Everitt B.S. The Cambridge Dictionary of Statistics

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Sequential importance sampling (SIS): A method for approximating the

posterior distribu-

tion

of the state vector for a possible nonlinear dynamic system. Used for prediction and

smoothing of nonlinear and non-Gaussian state space models. [Journal of the American

Statistical Association, 1998, 93, 1032–44.]

Sequential sums of squares: A term encountered primarily in regression analysis for the con-

tributions of variables as they are added to the model in a particular sequence. Essentially the

difference in the residual sum of squares before and after adding a variable. [ARA Chapter 4.]

Seri al corr el a tio n: The correlation often observed between pairs of measurements on the same

subject in a

longitudinal study

. The magnitude of such correlation usually depends on the

time separation of the measurements–typically the correlation becomes weaker as the time

separation increases. Needs to be properly accounted for in the analysis of such data if

appropriate inferences are to be made. [ARA Chapter 10.]

Serial interval: The period from the observation of symptoms in one case to the observation of

symptoms in a second case directly infected from the ﬁrst.

Serial measurements: Observations on the same subject collected over time. See also longitudi-

nal data . [SMR Chapter 14.]

Seri a tio n: The problem of ordering a set of objects chronologically on the basis of dissimilarities or

similarities between them. An example might be a

similarity matrix

for a set of archaeo-

logical graves, the elements of which are simply the number of varieties of artefacts shared

by each pair of graves. The number of varieties in common is likely to decrease as the time

gap between pairs of graves increases and a solution produced by say a

multidimensional

scaling

of the data might be expected to show a linear ordering of graves according to age. In

practice, however, the solution is more likely to show the

horseshoe shape

because for time

intervals above some threshold pairs of graves will have no artefacts in common. Figure 122

shows such a scaling solution. [MV1 Chapter 5.]

Shadeddistancematrix: A rough but simple way of graphically displaying a solution obtained

from a

cluster analysis

of a set of observations, so that the effectiveness of the solution

can be assessed. The individuals are rearranged so that those in the same cluster are

adjacent to one another in the distance matrix. Distances within a cluster will be small for

tight and well-separated clusters, while the distances between individuals in different

clusters will be large. Coding increasing distance by decreasing gray levels, should result

in a series of dark triangles under each tight well-separated cluster, while clusters that are

simply artifacts of the clustering procedure, will not exhibit such behaviour. See Fig. 123

for an example. [International Journal of Systematic Evolutionary Microbiology, 2004,

54,7–13.]

Shanno n , Cl aude Elwood ( 191 6^2001): Born in Petoskey, Michigan, Shannon obtained a

ﬁrst degree in mathematics and electrical engineering from the University of Michigan in

1936, and Ph.D. in mathematics from MIT in 1940. After graduating from MIT he took a

job at AT&T’s Bell Laboratories in New Jersey. It was in 1948 that Shannon published

one of the most signiﬁcant scientiﬁc contributions of the twentieth century, ‘A

Mathematical Theory of Communication’, which considered how to transmit messages

while keeping them from becoming garbled by noise. Shannon proposed that the infor-

mation content of a message had nothing to do with its contents but simply with the

number of 1’s and 0’s that it takes to transmit it. In 1958 he moved to MIT and his ideas

spread beyond communications engineering and computer science to the mathematics of

probability, DNA replication and cryptography. Shannon died in Medford, Massachusetts

on 24 February 2001.

389

Shannon’s information measure: A measure of the average information in an event with

probability P, given by

ðlog PÞ

The measure is intuitively reasonable in the sense that the more unlikely the event, the more

information is provided by the knowledge that the event has occurred. The presence of a

logarithm ensures that the information is additive. The logarithmic base is arbitrary and

determines the unit of information. Usually base 2 is used so that information is measured in

bits

. See also information theory and entropy index.[A First Course in Information

Theory, 2006, R. W. Yeung, Springer New York.]

Shape parameter: A general term for a parameter of a probability distribution function that

determines the ‘shape’ (in a sense distinct from location and scale) of the distribution within

a family of shapes associated with a speciﬁed type of variable. The parameter, γ,ofa

gamma

distribution

, for example.

Shapiro^Wilks W test: Tests that a set of random variables arise from a speciﬁed probability

distribution. Most commonly used to test for depatures from the normal distribution and the

exponential distribution

. For the latter, the

test statistic

Fig. 122 Horseshoe effect.

390

W ¼

n 1



x x

ð1Þ

i¼1

ðx

ðiÞ





xÞ

where x

ð1Þ

 x

ð2Þ

x

ðnÞ

are the ordered sample values and



x is their mean. Critical

values of W based on

simulation

studies are available in many statistical tables. [ARA

Chapter 10.]

Shattuck, Lemuel (1793^1859): Born in Ashby, Massachusetts, Shattuck studied in Detroit,

Michigan. At the age of 30 he returned to Concord, Massachusetts to operate a store, but

eventually moved to Boston where he became a bookseller and publisher. Shattuck became

interested in statistics while preparing a book, A History of the Town of Concord, one of the

chapters of which deals with statistical history. He became very inﬂuential in publicizing the

importance of high standard population statistics, and also founded the American Statistical

Association.

Shepard diagram: A type of plot used in

multidimensional scaling

in which observed

dissimilarity

coefﬁcients

are plotted against the distances derived from the scaling solution. By joining

together consecutive points in the diagram insight can be gained into the transformation

needed to convert the observed dissimilarities into distances. [Analysis of Proximity Data,

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

Sheppard’scorrections: Corrections to the sample

moments

when those are calculated from the

values of a grouped frequency distribution as if they were concentrated at the mid-points of

Fig. 123 An example of a shaded distance matrix.

391

the chosen class intervals. In particular the variance calculated from such data should be

corrected by subtracting from it h

2/12

where h is the length of the interval. [KA1 Chapter 2.]

Shewhart chart: A control chart designed to identify the time at which a signiﬁcant deviation in a

process occurs. See also cusum and exponentially weighted moving average control

chart.[Clinical Chemistry, 1981, 27, 493–501.]

Shewhart,W a lterA ( 1 891-1 967): Born in New Canton, Illinois, Shewhart received his Ph.D in

physics from the University of California, Berkeley in 1917. He began his professional life

as an engineer with the Western Electric Company and joined Bell Laboratories in 1925.

Most remembered for his invention of the statistical control of quality, Shewhart died on 13

March 1967.

Shi fted hats proced ure: A hybrid approach to classiﬁcation that uses both

kernel estimation

techniques and

ﬁnite mixture models

, essentially alternating between the two. [Pattern

Recognition, 1993, 26, 771–85.]

Shi ft outli er: An observation than needs a ‘shift’ of a particular magnitude for it to be consistent

with the rest of the sample. [Annals of the Institute of Statistical Mathematics, 1994, 46,

267–78.]

Shi ft tables: Tables used to summarize and interpret laboratory data usually showing, by treatment,

the number of patients who have ‘shifted’ from having ‘normal’ to having ‘abnormal’ values

during the trial and vice versa. [Analysis of Clinical Trials using SAS, A Practical Guide,

2005, A.Dmitrienko, G. Molenberghs, C. Chuang-Stein and W. Offen, SAS Publishing.]

Shock models: Probabilistic models of importance in the analysis of different risks in which the

effects of instantaneous harmful random events that represent danger to human beings, the

environment or to economic value are assessed. [Statistics and Probability Models, 2005,

74, 187–204.]

Shri nkage: The phenomenon that generally occurs when an equation derived from, say, a

multiple regression

, is applied to a new data set, in which the model predicts much less

well than in the original sample. In particular the value of the

multiple correlation

coefﬁcient

becomes less, i.e. it ‘shrinks’. See also shrinkage formulae .[Biometrics,

1976, 32,1–49.]

Shrinkage estimators: Estimators obtained by some common method of estimation such as

maximum likelihood estimation

least squares estimation

, modiﬁed in order to minimize

(maximize) some desirable criterion function such as

mean square error

. See also ridge

regression and James–Stein estimators.[Biometrics, 1976, 32,1–49.]

Shri nkage formulae: Usually used for formulae which attempt to estimate the amount of ‘shrink-

age’ in the

multiple correlation coefﬁcient

when a regression equation derived on one set of

data is used for prediction on another sample. Examples are Wherry’s formula,

¼ 1 ðn  1Þ=ðn k  1Þð1 R

and Herzberg’s formula,

¼ 1 

n 1

n  k  1



n 2

n  k  2



n þ 1



ð1 R

where R

is the multiple correlation coefﬁcient in the original sample, n is the sample size, k

is the number of variables in the equation and

is the estimated value of the multiple

392

correlation coefﬁcient when the regression equation is used for prediction in the new sample.

[Statistical Methods in Medical Research, 1997, 6, 167–83.]

Sibling estimators: Estimators based on sibling data that are designed to control for common

genetic and environmental family variables using sibling-speciﬁc effects. [International

Economic Review, 1975, 16, 422–449.]

Siegel^Tukey test: A

distribution free

test for the equality of variance of two populations having

the same median. See also Ansari–Bradley test, Conover test and Klotz test.[Handbook

of Parametic and Nonparametric Statistical Procedures, 3rd edn. D. J. Sheskin, Chapman

and Hall/CRC Press, Boca Raton.]

Sigmoid: A description of a curve having an elongated ‘S’-shape. [Modelling Binary Data, 2nd

edition, 2003, D. Collett, Chapman and Hall/CRC Press, London.]

Signedrank test: See Wilcoxon’s signed rank test .

Signed root transformation: A useful procedure for constructing

conﬁdence intervals

when the

observed

likelihood function

is noticeably non-normal. The transformation is

ðÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

½2fl

Þl

ðÞgsignð 

Þ

where l

ðÞ is the log-likelihood for n observations and

 is the maximum likelihood

estimator of . The quantity z

is approximately standard normal for n large for each ﬁxed .

[Biometrika, 1973, 60, 457–65.]

Significance level: The level of probability at which it is agreed that the null hypothesis will be

rejected. Conventionally set at 0.05. [SMR Chapter 8.]

Significance test: A statistical procedure that when applied to a set of observations results in a

p-value

relative to some hypothesis. Examples include

Student’s t-test

z-test

and

Wilcoxon’s signed rank test

. [SMR Chapter 8.]

Signtest: A test of the null hypothesis that positive and negative values among a series of observations

are equally likely. The observations are often differences between a response variable

observed under two conditions on a set of subjects. [SMR Chapter 9.]

Silhouette plot: A graphical method of assessing the relative compactness and isolation of groups

arising from a cluster analysis. For each object i an index s(i) [–1, 1] is deﬁned measuring

the (standardized) difference between the average dissimilarity (distance) of object i to all

other objects in its own cluster and the average dissimilarity (distance) of object i to all

objects in the nearest cluster to the one it is in. When s(i) is close to the value 1, object i is

nearer its own cluster than a neighbouring cluster and so is ‘well classi ﬁed’. When the

opposite is the case and s(i) is close to –1 the object i is taken to be ‘misclassiﬁed’. In the

silhouette plot the s(i) are dispalyed as horizontal bars, ranked in decreasing order for each

cluster. An example is shown in Fig. 124. See also icicle plot [Computational Statistics and

Data Analysis, 2006, 51, 526–544.]

Simes modified Bonferroni procedure: An improved version of the

Bonferroni correction

for conducting multiple tests of signiﬁcance. If H ¼fH

; H

; ...; H

g is a set of null

hypotheses with corresponding test statistics T

; T

; ...; T

, P-values, P

; ...; P

and

is the hypothesis that all H

; i ¼ 1; 2; ...; n are true, the suggested procedure rejects

if and only if there exists some value of j (1  j  n) such that P

ðjÞ

 jα=n where

ð1Þ

P

ðnÞ

are the ordered values of P

; ...; P

.[Biometrika, 1996, 83, 928–33.]

393

SIMEX: Abbreviation for simulation and extrapolation procedure.

Simi larity coefficient: Coefﬁcients ranging usually from zero to one used to measure the similarity

of the variable values of two observations from a set of multivariate data. Most commonly

used on binary variables. Example of such coefﬁcients are the

matching coefﬁcient

and

Jaccard’s coefﬁcient

. [MV1 Chapter 5.]

Simi larity matrix: A

symmetric matrix

in which values on the main diagonal are one and off-

diagonal elements are the values of some

similarity coefﬁcient

for the corresponding pair of

individuals. [MV1 Chapter 5.]

Simplerandom sampling: A form of

sampling design

in which n distinct units are selected from

the N units in a ﬁnite population in such a way that every possible combination of n units is

equally likely to be the sample selected. With this type of sampling design the probability

that the ith population unit is included in the sample is p

¼ n=N , so that the inclusion

probability is the same for each unit. Designs other than this one may also give each unit

equal probability of being included, but only here does each possible sample of n units have

the same probability. [SMR Chapter 8.]

Simple structure: See factor analysis.

Simplex algorithm: A procedure for maximizing or minimizing a function of several variables. The

basic idea behind the algorithm is to compare the values of the function being minimized at

the vertices of a simplex in the parameter space and to move this simplex gradually towards

the minimum during the iterative process by a combination of reﬂection, contraction and

expansion. See also Newton–Raphson method. [MV2 Chapter 12.]

Simplex models: Models for the analysis of relationships among variables that can be arranged

according to a logical ordering. [Psychometrika, 1962, 27, 155–62.]

Simpson’sparadox: The observation that a measure of association between two variables (for

example, type of treatment and outcome) may be identical within the levels of a third

variable (for example, sex), but can take on an entirely different value when the third

variable is disregarded, and the association measure calculated from the pooled data. Such

Fig. 124 Silhouette plot.

394

a situation can only occur if the third variable is associated with both of the other two

variables. As an example consider the following pair of

two-by-two contingency tables

giving information about amount of pre-natal care and survival in two clinics.

Clinic A

Infant’s survival

Died Survived Total

Less 3 176 179

Amount of Care

More 4 293 297

Total 7 469 476

Clinic B

Infant’s survival

Died Survived Total

Less 17 197 214

Amount of Care

More 2 23 25

Total 19 220 239

In both clinics A and B, the

chi-squared statistic

for assessing the hypothesis of independ-

ence of survival and amount of care leads to acceptance of the hypothesis. (In both cases the

statistic is almost zero.) If, however, the data are collapsed over clinics the resulting chi-

squared statistic takes the value 5.26, and the conclusion would now be that amount of care

and survival are related. See also collapsing categories and log-linear models.[Applied

Categorical Data Analysis, 1987, D. H. Freeman, Jr., Marcel Dekker, New York.]

Simulated annealing: An

optimization

technique, which is based on an analogy with the physical

process of annealing, the process by which a material undergoes extended heating and is

slowly cooled. Can be helpful in overcoming the local minimum problem by allowing some

probability of change in parameter values that lead to a local increase. [MV1 Chapter 4.]

Simulation: The artiﬁcial generation of random processes (usually by means of

pseudorandom

numbers

and/or computers) to imitate the behaviour of particular statistical models. See

also Monte Carlo methods. [SMR Chapter 7.]

Simulation and extrapolation (SIMEX) procedure: A procedure for calibration that is

computationally intensive but is applicable to highly

non-linear models

. The method can

be most clearly illustrated in simple

linear regression

when the explanatory variable is

subject to measurement error. If the regression model is

E ð Y jX Þ¼α þβX

but with W ¼ X þ U observed rather than X, where U has mean zero and variance 1 and

the measurement error variance 

is known. For any ﬁxed l

0 suppose that one repeatedly

‘adds on’ via simulation, additional error with mean zero and variance 

l to W, computes

the ordinary least squares slope each time and then takes the average. The simulation

estimator consistently estimates

gðlÞ¼



þ 

ð1 þlÞ

where 

is the variance of X. Plotting gðlÞ against l  0, ﬁtting a suitable model and then

extrapolating back to l ¼1 will yield a consistent estimate of β.[Measurement Error in

Nonlinear Models,: A Modern Perspective, 2nd edn, 2006, R. J. Carroll, D. Ruppert, L. A.

Stefanski, and C. Crainiceau, Chapman and Hall/CRC, Boca Raton,]

395

Simulation envelope: A

conﬁdence interval

about the expected pattern of

residuals

if the ﬁtted

model is true, constructed by simulating the distribution of the residuals taking estimated

model parameters as true values. [Statistical Inference, 2nd edn, 2002, P. H. Garthwaite, I. T.

Joliffe and B. Jones, Oxford. University Press, Oxford.]

Simultaneous confidence interval: A

conﬁdence interval

(perhaps more correctly a region)

for several parameters being estimated simultaneously. [KA2 Chapter 20.]

Simultaneous inference: Inference on several parameters simultaneously, when the parameters

are considered to constitute a family in some sense. See also multiple comparisons.

[Biometrical Journal, 2008, 50, 346–363.]

Sinclair, John(1754^1835): Born in Thurso Castle, Caithness, Scotland, Sinclair was educated at

the high school of Edinburgh and at Edinburgh, Glasgow and Oxford universities where he

read law. His lifelong enthusiasm for collecting ‘useful information’ led to his work The

Statistical Account of Scotland which was published between 1791 and 1799 in 21 volumes

each of 600–700 pages. One of the ﬁrst data collectors, Sinclair ﬁrst introduced the words

‘statistics’ and ‘statistical’ as now understood, into the English language. He died on 21

December 1835 in Edinburgh.

Single-blind: See blinding.

Single-case study: Synonym for N of 1 clinical trial.

Sing le index model: A generalization of the linear regression model and given by EðyjxÞ¼gðxÞ

where g is an unknown function. Often used in the ﬁnance industry to measure risk and return

of a stock. [Biometrika, 2007, 94,217–229.]

Sing l e-l inkage cl uste ring: A method of

cluster analysis

in which the distance between two

clusters is deﬁned as the least distance between a pair of individuals, one member of the

pair being in each group. [Cluster Analysis, 4th edition, 2001, B. S. Everitt, S. Landau and

M. Leese, Arnold, London.]

Sing le-masked: Synonym for single-blind.

Si ngl e sample t-test: See Student’s t-tests.

Singly censored data:

Censored observations

that occur in

clinical trials

where all the patients

enter the study at the same time point, and where the study is terminated after a ﬁxed time

period. See also progressively censored data.[Applied Life Data Analysis, 2004,

W. Nelson, Wiley, New York.]

Sing u la r mat rix: A square matrix whose determinant is equal to zero; a matrix whose inverse is not

deﬁned. See also Moore–Penrose inverse.

Singular value decomposition: The decomposition of an r  c matrix, A into the form

A ¼ USV

where U and V

are orthogonal matrices and S is a diagonal matrix. The basis of several

techniques of multivariate analysis including

correspondence analysis

. [MV1 Chapter 4.]

Sinh-normal distribution: The probability distribution, f(x), given by

f ðxÞ¼½2=ðα

ﬃﬃﬃﬃﬃﬃ

Þcosh½ðx  γÞ=expfð2=α

Þsinh

½ðx  γÞ=g

The distribution is symmetric about the location parameter γ, is strongly unimodal for α  2

and bimodal for α

2. The distribution of the logarithm of a random variable having a

396

Birnbaum–Saunders distribution

. Some examples are given in Fig. 125.[International

Maths Forum, 2006, 35, 1709–1727.]

Six sigma initiative: A programme aimed at the near elimination of defects from every product,

process and transaction within a company by adopting a highly disciplined and statistically

based approach. [Quality Progress, 1988, May, 60–4.]

SiZer map: A graphical device for use in association with smoothing methods in data analysis, that

helps to answer which observed features are ‘really there’ as opposed to being spurious

sampling artifacts. [Journal of the American Statistical Association, 1999, 94, 807–23.]

Skewness: The lack of symmetry in a probability distribution. Usually quantiﬁed by the index, s,

given by

s ¼



3=2

where µ

and µ

are the second and third

moments

about the mean. The index takes the value

zero for a

symmetrical distribution

. A distribution is said to have positive skewness when it

has a long thin tail to the right, and to have negative skewness when it has a long thin tail to

the left. Examples appear in Fig. 126. See also kurtosis. [KA1 Chapter 3.]

Skew -no rmal distr i buti o n: A probability distribution, f(x), given by

f(x)

−4 −2 0 2 4

0.0 0.2 0.4 0.6 0.8 1.0

alpha = 1.0,gamma = 0.5

alpha = 1.0,gamma = 3.0

alpha = 5.0,gamma = 1.0

Fig. 125 Sinh-normal distributions for several values of α and γ.

397

f ðxÞ¼2

ﬃﬃﬃﬃﬃﬃ



1

ﬃﬃﬃﬃﬃﬃ



du; 1

For l ¼ 0 this reduces to the standard normal distribution. [Biometrika, 1996, 83, 715–26.]

Skew-symmet ric matrix: A matrix in which the elements a

satisfy

¼ 0; a

¼a

i 6¼ j

An example of such a matrix is A given by

A ¼

013

10 2

3 20

Sl iced i nverse reg ressio n: A data-analytic tool for reducing the number of explanatory variables

in a regression modelling situation without going through any parametric or nonparametric

model-ﬁtting process. The method is based on the idea of regressing each explanatory

variable on the response variable, thus reducing the problem to a series of one-dimensional

regressions. [Journal of the American Statistical Association, 1994, 89, 141–8.]

Sli ding square plot: A graphical display of

paired samples

data. A scatterplot of the n pairs of

observations ðx

; y

Þ forms the basis of the plot, and this is enhanced by three

box-and-

whisker plots

, one for the ﬁrst observation in each pair (i.e. the control subject or the

measurement taken on the ﬁrst occasion), one for the remaining observation and one for

the differences between the pairs, i.e. x

 y

. See Fig. 127 for an example. [The American

Statistician, 1994, 48, 249–253.]

Sli me plot: A method of plotting

circular data

recorded over time which is useful in indicating changes

of direction. [Statistical Analysis of Circular Data, 1995, N. I. Fisher, Cambridge University

Press, Cambridge.]

Sloperatioassay: A general class of biological assay where the

dose–response

lines for the standard

test stimuli are not in the form of two parallel regression lines but of two different lines with

Distribution with positive skewness.

Distribution with ne

ative skewness.

Fig. 126 Examples of skewed

distributions.

398