Everitt B.S. The Cambridge Dictionary of Statistics

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Savage’s test: A

distribution free

procedure for the comparison of two samples of size n and m from

populations with cumulative probability distributions F

(x) and F

(x). For testing the null

hypothesis F

ðxÞ¼F

ðxÞ for all x against F

ðxÞF

ðxÞ for all x and F

ðxÞ

ðxÞ for

some x. The test statistic is

S ¼

i¼1

ðR

where R



are the ordered ranks associated with the sample of size n in the

combined sample, N = n + m and a

ðkÞ¼

j¼1

ðN j þ 1Þ

1

.Ifm and n are large and the

null hypothesis is true then S is approximately normal with mean equal to n and variance

given by

mnðN 1Þ

1

1 N

1

j¼1

ð1=jÞ

[Encyclopedia of Statistical Sciences, 2006, eds. S.Kotz, C. B. Read, N.Balakrishnan and B.

Vidakovic, Wiley, New York.]

Scaled deviance: A term used in

generalized linear models

for the

deviance

divided by the

scale

factor

. Equals the deviance unless

overdispersion

is being modelled. [GLM Chapter 2.]

Sca le factor: See generalized linear models.

Scale parameter: A general term for a parameter of a probability distribution that determines the

scale of measurement. For example, the parameter 

in a normal distribution.

Scaling: See attitude scaling, multidimensional scaling and optimal scaling.

Scanstatistic: A statistic for evaluating whether an apparent

disease cluster

in time or in space is due

to chance. The statistic employs a ‘moving window’ of a particular length and ﬁnds the

maximum number of cases revealed through the window as it moves over the entire time

period or spatial area. Approximations for an upper bound to the probability of observing a

certain size cluster under the null hypothesis of a

uniform distribution

are available. The

statistic has been applied to test for possible clustering of lung cancer at a chemical works,

rashes following injection of a varicella virus vaccine and trisomic spontaneous abortions.

See also geographical analysis machine.[International Journal of Health Geographics,

2005, 18,4–11.]

Scatte r: Synonym for dispersion.

Scatte r d i a gram: A two-dimensional plot of a sample of

bivariate observations

. The diagram is an

important aid in assessing what type of relationship links the two variables. An example is

shown in Fig. 118. See also correlation coefﬁcient and draughtsman’s plot. [SMR

Chapter 3.]

Scatte rg r am: Synonym for scatter diagram.

Scatte rplot: Synonym for scatter diagram.

Scatte rplot matr ix: An arrangement of the pairwise scatter diagrams of the variables in a set of

multivariate data in the form of a square grid. Such an arrangement may be extremely

useful in the initial examination of the data. Each panel of the matrix is a scatter plot

of one variable against another. The upper left-hand triangle of the matrix contains all

pairs of scatterplots and so does the lower right-hand triangle. The reasons for including

both the upper and lower triangles in the matrix, despite the seeming redundancy, is that

379

it enables a row and column to be visually scanned to see one variable against all others,

with the scales for the one variable lined up along the horizontal or the vertical. The plot

can be made even more useful by enhancing the individual scatterplots with, for example, a

ﬁtted linear regression or a

locally weighted regression

(or both) for the corresponding pair

of variables. Two examples of scatterplot matrices for data on sulphur dioxide concentration

and a number of other variables for some cities in the US are shown in Figures 119 and

120. The latter shows the ﬁtted linear regression for the two variables forming each panel in

the plot. [Visualizing Data, 1993, W. S. Cleveland, Hobart Press, Summit, New Jersey.]

Scenario analysis: A term used to describe a range of

time series

forecasts based on different

assumptions. [Journal of Forecasting, 1991, 10 , 549–64.]

Sceptical priors: See prior distributions.

Scheffe

,Henry(1907^1977):Born in New York City, Scheffé studied at the University of

Wisconsin, Madison receiving a Ph.D. in 1935 for his thesis entitled ‘Asymptotic solutions

of certain linear differential equations in which the coefﬁcient of the parameter may have a

zero’. Moved to Princeton in 1940 and then became Head of the Department of

Mathematical Statistics at Columbia University in 1951. His now classic work The

Analysis of Variance was published in 1959. Scheffé died on 5 July 1977 as a result of a

tragic cycling accident.

Mortality index

35 40

Mean annual temperature

45 50

60 70 80 90

100

Fig. 118 Scatter diagram for breast cancer mortality in a region against mean annual temperature in

the region.

380

Scheffe

’s test: A

multiple comparison test

which protects against a large

per-experiment error rate

In an

analysis of variance

of a

one-way design

, for example, the method calculates the

conﬁdence interval

for the difference between two means as









ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðg  1ÞF

g1;N g

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MSE



wherey

andy

are the observed means of groups i and j, g is the number of groups, MSE is the

mean square error in the

analysis of variance table

, n

and n

are the number of observations in

groups i and j,andF

g1;N g

is the F-value for some chosen signiﬁcance level. The total

sample size is represented by N.[Biostatistics: A Methodology for the Health Sciences,2nd

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

Schemper’smeasures:Two measures of explained variation in

Cox’s proportional hazards

model

, which correspond to the proportion of explained variation in the normal case. The

measures V

and V

are de ﬁned as follows

Fig. 119 Scatterplot matrix of seven variables recorded for a number of US cities.

381

1 V

1





1





l ¼ 1; 2

where S

=1 for individual i at all time points t

for which the individual is still alive, drops to

0.5 at the point at which the individual dies; and thereafter S

=0, K

is the total number of

deaths should individual i correspond to a death, otherwise it is the number of deaths before

the censoring time of the individual i,



is the

Kaplan–Meier estimate

of survival at time t

and



the estimate of survival for individual i at time point t

derived from the proportional

hazards model. [Biometrika, 1990, 77, 216–18.]

Schoenfieldre sidual: A diagnostic used in applications of

Cox’s proportional hazards model

. This

type of

residual

is not, in fact, a single value for each subject but a set of values, one for each

explanatory variable included in the ﬁtted model. Each such residual is the difference

between the jth explanatory variable and a weighted average of the values of the explanatory

variables over individuals in the risk set at the death time of the jth individual. The residuals

Fig. 120 Scatterplot matrix of variables recorded for a number of US cities with each panel enhanced

by the linear regression ﬁt.

382

have the property that in large samples their expected values are zero and that they are

uncorrelated with one another. [Modelling Survival Data in Medical Research, 2nd edition,

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

Schuster’s test: A test for non-zero periodic ordinates in the

periodogram

of a

time series

,which

assumes that the random component of the series is an independent zero mean normal series.

Under this assumption the probability distribution of each component of the periodogram Iðω

is proportional to

chi-squared distribution

with two degrees of freedom. Speciﬁcally for p 6¼ 0

Iðω

Þ=

 

where 

¼ varðX

Þ. Consequently the signiﬁcance of a given periodogram ordinate can be

tested. [Applications of Time Series Analysis in Astronomy and Meterology, edited by

E. Subba Rao, M. B. Priestley and O. Lessi, 1997, Chapman and Hall, London.]

Schwarz’s criterion: An index used as an aid in choosing between competing models. It is deﬁned as

 2L

þ m ln n

where n is the sample size, L

is the maximized

log-likelihood

of the model and m is the

number of parameters in the model. The index takes into account both the statistical good-

ness of ﬁt and the number of parameters that have to be estimated to achieve this particular

degree of ﬁt, by imposing a penalty for increasing the number of parameters. Lower values

of the index indicate the preferred model, that is, the one with the fewest parameters that still

provides an adequate ﬁt to the data. If n  8 this criterion will tend to favour models with

fewer parameters than those chosen by

Akaike’s information criterion

. See also Bayesian

information criterion (BIC), parsimony principle and Occam’s razor. [TMS Chapter 11.]

Score res id ua l: Synonym for Schoenﬁeld residual.

Score test: Typically (but not necessarily) a test for the hypothesis that a vector of parameters,

¼½

;

; ...;

, is the null vector. The test statistic is

s ¼ S

where S is the vector with elements @L=@

and L is the

log-likelihood

. V is the asymptotic

variance–covariance matrix

of the parameters. In contrast to the

likelihood ratio test

, the

score test only requires estimation under the null hypothesis. See also likelihood ratio test

and Wald’s test. [GLM Chapter 5.]

Scott, Eliz a beth Leona rd (1917^1988): Born in Fort Sills, Oklahoma, USA, Scott studied

astronomy at the University of California, Berkeley. During World War II she worked with

Jerzy Neyman

in the Statistical Laboratory at Berkeley on improving the precision of air

bombing. In 1962 Scott became Professor of Statistics in Berkeley. In her career she worked

on a variety of statistical problems including models for carcinogenesis and cloud seeding

experiments. She was President of the Institute of Mathematical Statistics in 1977. Scott died

on 20 December 1988 in Berkeley.

Scram d ata: Data produced by the nuclear power industry relating to unplanned shutting down of

nuclear plants by the reactor protection system following some transient event, such as the

loss of offsite power.

Scree d iag ram: A plot of the ordered eigenvalues of a

correlation matrix

, used to indicate the

appropriate number of factors in a

factor analysis

principal components analysis

.The

critical feature sought in the plot is an ‘elbow’, the number of factors then being taken as

383

the number of eigenvalues up to this point. An example of such a diagram is given in Fig. 121.

See also Kaiser’srule.[MV2Chapter 12.]

Screened-to-eligibleratio: The number of subjects that have to be examined in a

clinical trial

identify one protocol-eligible subject. [Critical Care Medicine, 2000, 28, 867–71.]

Screening experiments: Experiments in which many factors are considered with the purpose of

identifying those factors (if any) that have large effects. The factors that are identiﬁed as

important are then investigated more throughly in subsequent experiments. Such experi-

ments frequently employ

fractional factorial designs

. See also response surface method-

ology.[American Journal of Public Health, 2008, 98, 1354–1359.]

Screening studie s: Studies in which

diagnostic tests

are applied to a symptomless population in

order to diagnose disease at an early stage. Such studies are designed both to estimate disease

prevalence and to identify for treatment patients who have particular diseases. The proce-

dure is usually concerned with chronic illness and aims to detect disease not yet under

medical care. Such studies need to be carefully designed and analysed in order to avoid

possible problems arising because of

lead time bias

and

length biased sampling

. Most

suitable designs are based on

random allocation

. For example, in the

continuous screen

design

subjects are randomized either to a group that are given periodic screening through-

out the study, or to a group that do not get such screening but simply follow the usual medical

care practices. One drawback of this type of design is that the cost involved in screening all

the patients in the ‘intervention’ arm of the trial for the duration of the trial may be

prohibitive; if this is so an alternative approach can be used, namely the stop screen design,

in which screening is offered only for a limited time in the intervention group. [SMR

Chapter 14.]

SD: Abbreviation for standard deviation.

-distribution: See Johnson’s system of distributions.

Fig. 121 Scree diagram showing an ‘elbow’ at eigenvalue three.

384

-distribution: See Johnson’s system of distributions.

SE: Abbreviation for standard error.

Seasonally adjusted: A term applied to

time series

from which periodic oscillations with a

particular period, for example, one year have been removed. [SMR Chapter 4.]

Seaso nal varia tion: Although strictly used to indicate the cycles in a

time series

that occur yearly,

also often used to indicate other periodic movements. [SMR Chapter 7.]

Seber’smodel: A model for a ring recovery experiment in which R

birds are ringed in the ith year of

the study 1  i  k and m

of these are found and reported dead in the jth year i  j  k.

Letting 

be the probability that a bird survives its ith year of life conditional on it having

survived i-1 years, and λ be the probability that a bird, having died, is found and

reported, then given that a bird is ringed in year i the probability that it is recovered in

year i + j is l

...

ð1 

jþ1

Þ and the probability that it is never recovered is

1 lð1  



kþ1i

Þ; 1  i  k; 0  j  k  i.[Statistical Inference from

Capture Recapture Experiments, 1990, K. H. Pollock, J. D. Nicholls, C. Brownie and J. E.

Hines, The Wildlife Society.]

Seconda ry attack rate: The degree to which members of some collective or isolated unit, such as

a household, litter or colony, become infected with a disease as a result of coming into

contact with another member of the collective unit who became infected. [Statistics in

Medicine, 1996, 15, 2393–404.]

Second-level significance testing: See higher criticism .

Second-order stationarity: See stationarity.

Secretaryproblem: A problem in which n individuals applying for a job (for example, a secretarial

position) arrive sequentially in random order. Upon arrival each individual is measured on a

characteristic desirable for the position (for example, typing skill). The goal is to select the

applicant with the highest value for the characteristic assessed, but the individual must be

told immediately after the interview whether she or he has been hired. The optimal strategy is

to reject the ﬁrst s(n) candidates and to choose the ﬁrst, if any, among applicants

sðnÞþ1; ...; n who is the best seen so far, where sðnÞ=n  e

1

.[Chance Rules, 2nd edn,

2008, B. S. Everitt, Springer, New York.]

Secul ar tren d: The underlying smooth movement of a

time series

over a fairly long period of time.

[Proceedings of the Nutrition Society, 2000, 59, 317–324.]

Seemingly unrelated regression (SU R): A set of regression models that appear unrelated

because they have different response and explanatory variables but which are estimated from

the same set of data and so may have correlated error terms.

Ordinary least squares

will yield

unbiased and consistent estimates for the parameters of each separate model but because this

approach ignores the correlation of the error terms the estimates will not be efﬁcient.

[Seemingly Unrelated Regression Equations Models, 1987, V. K. Srivastava and D. E. A.

Giles, CRC, Boca Raton.]

Seg m e n tation: The division of an image into regions or objects. Often a necessary step before any

quantitative analysis can be carried out. Also used for ﬁnding market segments in market

research usually by some form of cluster analysis.[Statistics for Spatial Data, 1991,

N. A. C. Cressie, Wiley, New York.]

385

Segregation analysis: A method of collecting evidence for the presence of discrete phenotypic

classes that are probabilistically determined by the underlying

genotype

at a locus by

examining the joint distribution of

phenotype

in families. [Statistics in Human Genetics,

1998, P. Sham, Arnold, London.]

Selection bias: The bias that may be introduced into all types of scientiﬁc investigations

whenever a treatment is chosen by the individual involved or is subject to constraints

that go unobserved by the researcher. If there are unobserved factors inﬂuencing health

outcomes and the type of treatment chosen, any direct links between treatment and

outcome are confounded with unmeasured variables in the data. A classic example of

this problem occurred in the Lanarkshire milk experiment of the 1920s. In this trial

10 000 children were given free milk supplementation and a similar number received no

supplementation. The groups were formed by random allocation. Unfortunately, how-

ever, well-intentioned teachers decided that the poorest children should be given priority

for free milk rather than sticking strictly to the original groups. The consequence was

that the effects of milk supplementation were indistinguishable from the effects of

poverty. The term may also be used in the context of multiple regression when subsets

of variables chosen by the same procedure may be optimal for the original data but may

perform poorly in future data and in general when individuals included in a study are

not representative of the

target population

for the study. [Statistics in Medicine, 1988, 7,

417–22.]

Selection methods in regression: In regression analysis an

underﬁtted model

can lead to

severely biased estimation and prediction. In contrast, an

overﬁtted model

can seriously

degrade the efﬁciency of of the resulting parameter estimates and predictions. Consequently

a variety of techniques all with the aim of selecting the most important explanatory variables

for predicting the response variable and thereby obtaining a parsimonious and effectively

predictive model have been developed. In

multiple linear regression

, for example, three

commonly used methods are forward selection, backward elimination and a combination of

both of these known as stepwise regression. The criterion used for assessing whether or not a

variable should be added to an existing model in forward selection or removed from an

existing model in backward elimination is, essentially the change in the residual sum-of-

squares produced by the inclusion or exclusion of the variable. Speciﬁcally in forward

selection an ‘F-statistic’ known as the F-to-enter is calculated as

F ¼

RSS

 RSS

mþ1

RSS

mþ1

=ðn  m  2Þ

where RSS

and RSS

m+1

are the residual sums of squares when models with m and m+1

explanatory variables have been ﬁtted. The F-to-enter is then compared with a preset term;

calculated Fs greater than the preset value lead to the variable under consideration being

added to the model. In backward selection a calculated F less that a corresponding F-to-

remove leads to a variable being removed from the current model. In the stepwise procedure,

variables are entered as with forward selection, but after each addition of a new variable,

those variables currently in the model are considered for removal by the backward elimi-

nation process. In this way it is possible that variables included at some earlier stage might

later be removed, because the presence of new variables has made their contribution to the

regression model no longer important. It should be stressed none of these automatic

procedures for selecting variables is foolproof and they must be used with care.

Other methods that can be used for selecting variables are

all subsets regression

and

Mallow’s C

statistic. Akaike’s information criterion might also be useful in comparing

386

competing regression models. But all these methods suffer from a major drawback because

parameter estimation and model selection are considered as two different processes and this

can result in instability and complicated stochastic properties.. And when applying regres-

sion analysis to

high-dimensional data

special selection techniques that are computationally

efﬁcient and statistically accurate are needed for retaining relevant and deleting perhaps

thousands of irrelevant variables. Examples of such methods are the

Dantzing selector

, the

lasso, sure screening

and

smoothly clipped absolute deviation

. [ARA Chapter 6.]

Selection models: A class of models used to handle

missing data

. For a unit i, let y

be the response

variable of interest and M

a missing indicator taking the value 1 if the response is missing

and 0 if not. Let θ be the parameters of the model for y

, which are of main interest, and ψ the

parameters of the missingness mechanism. In a selection model, the joint distribution of y

and

is decomposed as

f ðy

; M

j; ψÞ¼f ðy

jÞf ðM

; ψÞ

where the ﬁrst distribution characterises the distribution of y

in the population and the

second distribution represents the probability of missing data as a function of y

, and the

parameter vectors  and ψ are distinct. The most famous model of this type is the

Heckman

selection model

. Selection models are more natural and easier to interpret than pattern

mixture models but have been criticised for being sensitive to distributional assumptions.

See also Pattern mixture model.[Statistical Analysis with Missing Data, 2002, R. J. A.

Little and D. B. Rubin, Wiley, New York.]

Self-modeling regression (SEMOR): A method developed to model groups of one-

dimensional response curves when the curves demonstrate a similar shape with individual

differences in scale and location. [Statistica, Sinica, 2004, 14, 695–711.]

Self- pairing: See paired samples.

Self-s imilarity: A term that applies to geometric shapes if the same geometric structures are observed

independently of the distance from which one looks at the shape. Also used to describe

certain

stochastic processes

, fX

g that are such that for every sequence of time points

; t

; ...; t

and for any positive constant c, then c

H

ðX

; X

; ...; X

Þ has the same

distribution as X

; X

; ...; X

, where H is known as the self-similarity parameter.[The

Fractal Nature of Geometry, 1982, B. B. Mandelbrot, Freeman, San Francisco.]

Self-simi larity parameter: See self-similarity.

Semi -i nter qua rti l e range: Half the difference between the upper and lower quartiles.

Semi-continuous variables: Variables that take on a single discrete value (often zero) with

positive probability but are otherwise continuously distributed. Such variable occur in a

wide variety of contexts; for example, variables in economic surveys relating to income or

expenditure where many individuals will have no income or expenditure of a certain type in

a given year. A commonly used approach to modelling such variables is one that involves

two stages: (a) a

logistic regression model

for predicting whether the response is zero or non-

zero and (b) a continuous, often

linear regression model

, for predicting the level of response

among those individuals whose response are non-zero. [Analysis of Incomplete Data, 1997,

J. Schafer, Wiley, New York.]

Semi-parametr ic efficiency bounds: Extensions of efﬁciency bounds such as the

Cramér-

Rao lower bound

and the

Gauss-Markov theorem

to statistical models with a nonparametric

component. [Journal of Applied Econometrics, 1990, 5,99–135.]

387

Semi-parametric regression: Regression models that are a compromise between parametric and

nonparametric models, which aim to offer the ﬂexibility of the latter whilst retaining a certain

amount of the parsimony and structure of the former. The most widely known semiparametric

regression model is

Cox’s proportional hazards model

.[Econometrica, 1996, 64,103–37.]

Semi-variogram: Synonym for variogram.

SEMOR: Acronym for self-modeling regression.

Sensitiv ity: An index of the performance of a

diagnostic test

, calculated as the percentage of

individuals with a disease who are correctly classiﬁed as having the disease, i.e. the

condi-

tional probability

of having a positive test result given having the disease. A test is sensitive

to the disease if it is positive for most individuals having the disease. See also speciﬁcity,

ROC curve and Bayes’ theorem. [SMR Chapter 14.]

Sensitiv ity ana lysis: See uncertainty analysis.

Sequence models: A class of statistical models for the simultaneous analysis of multiple ordered

events so as to identify their sequential patterns. An extension of

log-linear models

in which

a set of parameters is used to characterize marginal

odds

and

odds ratios

of frequencies

summed across sequence patterns for each combination of the occurrence/non-occurrence of

events. These parameters are used for the analysis of the occurrence and association of

events. Another set of parameters characterizes conditional odds and odds ratios among

sequence patterns within each combination of the occurrence/non-occurrence of events. The

parameters are used for the analysis of sequencing of events.

Sequential all ocation procedures: Procedures for allocating patients to treatments in a prospec-

tive clinical trial in which they enter the study sequentially. At the time of entry values of

prognostic factors that might inﬂuence the outcome of the trial are often known and procedures

for allocation that utilize this information have received much attention. One of the most

widely used of these procedures is the

permuted block allocation

in which strata are deﬁned in

terms of patients at allocation having the same values of all prognostic factors. In its simplest

form this method will randomly allocate a treatment to an incoming patient when balance

exists among the treatments within the stratum to which the new patient belongs. If balance

does not exist the treatment that will achieve balance will be allocated. A problem is that, in

principle, an investigator with access to all previous allocations can calculate, for a known set

of prognostic factors, the treatment allocation for the next patient and consequently makes

possible conscious selection bias. [Statistics in Medicine, 1986, 5,211–29.]

Sequentialanalysis: A procedure in which a statistical test of signiﬁcance is conducted repeatedly

over time as the data are collected. After each observation, the cumulative data are analysed

and one of the following three decisions taken:

stop the data collection, reject the null hypothesis and claim statistical signiﬁcance;

stop the data collection, do not reject the null hypothesis and state that the results are

not statistically signiﬁcant;

continue the data collection, since as yet the cumulated data are inadequate to draw a

conclusion.

In some cases, open sequential designs, no provision is made to terminate the trial with the

conclusion that there is no difference between the treatments, in others, closed sequential

designs, such a conclusion can be reached. In group sequential designs,

interim analyses

are

undertaken after each accumulation of a particular number of subjects into the two groups.

Suitable values for the number of subjects can be found from the overall signiﬁcance level,

the true treatment difference and the required power. [SMR Chapter 15.]

388