Everitt B.S. The Cambridge Dictionary of Statistics

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and the population mean is



X ¼

j¼1

Using this knowledge, we can improve on the estimators



Y and

Y by using the ratio

estimators



y

x

for the population mean and

¼ X



for the population total. The rationale of the ratio estimators is to exploit correlation between

and y

to increase precision compared to conventional simple random sampling estimators.

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

Ratio variable: A continuous variable that has a ﬁxed rather than an arbitrary zero point. Examples

are, height, weight and temperature measured in degrees Kelvin. See also categorical

variable and ordinal variable .

Rayleigh distribution: A form of

noncentral chi-squared distribution

encountered in mathematical

physics. The distribution is given speciﬁcally by

f ðxÞ¼



x

ð2β

1 β

The mean is βðp=2Þ

and the variance is ð2  p=2Þβ

. [STD Chapter 34.]

Rayleigh quotient: A function that arises in the conditional maximization of a quadratic form y

subject to the constraint y

y = c where c is a scalar and given by x

Ax/x

x where A is an n x n

matrix and x is an n x1 vector. [Journal of Sound and Vibration, 1997, 199, 155–164.]

Rayleigh’s test: A test of the null hypothesis that

circular random variable

, θ, has a

uniform

distribution

of the form

f ðÞ¼

0  p  2p

The test is based on a random sample of observed values 

;

; ...;

, and the

test statistic

i¼1

cos 

i¼1

sin 

;

Critical values of the test statistic are available. An example of the application of this test is to

determine if the arrangement of bird nest entrances for a particular type of bird in a particular

area differs signiﬁcantly from random. [MV2 Chapter 14.]

RBD: Abbreviation for randomized block design.

RCT: Abbreviation for randomized clinical trial.

RDS: Abbreviation for respondent-driven sampling.

Recall bias: A possible source of bias, particularly in a

retrospective study

, caused by differential

recall among cases and controls, in general by underreporting of exposure in the control

group. See also ascertainment bias. [SMR Chapter 5.]

359

R eceiver operati ng characteristic (R OC ) curve: A plot of the

sensitivity

of a

diagnostic

test

against one minus its

speciﬁcity

, as the cut-off criterion for indicating a positive test is

varied. Often used in choosing between competing tests, although the procedure takes no

account of the prevalence of the disease being tested for. As an example consider the

following ratings from 1 (deﬁnitely normal) to 5 (deﬁnitely diseased) arising from 50 normal

and 50 diseased subjects.

12345Total

Normal 417208 150

Diseased 3 3 17 19 8 50

If the rating of 5 is used as the cut-off for identifying diseased cases, then the sensitivity is

estimated as 8/50=0.16, and the speciﬁcity as 49/50=0.98. Now using the rating 4 as cut-off

leads to a sensitivity of 27/50=0.54 and a speciﬁcity of 41/50=0.82. The values of

(sensitivity, 1– speciﬁcity) as the cut-off decreases from 5 to 1 are (0.16,0.02), (0.54,0.18),

(0.88,0.58), (0.94,0.92), (1.00,1.00). These points are plotted in Fig. 112. This is the required

receiver operating characteristic curve. [SMR Chapter 14.]

Reciprocal causation: See recursive models.

Reciprocal model: The regression model deﬁned by E(y|x)=(α + β x)

−1

[Biometrics, 1979, 35,41–54.]

Reci procal transformation: A transformation of the form y =1/x, which is particularly useful for

certain types of variables. Resistances, for example, become conductances, and times

become rates. In some cases the transformation can lead to linear relationships, for example,

Sensitivity

1-Specificity

0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 112 An example

of an ROC curve for the

normal/diseased ratings

given.

360

airways resistance against lung volume is non-linear but airways conductance against lung

volume is linear. [SMR Chapter 7.]

Record linkage: The establishment of which records in a database relate to the same individual. A

straightforward task if there is some form of unique identiﬁer for each individual, but when

such identiﬁers are not available or not feasible, probabilistic matching may be needed.

[Handbook of Record Linkage, 1988, H. B. Newcombe, Oxford University Press, New

York.]

Rectangular distribution: Synonym for uniform distribution.

Recurrence risk: Usually the probability that an individual experiences an event of interest given

previous experience(s) of the event; for example, the probability of recurrence of breast

cancer in a woman who has previously had the disease. In medical genetics, however, the

term means the chance that a genetic disease present in the family will recur in that family

and affect another person (or persons).

Recurrent event data: Data in which recurrent events, for example, tumours, are ordered and each

event of interest is treated as being of the same nature. Other examples of such data are bugs

in software and the repeated ﬁxing of aircraft. A variety of models have been proposed for

analysing such data. [Technometrics, 2007, 49, 210–220.]

Recursive: A function or sequence is deﬁned recursively if

the value of f(0) and

the value of f (n + 1) given the value of f (n) are both stated.

For example, the factorial function may be deﬁned by

f (0) = 1,

f (n +1)=nf (n) for n =0,1,2,....

Recursive models: Statistical models in which causality ﬂows in one direction, i.e. models that

include only unidirectional effects. Such models do not include circular effects or

reciprocal

causation

, for example, variable A inﬂuences variable B, which in turn inﬂuences variable A;

nor do they permit feedback loops in which, say, variable A inﬂuences variable B, which

inﬂuences variable C, which loops back to inﬂuence variable A.[Structural Equations with

Latent Variables, 1989, K. A. Bollen, Wiley, New York.]

Recursive partitioning regression: Synonym for classiﬁcation and regression tree

technique.

Reduction of dimensionality: A generic term for the aim of methods of analysis which attempt to

simplify complex

multivariate data

to aid understanding and interpretation. Examples

include

principal components analysis

and

multidimensional scaling

Reed^Muench estimator: An estimator of the

median effective dose

bioassay

. See also

Dragstedt–Behrens estimator.[Probit Analysis, 1971, 3rd edition, D. J. Finney,

Cambridge University Press, Cambridge.]

R ef erence i nterval: A range of values for a variable that encompasses the values obtained from the

majority of normal subjects. Generally calculated as the interval between two predetermined

centiles (such as the ﬁfth and the 95th) of the distribution of the variable of interest. Often

used as the basis for assessing the results of

diagnostic tests

in the classiﬁcation of

individuals as normal or abnormal with respect to a particular variable, hence the commonly

used, but potentially misleading phrase ‘normal range’. [SMR Chapter 14.]

361

Reference population: The standard against which a population that is being studied can be

compared.

Reference priors: See prior distributions.

R egenerative cycles: See regenerative process.

Regenerative process: A term for a

stochastic process

for which there exist random time points

0 ¼ T

such that at each time point, the process ‘restarts’ probabilistically.

Formally this implies that the process {L

} can be split into independent identically

distributed (iid) probabilistic replicas, called regenerative cycles, of iid random lengths

¼ T

 T

i1

; i ¼ 1; 2; ....[Regenerative Inventory Systems, 1989, I. Sahin, Springer

Verlag, New York.]

Regression analysis: Essentially a synonym for regression modelling.

Regression calibration: A procedure of correcting the results from a regression analysis to allow

for measurement errors in the explanatory variables. See also attenuation.[American

Journal of Epidemiology, 2001, 154, 836–44.]

Regression coefficient: See multiple regression.

Regression diagnostics: Procedures designed to investigate the assumptions underlying partic-

ular forms of regression analysis, for example, normality, homogeneity of variance, etc., or

to examine the inﬂuence of particular data points or small groups of data points on the

estimated regression coefﬁcients. See also residuals, Cook’s distance, COVRATIO,

DFBETA, DFFIT and inﬂuence statistics. [ARA Chapter 10.]

Regression dilution: The term applied when a covariate in a model cannot be measured directly and

instead a related observed value must be used in the analysis. In general, if the model is

correctly speciﬁed in terms of the ‘true’ covariate, then a similar form of the model with a

simple error structure will not hold for the observed values. In such cases, ignoring the

measured values will lead to biased estimates of the parameters in the model. Often also

referred to as the errors in variables problem. See also attenuation, latent variables and

structural equation models. [ARA Chapter 9.]

Regression discontinuity design: A

quasi-experimental design

in which participants in, for

example, an intervention study, are assigned to treatment and control groups on the basis of

a cutoff value on a pre-intervention measure that affects the outcome, rather than by

randomization. If the treatment has an effect a discontinuity in the outcomes would be

expected at the cutoff. A weakness is that extrapolation of counterfactual outcomes for the

treated in absence of treatment is required, based on a regression model estimated for the

non-treated. See Figure 113 for an illustration where treatment is given to those below a

cutoff on a pretest. [Experimental and Quasi-Experimental Designs for Generalized Causal

Inference, 2002, W. R. Shadish, T. D. Cook and D. T. Campbell, Houghton-Mifﬂin,

Boston.]

Regression modelling: A frequently applied statistical technique that serves as a basis for studying

and characterizing a system of interest, by formulating a reasonable mathematical model of

the relationship between a response variable, y and a set of q explanatory variables,

; x

; ...; x

. The choice of the explicit form of the model may be based on previous

knowledge of the system or on considerations such as ‘smoothness’ and continuity of y as a

function of the x variables. In very general terms all such models can be considered to be of

the form.

362

y ¼ f ðx

; ...; x

Þþ

where the function f reﬂects the true but unknown relationship between y and the explan-

atory variables. The random additive error  which is assumed to have mean zero and

variance 



reﬂects the dependence of y on quantities other than x

; ...; x

. The goal is to

formulate a function

f ðx

; x

; ...; x

Þ that is a reasonable approximation of f. If the correct

parametric form of f is known, then methods such as

least squares estimation

maximum

likelihood estimation

can be used to estimate the set of unknown coefﬁcients. If f is linear in

the parameters, for example, then the model is that of

multiple regression

. If the experi-

menter is unwilling to assume a particular parametric form for f then nonparametric

regression modelling can be used, for example,

kernel regression smoothing

, recursive

partitioning regression or

multivariate adaptive regression splines

.[Regression Modeling

Strategies, 2001, F. E. Harrell, Jr., Springer, New York.]

R egressi on throug h the orig i n: In some situations a relationship between two variables

estimated by regression analysis is expected to pass through the origin because the true

mean of the dependent variable is known to be zero when the value of the explanatory

variable is zero. In such situations the linear regression model is forced to pass through the

origin by setting the intercept parameter to zero and estimating only the slope parameter.

[ARA Chapter 1.]

Regression to the mean: The process ﬁrst noted by

Sir Francis Galton

that ‘each peculiarity in

man is shared by his kinsmen, but on the average to a less degree.’ Hence the tendency, for

example, for tall parents to produce tall offspring but who, on average, are shorter than their

parents. The term is now generally used to label the phenomenon that a variable that is

extreme on its ﬁrst measurement will tend to be closer to the centre of the distribution for a

later measurement. For example, in a screening programme for hypertension, only persons

with high blood pressure are asked to return for a second measure. On the average, the

T = Treatment

C = Control

30 32 34 36 38 40

Pretest

Posttest

81012

Fig. 113 Hypothetical data from a regression discontinuity design.

363

second measure taken will be less than the ﬁrst. [Statistical Methods in Medical Research,

1997, 6, 103–14.]

Regression weight: Synonym for regression coefﬁcient.

Regularization: Approaches designed to prevent

overﬁtting

of statistical models, such as

ridge

regression

, the

lasso

and the L

-norm in

support vector machines

. The approach is also

used to refer to model selection procedures which penalize models according to their number

of parameters, such as

Akaike’s information criterion

(AIC) and the

Bayesian information

criterion

(BIC). [SIAM Review, 1998, 40, 636–666.]

R eg u la rized discr im i nant analysis: A method of

discriminant analysis

for small, high-

dimensional data sets, which is designed to overcome the degradation in performance of

the

quadratic discriminant function

when there are large numbers of variables. [Journal of

the American Statistical Association, 1989, 84, 165–75.]

Reification: The process of naming

latent variables

and the consequent discussion of such things as

quality of life and racial prejudice as though they were physical quantities in the same sense

as, for example, are length and weight. [MV1 Chapter 1.]

Reinsch spline: The function, g, minimizing

 gðx

Þg

over g such that

dt  C

where C is a smoothing parameter and Y

are observations at design points

; 0  x



 1 and

¼ gðx

Þþ

with the errors 

being independent and normally distributed with zero mean and variance

. See also spline function.[Nonparametric Regression and Generalized Linear Models,

1994, P. J. Green and B. W. Silverman, Chapman and Hall/CRC Press, London.]

Reinterviewing: A second interview for a sample of survey respondents in which the questions of

the original interview (or a subset of them) are repeated. The same methods of questioning

need not be used on the second occasion. The technique can be used to estimate components

in survey error models and to evaluate ﬁeld work.

Rejectable quality level: See quality control procedures.

R elati ve eff ici ency: The ratio of the variances of two possible estimates of a parameter or the ratio

of the sample sizes required by two statistical procedures to achieve the same

power

. See

also Pitman nearness criterion.

Relative frequency: See probability.

R elati ve poverty statistics: Statistics on the properties of populations falling below given

fractions of average income that play a central role in the discussion of poverty. The

proportion below half national median income, for example, has been used as the basis of

cross-country comparisons.

Relative risk: A measure of the association between exposure to a particular factor and risk or

probability of a certain outcome, calculated as

relative risk ¼

risk among exposed

risk among nonexposed

Thus a relative risk of 5, for example, means that an exposed person is 5 times as likely to have

the disease than one who is not exposed. Relative risk does not measure the probability that

364

someone with the factor will develop the disease. The disease may be rare among both the

nonexposed and the exposed. See also incidence,andattributable risk. [SMR Chapter 10.]

Relative survival: The ratio of the observed survival of a given group of patients to the survival the

group would have experienced based on the

life table

of the population from which they

were diagnosed. [Statistics is in Medicine, 2004, 23,51–64.]

Reliability: The extent to which repeated measurements on units (for instance people) yield similar

results. See also intraclass correlation coefﬁcient and kappa coefﬁcient.[Journal of

Applied Psychology, 1993, 78,98–104.]

Reliability data: A term generally applied to various types of data, for example, overall equipment failure

rate, common causes failure rates, fail-to-start probabilities etc., collected about some type of

engineering system. [The Reliability Data Handbook, 2005, T. R. Moss, Wiley, New York.]

Reliability theory: A theory which attempts to determine the reliability of a complex system from

knowledge of the reliabilities of its components. Interest may centre on either the lifetime or

failure-free operating time of a system or piece of equipment or on broader aspects of

system’s performance over time. In the former situation survival analysis techniques are

generally of most importance; in the latter case Markov processes are often used to model

system performance. [Statistical Theory of Reliability and Life Testing: Probability Models,

1975, R. E. Barlow and F. Proschan, Rinehart and Winston, New York.]

Relplot: Synonymous with bivariate boxplot.

Remedian: A

robust estimator

of location that is computed by an iterative process. Assuming that the

sample size n can be written as b

where b and k are integers, the statistic is calculated by

computing medians of groups of b observations yielding b

k−1

estimates on which the process

is iterated and so on until only a single estimate remains. [Journal of the American Statistical

Association, 1990, 85,97–104.]

Remington, Richard (1931^1992): Born in Nampa, Idaho, USA, Remington received both

B. A. and M. A. degrees in mathematics from the University of Montana and then in 1958

a Ph.D in biostatistics from the University of Michigan. He then joined the Department of

Statistics at the University of Michigan School of Public Health becoming professor in 1965.

In 1969 Remington moved to the University of Texas becoming Associate Dean for

Research, Professor and Head of Biometry. In 1975 he returned to the University of

Michigan; later he moved to the University of Iowa. For much of his distinguished career,

Remington was involved with inquiries into the public health impact of hypertension and in

1988 was the main author of the Institute of Medicine’s report, The Future of Public Health.

He died on July 26th, 1992 in Iowa City, Iowa, USA.

REML: Acronym for residual maximum likelihood estimation and restricted maximum likelihood

estimation.

Removal method: A method whereby the total number of animals in an enclosed area is estimated

from the numbers in successive ‘removals’ (i.e. captures without replacement). The assump-

tions behind the removal method are as follows:

(a) There is no immigration into or emigration from the enclosed area while the

removals are taking place.

(b) Each animal has an equal probability, p, of being caught.

to the rest.

If the above assumptions are valid, then the probability of catching c

; ...; c

animals in k

removals, given a total of N animals, can be written as f ðc; ...; c

jN; pÞ where

365

f ðc; ...; c

jN; pÞ¼

ðc

Þ!ðN  T Þ!

ð1 pÞ

kNY

where T ¼

i¼1

and Y ¼

i¼1

ðk  i þ 1Þc

. The maximum likelihood estimator of N

cannot be found exactly, but approximations can be made using

Stirling’s formula

. For two

removals it can be shown that

N ¼

 c

The approximate asymptotic variance of

N is

varð

NÞ¼

ð1 þ

qÞ

qð1 



 b

where

p ¼ðc

 c

Þ=c

;

q ¼ 1 

p and b ¼

qð1 þ

qÞ=

.[Biometrics, 1994, 50, 501–5.]

R enewal funct ion: See renewal theory.

R enewal process: See renewal theory.

R enewal theory: The study of

stochastic processes

counting the number of events that take place as a

function of time. The events could, for example, be the arrivals of customers at a waiting line or

the successive replacement of light bulbs. Formally , let X

; X

; ... be a sequence of non-

negative, independent random variables having a common cumulative probability distribution.

FðxÞ¼PrðX

 x

Þ; x  0; k ¼ 1; 2; ...

Let 

¼ Eðx

Þ and assume 0



1. The random variable X

denotes the interoccurrence

time between the (n − 1)th and nth event in some speciﬁc setting. Deﬁne

¼ 0; S

i¼1

n ¼ 1; 2; ...

then S

is the epoch at which the nth event occurs. Deﬁne for each t ≥ 0

NðtÞ¼ the largest integer n  0 for which S

 t

then the random variable N(t) represents the number of events up to time t. The

counting

process

fNðtÞ; t  0g is called the renewal process generated by the interoccurrence times

; x

; .... The expected value of N(t) is known as the renewal function.[Biometrical

Journal, 1996, 23, 155–64.]

nyi, Alfre

d (1921^1970): Born in Budapest, Hungary, Rényi studied mathematics and physics

at the University of Budapest until 1944. In 1950 he became the Director of the

Mathematical Institute of the Hungarian Academy of Sciences and also chairman of the

Department of Probability. Contributed to probability theory, random graphs and

informa-

tion theory

and published about 300 papers in a period of 25 years. Rényi died on 1 February

1970 in Budapest.

Repeatability: The closeness of the results obtained in the same test material by the same observer or

technician using the same equipment, apparatus and/or reagents over reasonably short

intervals of time.

Repeated measures data: See longitudinal data.

R eplicate observation: An independent observation obtained under conditions as nearly identical

to the original as the nature of the investigation will permit. [SMR Chapter 5.]

Reporting bias: Synonymous with recall bias.

366

R ep roduced matr ix: A matrix of correlations or covariances that is calculated from parameter

estimates from, for example, a

structural equation model

. Often used in the assessment of the

ﬁt of a model. [Causal Analysis: Models, Assumptions and Data, 1982, L. R. James, S. A.

Mulaik and J. M. Brett, Sage, Beverly Hills.]

Reproducibility : The closeness of results obtained on the same test material under changes of

reagents, conditions, technicians, apparatus, laboratories and so on. [SMR Chapter 12.]

Reproduction rate: See basic reproduction rate.

Research hypothesis: Speciﬁc testable predictions made generally about the response and

explanatory variables in a study. For example, if skin cancer is related to ultraviolet light,

then people with a high exposure to uv light will experience a greater

incidence

of skin

cancer. [Practical Research and Design, 7th edn, 2001, P. D. Leedy and J. E. Ormrod,

Merrill Prentice Hall, New York.]

Resentful demoralization: A possible phenomenon in

clinical trials

and intervention studies in

which comparison groups not obtaining a perceived desirable treatment become aware of

this potential inequity and become discouraged or retaliatory, and as a result perform worse

on the outcome measures. [Quasi-Experimentation: Design and Analysis Issues for Field

Setting, 1979, T. D. Cook and D. T. Campbell, Rand-McNally, Chicago.]

RESET test: See Ramsey regression equation speciﬁcation error test.

Residual: The difference between the observed value of a response variable (y

) and the value predicted

by some model of interest (

). Examination of a set of residuals, usually by informal

graphical techniques, allows the assumptions made in the model ﬁtting exercise, for

example, normality, homogeneity of variance, etc., to be checked. Generally, discrepant

observations have large residuals, but some form of standardization may be necessary in

many situations to allow identiﬁcation of patterns among the residuals that may be a cause

for concern. The usual standardized residual for observation y

, is calculated from



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 h

where s

is the estimated residual variance after ﬁtting the model of interest and h

is the ith

diagonal element of the

hat matrix

. An alternative deﬁnition of a standardized residual

(sometimes known as the Studentized residual), is



ðiÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 h

where now s

ðiÞ

is the estimated residual variance from ﬁtting the model after the exclusion

of the ith observation. See also regression diagnostics, Cook’s distance and inﬂuence.

[ARA Chapter 10.]

Residual maximum likelihood estimation (REM L): A method of estimation in which

estimators of parameters are derived by maximizing the

residual

restricted likelihood

rather than the

likelihood

itself. Most often used for estimating

variance components

in a

linear mixed models

.[Analysis of Longitudinal Data, 2nd edition, 2002, P. J. Diggle,

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

Residualplots: Plots of some type of

residual

that may be helpful in assessing the assumption made

by a ﬁtted model. In regression analysis there are various ways of plotting residual values

that can be helpful in assessing particular components of the regression model. The most

useful plots when diagnosing linear regression models are as follows;

367

boxplot

probability plot

of the residuals can be useful in checking for symmetry

and speciﬁcally for normality of the error terms in the regression model.

Plotting the residuals against the corresponding values of the explanatory variable.

Any sign of curvature in the plot might suggest that say a quadratic term in the

explanatory variable should be included in the model.

Plotting the residuals against the ﬁtted values of the response variable. If the variability

of the residuals appears to increase with the size of the ﬁtted values, a transformation

of the response variable prior to ﬁtting is indicated.

Figure 114 shows some idealized residual plots that indicate particular points about models.

Figure 114(a) is what is looked for to conﬁrm that the ﬁtted model is appropriate,

Figure 114(b) suggests that the assumption of constant variance is not justiﬁed so that

some transformation of the response variable before ﬁtting might be sensible,

Figure 114(c) implies that the model needs a quadratic term in the explanantory

variable.

See also added variable plot.[Regression Diagnostics, 1991, J.Fox, Sage, Newbury Park.]

Residual sum of squares: See analysis of variance.

Respondent-driven sampling (RDS): A form of

snowball sampling

which begins with the

recruitment of a small number of people in the target population to serve as seeds. After

participating the seeds are asked to recruit other people they know in the target population. The

sampling continues in this way with current sample members recruiting the next wave of

3020100–10–20

(c)

Residuals

0 1020304050

40200–20

(b)

Residuals

20 40 60 80

Predicted value of y

(a)

Residuals

–5–10–15

0 1020304050

Fig. 114 Idealized residual plots.

368