Everitt B.S. The Cambridge Dictionary of Statistics

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sample members until the desired sample size is reached. By using a mathematical model that

weights the sample to compensate for the fact that the sample was collected in a non-random

way, the data provided by such a sampling scheme can be used to provide asymptotically

unbiased estimates about the target population. An example of the use of this approach is the

estimation of drug users in New York who have HIV. [AIDS and Behavior, 2008, 12, 105–130.]

Responders versus non-responders analysis: A comparison of the survival experience of

patients according to whether or not there is some observed response to treatment. In general

such analyses are invalid because the groups are deﬁned by a factor not known at the start of

treatment. [Psychiatric Genetics, 2001, 11,41–3.]

R esponse bias: The systematic component of the difference between information provided by

survey respondent and the ‘truth’.

Response feature analysis: An approach to the analysis of

longitudinal data

involving the

calculation of suitable summary measures from the set of repeated measures on each subject.

For example, the mean of the subject ’s measurements might be calculated or the maximum

value of the response variable over the repeated measurements, etc. Simple methods such as

Student’s t-tests

Mann–Whitney tests

are then applied to these summary measures to

assess differences between treatments. [Analysis of Longitudinal Data, 2nd edition, 2002,

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

Response rate: The proportion of subjects who respond to, usually, a postal questionnaire.

Response surface methodology (RSM): A collection of statistical and mathematical techni-

ques useful for developing, improving and optimizing processes with important applications

in the design, development and formulation of new products, as well as in the improvement

of existing product designs. The most extensive applications of such methodology are in the

industrial world particularly in situations where several input variables potentially inﬂuence

some performance measure or quality characteristic of the product or process. The main

purpose of this methodology is to model the response based on a group of experimental

factors, and to determine the optimal settings of the experimental factors that maximize or

minimize the response. Most applications involve ﬁtting and checking the adequacy of

models of the form

EðyÞ¼β

þ x

β þ x

where x ¼½x

; x

; ...; x



is a vector of k factors, β ¼½β

; β

; ...; β



and

B ¼



 β

The vector β and the matrix B contain the parameters of the model. See also criteria of

optimality, design matrix and multiple regression.[Empirical Model-Building with

Response Surfaces, 1987, G. E. P. Box and N. D. Draper, Wiley, New York.]

R esponse var iab le: The variable of primary importance in investigations since the major objective

is usually to study the effects of treatment and/or other explanatory variables on this variable

and to provide suitable models for the relationship between it and the explanatory variables.

[SMR Chapter 11.]

Restricted maximum likelihood estimation: Synonym for residual maximum likelihood

estimation.

369

R esubstituti on error rate: The estimate of the proportion of subjects misclassiﬁed by a rule

derived from a

discriminant analysis

, obtained by reclassifying the

training set

using the

rule. As an estimate of likely future performance of the rule, it is almost always optimistic,

i.e. it will underestimate the ‘true’ misclassiﬁcation rate. See also jackknife and b632

method. [MV2 Chapter 9.]

Resubstitution method: See error-rate estimation.

Retrospective cohort study: See retrospective study.

Retrospective study: A general term for studies in which all the events of interest occur prior to the

onset of the study and ﬁndings are based on looking backward in time. Most common is the

traditional case-control study, in which comparisons are made between individuals who have

a particular disease or condition (the cases) and individuals who do not have the disease (the

controls). A sample of cases is selected from the population of individuals who have the

disease of interest and a sample of controls is taken from among those individuals known not

to have the disease at the end of the observation period. Information about possible

risk factors

for the disease is then obtained retrospectively for each person in the study by examining past

records, by interviewing each person and/or interviewing their relatives, or in some other way.

In order to make the cases and controls otherwise comparable, they are frequently matched on

characteristics known to be strongly related to both disease and exposure leading to a matched

case-control study. Age, sex and socioeconomic status are examples of commonly used

matching variables. Also commonly encountered is the retrospective cohort study,inwhich

a past cohort of individuals are identiﬁed from previous information, for example, employ-

ment records, and their subsequent mortality or morbidity determined and compared with the

corresponding experience of some suitable control group. [SMR Chapter 5.]

Reversal designs: Synonym for switchback designs.

Reverse-causality: Synonym for protopathic bias.

Reversible jump sampler: An improved

Markov chain Monte Carlo method

which allows

‘jumps’ between parameter subspaces of differing dimensionality. [Biometrika, 1995, 82,

711–32.]

Reversibility: A term used in the context of

time series

to imply that all modelling and statistical

analyses can be performed equally on the reversed time-ordered values as on the original

sequence. [International Statistical Review, 1991, 59,67–79.]

RHC: Abbreviation for Rao–Hartley–Cochran method.

Rice distribution: Synonym for the noncentral chi-squared distribution.

R idge esti mator: See ridge regression.

R idge regressi on: A method of

multiple linear regression

designed to overcome the possible

problem of

multicollinearity

among the explanatory variables. Such multicollinearity

often makes it difﬁcult to estimate the separate effects of variables on the response. In

such regression, the regression coefﬁcients are obtained by solving



ðÞ¼ðX

X þIÞ

1

where θ ≥ 0 is a constant. Generally values in the interval 0    1 are appropriate. The

ridge estimator b



ðÞ is not an unbiased estimator of β as is the ordinary least squares

estimator b. This form of regression seeks to ﬁnd a set of regression coefﬁcients that is more

370

‘stable’ in the sense of having a small mean square error. To obtain the ridge estimator

explicitly a value for θ must be speciﬁed. This is usually done by plotting the values of b



ðÞ

against θ, This display is called the ridge trace. An example is given in Fig. 115. See also

biased estimator. [ARA Chapter 12.]

Ridge trace: See ridge regression.

Ridit analysis: A method of analysis for ordinal variables that proceeds from the assumption that the

ordered categorical scale is an approximation to an underlying, but not directly measurable,

continuous variable. The successive categories are assumed to correspond to consecutive

intervals on the underlying continuum. Numerical values called ridits are calculated for each

category, these values being estimates of the probability that a subject’s value on the

underlying variable is less than or equal to the midpoint of the corresponding interval.

These scores are then used in subsequent analyses involving the variable. [Journal of Dental

Research, 1979, 58, 2080–2084.]

Ridits: See ridit analysis.

Riemann integral: See Riemann–Stieltjes integral.

R iemann ^ Stieltjes i nteg ral: A more general type of integration than the usual

Cauchy integral

which involves the sum;

¼ ψð

ÞfFðx

ÞFðaÞg þψð

ÞfFðx

ÞFðx

Þgþþψð

nþ1

ÞfFðbÞFðx

Þg

where ψ (x) is a continuous function of x and θ

is in the range a to x

, θ

is in the range x

to x

andsoon.AsthelengthoftheintervalstendtozerouniformlyS

tends to a limit that is

independent of the location of the points θ or the boundary points of the intervals. This limit is

written as

ψðxÞdF

403020100–10–20

Theta

Regression coeffecients

0.0 0.5 1.0 1.5 2.0 2.5

Pop

Manuf

Temp

Fig. 115 Ridge trace for regression example with three explanatory variables, Manuf, Pop and Temp.

371

and is known as the Stieltjes integral of ψ (x) with respect to F (x). When F(x)=x it becomes

the Riemann integral. [KA1 Chapter 1.]

Right censored: See censored observations.

Rim estimation: Synonym for raking adjustment.

Risk : A term often used in medicine for the probability that an event will occur, for example, that a

person will become ill, will die, etc.

Risk factor: An aspect of personal behaviour or lifestyle, an environmental exposure, or an inborn or

inherited characteristic which is thought to be associated with a particular disease or condition.

Risk measure: A function that assigns real numbers to random variables representing uncertainty

payoffs, for example, insurance losses. [Insurance Mathematics and Economics , 1999, 25,

337–347.]

Risk set: A term used in

survival analysis

for those individuals who are alive and uncensored at a time

just prior to some particular time point.

Robbins, Herbert (1915^2001): Robbins obtained a ﬁrst degree from Harvard in 1935 and a

Ph.D. in pure mathematics in 1938. After the war he joined the newly formed department at

the University of North Carolina, moving in 1953 to Columbia University as the Higgins

Professor of Mathematical Statistics. Robbins made important contributions to empirical

Bayes methods and stochastic approximation, but was perhaps best known for his very

popular book What is Mathematics? Robbins died in Princeton, NJ, on 12 February 2001.

Robinson matrix : A matrix whose elements are monotonically non-decreasing as one moves from

the diagonal within each row and within each column. Such matrices are important in some

areas of

cluster analysis

.[Journal of Classiﬁcation, 1984, 1,75–92.]

R obust esti mat i on: Methods of estimation that work well not only under ideal conditions, but also

under conditions representing a departure from an assumed distribution or model. See also

high breakdown methods.[Introduction to Robust Estuiation and Hypotresis Testing,

2005, R. R. Wilcox, Academic Press.]

Robustregression: A general class of statistical procedures designed to reduce the sensitivity of the

parameter estimates to failures in the assumption of the model. For example,

least squares

estimation

is known to be sensitive to

outliers

, but the impact of such observations can be

reduced by basing the estimation process not on a sum-of-squares criterion, but on a sum of

absolute values criterion. See also M-estimators.[Robust Statistics, 2nd edition, 2003, P. J.

Huber, Wiley, New York.]

Robust statistics: Statistical procedures and tests that still work reasonably well even when the

assumptions on which they are based are mildly (or perhaps moderately) violated.

Student’s

t-test

, for example, is robust against departures from normality. See also high breakdown

methods and robust estimation.[Robustness in Statistics, 1979, R. Launer and

G. Wilkinson, Academic Press, New York.]

ROC curve: Abbreviation for receiver operating characteristic curve.

Rootogram: A diagram obtained from a

histogram

in which the rectangles represent the square roots

of the observed frequencies rather than the frequencies themselves. The idea behind such a

diagram is to remove the tendency for the variability of a count to increase with its typical

size. See also hanging rootogram.[Exploratory Data Analysis, 1977, J. W. Tukey, Addison

Wesley, Reading, Massachusetts.]

372

Rose diagram: A form of

angular histogram

in which each group is displayed as a sector. The radius

of each sector is usually taken to be proportional to the square root of the frequency of the

group, so that the area of the sector is proportional to the group frequency. Figure 116 shows

such a diagram for arrival times on a 24 hour clock of 254 patients at an intensive care unit,

over a period of 12 months. [Statistical Analysis of Circular Data, 1995, N. I. Fisher,

Cambridge University Press, Cambridge.]

Rosenbaum’s test: A

distribution free method

for testing the equality of the scale parameters of

two populations known to have the same median. The test statistic is the total number of

values in the sample from the ﬁrst population that are either smaller than the smallest or

larger than the largest values in the sample from the second population. [Model Assisted

Statistics and Applications, 2008, 3,59–69.]

Rosenthal effect: The observation that investigators often ﬁnd what they expect to ﬁnd from a

study. For example, in a reliability study in which auscultatory measurements of the foetal

heart rate were compared with the electronically recorded rate, it was found that when the

true rate was under 130 beats per minute the hospital staff tended to overestimate it, and

when it was over 150 they tended to underestimate it. See also Hawthorne effect.[Journal

of Educational Psychology, 1984, 76,85–97.]

Rotata ble desi gn: See design rotatability.

Rotation sampling: Sampling a given population at several regular intervals of time in order to

estimate the manner in which the population is changing. Essentially equivalent to

panel studies

longitudinal studies

.[Jour nal of Statistical Planning and Inferen ce, 2007, 137, 549–610.]

Rothamsted Park Grass Experiment: A long term experiment started at Rothamsted

Experimental Station in 1865 to study the effect of different fertilizers on the yield of hay.

Rounding: The procedure used for reporting numerical information to fewer decimal places than used

during analysis. The rule generally adopted is that excess digits are simply discarded if the

ﬁrst of them is less than ﬁve, otherwise the last retained digit is increased by one. So

rounding 127.2492341 to three decimal places gives 127.249.

Rose plot for uniform circular data

3π

Fig. 116 Rose diagram for arrival times at

an intensive care unit. (Reproduced from

Statistical Analysis of Circular Data by

permission of the publishers, Wiley.)

373

Round robin study: A term sometimes used for interlaboratory comparisons in which samples of a

material manufactured to well-deﬁned speciﬁcations are sent out to the participating labo-

ratories for analysis. The results are used to assess differences between laboratories and to

identify possible sources of incompatibility or other anomalies. See also interlaboratory

studies.[Organohalogen Compounds, 2004, 66, 519–24.]

Roy’s largest root criterio n: See multivariate analysis of variance.

R-techniques: The class of data analysis methods that look for relationships between the variables in

a set of

multivariate data

. Includes

principal components analysis

, and

factor analysis

. See

also Q-techniques.

Ruben, Harold (1923^2001): Born in Rawa, Poland, Ruben’s family moved to England when he

was quite young, and he studied mathematics at Imperial College, London. In 1950 he was

awarded a Doctor of Philosophy degree in mathematical statistics. After this Ruben worked

in Aberdeen and then in the Department of Genetics at Cambridge. In 1953 he moved to the

University of Manchester and then four years later to Columbia University in New York.

Further moves between the USA and the UK led to posts at the University of Shefﬁ eld and

the University of Minnesota. In 1969 Ruben moved to Montreal as Professor in the

Department of Mathematics at McGill University where he stayed until his retirement in

1988. Ruben made important contributions in

multivariate analysis

, geometric probability

and statistical physics. He died in Harlow, Essex, UK, on 30 November 2001.

Rudas^Clogg^Lindsayindexof fit: A measure for evaluating goodness of ﬁt in the analysis of

contingency tables

, which gives a directly interpretable quantitative measure of lack of ﬁt,

namely the proportion of cases incompatible with the model being tested. In this way the

index allows immediate comparisons across samples or studies which is not sensitive to

sample size. [Journal of the Royal Statistical Society, Series B, 1994, 56, 623–39.]

Rugplot: A method of displaying graphically a sample of values on a continuous variable by indicating

their positions on a horizontal line. See Fig. 117 for an example.

Ruin theory: A method for examining the solvency of an insurance company by modelling the

insurer’s surplus or reserve for an insurance portfolio over time, by a

stochastic process

Of central importance are the time to ruin at which the surplus becomes negative for the ﬁrst

time, the surplus immediately before the time of ruin and the deﬁcit at the time of ruin. [Ruin

Probabilities, 2000, S. Asmussen, World Scientiﬁc, Singapore.]

Rule of three: A rule which states that if in n trials, zero events of interest are observed, an upper 95%

conﬁdence bound on the underlying rate is 3/n.[The American Statistician, 1997, 51,137–9.]

Run-in: A period of observation prior to the formation of treatment groups in a

clinical trial

during

which subjects acquire experience with the major components of a study

protocol

. Those

subjects who experience difﬁculty complying with the protocol are excluded while the group

oooo oooooo oooooo oo

Percenta

e body fat

01020304050

Fig. 117 A rugplot of

percentage body fat in a

number of individuals.

374

of proven compliers are randomized into the trial. The rationale behind such a procedure is

that, in general, a study with higher compliance will have higher

power

because the observed

effects of the difference between treatment groups will not be subjected to the diluting

effects of non-compliance. [Statistics in Medicine, 1990, 9,29–34.]

Run: In a series of observations the occurrence of an uninterrupted sequence of the same value. For

example, in the series

111224333333

there are four ‘runs’, the single value, 4, being regarded as a run of length one.

Runs test: A test frequently used to detect

serial correlations

. The test consists of counting the number

of runs, or sequences of positive and negative residuals, and comparing the result to the

expected value under the hypothesis of independence. If the sample observations consist of

positive and n

negative residuals with both n

and n

greater than 10, the distribution of

the length of runs under independence can be approximated by a normal distribution with

mean, µ, and variance, σ

given by

 ¼

þ n

þ 1



ð2n

 n

ðn

þ n

ðn

þ n

 1Þ

[ARA Chapter 10.]

RV-coefficient: A measure of the similarity of two conﬁgurations of n data points given by

RVðX; YÞ¼

trðXY

ftrðXX

trðYY

where X and Y are the matrices describing the two conﬁgurations. Several techniques of

multivariate analysis can be expressed in terms of maximizing RV (X, Y) for some deﬁnition

of X and Y.[Applied Statistics, 1976, 25, 257–65.]

375

Saddl e p ointmethod: An approach to providing extremely accurate approximations to probability

distributions based on the corresponding

cumulative generating function

. For a variable y

with cumulative generating function c(t), the ﬁrst-order approximation is

f ðyÞð2pÞ



Þj



expfcð

Þ



where

 is determined by c

Þ¼y and c

and c

denote ﬁrst derivative vector and second

derivative matrix. [KA1 Chapter 11.]

Saint Petersburg paradox: Synonym for Petersburg game.

Sakoda coeff i ci ent: A measure of association, S, of the two variables forming an r × c two-

dimensional contingency table, given by

S ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðq 1ÞðX

þ NÞ

where X

is the usual chi-squared statistic for testing the independence of the two variables,

N is the sample size and q ¼ minðr; cÞ. See also contingency coefﬁcient, phi-coefﬁcient

and Tschuprov coefﬁcient.[An Introduction to Categorical Data Analysis, 1996,

A. Agresti, Wiley, New York.]

Sample: A selected subset of a population chosen by some process usually with the objective of

investigating particular properties of the parent population.

Sample size: The number of individuals to be included in an investigation. Usually chosen so that the

study has a particular power of detecting an effect of a particular size. Software is available

for calculating sample size for many types of study, for example,

SOLO

and

nQuery advisor

See also nomogram. [SMR Chapter 8.]

Sample size fo rmulae: Formulae for calculating the necessary sample size to achieve a given

power of detecting an effect of some ﬁxed magnitude. For example, the formula for

determining the number of observations needed in each of two samples when using a one-

tailed z-test to test for a difference in the means of two populations is

n ¼

2ðz

 z



ð

 

where σ

is the known variance of each population, µ

and µ

the population means, z

the

normal deviate for the chosen signiﬁcance level α and z

the normal deviate for β where 1 − β

is the required power. See also SOLO and nQuery advisor. [SMR Chapter 15.]

Sample space: The set of all possible outcomes of a probabilistic experiment. For example, if two

coins are tossed, the sample space is the set of possible results, HH, HT, TH and TT, where H

indicates a head and T a tail.

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Sa m p le survey: A study that collects planned information from a sample of individuals about their

history, habits, knowledge, attitudes or behaviour in order to estimate particular population

characteristics. See also opinion survey, random sample, and quota sample.[The

American Census: a Social History, 1988, M. Anderson, Yale University Press, New

Haven.]

Sampling: The process of selecting some part of a population to observe so as to estimate something of

interest about the whole population. To estimate the amount of recoverable oil in a region,

for example, a few sample holes might be drilled, or to estimate the abundance of a rare and

endangered bird species, the abundance of birds in the population might be estimated on the

pattern of detections from a sample of sites in the study region. Some obvious questions are

how to obtain the sample and make the observations and, once the sample data are to hand,

how best to use them to estimate the characteristic of the whole population. See also simple

random sampling and cluster sampling. [SMR Chapter 5.]

Sampling design: The procedure by which a sample of units is selected from the population. In

general a particular design is determined by assigning to each possible sample S the

probability Pr(S) of selecting that sample. See also random sample.

Sampling distribution: The probability distribution of a statistic calculated from a random sample

of a particular size. For example, the sampling distribution of the arithmetic mean of samples

of size n taken from a normal distribution with mean µ and standard deviation σ, is a normal

distribution also with mean µ but with standard deviation =

ﬃﬃﬃ

. [SMR Chapter 8.]

Sampling error: The difference between the sample result and the population characteristic being

estimated. In practice, the sampling error can rarely be determined because the population

characteristic is not usually known. With appropriate sampling procedures, however, it can be

kept small and the investigator can determine its probable limits of magnitude. See also

standard error.[Sampling Techniques, 1977, 3rd edition, W. G. Cochran, Wiley, New York.]

Sampling frames: The portion of the population from which the sample is selected. They are usually

deﬁned by geographic listings, maps, directories, membership lists or from telephone or

other electronic formats. [Survey Sampling, 1995, L. Kish, Wiley, New York.]

Sampling units: The entities to be sampled by a

sampling design

. In many surveys these entities will

be people, but often they will involve larger groupings of individuals. In

institutional surveys

for example, they may be hospitals, businesses, etc. Occasionally it may not be clear what the

units should be. In a survey of agricultural crops in a region, the region might be divided into a

set of geographic areas, plots or segments, and a sample of units selected using a map. But such

units could obviously be made alternative sizes and shapes, and such choices may affect both

the cost of the survey and the precision of the estimators. [SMR Chapter 14.]

Sampling variation: The variation shown by different samples of the same size from the same

population. [SMR Chapter 7.]

Samp l ing with and with out repl acement: Terms used to describe two possible methods of

taking samples from a

ﬁnite population

. When each element is replaced before the next one

is drawn, sampling is said to be ‘with replacement’. When elements are not replaced then the

sampling is referred to as ‘without replacement’. See also bootstrap, jackknife and hyper-

geometric distribution. [KA1 Chapter 9.]

Sampling zeros: Zero frequencies that occur in the cells of

contingency tables

simply as a result of

inadequate sample size. See also structural zeros.[The Analysis of Contingency Tables, 2nd

edition, 1992, B. S. Everitt, Chapman and Hall/CRC Press, London.]

377

Sandwi ch estimat o rs: Estimators of

variance–covariance matrices

of estimators encountered for

instance in the analysis of clustered data by

generalized estimating equations

and when using

the

pseudolikelihood

for

complex survey data

. The estimators are of the form

ABA

where the outside pieces of the ‘sandwich’ (the As) correspond to a model-based covariance

matrix assuming a correct model and the centre term (the B) corresponds to an empirical

covariance matrix. [Advances in Econometrics, 2003, 17,45–73.]

San ghv i’sdistance: A measure of the distance between two populations described by a set of

categorical variables; given by

¼ 100

j¼1

þ1

k¼1

ðp

1jk

 p

ðp

2jk

 p

j¼1

where p

1jk

and p

2jk

are the proportions in the kth class for the jth variable in populations P

and P

, respectively, P

ðp

1jk

þ p

2jk

Þ and s

j+1

is the number of classes for the jth

character. [Biometrics, 1970, 26, 525–34.]

Sargan distribution: A probability distribution, f (x), given by

f ðxÞ¼

Kαe

αjxj

1 þ



j¼1

jxj

where α  0; γ

0ðj ¼ 1; 2; ...;Þ and K ¼



j¼1

ð1 þj!γ

1

. The parameter ρ is usually

referred to as the order of the distribution. Order zero corresponds to a

Laplace distribution

Used as a viable alternative to the normal distribution in econometric models. [Journal of

Econometrics, 1987, 34, 349–54.]

Sargantest: A test of the validity of

instrumental variables

, namely that they are uncorrelated with

the appropriate set of error terms. [Applied Economic Letters , 2008, 15, 349–353.]

SAS: An acronym for Statistical Analysis System, a large computer software system for data processing

and data analysis. Extensive data management, ﬁle handling and graphics facilities. Can be

used for most types of statistical analysis including

multiple regression

log-linear models

principal components analysis

, etc. [http://www.sas.com.]

Satterthw ait e’sapproximation: A useful general method of approximating the probability

distribution of a nonnegative variable which has a known (or well-estimated) mean and

variance. [ARA Chapter 16.]

Saturated model: A model that contains all main effects and all possible interactions between

factors. Since such a model contains the same number of parameters as cells (categorical

data) or means (continuous data) it results in a perfect ﬁt for a data set.

Sav a g e, Leonar d J ames (1917^1971) : Born in Detroit, Savage studied at Wayne State and

later University of Michigan, eventually gaining a Ph.D. on the application of vector

methods to metric geometry. In 1944 he became a member of the Statistical Research

Group at Columbia. In 1949 Savage helped to form the statistics department at Chicago

and in 1954 produced The Foundations of Statistics a book which had great inﬂuence

on later developments in the subject. He was President of the Institute of Mathematical

Statistics for the period 1957–58. Savage died on 1 November 1971, in New Haven.

378