Everitt B.S. The Cambridge Dictionary of Statistics

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important role in the development of

factor analysis

and the theory of intelligence and

apparently formalized the idea of

maximum likelihood estimation

earlier than Fisher.

Thomson was knighted in 1949 and died in Edinburgh on 9

of February 1955.

Three-g rou p resistant l i n e: A method of

linear regression

that is resistant to

outliers

and

observations with large inﬂuence. Basically the method involves dividing the data into

three groups and ﬁnding the median of each group. A straight line is then ﬁtted through these

medians.

Three-mode ana lysis: A term for techniques of

multivariate analysis

that can be used to analyse

data that can be classiﬁed in three ways, for example, by subjects, variables and time. The

general aim of such techniques is to ﬁt a model to the data with a low-dimensional

representation so that the basic underlying structure can be more readily discerned and

interpreted.

Individual differences scaling

is an example of such a technique. [Analysis of

Proximity Data, 1997, B. S. Everitt and S. Rabe-Hesketh, Edward Arnold, London.]

Three-period crossover design: A design in which two treatments, A and B, are given to

subjects in the order A:B:B or B:A:A. Two sequence groups are formed by random

allocation. The additional third observation period alleviates many of the problems asso-

ciated with the analysis of the

two-period crossover design

. In particular, an appropriate

three-period crossover design allows for use of all the data to estimate and test direct

treatment effects even when

carry-over effects

are present. [Statistics in Medicine, 1996,

15, 127–44.]

Three-stage sa mpli ng: A process that involves sampling the subunits from a two-stage sampling

instead of enumerating them completely. [Sampling Techniques, 3rd edition, 1977,

W. G. Cochran, Wiley, New York.]

Threshold autoregressive model: A non-linear model for

time series

capable of capturing

asymmetric cyclical be haviour, for example, an unemploy ment rate which exhib its

a tenden cy to in crease at a faste r rate than it decreases. A simple example of such a

model is

¼1:5x

t1

þ ε

if x

t1

 0

¼ 0:6x

t1

þ ε

if x

t1

[Threshold Models in Nonlinear Time Series Analysis, 1983, H. Tong, Springer-Verlag,

New York.]

Threshold-crossing data: Measurements of the time when some variable of interest crosses a

threshold value. Because observations on subjects occur only periodically, the exact time of

crossing the threshold is often unknown. In such cases it is only known that the time falls

within a speci ﬁed interval, so the observation is

interval censored

.[Statistics in Medicine,

1993, 12, 1589–1603.]

Threshold-crossing model: A model for a response that is a binary variable, z, which assumes

that the value of the variable is determined by an observed random variable, x, and an

unobservable random variable u, such that

z ¼ 1ifx

β þu  0

¼ 0 otherwise

where β is a vector of parameters. In medical research z might be an observable binary

indicator of health status, and x

β þu a latent continuous variable determining health status.

[Network Computation in Neural System, 1996, 7, 325–33.]

429

Threshold limit value: The maximum permissible concentration of a chemical compound present

in the air within a working area (as a gas, vapour or particulate matter) which, according to

current knowledge, generally does not impair the health of the employee or cause undue

annoyance. [Occupational Medicine, 1989, 31, 910.]

Thresh ol d model: A model that postulates that an effect occurs only above some threshold value.

For example, a model that assumes that the effect of a drug is zero below some critical dose

level.

Thurstone, Louis Leon (1887^1955): Thurstone (an anglization of the Swedish name Torsten)

was born in 1887 in Chicago. He received a masters degree in mechanical engineering from

Cornell in 1912 and was offered a brief assistantship in the laboratory of Thomas Edison.

After two years as an instructor of geometry and drafting at Minnesota he moved to the

University of Chicago. Thurstone completed a PhD in Psychology in 1917 and worked at

Chicago between 1924 and 1952. A major contributor to the development of

factor analysis

psychophysics and the theory of intelligence, he was one of the founders of the journal

Psychometrika and the Psychometric Society, becoming the society’s ﬁrst President in 1936.

Thurstone died in 1955.

TIC: Abbreviation for Takeuchi’s information criterion.

Tiedobservations: A term usually applied to ordinal variables to indicate observations that take the

same value on a variable. Also often used for identical

survival times

Ti me-by -t i me AN OVA: A ﬂawed approach to the analysis of

longitudinal data

in which groups are

compared by

analysis of variance

at each time point. Ignores correlations between the

repeated measurements and gives no overall answer to the question of whether or not

there is a group difference. [Encyclopedia of Statistical Sciences, 2006, eds S.Kotz, C. B.

Read, N.Balakrishnan and B.Vidakovic, Wiley, New York.]

Ti me-dependent cova r iates: Covariates whose values change over time as opposed to cova-

riates whose values remain constant over time (time-independent covariates). A pre-

treatment measure of some characteristic is an example of the latter, age and weight,

examples of the former. [Journal of the Royal Statistical Society, Series B, 1972, 34,

187–220.]

Ti me-frequency dist ri but i on: A description of the energy or power of a signal as a two-

dimensional function of both time and frequency. Such distributions are used to study

nonstationary time series

. See also Chow-Williams distribution and Wigner-Ville distri-

bution.[Proceedings of the IEEE, 1989, 77, 941–981.]

Time-homogeneous Markov chain: See Markov chain.

Ti me-i ndependent cova r iates: See time-dependent covariates.

Ti me ser ies: Values of a variable recorded, usually at a regular interval, over a long period of time.

The observed movement and ﬂuctuations of many such series are composed of four different

components, secular trend, seasonal variation, cyclical variation, and irregular variation. An

example from medicine is the incidence of a disease recorded yearly over several decades

(see Fig. 138). Such data usually require special methods for their analysis because of the

presence of

serial correlation

between the separate observations. Most often time series are

analysed by linear models such the classic family of

autoregressive moving average models

But there are many observable phenomena that cannot be accounted for adequately by linear

models and which give rise to nonlinear time series, for which special models have been

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developed, for example, autoregressive conditional heteroscedastic models. See also

autocorrelation, harmonic analysis, spectral analysis and fast Fourier transformation.

[TMS.]

Ti me-to-event data: Synonym for survival data.

Ti me t rade-o ff tech n iq ue: See Von Neumann–Morgenstern standard gamble.

Ti me-varyi ng cova r iates: Synonym for time-dependent covariates.

max

: A measure traditionally used to compare treatments in

bioequivalence trials

. The measure is

simply the time at which a patient’s highest recorded value occurs. See also C

max

To b i t m o d e l : Synonym for censored regression model.

Toeplitz matrix: A matrix in which the element in the ith row and jth column depends only on ji  jj.

The

variance–covariance matrices

stationary series

have this property. [SIAM Journal on

Matrix Analysis and Applications, 1996, 16,40–57.]

To l e r a n c e : A term used in

stepwise regression

for the proportion of the sum-of-squares about the mean

of an explanatory variable not accounted for by other variables already included in the

regression equation. Small values indicate possible

multicollinearity

problems.

Number of deaths

1974 1975 1976 1977 1978 1979 1980

1500 2000 2500 3000 3500

Fig. 138 A time series of monthly deaths from lung cancer in the UK 1974–1979.

431

Tole rance interval: Statistical intervals that contain at least a speciﬁed proportion of a population

either on average, or else with a stated conﬁdence value. Used to summarize uncertainty

about values of a random variable, usually a future observation. [Statistical Tolerance

Regions, 1970, I. Guttman, Hafner, Darien.]

Totalmatrixofsumsofsquaresandcross-products:See multivariate analysis of

variance.

Total sum of squares: The sum of the squared deviations of all the observations from their

mean.

To u r : A motion graphic designed to study the joint distribution of

multivariate data

in search of

relationships that may involve several variables. A tour is created by generating a sequence

of low-dimensional projections of

high-dimensional data

. See also brushing scatterplots,

dynamic graphics and projection pursuit.[Interactive and Dynamic Graphics for Data

Analysis, 2007, D. Cook and D. F. Swayne, Springer, New York.],

Townsend i nde x: An index designed to measure material deprivation of an area. The index is

calculated as an unweighted sum of four standardized census variables corresponding to

area-level proportions of unemployment, car ownership, crowding and home ownership.

[International Journal of Health Services, 1985, 15, 637–63.]

Trace of a matrix: The sum of the elements on the main diagonal of a

square matrix

; usually denoted

as tr(A). So, for example,

if A ¼



then trðAÞ¼4

Tr a c k i n g : A term sometimes used in discussions of data from a

longitudinal study

, to describe the

ability to predict subsequent observations from earlier values. Informally this implies that

subjects that have, for example, the largest values of the response variable at the start of the

study tend to continue to have the larger values. More formally a population is said to track

with respect to a particular observable characteristic if, for each individual, the expected

value of the relevant deviation from the population mean remains unchanged over time.

[Statistics in Medicine, 1992, 11, 2027.]

Tr a i n i n g s e t : See discriminant analysis.

Transfer functi o n model: A model relating two

time series

, fx

g and fy

g and given by

¼ a

þ a

t1

þþε

where a

; a

; ...are a set of parameters and fε

gis a

white noise sequence

. In general fx

gis

referred to as the ‘input’ series and fy

gas the ‘output’ series. An example might be monthly

employment statistics related to inﬂation rate. Such models have proved very useful

for prediction and forecasting. [Time Series Analysis: Forecasting and Control, 1994,

G. E. Box, G. M. Jenkins and G. Reinsel, Prentice Hall, San Francisco.]

Transformation: A change in the scale of measurement for some variable(s). Examples are the

square root transformation

and

logarithmic transformation

. [SMR Chapter 3.]

Tr a n s f o r m - e x p a n d s a m p l e ( T E S ) : A versatile methodology speciﬁcally designed to ﬁt models

to autocorrelated empirical observations with general marginal distribution. With this

approach the aim is to ﬁt both the empirical marginal distribution as well as the empirical

autocorrelation function

.[Simulation, 1995, 64, 353–70.]

432

Transition matrix: See Markov chain.

Tr a n s i t i o n m o d e l s : Models used in the analysis of

longitudinal data

in which the distribution of the

observations at each time point is considered conditional on previous outcomes. Unlike

marginal models

, these models have no representation in terms of cross-sectional data. The

models are comparatively straightforward to use in practice and applications with binary

data, for which the models represent

Markov chains

, are widespread. See also ante-

dependence model.[Analysis of Longitudinal Data, 2nd edition, 2002, P. J. Diggle,

P. J. Heagerty, K.-Y. Liang and S. Zeger, Oxford Science Publications, Oxford.]

Transition probabil ity: See Markov chain.

Transmission probabil ity: A term used primarily in investigations of the spread of AIDS for the

probability of contracting infection from an HIV-infected partner in one intercourse.

[Journal of the Royal Statistical Society, Series C, 2001, 50,1–14.]

Tr a n s m i s s i o n r a t e : The rate at which an infectious agent is spread through the environment or to

another person.

Tr a p e z i u m r u l e : A simple rule for approximating the integral of a function, f(x), between two limits,

using the formula

aþh

f ðxÞdx ¼

h½f ðaÞþf ða þ hÞ

See also Gaussian quadrature. [SMR Chapter 14.]

Tr a p e z o i d a l r u l e : Synonym for trapezium rule.

Tr a p p i n g w e b d e s i g n : A technique for obtaining density estimates for small mammal and arthropod

populations which consists of a number of lines of traps emanating at equal angular incre-

ments from a centre point, like the spokes of a wheel. Such a design provides the link

between capture data and distance sampling theory. The estimator of density is

D ¼ M

tþ1

f ð0Þ, where M

t+1

is the number of individuals captured and f(0) is calculated

from the M

t+1

distances from the web centre to the traps in which those individuals were ﬁrst

captured. [Biometrics, 1994, 50, 733–45.]

Treatment allocation ratio: The ratio of the number of subjects allocated to the two treatments in

clinical trial

. Equal allocation is most common in practice, but it may be advisable to

allocate patients randomly in other ratios when comparing a new treatment with an old one,

or when one treatment is much more difﬁcult or expensive to administer. The chance of

detecting a real difference between the two treatments is not reduced much as long as the

ratio is not more extreme than 2:1. [SMR Chapter 15.]

Treatment cross-contamination: Any instance in which a patient assigned to receive a partic-

ular treatment in a

clinical trial

is exposed to one of the other treatments during the course of

the trial. [Genetic Molecular Research, 2004, 3, 456–62.]

Treatment received analysis: Analysing the results of a

clinical trial

by the treatment received

by a patient rather than by the treatment allocated at randomization as in

intention-to-treat

analysis

. Not to be recommended because patient

compliance

is very likely to be related to

outcome. [SMR Chapter 15.]

Treatment trial: Synonym for clinical trial.

433

Tree: A term from a branch of mathematics known as

graph theory

, used to describe any set of straight-

line segments joining pairs of points in some possibly multidimensional space, such that

every point is connected to every other point by a set of lines,

each point is visited by at least one line,

no closed circuits appear in the structure.

If the length of any segment is given by the distance between the two points it connects, the

length of a tree is deﬁned to be the sum of the lengths of its segments. Particularly important

in some aspects of

multivariate analysis

is the

minimum spanning tree

of a set of n points

which is simply the tree of shortest length among all trees that could be constructed

from these points. See also dendrogram.[Tree Models of Similarity and Association,

1996, J. E. Corter, Sage University Papers 112, Sage Publications, London.]

Tree-top alternat ive: A term used in multivariate ‘one-sided’ testing problems in which the null

hypothesis is that the means of k independent normal populations are equal, and the

alternative hypothesis is that all the ﬁrst (k − 1) means are greater than or equal to the

mean of the last population, with strict inequality for at least one.

Trellis graphics: Graphical displays in which scatterplots, contour plots, perspective plots etc., are

constructed for a series of ranges of a separate variable; the component plots are essentially

‘conditioned’ on the values of the latter. An example is shown in Fig. 139. See also coplots

and S-PLUS.

Tr e n d : Movement in one direction of the values of a variable over a period of time. See also linear

trend. [SMR Chapter 9.]

pollution pollution

Wind

Precip

Temp

Wind

Precip

Temp

Wind

Precip

Temp

Wind

Precip

Temp

Fig. 139 An example of a trellis graphic.

434

Tria ngu l a r contingency tabl e: A

contingency table

in which there are

structural zeros

a particular portion of the table. The table below, for example, gives stroke patient’s

initial and ﬁnal disability states graded on a ﬁve point scale, A–E, of increasing

severity. If a hospital never discharges a patient whose condition is worse than at

admiss ion, then no discharged patient’s ﬁnal state can be worse than their initial state.

Such tables are generally investi gated for quasi-independence. [Biometrics, 1975, 31 ,

233–8.]

Initial rating Number with following ﬁnal rating Totals

AB CDE

E11231215869

D91041– 24

C644––14

B45–––9

A5––––5

Totals 35 42 20 16 8 121

Tria ngu l a r d istr i buti o n: A probability distribution of the form

f ðxÞ¼2ðx  aÞ= ½ðβ  αÞðγ  αÞ α  x  γ

¼ 2ðβ  xÞ=½β  αÞðβ  γÞ γ  x  β; α  x  β

The mean of the distribution is ðα þ β þ γÞ=3 and its variance is ðα

þ β

 αβ  αγ  βγÞ=18. [STD Chapter 40.]

Triangular test: A term used for a particular type of

closed sequential design

in which the boundaries

that control the procedure have the shape of a triangle as shown in Fig. 140.[Statistics in

Medicine, 1994, 13, 1357–68.]

Trimmed mean: See alpha-trimmed mean.

Tr i o g r a m m e t h o d : A method for function estimation using piecewise linear, bivariate splines based

on an adaptively constructed triangulation. [Journal of the American Statistical Association,

2004, 93, 101–19.]

Experimental

significantly better

No significant

difference

Experimental

nificantly worse

Fig. 140 The triangular region of a triangular test.

435

Triograms: A family of continuous piecewise linear functions deﬁned over adaptively selected

triangulations of the plane and used in a general approach to the statistical modelling of

bivariate densities, to regression and to the estimation of

hazard functions

.[Journal of the

Royal Statistical Society, Series B, 66, 145–163.]

Tr i p l e s c a t t e r p l o t : Synonym for bubble plot.

Triples test: An asymptotically

distribution free method

for testing whether a univariate population is

symmetric about some unknown value versus a broad class of asymmetric distribution

alternatives. [Statistical Methodology, 2007, 4, 423–433.]

Trohoc study: A term occasionally used for

retrospective study

, derived from spelling cohort back-

wards. To be avoided at all costs!

Tr o j a n s q u a r e s : Row-and-columns designs that arrange n replicates of nk treatments in n complete

rows and n complete columns. They are constructed from k mutually orthogonal n × n

Latin

squares

. Often used in crop research. [Agricultural Science Cambridge, 1998, 131, 135–42.]

Trou g h -t o-peak ratio: A measure most often used in

clinical trials

of anti-hypertensive drugs and

their effect on blood pressure. This usually achieves a maximum (the peak effect P) and then

decreases to a minimum (the trough effect T). The FDA recommends that the ratio should be

at least 0.5. Statistical properties of the ratio are complicated by the known correlation of

trough and peak effects. [Hypertension, 1995, 26, 942–9.]

TR U E EPIST A T: Software for

epidemiology

and

clinical trials

. [Epistat Services, 2011 Cap Rock

Circle, Richardson, TX 75080-3417, USA.]

Truncatedbinarydata: Data arising when a group of individuals, who each have a binary response,

are observed only if one or more of the individuals has a positive response. [Biometrics,

2000, 56, 443 –50.]

Truncateddata: Data for which sample values larger (truncated on the right) or smaller (truncated on

the left) than a ﬁxed value are either not recorded or not observed. See also censored

observations and truncated binary data .

Truncated Poisson distribution: Synonymous with positive Poisson distribution.

Tschuprov coefficient: A measure of association, T, of the two-variables forming an r × c

contingency table

, given by

T ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðr  1Þðc 1Þ

where X

is the usual

chi-squared statistic

for testing the independence of the two variables

and N is the sample size. See also contingency coefﬁcient, phi-coefﬁcient and Sakoda

coefﬁcient.

T-squaretest: A test for randomness of a

spatial process

which is based on comparing the distance of

the nearest event of the process from a randomly chosen point with the nearest neighbour

distance between randomly chosen events.

Tukey, John (1915^2000): Born in New Bedford, Massachusetts, Tukey was educated largely at

home before entering Brown University to study chemistry. On obtaining his degree he

moved to Princeton to take up mathematics and was awarded a doctorate in 1939. He spent

his entire academic career at Princeton becoming the founding Chair of the Statistics

Department. Tukey’s contributions to statistics were immense and he carried out

436

fundamental work in

time series

robustness

analysis of variance

experimental design

and

multiple comparisons

. He invented

exploratory data analysis

and was forever devising new

terms for his new ideas. Tukey is credited with inventing the word ‘software’. His collected

works ﬁll eight volumes and in 1960 he was made President of the Institute for Mathematical

Statistics. Tukey died on 26 July 2000.

Tu k e y ’s lambda distributions: A family of probability distributions deﬁned by the

transformation

Z ¼½U

ð1  UÞ

=l

where U has a

uniform distribution

in (0,1). [Technometrics, 1985, 27,81–4.]

Tu k e y ’s quick test: A two-sample

distribution free method

test of location. The test statistic T is

derived from merging the two samples, A and B, and arranging the values in ascending

order; if there is an A observation at either end of the combined sample and a B at the other,

then the number of A’s at the ﬁrst-mentioned end that have values more extreme than all the

B’s is added to the number of B’s at the other end that are more extreme than all the A’sto

give the value of T. If, on the other hand, there are A’s at both ends or B’s at both ends, we put

T = 0. The test rejects the null hypothesis of no difference in location for large values of T.

[Technometrics, 1959, 1,31–48.]

Tu k e y ’s test for nonadditivity: A method of testing for interaction in a two-way design with

only one observation per treatment combination. Based on the sum of squares S

given by

i¼1

j¼1

ðy

y



Þðy

y



Þg

where r is the number of rows, c is the number of columns, y

is the observation in the ijth

cell,



is the mean of the ith row,



is the mean of the jth column and



is the mean of all

observations. S

and S

are the main effects sums of squares. [Encyclopedia of Statistical

Sciences, 2006, eds. S. Kotz, C. B. Read, N. Balakrishnan and B. Vidakovic, Wiley,

New York.]

Tumorigenicdose50(TD50): The daily dose of a chemical of interest that gives 50% of the test

animals tumours by some ﬁxed age. See also lethal dose 50.[Biometrics, 1984, 40,27–40.]

Turn bu l l esti mato r: A method for estimating the

survival function

for a set of

survival times

when

the data contain

interval censored observations

. See also product limit estimator.[Journal

of the Royal Statistical Society, Series B, 1976, 38, 290–5.]

Tweedie, Richard (1 947^2001): Born in Leeton, New South Wales, Australia, Tweedie won a

scholarship to attend the Australian National University (ANU), obtaining a ﬁrst class

honours degree in statistics in 1969. From Australia he travelled to Cambridge to undertake

graduate work with David Kendall. Returning to Australia in 1972 Tweedie became Post

Doctoral Fellow in Statistics at the Institute of Advanced Studies at the ANU. In 1978 he

took up the post of Director of Victorian Regional Ofﬁce of the Commonwealth Scientiﬁc

and Research Organization. Three years later he became head of the software consultancy

company, Siromath. Despite the demands of this post, Tweedie managed to continue his

fundamental research in Markov chains, for which he was awarded a Doctor of Science

degree from the ANU in 1986. The ﬁnal stages of his career were spent in the USA, ﬁrst in

the Department of Statistics at Colorado State University and then at the University of

Minnesota, continuing his work on the convergence theory of

Markov chains

. Tweedie died

on 7 June 2001.

437

Twin analysis: The analysis of data on identical and fraternal twins to make inferences about the

extent and overlaps of genetic involvement in the determinants of one or more traits. It

usually makes the assumption that the shared environment experiences relevant to the traits

in question are equally important for identical and fraternal twins. [Statistics in Human

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

Two-armed band it al l ocati o n: A procedure for forming treatment groups in a

clinical trial

,in

which the probability of assigning a patient to a particular treatment is a function of the

observed difference in outcomes of patients already enrolled in the trial. The motivation

behind the procedure is to minimize the number of patients assigned to the inferior treatment.

See also minimizationand play-the-winner rule.[Journal of the Royal Statistical Society,

Series B, 1979, 41, 148–77.]

Two-bend transf o rmat i o n: Transformation which straightens out more than a single bend in the

relationship between two variables. Most common of such transformations are the

logistic

transformation

arcsin transformation

and the

probit transformation

. See also one-bend

transformation.

Two-by -two (2  2) contingency tab le: A

contingency table

with two rows and two columns

formed from cross-classifying two binary variables. The general form of such a table is

Variable 1

0ab

Variable 2

1cd

[SMR Chapter 10.]

Two-by-two (2  2) crossover design: See crossover design.

Two-di mensi o na l tabl es: See contingency tables.

Two-parameter exponential distribution: A probability distribution, f(t), given by

f ðtÞ¼le

lðtG Þ

; t  G

The term G is known as the

guarantee time

and corresponds to the time during which no

events can occur. In the analysis of survival times, for example, it would represent

the minimum survival time. [Journal of Statistical Planning and Inference, 1997, 59,

279–89.]

T wo-phase sampling: A sampling scheme involving two distinct phases, in the ﬁrst of which

information about particular variables of interest is collected on all members of the sample,

and in the second, information about other variables is collected on a sub-sample of the

individuals in the original sample. An example of where this type of sampling procedure

might be useful is when estimating prevalance on the basis of results provided by a fallible,

but inexpensive and easy to use, indicator of the true disease state of the sampled individuals.

The diagnosis of a subsample of the individuals might then be validated through the use of

an accurate diagnostic test. This type of sampling procedure is often wrongly referred to as

two-stage sampling

which in fact involves a completely different design. [Statistical

Methods in Medical Research, 1995, 4,73–89.]

Two-sided test: A test where the alternative hypothesis is not directional, for example, that one

population mean is either above or below the other. See also one-sided test. [SMR

Chapter 8.]

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