Everitt B.S. The Cambridge Dictionary of Statistics

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q ¼



largest





smallest

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MSE=n

where



largest

and



smallest

are the largest and smallest means among the means of k groups and

MSE is the

error mean square

from an

analysis of variance

of the groups. [Biostatistics, 2nd

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

Studentized residua l: See residual.

Stud ent’s t-distribution: The distribution of the variable

t ¼



x 

ﬃﬃﬃ

where



x is the arithmetic mean of n observations from a normal distribution with mean  and

s is the sample standard deviation. Given explicitly by

f ðtÞ¼

ð þ 1Þg

ðpÞ

Gð

Þ

ð1 þ





ðþ1Þ

where  ¼ n  1. The shape of the distribution varies with  and as  gets larger the shape of

the t-distribution approaches that of the standard normal distribution. Some examples of

such distributions are shown in Fig. 136. A mutivariate version of this distribution arises

from considering a q-dimensional vector x

¼½x

; x

; ...; x

 having a

multivariate normal

distribution

with mean vector m and

variance–covariance matrix

S and deﬁning the elements

of a vector u as u

¼ 

þ x

1=2

; i ¼ 1; 2; ...; q where y  



. Then u has a

multi-

variate Student’s t-distribution

given by

gðuÞ¼

ð þ qÞg

ðpÞ

q=2

Gð

Þ

jSj



½1 þ



ðu mÞ

1

ðu  mÞ



ðþqÞ

This distribution is often used in

Bayesian inference

because it provides a multivariate

distribution with ‘thicker’ tails than the multivariate normal. For  ¼ 1 the distribution is

known as the multivariate Cauchy distribution. [STD Chapter 37.]

Fig. 136 Examples of Student’s distributions at various values of .

419

Student’s t-tests: Signiﬁcance tests for assessing hypotheses about population means. One version

is used in situations where it is required to test whether the mean of a population takes a

particular value. This is generally known as a single sample t-test. Another version is

designed to test the equality of the means of two populations. When independent samples

are available from each population the procedure is often known as the independent samples

t-test and the test statistic is

t ¼







ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

where



and



are the means of samples of size n

and n

taken from each population, and

is an estimate of the assumed common variance given by

ðn

 1Þs

þðn

 1Þs

þ n

 2

If the null hypothesis of the equality of the two population means is true t has a

Student’s

t-distribution

with n

þ n

 2 degrees of freedom allowing P-values to be calculated. The

test assumes that each population has a normal distribution but is known to be relatively

insensitive to departures from this assumption. See also Behrens–Fisher problem and

matched pairs t-test. [SMR Chapter 9.]

Sturdy statistics: Synonym for robust statistics.

Stu rges’ ru le: A rule for calculating the number of classes to use when constructing a

histogram

and

given by

no: of classes ¼ log

n þ1

where n is the sample size. See also Doane’s rule.[Encyclopedia of Statistical Sciences,

2006, eds. S.Kotz, C. B.Read, N.Balakrishnan and B.Vidakovic, Wiley, New York.]

Subgroup analysis: The analysis of particular subgroups of patients in a

clinical trial

to assess

possible treatment–subset interactions. An investigator may, for example, want to under-

stand whether a drug affects older patients differently from those who are younger.

Analysing many subgroupings for treatment effects can greatly increase overall type I

error rates. See also ﬁshing expedition and data dredging. [SMR Chapter 15.]

Subjective endpoints: Endpoints in

clinical trials

that can only be measured by subjective clinical

rating scales. [Nature Medicine, 2004, 10, 909–15.]

Subjective probability: Synonym for personal probability.

Subject-specific models: See population averaged models.

SUDAAN: A software package consisting of a family of procedures used to analyse data from

complex

survey data

and other observational and experimental studies including

clustered data

[www.rti.org/SUDAAN/]

Sufficiency principle: If two different observations x and y have the same values T(x)=T(y) of a

sufﬁcient statistic

for a family of distributions f(.|), then the inference about  based on x and

y should be the same. [Journal of the American Statistical Association, 1962, 57, 269–306.]

Sufficient statistic: A statistic that, in a certain sense, summarizes all the information contained in

a sample of observations about a particular parameter. In more formal terms this can be

expressed using the conditional distribution of the sample given the statistic and the

parameter f ðyjs;Þ, in the sense that s is sufﬁcient for  if this conditional distribution

420

does not depend on . As an example consider a random variable x having the following

gamma distribution;

f ðxÞ¼x

γ1

expðxÞ=GðγÞ

The

likelihood

for γ given a sample x

; x

; ...; x

is given by

exp 

i¼1

γ1

=½GðγÞ

Thus in this case the geometric mean of the observations is sufﬁcient for the parameter. Such

a statistic, which is a function of all other such statistics, is referred to as a minimal sufﬁcient

statistic. [KA2 Chapter 17.]

Sukh atme, Panu ra ng Vasudeo ( 1911^1 997): Born in Budh, India, Sukhatme graduated in

mathematics in 1932 from Fergusson College in Pine. He received a Ph.D. degree from

London University in 1936 and D.Sc. in statistics from the same university in 1939.

Sukhatme started his career as statistical advisor to the Indian Council of Agricultural

Research, and then became Director of the Statistics Division of the Food and Agriculture

Organisation in Rome. He played a leading role in developing statistical techniques that

were relevant to Indian conditions in the ﬁelds of agriculture, animal husbandry and ﬁshery.

In 1961 Sukhatme was awarded the Guy medal in silver by the Royal Statistical Society. He

died in Pun on 28 January 1997.

Summary measure analysis: Synonym for response feature analysis.

Sun f l ower plot: A modiﬁcation of the usual

scatter diagram

designed to reduce the problem of

overlap caused by multiple points at one position (particularly if the data have been rounded

to integers). The scatterplot is ﬁrst partitioned with a grid and the number of points in each

cell of the grid is counted. If there is only a single point in a cell, a dot is plotted at the centre

of the cell. If there is more than one observation in a cell, a ‘sunﬂower’ icon is drawn on

which the number of ‘petals’ is equal to the number of points falling in that cell. An example

of such a plot is given in Fig. 137.[Journal of the American Statistical Association, 1984,

79, 807–822.]

Fig. 137 A sunﬂower plot for height and weight.

421

Supercomputers: Various deﬁnitions have been proposed, some based on price, ‘any computer

costing more than 10 million dollars’, others on performance, ‘any computer where perform-

ance is limited by input/output rather than by CPU’. In essence supercomputing relates to

mass computing at ultra high speeds.

Super-Duper: A random number generator arising from combining a number of simpler techniques.

[Computer Science and Statistics, edited by W. J. Kennedy, 1973, Statistical Laboratory,

Iowa State University.]

Superefficient: A term applied to an estimator which for all parameter values is asymptotically

normal around the true value with a variance never exceeding and sometimes less than the

Cramér–Rao lower bound

Superiority trial: A

clinical trial

conducted to establish that an experimental treatment is superior to

a control treatment by some measure of treatment effect or efﬁcacy. An example of a

superiority trial is the Salk vaccine trial of 1954. A noninferiority trial is typically conducted

to establish that the treatment effect of an experimental treatment is not much worse than that

of an accepted standard treatment. [Federal Register, 1998, 63, 49583–49598.]

Superleverage: A term applied to observations in a

non-linear model

which have

leverage

exceed-

ing one.

Supernormality: A term sometimes used in the context of

normal probability plots

residuals

from, for example, a regression analysis. Because such residuals are linear functions of

random variables they will tend to be more normal than the underlying error distribution if

this is not normal. Thus a straight plot does not necessarily mean that the error distribution is

normal. Consequently the main use of probability plots of this kind should be for the

detection of unduly inﬂuential or outlying observations.

Superpopulation: A hypothetical inﬁnite population from which a ﬁnite population is a sample. A

superpopulation model provides an alternative framework for inference in sampling by

focusing not on the target ﬁnite-population parameters but on the superpopulation param-

eters associated with a stochastic mechanism hypothesized to generate the observations in

the population, parameters that are usually more relevant for scientiﬁc questions. [Statistical

Science, 2002, 17,73–96.]

Supersaturated design: A

factorial design

with n observations and k factors with k

n 1. If a

main-effects model is assumed and if the number of signiﬁcant factors is expected to be

small, such a design can save considerable cost. [Technometrics, 1995, 37, 213–25.]

Supervised learning: See pattern recognition.

Supervised pattern recognition: See pattern recognition.

Sup port: A term sometimes used for the

likelihood

to stress its role as a measure of the evidence

produced by a set of observations for a particular value(s) of the parameter(s) of a model.

Sup port functi on: Synoymous with likelihood.

Support vector machi nes (SVM ): A group of supervised learning methods that can be applied

to classiﬁcation or regression. For classiﬁcation, for example, the SVM constructs a hyper-

plane that optimally separates the data into two categories. Closely related to

neural networks

[Support Vector Machines, 2008, I. Steinwart and A. Christmann, Springer, New York.]

Suppressor variabl es: A variable in a regression analysis that is not correlated with the dependent

variable, but that is still useful for increasing the size of the

multiple correlation coefﬁcient

422

by virtue of its correlations with other explanatory variables. The variable ‘suppresses’

variance that is irrelevant to prediction of the dependent variable. [Encyclopedia of

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

Wiley, New York.]

SUR: Abbreviation for seemingly unrelated regression.

Sure screeningmethods: Methods for

selection of variables

in regression that have the property

that all the important variables survive with probability tending to one after the method has

been applied. Such methods are important when ﬁtting regression models to

high-

dimensional data

. See also lasso, oracle property and smoothly clipped absolute devia-

tion.[Journal of the Royal Statistical Society, Series B, 2008, 70, 849–911.]

Su rface models: A term used for those models for

screening studies

that consider only those events

that can be directly observed such as disease incidence, prevelance and mortality. See also

deep models.

Surrogate endpoi nt: A term often encountered in discussions of

clinical trials

to refer to an

outcome measure that an investigator considers is highly correlated with an endpoint of

interest but that can be measured at lower expense or at an earlier time. In some cases ethical

issues may suggest the use of a surrogate. Examples include measurement of blood pressure

as a surrogate for cardiovascular mortality, lipid levels as a surrogate for arteriosclerosis,

and, in cancer studies, time to relapse as a surrogate for total survival time. Considerable

controversy in interpretation can be generated when doubts arise about the correlation of the

surrogate endpoint with the endpoint of interest or over whether or not the surrogate

endpoint should be considered as an endpoint of primary interest in its own right. [The

Evaluation of Surrogate Endpoints, 2005, eds. T.Burzykowski, G.Molenberghs and

M.Buyse, Springer, New York.]

Su rvey: Synonym for sample survey.

Survivalcurve: See survival function.

Su rviv al functi o n: The probability that the

survival time

of an individual is longer than some

particular value. A plot of this probability against time is called a

survival curve

and is a

useful component in the analysis of such data. See also product-limit estimator and hazard

function. [SMR Chapter 13.]

Su rviva l t i me: Observations of the time until the occurrence of a particular event, for example,

recovery, improvement or death. [SMR Chapter 13.]

Su rvivo r functi o n: Synonym for survival function.

Suspended rootog ram: Synonym for hanging rootogram.

Sverdrup, Erl ing (1 917^1 994): Born in Bergen, Norway, Sverdrup’s studies in actuarial mathe-

matics were interrupted by the outbreak of World War II and were only completed in 1945.

He was recruited to work in the econometrics group of the University of Oslo and in 1948 he

was appointed research assistant at the Seminar of Insurance Mathematics also at the

University of Oslo. In 1952 Sverdrup obtained his doctoral degree with a thesis on minimax

methods and as Professor of Insurance Mathematics and Mathematical Statistics at the

University of Oslo he built a modern educational program in mathematical statistics. He

died on March 15

, 1994 in Oslo.

423

Switch -back desig ns:

Repeated measurement designs

appropriate for experiments in which the

responses of experimental units vary with time according to different rates. The design

adjusts for the different rates by switching the treatments given to units in a balanced way.

Switchingeffect: The effect on estimates of treatment differences of patients’ changing treatments

in a

clinical trial

. Such changes of treatment are often allowed in, for example, cancer trials,

due to lack of efﬁcacy and/or disease progression. [Statistics in Medicine, 2005, 24,

1783–90.]

Switchingregression model: A model linking a response variable y and q explanatory variables

; ...; x

and given by

y ¼ β

þ β

þþβ

þ 

with probability l

y ¼ β

þ β

þþβ

þ 

with probability 1  l

where 

 N ð0;

Þ, 

 N ð0;

Þ and l;

;

and the regression coefﬁcients are

unknown. See also change point problems.[Sociological Methods & Research, 1993,

22, 248–273.]

Sylvan’sestimator: An estimator of the

multiple correlation coefﬁcient

when some observations of

one variable are missing. [Continuous Univariate Distributions, Volume 2, 2nd edition,

1995, N. L. Johnson, S. Kotz and N. Balakrishnan, Wiley, New York.]

Symmetrical distribution: A probability distribution or frequency distribution that is symmet-

rical about some central value. [KA1 Chapter 1.]

Symmetric matrix: A square matrix that is symmetrical about its leading diagonal, i.e. a matrix

with elements a

such that a

¼ a

. In statistics,

correlation matrices

and

variance–

covariance matrices

are of this form.

Symmetry i n squa re cont i ngency tabl es: See Bowker’s test for symmetry.

Synchronicity: A term used by some people (including Carl Jung and Arthur Koestler) for what they

see as an acausal connecting principle needed to explain

coincidences

, arguing that such

events occur far more frequently than can be explained by chance. But Jung and Koestler get

little support from Fisher who suggested that the one chance in a million will undoubtedly

occur with no more and no less that its appropriate frequency, however surprised we may be

that it should occur to us. [Chance Rules, 2nd edn, 2008, B. S. Everitt, Springer, New York.]

Synergism: A term used when the joint effect of two treatments is greater than the sum of their effects

when administered separately (positive synergism), or when the sum of their effects is less

than when administered separately (negative synergism).

Synthesis ana lysis: A method that can be used to estimate the multivariate relations between

multiple predictors (Xs) and an outcome variable (Y) from the univariate relation of each

X with Y and the two-way correlations between each pair of Xs. [Statistics in Medicine,

2009, 28, 1620–1635.]

Synthetic estimator: A

small-area estimator

is called synthetic if a reliable direct estimator for a

large area, covering several small areas, is used to derive an indirect estimator for a small

area under the assumption that the small areas have the same characteristics as the large area.

[Small Area Estimation, 2003, J. N. K. Rao, Wiley, New York.]

Synthetic risk maps: Plots of the scores derived from a

principal components analysis

of the

correlations among cancer mortality rates at different body sites. [Statistics in Medicine,

1993, 12, 1931–42.]

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SYST A T: A comprehensive statistical software package with particularly good graphical facilities.

[Systat Software Inc, 501 Canal Blvd Suite E, Point Richmond, CA 94804-2028,

USA; Systat Software UK Ltd., 24 Vista Centre, 50 Salsbury Rd., Hounslow, London

TW4 6JQ, UK.]

Systematic all ocati o n: Procedures for allocating treatments to patients in a

clinical trial

that

attempt to emulate random allocation by using some systematic scheme such as, for

example, giving treatment A to those people with even birth dates, and treatment B to

those with odd dates. While in principle unbiased, problems arise because of the openness of

the allocation system, and the consequent possibility of abuse. [SMR Chapter 15.]

Systematic erro r: Errors that produce results that differ from the true value of some quantity by a

ﬁxed amount. Such errors can occur because of biases in the measuring instrument or by the

incorrect use of the instrument by and investigator or both.

Systematic review: Synonym for meta-analysis.

425

Taboo pr o ba bilit ies: Probabilities P

ðnÞ

ij;H

, associated with

discrete time Markov chains

, that give the

conditional probability that the chain is in state j at time n and that, in moving from initial

state i to j, it has not visited the set of states H.[Journal of Applied Probability, 1994, 31A,

251–67.]

Ta g u c h i ’s id le column method: A procedure that enables a two-level orthogonal array to be used

as the bases of a mixed-level fractional factorial experiment. The resulting designs are not

orthogonal but can in some circumstances be described as nearly orthogonal. [Systems of

Experimental Design (Volume 1), 1987, G. Taguchi, Kraus International, New York.]

Ta g u c h i ’s minute accumulating analysis method: A method for the analysis of

interval-

censored observations

.[Systems of Design , 1987, G. Taguchi, Krause International,

New York.]

Ta g u c h i ’s parameter design: An approach to reducing variation in products and processes that

emphasizes proper choice of levels of controllable factors in a process for the manufacture of

a product. The principle of choice of levels focuses to a great extent on variability around a

prechosen target for the process response. [Introduction to Off-line Quality Control, 1980,

G. Taguchi and Y. I. Wu, Central Japan Quality Control Association, Nagnja, Japan.]

Ta k e u c h i ’s information criterion (TIC): An index that can be used as an aid in choosing

between competing models. It is deﬁned as

TIC ¼2L

þ 2trace½JðÞIðÞ

1



where L

is the maximized

log-likelihood

and m is the number of parameters in the model.

The matrices J(θ) and I(θ) contain the ﬁrst and second-order mixed partial derivatives of the

log-likelihood function. Thought to be useful in cases where the candidate models are not

particularly close to the ‘true’ model. See also Akaike’s information criterion and

Schwartz’s criterion.[Model Based inference in the Life Sciences: A Primer on

Evidence, 2007, D. R. Anderson, Springer, New York.]

Ta n g o ’sindex: An index for summarizing the occurrences of cases of a disease in a stable geo-

graphical unit when these occurrences are grouped into discrete intervals. The index is given

C ¼ r

where r

¼½r

; ...; r

 is the vector of relative frequencies of cases in successive periods,

and A is an m × m matrix, the elements of which represent a measure of the closeness of

interval i and j. Can be used to detect

disease clusters

occurring over time. See also scan

statistic.[Statistics in Medicine, 1993, 12, 1813–28.]

Tapering: A method used in the analysis of

time series

whereby the series is transformed in some way

prior to

spectral analysis

, in an effort to reduce bias; the price of less bias is generally an

increase in variance. [Practical Time Series, 2001, G. Janacek, Arnold, London.]

426

TAR: Acronym for threshold autoregression

Target population: The collection of individuals, items, measurements, etc., about which it is

required to make inferences. Often the population actually sampled differs from the target

population and this may result in misleading conclusions being made. The target population

requires a clear precise deﬁnition, and that should include the geographical area (country,

region, town, etc.) if relevant, the age group and gender. [Sampling Techniques, 3rd edition,

1977, W. G. Cochran, Wiley, New York.]

Tarone-Ware test: See logrank test.

Ta y l o r ’s expansion: The expression of a function, f(x), as the sum of a polynomial and a remainder.

Speciﬁcally given by

f ðxÞ¼f ðaÞþf

ðaÞðx aÞþf

ðx aÞ

=2! þf

000

ðx aÞ

=3! þ

where primes on f denote differentiation. Used in the

delta method

for obtaining variances of

functions of random variables. [Introduction to Optimization Methods and Their

Application in Statistics, 1987, B. S. Everitt, Chapman and Hall/CRC Press, London.]

Ta y l o r ’spowerlaw: A convenient method for ﬁnding an appropriate transformation of grouped data

to make them satisfy the homogeneityof variance assumption oftechniquessuch as the

analysis

of variance

. The method involves calculating the slope of the regression line of the logarithm

of the group variances against the logarithm of the group means, i.e. b in the equation

log

¼ a þ b log

x

The value of 1 − b/2 indicates the transformation needed, with non-zero values correspond-

ing to a particular

power transformation

, and zero corresponding to a

log transformation

Most often applied in ecology studies in investigations of the relationship bertween the

abundance of a species and its variability. [Nature, 2003, 422,65–8.]

TD50: Abbreviation for tumorigenic dose 50.

Telephone i nterview surveys: Surveys which generally involve some form of random digit

dialling. For example, in the Mitofsky–Waksberg

scheme

, telephone exchanges are ﬁrst

sampled. Then telephone numbers within sampled exchanges are sampled and called. If

there is success in locating a residential telephone number on the ﬁrst call for an exchange,

additional numbers from the exchange are sampled and called—otherwise the next sample

exchange is tried. Since some exchanges tend to contain all non-residential numbers, there

are efﬁciency gains in using this scheme over selecting phone numbers completely at

random (random digit dialling). [Survey Response by Telephone, 1983, J. H. Frey, Sage

Publications, Beverley Hills, California.]

Telescoping: A term used in the collection of

event history data

for an apparent shifting of events

from the near distant past towards the present or more precisely towards the time at which the

recollection was made.

TES: Abbreviation for transform-expand sample.

Test statistic: A statistic used to assess a particular hypothesis in relation to some population. The

essential requirement of such a statistic is a known distribution when the null hypothesis is

true. [SMR Chapter 8.]

Tetrachoric correlation: The correlation between two variables that originally arise from a

bivariate normal distribution but are only observed as variables that have been dichotomized

427

at some threshold value, leading to a data set that is simply a 2 × 2 table of counts. From the

four frequencies in this table the tetrachoric correlation can be estimated by

maximum

likelihood

, the likelihood function involving bivariate-normal probability integrals. Such

correlations are widely used in behavioural genetics research and as the basis of the factor

analysis of multivariate binary data. [MV2 Chapter 12.]

Thei l -Sen esti mato r: An estimator for the slope of a linear regression equation linking two

variables which is based on the median of the slopes of all pairs of points. [Fundamentals

of Modern Statistical Methods, 2001, R. R. Wilcox, Springer, New York.]

Thei l’s test: A

distribution free

test that the slope parameter in a simple

linear regression

model

¼ β

þ β

þ ε

takes a particular value β

ð0Þ

where it is assumed only that the error terms

follow a distribution with zero median. The test statistic T is deﬁned as

T ¼

cðD

 D

where

cðxÞ¼1ifx

¼ 0ifx ¼ 0

¼1ifx

and D

¼ y

 β

ð0Þ

. The approximate variance of T under the null hypothesis is

var

ðTÞ¼

nðn 1Þð2n þ 5Þ

The statistic T =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

var

ðTÞ

can be compared to a stand ard normal variable to assess the

null hypothesis. A p oint estimate of β

is given by the median of n(n−1)/2 es timates of

the slope

 y

 x

[Biometrika, 1984, 71,51–6.]

Therapeutic tria l: Synonym for clinical trial.

Th ie l , Tho rva ld N ico la i ( 1838^1910) : Born in Copenhagen, Denmark, Thiel studies

Astronomy at the U niversity of Copenhagen and beca me Professor of Astronomy and

Director of the Copenhagen Obs ervatory in 1875. He founded the Danish Society of

Actuaries in 1901 and served as its president. Thiel’s main contributi ons to statis tics were

in the areas of

skew distributions

and

cumulants

. He died on September 26

, 1910 in

Copenhagen.

Tho mas d ist ri but i on: A probability distribution used in constructing models for the distribution of

number of plants of a given species in randomly placed quadrants. [Biometrika 1949, 36,

18–25.]

Thomson, Sir Godfrey H ilton (1 881^1955): Born in Carlisle, England on March 27, 1881

Thomson studied for a MSc in mathematics and physics at Durham University and

completed a PhD at Strasbourg University in Germany in 1906. His interests later turned

from science to psychology and he returned to obtain a DSc in Psychology in 1913 at

Durham University where he became a professor of education in 1920. From 1925 until his

retirement in 1951 Thomson served as professor of education in Edinburgh. He played an

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