬁcient between two variables.

Gives the proportion of the variation in one variable that is accounted for by the other. [ARA

Chapter 7.]

Coeffic i ent of rac ia l li k eness (C RL ) : Developed by

Karl Pearson

for measuring resemblances

between two samples of skulls of various origins. The coefﬁcient is deﬁned for two samples I

and J as:

CRL ¼

k¼1

ðX

 X

ðs

Þþðs

()



where

stands for the sample mean of the kth variable for sample I and s

stands for the

variance of this variable. The number of variables is p.[American Journal of Physical

Anthropology, 2005, 23, 101–109.]

Coeffic ient o f va r iat io n: A measure of spread for a set of data de ﬁned as

100  standard deviation=mean

Originally proposed as a way of comparing the variability in different distributions, but

found to be sensitive to errors in the mean. [KA1 Chapter 2.]

Coefficientsignpredictionmethods: Methods used for predicting the signs of the coefﬁcients

in a regression model; this is useful when the main purpose in selecting a model is

interpretation, because the sign of a coefﬁcient is often of primary importance for this

task. [Journal of the Royal Statistical Society, Series B, 2007, 69, 447–461.]

Coherence: A term most commonly used in respect of the strength of association of two

time series

Figure 37, for example, shows daily mortality and SO

concentration time series in London

during the winter months of 1958, with an obvious question as to whether the pollution was

in any way affecting mortality. The relationship is usually measured by the time series

analogue of the correlation coefﬁcient, although the association structure is likely to be more

complex and may include a leading or lagging relationship; the measure is no longer a single

number but a function, ρ(ω), of frequency ω.Deﬁned explicitly in terms of the Fourier

transforms f

(ω), f

(ω) and f

(ω) of the

autocovariance functions

of X

and Y

and their

cross-

covariance function



ðωÞ¼

ðωÞj

ðωÞf

ðωÞ

The squared coherence is the proportion of variability of the component of Y

with frequency

ω explained by the corresponding X

component. This corresponds to the interpretation of

the squared

multiple correlation coefﬁcient

multiple regression

. See also multiple time

series. [TMS Chapter 8.]

Cohort: See cohort study.

Cohort component method: A widely used method of forecasting the age- and sex-speciﬁc

population to future years, in which the initial population is stratiﬁed by age and sex and

projections are generated by application of survival ratios and birth rates, followed by an

additive adjustment for net migration. The method is widely used because it provides

detailed information on an area’s future population, births, deaths, and migrants by age,

sex and race, information that is uesful for many areas of planning and public administration.

[Demography, 1977, 14, 363–8.]

Cohort study: An investigation in which a group of individuals (the cohort) is identiﬁed and

followed prospectively, perhaps for many years, and their subsequent medical history

recorded. The cohort may be subdivided at the onset into groups with different character-

istics, for example, exposed and not exposed to some risk factor, and at some later stage a

comparison made of the incidence of a particular disease in each group. See also prospective

study. [SMR Chapter 5.]

Coincidences: Surprising concurrence of events, perceived as meaningfully related, with no appa-

rent causal connection. Such events abound in everyday life and are often the source of some

amazement. As pointed out by

Fisher

, however, ‘the one chance in a million will undoubt-

edly occur, with no less and no more than its appropriate frequency, however surprised we

Fig. 37 Time series for daily mortality and sulphur dioxide concentration in London during the winter

months of 1958.

may be that it should occur to us’. See also synchronicity [Chance Rules, 2nd edn, 2008,

B. S. Everitt, Springer, New York.]

Co in t egrat io n: Avector of

nonstationary time series

is said to be cointegrated if a linear combination

of the individual series is stationary. Facilitates valid testing of the hypothesis that there is a

relationship between the nonstationary series. [Econometrica, 1987, 55, 251–276.]

Co llaps ingcat egori es: A procedure often applied to

contingency tables

in which two or more row

or column categories are combined, in many cases so as to yield a reduced table in which

there are a larger number of observations in particular cells. Not to be recommended in

general since it can lead to misleading conclusions. See also Simpson’s paradox.

[Multivariable Analysis, 1996, A. R. Feinstein, New York University Press, New Haven.]

Collective risk models: Models applied to insurance portfolios which do not make direct reference

to the risk characteristics of the individual members of the portfolio when describing the

aggregate claims experience of the whole portfolio itself. To model the total claims of a

collection of risks over a ﬁxed period of time in the future, the collective approach

incorporates both claim frequency and claim severity components into the probability

distribution of the aggregate. [Encyclopedia of Quantitative Risk Analysis and

Assessment, Volume 1, 2008, eds. E. L. Melnick and B. S. Everitt, Wiley, Chichester.]

Collector’sproblem: A problem that derives from schemes in which packets of a particular brand

of tea, cereal etc., are sold with cards, coupons or other tokens. There are say n different

cards that make up a complete set, each packet is sold with one card, and a priori this card is

equally likely to be any one of the n cards in the set. Of principal interest is the distribution of

N, the number of packets that a typical customer must buy to obtain a complete set. It can be

shown that the

cumulative probability distribution

of N ; F

ðr; nÞ is given by

ðr; nÞ¼

t¼0

ð1Þ

n!ðn tÞ

t!ðn tÞ!

; r  n

The expected value of N is

E ð N Þ¼n

t¼1

[The American Statistician, 1998, 52, 175–80.]

Collinearity: Synonym for multicollinearity.

Collision test: A test for randomness in a sequence of digits. [The Art of Computer Programming,

2nd edition, 1998, D. K. Knuth, Addison-Wesley, Reading, Maine.]

Comb i n i ng cond iti o na l ex pectatio ns and resi du als pl ot (CE RES ) : A generalization of

the partial residual plot that is often useful for identifying curvature in regression models. If,

for example, it is suspected that for one of the explanatory variables, x

, a term in x

needed, then this term is added to the usual residuals from the regression model and the result

is plotted against x

. Figure 38 shows an example from studying the relationship between

pollution (sulphur dioxide level) and annual rainfall and wind speed in a number of cities in

the USA. The plot for wind speed showing the linear and locally weighted ﬁts indicates that

a quadratic term for this variable might be needed in the model. [Linear Regression Analysis,

2003, G. A. F. Seber and A. J. Lee, Wiley, New York]

Combining p-values: The combining of a number of p-values obtained from testing hypotheses

that are all related to a common question. In a

clinical trial

, for example, there may be several

outcome measures that are used in trying to determine whether or not a treatment is effective;

combining the p-values from the tests on each outcome may give stronger evidence than is

obtained from considering each p-value individually. The most common method of combin-

ing p-values is the Fisher combination test where the combined test statistic is

¼2

i¼1

logðp

Þ where p

; i ¼ 1; ...k, are the separate p-values; C

is tested as a

chi-square with 2k degrees of freedom. See also meta-analysis.[Computational Statistics

and Data Analysis, 2004, 47, 467–485.]

Comingling analysis: Essentially a synonym for the ﬁtting of ﬁnite mixture distributions.

Commensurat e va ri ab les: Variables that are on the same scale or expressed in the same units, for

example, systolic and diastolic blood pressure.

Common cause fail ures (CC F): Simultaneous failures of multiple components due to a common

cause. A cause can be external to the components, or it can be a single failure that propagates

to other components in a cascade. Even if such events are rare compared to single failures of

a component, they can dominate system unreliability or unavailability. A variety of models

for predicting CCFs have been suggested. [Reliability Engineering and System Safety, 2005,

90, 186–195.]

Common factor: See factor analysis .

Common facto r var iance: A term used in

factor analysis

for the proportion of the variance of a

manifest variable

that is attributed to the

common factors

. [MV2 Chapter 12.]

Commonprincipalcompon ents analysis: A generalization of

principal components analysis

to several groups. The model for this form of analysis assumes that the

variance–covariance

matrices

, S

of k populations have identical

eigenvectors

, so that the same orthogonal

matrix diagonalizes all S

simultaneously. A modiﬁcation of the model, partial common

Wind speed

6789101112

CERES residual

280260240220200

Fig. 38 CERES plot for wind speed in multiple regression model that includes wind speed and

precipitation with the dependent variable being sulphur dioxide concentration.

principal components, assumes that only r out of q eigenvectors are common to all 

while

the remaining q  r eigenvectors are speciﬁc in each group. [MV1 Chapter 4.]

Co m munality: Synonym for common factor variance.

Community controls: See control group.

Commun ity i nterventio n study: An

intervention study

in which the experimental unit to be

randomized to different treatments is not an individual patient or subject but a group of

people, for example, a school, or a factory. See also cluster randomization.[Statistics in

Medicine, 1996, 15, 1069–92.]

Co-morbidity: The potential co-occurrence of two disorders in the same individual, family etc.

[Statistics in Medicine, 1995, 14, 721–33.]

Co m parativ e ca lib ra t io n: The statistical methodology used to assess the calibration of a number

of instruments, each designed to measure the same characteristic on a common group of

individuals. Unlike a normal calibration exercise, the true underlying quantity measured is

unobservable. [Statistics in Medicine, 1997, 16, 1889–95.]

Comparative exposure rate: A measure of association for use in a

matched case–control study

deﬁned as the ratio of the number of case–control pairs, where the case has greater exposure

to the risk factor under investigation, to the number where the control has greater exposure.

In simple cases the measure is equivalent to the

odds ratio

or a weighted combination of odds

ratios. In more general cases the measure can be used to assess association when an odds

ratio computation is not feasible. [Statistics in Medicine, 1994, 13, 245–60.]

Comparative trial: Synonym for controlled trial.

Co m pari so n g r o u p: Synonym for control group.

Comparison wise error rate: Synonym for per-comparison error rate.

Co m partment models: Models for the concentration or level of material of interest over time. A

simple example is the washout model or one compartment model given by

E ½Y ðtÞ ¼ α þ β expðγtÞ; ðγ

0Þ

where Y(t) is the concentration at time t  0[Compartmental Analysis in Biology in

Medicine, 1972, J. A. Jacquez, Elsevier, New York.]

Compensatory equalization: A process applied in some

clinical trials

and intervention studies,

in which comparison groups not given the perceived preferred treatment are provided with

compensations that make these comparison groups more equal than originally planned.

[Critically Evaluating the Role of Experiments in Program Evaluation, 1994, ed. K. J.

Conrad, Jossey-Bass, San Francisco.]

Competingrisks: A term used particularly in

survival analysis

to indicate that the event of interest

(for example, death), may occur from more than one cause. For example, in a study of

smoking as a risk factor for lung cancer, a subject who dies of coronary heart disease is no

longer at risk of lung cancer. Consequently coronary heart disease is a competing risk in this

situation. [Statistics in Medicine, 1993, 12, 737–52.]

Comp lementary events:

Mutually exclusive

events A and B for which

PrðAÞþPrðBÞ¼1

where Pr denotes probability.

Comp lementary l og ^log tra nsfo rmati o n: A transformation of a proportion, p, that is often a

useful alternative to the

logistic transformation

. It is given by

y ¼ ln½lnð1  pÞ

This function transforms a probability in the range (0,1) to a value in (1; 1), but unlike

the logistic and

probit transformation

it is not symmetric about the value p = 0.5. In the

context of a

bioassay

, this transformation can be derived by supposing that the tolerances of

individuals have the

Gumbel distribution

. Very similar to the logistic transformation when p

is small. [Modelling Survival Data in Medical Research, 2nd edn, 2003, D. Collett,

Chapman and Hall/CRC, London.]

Complete case analysis: An analysis that uses only individuals who have a complete set of

measurements. An individual with one or more missing values is not included in the

analysis. When there are many individuals with missing values this approach can reduce

the effective sample size considerably. In some circumstances ignoring the individuals with

missing values can bias an analysis. See also available case analysis and missing values.

[Journal of the American Statistical Association, 1992, 87, 1227–37.]

Complete ennumeration: Synonym for census.

Complete e stimator: A weighted combination of two (or more) component estimators. Mainly

used in sample survey work. [Journal of the American Statistical Association, 1963, 58,

454–67.]

Complete li nkage cl uster analysis: An

agglomerative hierarchical clustering method

in which

the distance between two clusters is deﬁned as the greatest distance between a member of

one cluster and a member of the other. The between group distance measure used is

illustrated in Fig. 39. [MV2 Chapter 10.]

Completely randomized design: An experimental design in which the treatments are allocated

to the experimental units purely on a chance basis.

Completeness: A term applied to a statistic t when there is only one function of that statistic that can

have a given expected value. If, for example, the one function of t is an unbiased estimator of

a certain function of a parameter, θ, no other function of t will be. The concept confers a

uniqueness property upon an estimator. [Kendall’s Advanced Theory of Statistics, Volume

2A, 6th ed., 1999, A. Stuart, K. Ord and S. Arnold, Arnold, London.]

Complete spatial randomness: A

Poisson process

in the plane for which:

the number of events N(A) in any region A follows a

Poisson distribution

with mean

ljAj;

given N(A)=n, the events in A form an independent random sample from the uniform

distribution on A.

Cluster A

Cluster B

Fig. 39 Complete linkage distance for two clusters

A and B.

Here |A| denotes the area of A, and λ is the mean number of events per unit area. Often used

as the standard against which to compare an observed spatial pattern. [Spatial Data Analysis

by Example, Volume 1, 1985, G. Upton and B. Fingleton, Wiley, New York.]

Complex Bingham distributions: An exponential family with a

sufﬁcient statistic

which is

quadratic in the data. Such distributions provide tractable models for landmark-based shape

analysis. [Journal of the Royal Statistical Society, Series B, 1994, 56, 285–299.]

Co m p lex su rvey da ta: Sample survey data obtained by using

cluster sampling

, unequal sampling

probabilities and stratiﬁcation instead of

simple random sampling

.[Analysis of Complex

Surveys, 1989, C. J. Skinner, D. Holt and T. M. F. Smith, eds., Wiley, New York.]

Compliance: The extent to which participants in a

clinical trial

follow the trial protocol, for example,

following both the intervention regimen and trial procedures (clinical visits, laboratory

procedures and ﬁlling out forms). For clinical trial investigators it is an inescapable fact of

life that the participants in their trials often make life difﬁcult by missing appointments,

forgetting to take their prescribed treatment from time to time, or not taking it at all but

pretending to do so. Because poor participant compliance can adversely affect the outcome

of a trial, it is important to use methods both to improve and monitor the level of compliance.

See also complier average causal effect and intention-to-treat analysis.[Clinical Trials in

Psychiatry, 2nd edn, 2008, B. S. Everitt and S. Wessely, Wiley.]

Complier average causal effect (CAC E ): The treatment effect among true compliers in a

clinical trial

. For a suitable response variable, the CACE is given by the mean difference in

outcome between compliers in the treatment group and those controls who would have

complied with treatment had they been randomized to the treatment group. The CACE may

be viewed as a measure of ‘efﬁcacy’ as opposed to ‘effectiveness’.[Review of Economic

Studies, 1997, 64, 555–74.]

Co m ponen tbarchart: A

bar chart

that shows the component parts of the aggregate represented by

the total length of the bar. The component parts are shown as sectors of the bar with lengths

in proportion to their relative size. Shading or colour can be used to enhance the display.

Figure 40 gives an example.

Component -pl us-residua l plot: Synonym for partial-residual plot.

Composite estimators: Estimators that are a weighted combination of two or more component

estimators. Often used in sample survey work. [American Journal of Physical Anthropology,

2007, 133, 1028–1034.]

Composite hypothesis: A hypothesis that speciﬁes more than a single value for a parameter. For

example, the hypothesis that the mean of a population is greater than some value.

Composite indicators: Indicators of multidimensional concepts, for example, sustainability, sin-

gle market policy and globalization, which cannot be captured by a single indicator. Such

indicators (or indices) are formed from combining a number of sub-indicators on the basis of

an underlying model of the multidimensional concept that is to be measured. [Handbook on

Constructing Composite Indicators: Methodology and User Guide, 2008, M. Nardo, M.M

Saisana, A. Saltelli and S. Tarantola, OECD publication.]

Co m pos it e l i ke lihoods: Pseudo-likelihoods constructed by pooling

likelihood

components, with

each component corresponding to a marginal or conditional event. Such likelihoods are used

to counter the problem of large computational demands produced in particular by the need to

evaluate integrals in many dimensions. [Biometrika, 2005, 92, 519–528.]

Composite sampling: A procedure whereby a collection of multiple sample units are combined in

their entirety or in part, to form a new sample. One or more subsequent measurements are

taken on the combined sample, and the information on the sample units is lost. An example

Dorfman’s scheme

. Because composite samples mask the respondent’s identity their use

may improve rates of test participation in senstive areas such as testing for HIV.

[Technometrics, 1980, 22, 179–86.]

Compositi onal data: A set of observations, x

; x

; ...; x

for which each element of x

is a

proportion and the elements of x

are constrained to sum to one. For example, a number of

blood samples might be analysed and the proportion, in each, of a number of chemical

elements recorded. Or, in geology, percentage weight composition of rock samples in terms

of constituent oxides might be recorded. The appropriate

sample space

for such data is a

simplex and much of the analysis of this type of data involves the

Dirichlet distribution

[MV2 Chapter 15.]

Compound bino mi al distr i buti o n: Synonym for beta binomial distribution

Compounddecisi o n pro bl em: The simultaneous consideration of n decision problems that have

an identical formal structure. [Annals of Statistics, 1993, 21, 736–745.]

Compound distributi o n: A type of probability distribution arising when a parameter of a

distribution is itself a random variable with a corresponding probability distribution.

See, for example,

beta-binomial distribution

. See also contagious distribution. [KA1

Chapter 5.]

Compound symmetry: The property possessed by a

variance–covariance matrix

of a set of

multivariate data

when its main diagonal elements are equal to one another, and additionally

its off-diagonal elements are also equal. Consequently the matrix has the general form;

Not good

Fairly good

Good

Southampton Swindon Jersey Guernsey

150010005000

Number of people

Fig. 40 Component bar

chart showing subjective

health assessment in four

regions.

 ¼











 



 

where ρ is the assumed common correlation coefﬁcient of the measures. Of most impor-

tance in the analysis of

longitudinal data

since it is the correlation structure assumed by the

random intercept model

often used to analyse such data. See also Mauchly test. [MV2

Chapter 13.]

Computatio nal compl exity: The number of operations of the predominant type in an algorithm.

Investigations of how computational complexity increases with the size of a problem are

important in keeping computational time to a minimum. [Communications of the ACM,

1983, 26, 400–8.]

Co m p ut e r-ai ded d i agnos is: Computer programs designed to support clinical decision making.

In general, such systems are based on the repeated application of

Bayes’ theorem

. In some

cases a reasoning strategy is implemented that enables the programs to conduct clinically

pertinent dialogue and explain their decisions. Such programs have been developed in a

variety of diagnostic areas, for example, the diagnosis of dyspepsia and of acute abdominal

pain. See also expert system.[Biostatistics, A Methodology for the Health Sciences, 2nd

edn, 2004, G. Van Belle, L. D. Fisher P. J. Heagerty and T. S. Lumley, Wiley, New York.]

Computer al gebra: Computer packages that permit programming using mathematical expressions.