Everitt B.S. The Cambridge Dictionary of Statistics

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Two-stage least squares: A way of implementing the

instrumental variable estimator

. In the ﬁrst

stage, each endogenous covariate is regressed on all exogenous variables in the model,

including both exogenous covariates in the equation of interest and excluded instruments.

The predicted values from these regressions are then obtained. In the second stage, the

regression model of interest is estimated as usual, except that each endogenous covariate

is replaced with the predicted values from the ﬁrst stage. [Microeconometrics, 2005,

A. C. Cameron and P. K. Trivedi, Cambridge University Press, Cambridge.]

Two-stage sampling: Sampling of units from a population where the ﬁrst stage consists of

sampling clusters of units from the population and the second stage consists of sampling

units from each cluster. [Practical Methods for Design and Analysis of Complex Surveys,

2nd edition, 2004, R. Lehtonen and E. Pahkinen, Wiley, New York.]

Two-stage stoppingrule: A procedure sometimes used in

clinical trials

in which results are ﬁrst

examined after only a fraction of the planned number of subjects in each group have

completed the trial. The relevant test statistic is calculated and the trial stopped if the

difference between the treatments is signiﬁcant at Stage-1 level α

. Otherwise, additional

subjects in each treatment group are recruited, the test statistic calculated once again and the

groups compared at Stage-2 level α

, where α

and α

are chosen to give an overall level of α.

[Biometrics, 1984, 40, 791–5.]

T wo-way classification: The classiﬁcation of a set of observations according to two criteria, as,

for example, in a

contingency table

constructed from two variables.

Type H matrix: A

variance–covariance matrix

Σ having the form

S ¼

l þ 2α

þ α

 α

þ α

l þ2α

 α

þ α

 l þ 2α

where α

; α

; ...; α

and λ are unknown parameters. Such a structure is a necessary and

sufﬁcient condition for the F-tests used in a univariate analysis of variance of repeated

measures designs to be valid.

Compound symmetry

is a special case of this type with

¼ α

¼¼α

. See also sphericity.

Type I error: The error that results when the null hypothesis is falsely rejected. [SMR Chapter 8.]

Ty p e II e r ro r : The error that results when the null hypothesis is falsely accepted. [SMR Chapter 8.]

T ype III error: It has been suggested by a number of authors that this term be used for identifying the

poorer of two treatments as the better. [American Journal of Public Health, 1999, 89,

1175–80.]

439

UK retail prices index: The average measure of change in the prices of goods and services bought

for the purpose of consumption by the vast majority of households in the UK. [The Retail

Prices Index Technical Manual, 1998, M. Baxter, The Stationery Ofﬁce, London.]

Ultrametric i nequality: See ultrametric tree.

Ultrametric tree: An

additive tree

in which each terminal node is equidistant from some speciﬁc

node called the root of the tree. Each level of such a tree deﬁnes a partition of the objects and

in terms of a series of partitions, the distance between two terminal nodes i and j can be

deﬁned as the height corresponding to the smallest subset containing both i and j. These

distances, d

, satisfy the ultrametric inequality

 max½d

; d



An example is shown in Fig. 141.[Tree Models of Similarity and Association, 1996, J. E.

Corter, Sage University Paper 112, Sage Publications, London.]

U mbrel la ordering: A commonly observed response pattern in a one-factor design with ordered

treatment levels, in which the response variable increases with an increase in treatment level

up to a point, then decreases with further increase in the treatment level. [Journal of the

American Statistical Association, 1981, 76, 175–81.]

U mbrel la problem: A problem, often used to demonstrate the principles of decision making under

uncertainty, which refers to the situation in which an individual must decide whether to take

an umbrella in the face of uncertainty concerning whether it will rain today. [The American

Statistician, 1987, 41, 187–9.]

Unanimity rule: A requirement that all of a number of

diagnostic tests

yield positive results before

declaring that a patient has a particular complaint. See also majority rule.[Statistics in

Medicine, 1988, 7, 549–58.]

Unbalanced design: Synonym for nonorthogonal design and also used for longitudinal data

where the number of measurement occasions differs between individuals.

Unbiased: See bias.

Uncer tainty analysis: A method for assessing the variability in an outcome variable that is due to

the uncertainty in estimating the values of the input parameters. A

sensitivity analysis

can

extend an uncertainty analysis by identifying which input parameters are important in

contributing to the prediction imprecision of the outcome variable. Consequently a sensi-

tivity analysis quantiﬁes how changes in the values of the input parameters alter the value of

the outcome variable. [SMR Chapter 14.]

Underfit ted models : Models where some parameters or terms that would appear in a correctly

speciﬁed model are missing either by mistake or by design. See also overﬁtted models.

Undirected graph: See graph theory.

440

Unequal probability sampling: A

sampling design

in which different

sampling units

in the

population have different probabilities of being included in the sample. The differing

inclusion probabilities

may result from some inherent feature of the sampling process, or

they may be deliberately imposed in order to obtain better estimates by including ‘more

important’ units with higher probability. The unequal inclusion probabilities need to be

accounted for in order to come up with reasonable estimates of population quantities. An

example of such a design is

line-intercept sampling

of vegetation cover, in which the size of

a patch of vegetation is measured whenever a randomly selected line intersects it. The larger

the patch, the higher the probabilities of inclusion in the sample. This type of sampling may

be carried out by assigning to each unit an interval whose length is equal to the desired

probability and selecting random numbers from the

uniform distribution

: a unit is included if

the random number is in its interval. [Biometrika, 1998, 85,89–101.]

Unfolding: An approach to simultaneous scaling of subjects and stimuli based on subjects’ rankings of

the stimuli. The subjects’ ideal points and the points representing the stimuli are thought of

as arranged along an unobserved scale where distances between ideal points and stimuli

would reproduce the observed rankings. Called unfolding because the analyst must “unfold”

the rankings to obtain the scale. See also multidimensional unfolding.[Applied

Psychological Measurement, 1989, 13, 193–296.]

U nidentified model: See identiﬁcation.

Uniform distribution: The probability distribution, f (x), of a random variable having constant

probability over an interval. Speciﬁcally given by

f ðxÞ¼

β  α

The mean of the distribution is ðα þ βÞ=2 and the variance is ðβ  αÞ

=12. The most

commonly encountered such distribution is one in which the parameters α and β take the

values 0 and 1, respectively. [STD Chapter 35.]

U nif ormity trial: A method often used in agronomy to determine the ‘best’ size of a plot, i.e. a size

such that the efﬁciency of a ﬁeld experiment is optimized. Basically the aim is to obtain the

maximal information for the lowest cost. [Agriculture Research, 1997, 28, 621–7.]

a. Dissimilarities

b. Ultrametric Tree

Iying

property

crimes

destructive acts

Arson

Vandalism

Arson

- -

10 10

Bu Pe Sh Va

Burglary

Perjury

Shoplifting

Vandalism

Fig. 141 An example of

an ultrametric tree. (Taken

from Tree Models of

Similarity and Association,

1996, J. E. Corter, by

permission of the

publishers, Sage.)

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U niformly most accurate interval: A

conﬁdence interval

that has the smallest probability of

covering every possible value of a parameter except the true value. [ Journal of

Econometrics, 2001, 103, 155–181.]

Uniformly mostpowerful test: A test of a given hypothesis that is least as powerful as another

for all values of the parameter under consideration, and more powerful for at least one value

of the parameter. [KA2 Chapter 21.]

Unimodal: A probability distribution or frequency distribution having only a single mode. A normal

distribution is an example. [KA1 Chapter 1.]

Unimodal ordering: Synonym for umbrella ordering.

Union-intersection principle: A procedure for constructing statistical tests for

multivariate

data

. [MV1 Chapter 6.]

Unit normal variate: Synonym for standard normal variate.

Unitroot test: A test applied to

time series

to determine whether a stochastic or a deterministic trend

is present in the series. The test is based on the asymptotic distribution of the  obtained by

least squares estimation

in the model

¼  þ y

t1

þ 

where 

is a stationary process. The null hypothesis is that  ¼ 1. [Journal of the American

Statistical Association, 1995, 90, 268–81.]

Univariate data: Data involving a single measurement on each subject or patient.

U nivariate d irectio nal orderi ngs: A term applied to the question of whether one distribution is

in some sense to the right of another. In this connection many different ordering concepts

have been proposed, for example,

stochastic ordering

where the random variable Y is said to

be stochastically larger then X if their

cumulative probability distributions

F and G satisfy

FðxÞGðxÞ for all x

[Annals of Statistics, 1992, 20, 2100–2110.]

Universe: A little used alternative term for population.

Unreplicated factorial: See factorial designs.

Unsupervisedlearning: See pattern recognition.

Unsupervised pattern recognition: See pattern recognition.

U n wei g hted means analysis: An approach to the analysis of two-way and higher-order

factorial

designs

when there are an unequal number of observations in each cell. The analysis is based

on cell means, using the

harmonic mean

of all cell frequencies as the sample size for all cells.

Up-and-down method: A method sometimes use for estimating the

lethal dose 50

. The method

consists of the following steps; after a series of equally spaced dosage levels is chosen, the

ﬁrst trial is performed at some dosage level and then trials take place sequentially. Each

subsequent trial is performed at the next lower or the next higher dosage level according as

the immediately preceding trial did or did not evoke a positive response. [Clinical

Chemistry, 1995, 41, 198.]

U-shaped distribution: A probability distribution or frequency distribution shaped more or less

like a letter U, though not necessarily symmetrical. Such a distribution has its greatest

442

frequencies at the two extremes of the range of the variable. See Fig. 142 for an example.

[KA1 Chapter 1.]

Utility analysis: A method for decision-making under uncertainty based on a set of axioms of

rational behaviour. [Utility Theory for Decision Making, 1970, P. C. Fishburn, Wiley,

New York.]

Fig. 142 An example of a U-shaped distribution. Distribution of the Barthel index.

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Vague prior : A term used for the

prior distribution

Bayesian inference

in the situation when there is

complete ignorance about the value of a parameter. See also non-informative prior.

[Subjective and Objective Bayesian Statistics, 2002, S. J. Press, Wiley, New York.]

Validity: The extent to which a measuring instrument is measuring what was intended.

Validitychecks: A part of

data editing

in which a check is made that only allowable values or codes

are given for the answers to questions asked of subjects. A negative height, for example,

would clearly not be an allowable value.

VALT: Abbreviation for variable-stress accelerated life testing trials.

V an Dantzig, David (1 900^1 959): One of Holland’s foremost mathematical statisticians who

was head of the Department of Mathematical Statistics at the University of Amsterdam. A

severe critic of

Bayesian inference

his major contribution was an extension of the theory of

characteristic functions

Van der W aerden’srankstatistics: A

distribution free method

for assessing whether two

populations have the same location. The test statistic, c , is given by

c ¼

j¼1



1

N þ 1



where S

; S

; ...; S

are the ranks of the observations in the smaller of the two samples

having n observations, and N is the total number of observations in the two samples. 

1

ðt Þ

is the tth percentile of the standard normal distribution. Tables of critical values of c are

available. An alternative to

Wilcoxon’s rank sum test

.[Nonparametric Statistical Inference,

2003, J. Dickinson and S. Chakraborti, CRC, Boca Raton.]

Variable : Some characteristic that differs from subject to subject or from time to time. [SMR

Chapter 3.]

Va riab le sa mpl i ng interval cusum charts: A

cusum

in which the time between samples is

varied as a function of the value of the cusum statistic. If there is an indication that the

process mean may have shifted the chart uses short sampling intervals, if there is no

indication of such a shift longer sampling intervals are used. In many circumstances this

type of chart is more suitable than the standard version. [Journal of Quality Technology,

2001, 33,66–81.]

Variable selection: The problem of selecting subsets of variables, in regression analysis, that

contain most of the relevant information in the full data set. See also adequate subset, all

subsets regression and selection methods in regression.[Subset Selection in Regression,

1990, A. Miller, Chapman and Hall/CRC Press, London.]

Variable-stress accelerated life testing trials (VALT): Experiments in which each of the

units in a random sample of units of a product is run under increasingly severe conditions to

444

get information quickly about the distribution of lifetimes of the units. Each unit in the

experiment is subjected to a certain pattern of speciﬁed stresses, each for a speciﬁed time

interval, until the unit fails. See also fatigue models.[TVP, 1996, 37, 139–42.]

Variance: In a population, the second

moment

about the mean. An unbiased estimator of the pop-

ulation value is provided by s

given by

n 1

i¼1

ðx





xÞ

where x

; x

; ...; x

are the n sample observations and



x is the sample mean. [SMR

Chapter 3.]

Var iance-balanced designs: Designs for

crossover trials

with more than two treatments which

are such that all pairwise difference between the treatment are estimated with the same

precision. [Design and Analysis of Cross-Over Trials, 1989, B. Jones and M. Kenward,

Chapman and Hall/CRC Press, London.]

Variance components: Variances of

random effect

terms in

multilevel models

. For example, in a

simple

random intercept model

for

longitudinal data

, both subject effects and error terms are

random and estimation of their variances is of some importance. In the case of a

balanced

design

, estimation of these variances is usually achieved directly from the appropriate

analysis

of variance table

by equating mean squares to their expected values. When the data are

unbalanced a variety of estimation methods might be used, although

maximum likelihood

estimation

and

restricted maximum likelihood estimation

are most often used. [Var ia n ce

Components, 1992, S.R. Searle, G. Casella and C. E. McCulloch, Wiley, New York.]

V ariance ^ covariance matrix: A

symmetric matrix

in which the off-diagonal elements are the

covariances (sample or population) of pairs of variables and the elements on the main

diagonal are the variances (sample or population) of the variables. [MV1 Chapter 2.]

Vari ance i n fl at i on factor: An indicator of the effect the other explanatory variables have on the

variance of a regression coefﬁcient of a particular variable, given by the reciprocal of the

square of the

multiple correlation coefﬁcient

of the variable with the remaining variables.

[Regression Analysis, Volume 2, 1993, edited by M. S. Lewis-Beck, Sage Publications,

London.]

Variance ratio distribution: Synonym for F-distribution.

Variance ratio test: Synonym for F-test.

Var iance stabil izing transformatio ns: Transformations designed to give approximate inde-

pendence between mean and variance as a preliminary to, for example,

analysis of variance

The

arc sine transformation

is an example. [Bioinformatics, 2004, 20, 660–7.]

V ariety trials: A term use d for agricultural ﬁeld experiments carried out as part of a plan t breeding

programme. They range from the i nitial studies involving small plots (single pl ants or

single rows) to replicated yield trials involving fairly large plots. [Progress Report 407,

Kent ucky Hybrid Corn Performance Tests, College of Agriculture, University of

Kent ucky, USA.]

Vari maxrotati o n: A method for

factor rotation

that by maximizing a particular function of the initial

factor loadings

, attempts to ﬁnd a set of orthogonal factors satisfying, approximately at least,

simple structure

. [MV2 Chapter 12.]

Variogram: A function, V(u), of a

time series

g given by

445

V ðuÞ¼E

 x

tu



Gives an alternative description to the

autocovariance function

of the series that may have

advantages particularly when the series is observed at irregular time-points. The

empirical

variogram

is the set of points ðu

ijk

; v

ijk

Þ : k

j : i ¼ 1; ...; m where u

ijk

¼jt

 t

j and

ijk

ðy

 y

; t

; j ¼ 1; ...; h

are the n

times at which observations are made on subject

i, with the corresponding observations being y

. A scatterplot of the points ðu

ijk

; v

ijk

Þ is called

the variogram cloud, which can be used to suggest a parametric model for V(u). Examples of

an empirical variogram and a variogram cloud are shown in Fig. 143. [TMS Chapter 2.]

Variogram cloud: See variogram

Varying-coeff icient models: Models that are linear in the explanatory variables but whose

coefﬁcients are allowed to change smoothly with the value of other variables. For example,

if the relationship between maternal weight change and birth weight is linear but with

differing shapes and intercepts that correspond to differing levels of maternal pre-pregnancy

weight, then a suitable model might be:

BW ¼ β

ðMPPWÞþMWCβ

ðMPPWÞþ

Thus birth weights are modelled linearly in maternal weight changes but with coefﬁcients that

vary as functions of maternal pre-pregnancy weight. Often these functions are ﬁtted non-

parametrically by, for example,

cubic splines

.[Statistics in Medicine, 1997, 16, 1603–16.]

Ve c op erator : An operator that creates a column vector from a matrix A by simply stacking the

column vectors of A below one another. Hence for the m  n matrix A ¼½a

...a

 with

column vectors a

; ...; a

vecðAÞ¼