Everitt B.S. The Cambridge Dictionary of Statistics

Подождите немного. Документ загружается.

Modeller and Predictor, 1995, S. J. Koopman, A. C. Harvey, J. A. Doornik and N. Shephard,

Chapman and Hall/CRC Press, London.]

Standa rd curve: The curve which relates the responses in an assay given by a range of standard

solutions to their known concentrations. It permits the analytic concentration of an unknown

solution to be inferred from its assay response by interpolation.

Standard design: Synonym for Fibonacci dose escalation scheme.

Standard deviation: The most commonly used measure of the spread of a set of observations.

Equal to the square root of the variance. [SMR Chapter 3.]

Standard error: The standard deviation of the

sampling distribution

of a statistic. For example, the

standard error of the sample mean of n observations is =

ﬃﬃﬃ

, where 

is the variance of the

original observations. [SMR Chapter 8.]

Standard gamble: An alternative name for the von Neumann–Morgensten standard gamble.

Standa rd ga mma d ist ri buti o n: See gamma distribution.

Standa rd izati on: A term used in a variety of ways in medical research. The most common usage is

in the context of transforming a variable by dividing by its standard deviation to give a new

variable with standard deviation 1. Also often used for the process of producing an index of

mortality, which is adjusted for the age distribution in a particular group being examined.

See also standardized mortality rate, indirect standardization and direct standardiza-

tion. [SMR Chapter 2.]

Standardizedmortalityrate (SMR): The number of deaths, either total or cause- speciﬁc, in a

given population, expressed as a percentage of the deaths that would have been expected if

the age and sex-speciﬁc rates in a ‘standard’ population had applied. [SMR Chapter 2.]

Standard ized range p roced ure: See many outlier detection procedures.

Observation

1 4 7 10 13 16 19 22 25 28 31 34 4037

41 38 3235 29 26 23 20 17 14 811

N of observations used for estimation

Fig. 131 Stalactite plot of seven climate and ecology variables measured on 41 cities in the USA.

409

Standar d ized reg ressio n coeff ic ient: See beta coefﬁcient.

Standardizedresidual: See residual.

Standard logistic distribution: See logistic distribution.

Standard normal variable: A variable having a normal distribution with mean zero and variance

one. [SMR Chapter 4.]

Standard scores: Variable values transformed to zero mean and unit variance.

Star pl ot: A method of representing

multivariate data

graphically. Each observation is represented

by a ‘star’ consisting of a sequence of equiangular spokes called radii, with each spoke

representing one of the variables. The length of a spoke is proportional to the value of the

variable it represents relative to the maximum value of the variable across all the

observations in the sample. The star plots for each of the 41 cities in the USA constructed

from seven climate and ecology variables measured on each city are shown in Fig. 132.

Chicago is clearly identiﬁed as being very different from the other four cities.

STATA: A comprehensive software package for many forms of statistical analysis; particularly useful

for epidemiological and

longitudinal data

. [STATA Corporation, 4905 Lakeway Drive,

College Station, TX 77845, USA; Timberlake Consultants, Unit B3, Broomsley Business

Park, Worsley Bridge Rd., London SE26 5BN, UK; www.stata.com.]

Phoenix

Washington

Des Moines

Wichita

Kansas City

St. Louis

Buffalo

Providence

Norfolk

Memphis

Richmond

Seattle

Charleston

Milwaukee

Nashville

Dallas

Houston

Salt Lake Cit

Cincinnati

Columbus

Pittsburgh

PhiladelphiaCleveland

Minneapolis-St. Paul

Louisville

New Orleans

Baltimore

Detroit

AlbanyOmaha

Alburquerque

Jacksonville

Miami

Atlanta

Chicago

Indianapolis

Little Rock

San Francisco

Denver

Wilmington

Haitford

Fig. 132 Star plots for 41 cities in the United States.

410

State space: See stochastic process.

State-space representati o n o f ti me ser ies: A compact way of describing a

time series

based on the result that any ﬁnite-order linear difference equation can be rewritten as a ﬁrst-

order vector difference equation. For example, consider the following autoregressive model

þ a

t1

þ a

t2

¼ 

and write X

ð2Þ

¼ X

, X

ð1Þ

¼a

t1

ð¼ a

ð2Þ

t1

Þ then the model may be rewritten as

ð1Þ

ð2Þ

0 a

1 a



ð1Þ

t1

ð2Þ

t1





To recover X

from the vector ½X

ð1Þ

; X

ð2Þ



we use

¼½0; 1

ð1Þ

ð2Þ

The original model involves a two-stage dependence but the rewritten version involves only

a (vector) one-stage dependence. [Applications of Time Series Analysis in Astronomy and

Meterology, 1997, edited by T. Subba Rao, M. B. Priestley and O. Lessi, Chapman and Hall/

CRC Press, London.]

Stationarity: A term applied to

time series

spatial data

to describe their equilibrium behaviour. For

such a series represented by the random variables, X

; X

; ...; X

, the key aspect of the term

is the invariance of their joint distribution to a common translation in time. So the require-

ment of strict stationarity is that the joint distribution of fX

; X

; ...; X

g should be

identical to that of fX

þh

; X

þh

; ...; X

þh

g for all integers n and all allowable h,

1

1. This form of stationarity is often unnecessarily rigorous. Simpler forms are

used in practice, for example, stationarity in mean which requires that EfX

g does not

depend on t. The most used form of stationarity, second-order stationarity, requires that the

moments up to the second order, EfX

g; varfX

g and covfX

þh

; X

þh

g,1 i; j  n do not

depend on translation time. [TM2 Chapter 2.]

Stationary distribution: See Markov chain.

Stationary point process: A

stochastic process

deﬁned by the following requirements:

(a) The distribution of the number of events in a ﬁxed interval ðt

; t

Þ is invariant under

translation, i.e. is the same for ð t

þ h; t

þ h for all h.

(b) The joint distribution of the number of events in ﬁxed intervals ðt

; t

; ðt

; t

 is

invariant under translation, i.e. is the same for all pairs of intervals

ðt

þ h; t

þ h; ðt

þ h; t

þ h for all h.

Consequences of these requirements are that the distribution of the number of events in an

interval depends only on the length of the interval and that the expected number of events in

an interval is proportional to the length of the interval. [Spatial Statistics, 2nd edition, 2004,

B. D. Ripley, Wiley, New York.]

Statistic: A numerical characteristic of a sample. For example, the sample mean and sample variance.

See also parameter.

Statistical d iscl osure l im itati o n: Procedures whose purpose is to ensure that the risk of

disclosing conﬁdential information about identiﬁable persons, businesses etc., will be very

small. The goal of disclosure limitation is to achieve an acceptable balance between data

utility and disclosure risk. Data utility is a measure of the usefulness of a dataset for making

accurate inferences and disclosure risk measures the degree to which a dataset and its

411

realised statistics reveal sensitive information. A variety of methodologies have been

developed for this data protection task, for example, adding random noise to the original

data (data perturbation) and choosing random pairs of respondents and exchanging a

fraction of their data (data swapping). [Statistical Science, 2006, 21, 143–154.]

Statistical expert system: A computer program that leads a user through a valid statistical

analysis, choosing suitable tools by examining the data and interrogating the user, and

explaining its actions, decisions, and conclusions on request.

Statist icalgra p h ics: Graphics that display measured quantities by means of the combined use of

points, lines, a coordinate system, numbers, symbols, words, shading and colour.

Graphical displays are very popular; it has been estimated that between 900 billion

(9 × 10

) and 2 trillion (2 × 10

) images of statistical graphics are printed each year.

Perhaps one of the main reasons for such popularity is that graphical presentation of data

often provides the vehicle for discovering the unexpected; the human visual system is very

powerful in detecting patterns, although the following caveat from the late Carl Sagan

should be kept in mind namely, ‘Humans are good at discerning subtle patterns that are

really there, but equally so at imagining them when they are altogether absent.’ The prime

objective of a graphical display is to communicate to ourselves and others. Graphic design

must do everything it can to help people understand. In some cases a graphic is required to

give an overview of the data and perhaps to tell a story about the data. In other cases a

researcher may want a graphical display to suggest possible hypotheses for testing on new

data and after some model has been ﬁtted to the data a graphic that criticizes the model

may be what is needed (for example, a plot of

residuals

.) Examples of statistical graphics

are given in the following entries;

histogram, bar chart, pie chart, dot plot, scatterplot,

scatterplot matrix

, and

coplot

. See also graphical deception.[Visual Revelations, 1997,

H. Wainer, Springer, New York.]

Statist ical j ou rn al s: A list of journals which publish articles in statistical science is given on the

web site, www.statsci.org/jourlist.html. Amongst these are journals that publish primarily

theoretical papers, for example, Biometrika, Journal of the Royal Statistical Society, Series B

and Annals of Statistics, journals that also publish more applied papers like Journal of the

Royal Statistical Society, Series C and Journal of the American Statistical Association and

journals that publish papers in one particular area, for example medical statistics with

journals like Statistics in Medicine and Statistical Methods in Medical Research.

Statistical quality control (SPC): The inspection of samples of units for purposes relating to

quality evaluation and control of production operations, in particular to:

(1) determine if the output from the process has undergone a change from one point in

time to another;

(2) make a determination concerning a ﬁnite population of units concerning the overall

quality;

(3) screen defective items from a sequence or group of production units to improve the

resulting quality of the population of interest.

[Statistical Methods for Quality Improvement, 1989, T. P. Ryan, Wiley, New York.]

Statisti calquotat i o ns: These range from the well known, for example, ‘a single death is a tragedy,

a million deaths is a statistic’ (Joseph Stalin) to the more obscure ‘facts speak louder than

statistics’ (Mr Justice Streatﬁeld). Other old favourites are ‘I am one of the unpraised,

unrewarded millions without whom statistics would be a bankrupt science. It is we who

are born, marry and who die in constant ratios.’ (Logan Pearsall Smith) and ‘thou shalt not sit

with statisticians nor commit a Social Science’ (W. H. Auden).

412

Statistical societies: A list of professional societies with statistical interests and links to their

respective web sites is given on www.statsci.org/soc.html.

Statistical software: Computer programs that implement a wide range of statistical techniques

and which are reasonably easy to use even for non-statisticians. The use of statistical

computer packages by statisticians and other started in the 1950s and today there is a

plethora of packages, all of which can be used on a researchers PC. Perhaps the most widely

used are SAS, STATA, SPSS and STATISTICA, although R is favoured by many statis-

ticians. Such software enables even naive researchers to apply sophisticated statistical

methodology to their data which is, of course, not without its dangers. In all current

statistical software there is a heavy emphasis on graphical methods. See also BMDP,

GLIM, GENSTAT, MINITAB, S-PLUS, EGRET, BUGS, STATXACT and

LOGXACT .[American Statistician, 1994, 48, 254–255.]

Statistical surveillance: The continual observation of a

time series

with the goal of detecting an

important change in the underlying process as soon as possible after it has occurred. An

example of where such a procedure is of considerable importance is in monitoring foetal heart

rate during labour. [Journal of the Royal Statistical Society, Series A, 1996, 159,547–63.]

Statistics: Either the plural of statistic or the name of a discipline that many have tried to deﬁne; some

examples are

Statistics may be regarded as (i) the study of populations, (ii) as the study of variation,

(iii) as the study of methods for the reduction of data.

Statistics is concerned with the inferential process, in particular with the planning and

analysis of experiments or surveys, with the nature of observational errors and sources

of variability that obscure underlying patterns, and with the efﬁcient summarizing of

sets of data.

The technology of the scientiﬁc method

Statistics is a general intellectual method that applies wherever data, variation, and

chance appear. It is a fundamental method because data, variation and chance are

omnipresent in modern life. It is an independent discipline with its own core ideas,

rather than, for example, a branch of mathematics...Statistics offers general, funda-

mental and independent ways of thinking.

There is clearly no consensus but certain elements appear in most deﬁnitions namely, variation,

uncertainty, and inference. One thing that statistics is not is simply a branch of mathematics.

STATISTI CA: A comprehensive package for many forms of statistical analysis. [StatSoft, Inc., 2300

East 14th Street, Tulsa OK 74104, USA; www.statsoft.com.]

Statistics fo r the terri fied: A computer-aided learning package for statistics. See also Activ

Stats.[Statistics for the Terriﬁed, version 3.0, 1998, Radcliffe Medical Press, Oxford.]

STAT/TRA N SFE R: Software for moving data from one proprietary format to another. [Circle

System, 1001 Fourth Ave., Suite 3200, Seattle, WA 98154.]

STATXACT: A specialized statistical package for analysing data from

contingency tables

that pro-

vides exact p-values, which, in the case of sparse tables may differ considerable from the

values given by

asymptotic statistics

such as the

chi-squared statistic

. [www.Cytel.com/

products/statxact/]

Steepest descent: A procedure for ﬁnding the maximum or minimum value of several variables by

searching in the direction of the positive (negative) gradient of the function with respect to the

parameters. See also simplex method and Newton–Raphson method. [MV1 Chapter 4.]

413

Stem-and -leaf plot: A method of displaying data in which each observation is split into two parts

labelled the ‘stem’ and the ‘leaf’. A tally of the leaves corresponding to each stem has the

shape of a

histogram

but also retains the actual observation values. See Fig. 133 for an

example. See also back-to-back stem-and-leaf plot. [SMR Chapter 3.]

Stepwise reg ressi o n: See selection methods in regression.

Stereology: The science of inference about three-dimensional structure based on two-dimensional or

one-dimensional probes. Has important applications in mineralogy and metallurgy.

[Quantitative Stereology, 1970, E. E. Underwood, Addison-Wesley, Reading.]

Stieltjes integral: See Riemann–Stieltjes integral.

Stirling’sformula: The formula

n! ð2pÞ

nþ

n

The approximation is remarkably accurate even for small n. For example 5! is approximated

as 118.019. For 100! the error is only 0.08%. [Handbook of Mathematical and

Computational Science, 1998, J. W. Harris and H. Stocker, Springer, New York.]

Stochastic approximation: A procedure for ﬁnding roots of equations when these roots are

observable in the presence of statistical variation. [Stochastic Approximation, 2004, M. T.

Wasson, Cambridge University Press, Cambridge.]

Stochasticfrontiermodels: Models that postulate a function hðÞrelating a vector of explanatory

variables, x, to an output, y

y ¼ hðxÞ

where the function hðÞ is interpreted as reﬂecting best practice, with individuals typically

falling short of this benchmark. For an individual i , who has a measure of this shortfall, τ

with 0

¼ hðxÞτ

Fig. 133 A stem-and-leaf plot for the heights of 351 elderly women.

414

The model is completed by adding measurement error (usually assumed to be normal)

choosing a particular functional form for hðÞ and a distribution for τ

.[Journal of

Econometrics, 1977, 6,21–37.]

Stochastic matrix: See Markov chain.

Stochastic ordering: See univariate directional ordering.

Stochastic process: A series of random variables, fX

g, where t assumes values in a certain range

T. In most cases x

is an observation at time t and T is a time range. If T ¼f0; 1; 2; ...g the

process is a discrete time stochastic process and if T is a subset of the nonnegative real

numbers it is a continuous time stochastic process. The set of possible values for the process,

T, is known as its state space. See also Brownian motion, Markov chain and random

walk.[Theory of Stochastic Processes, 1977, D. R. Cox and H. D. Miller, Chapman and

Hall/CRC Press, London.]

Stoppi ng rules: Procedures that allow

interim analyses

clinical trials

at predeﬁned times, while

preserving the type I error at some pre-speciﬁed level. See also sequential analysis.

Stop screen design: See screening studies.

Strata: See stratiﬁcation.

Stratification: The division of a population into parts known as strata, particularly for the purpose of

drawing a sample. In addition in

epidemiology

for example, stratiﬁcation may be used to

asses associations between exposure and disease in strata of a third variable or possibly strata

deﬁned by several variables in an investigation of confounding and effect modiﬁcation.

Strat i f i ed Cox models: An extension of

Cox’s proportional hazards model

which allows for

multiple strata which divide the units into distinct groups, each of which has a distinct

baseline hazard function but common values for the coefﬁcient vector β.[Modelling

Survival Data, 2000, T. M. Therneau and P. M. Grambsch, Springer, New York.]

Stratifiedlogrank test: A method for comparing the survival experience of two groups of subjects

given different treatments, when the groups are stratiﬁed by age or some other prognostic

variable. [Modelling Survival Data in Medical Research, 2nd edition, 2003, D. Collett,

Chapman and Hall/CRC Press, London.]

Stratifiedrandomization: A procedure designed to allocate patients to treatments in

clinical trials

to achieve approximate balance of important characteristics without sacriﬁcing the advan-

tages of random allocation. See also minimization.[Journal of Clinical Epidemology, 1999,

52,19–26.]

Stratifiedrandom sampling: Random sampling from each strata of a population after

stratiﬁca-

tion

.[Survey Sampling Principles, 1991, E. K. Foreman, Marcel Dekker, New York.]

Streaky hypothesis: An alternative to the hypothesis of independent

Bernoulli trials

with a constant

probability of success for the performance of athletes in baseball, basketball and other sports. In

this alternative hypothesis, there is either nonstationarity where the probability of success does

not stay constant over the trials or autocorrelation where the probability of success on a given trial

depends on the player’s success in recent trials. [The American Statistician, 2001, 55,41–50.]

Stress : A term used for a particular measure of goodness-of-ﬁtin

multidimensional scaling

.[Analysis

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

Strict stationarity: See stationarity.

415

Stripes plots: A simple but effective plot for visualizing the distance of each object for its closest and

second-closest cluster centroids after applying a

k-means cluster analysis

. For each cluster,

k ¼ 1; ...; K there is a rectangular area which is vertically divided into K smaller rectangles

and horizontal lines are plotted for the distances of each object to its own cluster and then for

the distances to the nearest other cluster. An example where the clusters are well separated is

show in Figure 134 and one where the clusters are not so well separated in Figure 135

[Computational Statistics and Data Analysis, 2006, 51, 526–544.]

Stri p-pl ot designs: A design sometimes used in agricultural ﬁeld experiments in which the levels of

one factor are assigned to strips of plots running through the block in one direction. A

separate randomization is used in each block. The levels of the second factor are then applied

to strips of plots that are oriented perpendicularly to the strips for the ﬁrst factor. [The Design

of Experiments, 1988, R. Mead, Cambridge University Press, Cambridge.]

Structural equation model: A statistical model where exogenous variables (explanatory varia-

bles) can potentially affect endogenous variables (response variables) both directly and

indirectly via intervening variables. There could also be feedback effects although this is

uncommon. A structural equation model for a vector of observed endogenous variables y

and a vector of observed exogenous variables x is usually speciﬁed as

y ¼ By þGx þ 

Here B, which governs the relations among the endogenous variables, has zeros on the

diagonal and is of full rank. The regression parameter matrix G governs the regressions of

endogenous on exogenous variables and  is a vector of random disturbances. The “reduced

form” of a structural equation model is obtained by solving for endogenous variables in

terms of exogenous variables to give

y¼ðI  BÞ

1

Gx þðI  BÞ

1



distance from centroid

Fig. 134 Stripes plot for data containing three well separated clusters.

416

where P ¼ðI  BÞ

1

G gives the reduced form regression parameters.

Structural equation models with latent rather than observed variables are also of great

importance. In this case the model is composed of two submodels:

(1) a “structural model” specifying how different latent variables affect one another;

explicitly the submodel is

h ¼ Bh þ G þ 

where now h is a vector of latent endogenous variables,  is a vector of latent exogenous

variables and  is again a vector of random disturbances.

(2) a “measurement model” specifying how the latent variables are measured by

observed variables or indicators x and y; explicitly this second submodel is

speciﬁed as

x ¼ L

x þ d

y ¼ L

h þ

i.e., two con ﬁrmatory

factor models

in which L

and L

are factor loading matrices and d

and  vectors of unique factors. [Structural Equations with Latent Variables, 1989, K. A.

Bollen, Wiley, New York]

Structural nested model: A model for estimation of the causal effect of a time-dependent

exposure in the presence of

time-dependent covariates

that may simultaneously be con-

founders and intermediate variables. Causal effects are estimated using the

G-estimator

More complex but also more generally applicable approach than

marginal structural model

[Epidemiology, 1992, 3, 319–336.]

distance from centroid

Fig. 135 Stripes plot for data where the clusters are not so well separated.

417

Structural time series models: Regression models in which the explanatory variables are

functions of time, but with coefﬁcients which change over time. Thus within a regression

framework a simple trend would be modelled in terms of a constant and time with a random

disturbance added on, i.e.

¼ α þ βt þ 

; t ¼ 1; ...; T

This model suffers from the disadvantage that the trend is deterministic, which is too

restrictive in general so that ﬂexibility is introduced by letting the coefﬁcients α and β

evolve over time as

stochastic processes

. In this way the trend can adapt to underlying

changes. The simplest such model is for a situation in which the underlying level of the series

changes over time and is modelled by a

random walk

, on top of which is superimposed a

white noise

disturbance. Formally the proposed model can be written as

¼ 

þ 



¼ 

t1

þ 

for t ¼ 1; ...; T;

 N ð0;



Þand 

 N ð0;



Þ. Such models can be used for

forecasting

and also for providing a description of the main features of the series. See also STAMP.

[Statistical Methods in Medical Research, 1996, 5,23–49.]

Structural zeros: Zero frequencies occurring in the cells of

contingency tables

which arise because

it is theoretically impossible for an observation to fall in the cell. For example, if male and

female students are asked about health problems that cause them concern, then the cell

corresponding to say menstrual problems for men will have a zero entry. See also sampling

zeros.[The Analysis of Contingency Tables, 2nd edition, 1992, B. S. Everitt, Chapman and

Hall/CRC Press, London.]

Stua rt, Ala n ( 1922^1 998): After graduating from the London School of Economics (LSE), Stuart

began working there as a junior research ofﬁcer in 1949. He spent most of his academic

career at the LSE, working in particular on nonparametric tests and sample survey theory.

Stuart is probably best remembered for his collaboration with

Maurice Kendall

on the

Advanced Theory of Statistics.

Stuart^Maxwell test: A test of

marginal homogeneity

in a

square contingency table

. The test

statistic is given by

¼ d

1

where d is a column vector of any r–1 differences of corresponding row and column marginal

totals with r being the number of rows and columns in the table. The (r–1)×(r–1) matrix V

contains variances and covariances of these differences, i.e.

¼ n

þ n

 2n

¼ðn

þ n

where n

are the observed frequencies in the table and n

and n

are marginal totals. If the

hypothesis of marginal homogeneity is true then X

has a

chi-squared distribution

with r-1

degrees of freedom. [SMR Chapter 10.]

Studentization: The removal of a

nuisance parameter

by constructing a statistic whose

sampling

distribution

does not depend on that parameter.

Studentizedrange statistic: A statistic that occurs most often in

multiple comparison tests

.Itis

deﬁned as

418