Everitt B.S. The Cambridge Dictionary of Statistics

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latter will be termed the cumulative probability distribution and probability distribution and

probability density used synonymously.) [KA1 Chapter 8.]

Probabil ity generati ng function: A function, P(t), which when expanded gives the probability

that a discrete random variable takes a particular value r as the coefﬁcient of t

. Often a useful

summarization of the probability distribution of such a variable. The function can be found from:

PðtÞ¼

r¼0

PrðX ¼ rÞt

where PrðX ¼ rÞ is the probability that the random variable X takes the value r. For a

variable with a

binomial distribution

, for example,

PðtÞ¼

j¼0



ð1 pÞ

nj

¼fpt þð1  pÞg

[KA1 Chapter 2.]

Probabil ity judgements: Human beings often need to assess the probability that some event will

occur and the accuracy of these probability judgements often determines the success of our

actions and decisions. There is strong evidence that human probability judgements are

biased with overestimation of low probabilities. [Judgment under Uncertainty: Heuristics

and Biases, 1982, D. Kahneman, P. Slovic, and A. Tversky, eds., Cambridge University

Press, Cambridge.]

P r o bability-o f-being-in- respo nse functio n: A method for assessing the ‘response experi-

ence’ of a group of patients, by using a function of time, P(t), which represents the

probability of being ‘in response’ at time t. The purpose of such a function is to synthesize

the different summary statistics commonly used to represent responses that are

binary

variables

, namely the proportion who respond and the average duration of response. The

aim is to have a function which will highlight the distinction between a treatment that

produces a high response rate but generally short-lived responses, and another that

produces a low response rate but with longer response durations. [Biometrics, 1982, 38,

59–66.]

Probabilitypaper: Graph paper structured in such a way that the values in the

cumulative frequency

distribution

of a set of data from a normal distribution fall on a straight line. Can be used to

assess sample data for normality. See also probability plot.

Probabil ity plot: Plots for comparing two probability distributions. There are two basic types, the

probability–probability plot and the quantile–quantile plot. A plot of points whose

coordinates are the cumulative probabilities fp

ðqÞ; p

ðqÞg for different values of q is a

probability–probability plot, while a plot of the points whose coordinates are the quantiles

ðpÞ; q

ðpÞg for different values of p is a quantile–quantile plot. The latter is the more

frequently used of the two types and its use to investigate the assumption that a set of data is

from a normal distribution, for example, would involve plotting the ordered sample values

ð1Þ

; ...; y

ðnÞ

against the quantiles of a standard normal distribution, i.e.



1

½p

ðiÞ



where usually

i 

and ðxÞ¼

1

ﬃﬃﬃﬃﬃﬃ



339

(This is usually known as a

normal probability plot

, an example of which is shown in

Fig. 107.) [Methods for Statistical Analysis of Multivariate Observations, 1977,

R. Gnandesikan, Wiley, New York.]

Probability ^ probability (P ^ P) plot: See probability plot.

Probabilitysample: A sample obtained by a method in which every individual in a

ﬁnite population

has a known (but not necessarily equal) chance of being included in the sample. [ Sampling of

Populations: Methods and Applications, 3rd edition, 1999, P. S. Levy and S. Lemeshow,

Wiley, New York.]

Probability weighting: The process of attaching weights equal to the inverse of the probability of

being selected, to each respondent’s record in a sample survey. These weights are used to

compensate for the facts that sample elements may be selected at unequal sampling rates and

have different probabilities of responding to the survey, and that some population elements

may not be included in the list or frame used for sampling. [Journal of Risk and Uncertainty,

2001, 1,21–33.]

Pr obit ana lysis: A technique most commonly employed in bioassay, particularly toxicological

experiments where sets of animals are subjected to known levels of a toxin and a model is

required to relate the proportion surviving at a particular dose, to the dose. In this type of

analysis the

probit transformation

of a proportion is modelled as a

linear function

of the dose

or more commonly, the logarithm of the dose. Estimates of the parameters in the model are

found by

maximum likelihood estimation

.[Modelling Binary Data , 2nd edition, 2003,

D. Collett, Chapman and Hall, London.]

Probit transformation: A transformation y of a proportion p derived from the equation

p ¼ 5 þ

ﬃﬃﬃﬃﬃﬃ

1

exp 



Theoretical Quantiles

–2 –1

–10 0 10 20 30

012

Sample Quantiles

Fig. 107 An example of a normal probability plot.

340

The ‘5’ in the equation was introduced by

Fisher

to prevent negative values in the days when

biologists were even more resistant to doing arithmetic than they are today! The transformation

is used in the analysis of dose-response curves. See also probit analysis.[GLMChapter 10.]

Procrustes analysis: A method of comparing alternative geometrical representations of a set of

multivariate data or of a

proximity matrix

, for example, two competing

multidimensional

scaling

solutions for the latter. The two solutions are compared using a residual sum of

squares criterion, which is minimized by allowing the coordinates corresponding to one

solution to be rotated, reﬂected and translated relative to the other. Generalized Procrustes

analysis allows comparison of more than two alternative solutions by simultaneously trans-

lating, rotating and reﬂecting them so as to optimize a predeﬁned goodness-of-ﬁt measure.

(The name of this type of analysis arises from Procrustes, ‘The Smasher’, who, according to

Greek mythology, caught travellers on the borders of Athens and by stretching or pruning

them using rack or sword, made them an exact ﬁt to his lethal bed. Procrustes eventually

experienced the same fate as that of his guests at the hands of Theseus.) [MV2 Chapter 14.]

Product -limit estimator: A procedure for estimating the

survival function

for a set of survival

times, some of which may be

censored observations

. The idea behind the procedure involves

the product of a number of conditional probabilities, so that, for example, the probability of a

patient surviving two days after a liver transplant can be calculated as the probability of

surviving one day, multiplied by the probability of surviving the second day given that the

patient survived the ﬁrst day. Speciﬁcally the estimator is given by

SðtÞ¼

jjt

ðjÞ

t

1 



where

SðtÞ is the estimated survival function at time t, t

ð1Þ

 t

ð2Þ

t

ðnÞ

are the ordered

survival times

, r

is the number of individuals at risk at time t

(j)

and d

is the number of

individuals who experience the event of interest at time t

(j)

. (Individuals censored at time t

(j)

are included in r

.) The resulting estimates form a step function that can be plotted to give a

graphical display of survival experience. An example appears in Fig. 108. See also

Alshuler’s estimator and Greenwood’s formula. [SMR Chapter 13.]

Fig. 108 Survival curves estimated by product-limit estimator: leukemia study of under and over 45s.

341

Profi le analysis: A term sometimes used for the

analysis of variance

longitudinal data

P r ofile l ik e lihood: See likelihood.

Profileplots: A method of representing

multivariate data

graphically. Each observation is represented

by a diagram consisting of a sequence of equispaced vertical spikes, with each spike

representing one of the variables. The length of a given spike is proportional to the

magnitude of the variable it represents relative to the maximum magnitude of the variable

across all observations. As an example, consider the data below showing the level of air

pollution in four cities in the United States along with a number of other climatic and human

ecologic variables. The proﬁle plots representing these data are shown in Fig. 109(a).

Chicago is identiﬁed as being very different from the other three cities. Another example

of a proﬁle plot is shown in Fig. 109(b); here the weight proﬁles over time for rats in three

groups are all shown on the same diagram. See also star plot.

Air pollution data for four cities in the USA

SO2 Temp Manuf Pop Wind Precip Days

Atlanta 24 61.5 368 497 9.1 48.34 115

Chicago 110 50.6 3344 3369 10.4 34.44 122

Denver 17 51.9 454 515 9.0 12.95 86

San Francisco 12 56.7 453 716 8.7 20.66 67

SO2 = Sulphur dioxide content of air (microgram per cubic metre)

Temp = Average annual temperature (F)

Manuf = Number of manufacturing enterprises employing 20 or more workers

Pop = Population size

Wind = Average annual wind speed (miles per hour)

Precip = Average annual precipitation (miles per hour)

Days = Average number of days with precipitation per year

1 2 3456 7

Variable

1 2 34 5 6 7

Variable

1234 567

Variable

Days

Precip

Wind

Pop

Manuf

Temp

1234567

Variable

Days

Precip

Wind

Pop

Manuf

Te mp

Days

Precip

Wind

Pop

Manuf

Te mp

Atlanta

Denver

Chicago

San Francisco

Fig. 109(a) Proﬁle plots for air pollution data from four cities in the United States.

342

Profile t plots: A graphical method used to construct

conﬁdence intervals

for components of a

parameter vector when ﬁtting a

non-linear model

.[Journal of the American Statistical

Association, 1995, 90, 1271–6.]

Prognostic scoring system: A method of combining the prognostic information contained in a

number of risk factors, in a way which best predicts each patient’s risk of disease. In many

cases a linear combination of scores is used with the weights being derived from, for

example, a

logistic regression

. An example of such a system developed in the British

Regional Heart Study for predicting men aged 40–59 years to be at risk of ischaemic heart

disease over the next ﬁve years is as follows:

51  total serum cholesterol(mmol/l)

þ5  total time man has smoked(years)

þ3  systolic blood pressure(mmHg)

þ100 if man has symptoms of angina

þ170 if man can recall diagnosis of IHD

þ50 if either parent died of heart trouble

þ95 if man is diabetic

See also point scoring . [SMR Chapter 12.]

Prognostic survival model: A quantiﬁcation of the survival prognosis of patients based on

information at the start of follow-up. [Statistics in Medicine, 2000, 19, 3401–15.]

Prognostic variab les: In medical investigations, an often used synonym for explanatory variables.

[SMR Chapter 5.]

Programming: The act of planning and producing a set of instructions to solve a problem by

computer. See also algorithm.

Progressively censored data:

Censored observations

that occur in

clinical trials

where the

period of the study is ﬁxed and patients enter the study at different times during that period.

Fig. 109(b) Weight proﬁle for rats in different groups.

343

Since the entry times are not simultaneous, the censored times are also different. See also

singly censored data.[Technometrics, 1994, 36,84–91.]

Progressively typeII censoredsample: The sample that arises when n experimental units are

placed on a life test, and at the time of the ﬁrst failure, R

of the n  1 surviving units are

randomly withdrawn from the experiment. At the time of the second failure, R

of the

n  R

 2 surviving units are randomly withdrawn. Finally at the time of the mth failure (m

being set by the investigator), all of the remaining n  m  R

R

m1

surviving units

are withdrawn from the experiment. [Technometrics, 2004, 46, 470–81.]

Projection: The numerical outcome of a speciﬁc set of assumptions regarding future trends. See also

forecast.

Projection plots: A general term for any technique that can be used to produce a graphical display

of multivariate data by projecting the data into two dimensions enabling a scatterplot to be

drawn. Examples of such techniques are

principal components analysis

multidimensional

scaling

and

projection pursuit

Projection pursuit: A procedure for obtaining a low-dimensional (usually two-dimensional) rep-

resentation of multivariate data, that will be particularly useful in revealing interesting

structure such as the presence of distinct groups of observations. The low-dimensional

representation is found by optimizing some pre-deﬁned numerical criterion designed to

reveal ‘interesting’ patterns. [Pattern Recognition and Neural Networks, 1996, B. D. Ripley,

Cambridge University Press, Cambridge.]

Proofofconcept trials:

Clinical trials

carried out to determine if a treatment is biologically active

or inactive. [Statistics in Medicine, 2005, 24, 1815–35.]

Propensityscore: The conditional probability of assignment to a particular treatment given a vector

of covariates. Used for causal inference in observational studies by forming the basis for

matching to

balance

observed confounders. [Biometrika, 1983, 70,41–55.]

Proportional all ocation: In

stratiﬁed sampling

, the allocation of portions of the total sample to the

individual strata so that the sizes of these subsamples are proportional to the sizes of the

corresponding strata in the total sample. [Applied Sampling, 1976, S. Sudman, Academic

Press, New York.]

Proportional hazards model: See Cox’s proportional hazards model.

Proportional mortality ratio: A ratio that consists of a numerator that is the number of deaths

from a particular cause and a denominator of total deaths from all causes. Simply the fraction of

deaths from a particular cause. [American Journal of Individual Medicine, 1986, 10,127–41.]

Proportional -odds model: A model for investigating the dependence of an ordinal variable on a

set of explanatory variables. In the most commonly employed version of the model the

cumulative probabilities, Prðy  kÞ, where y is the response variable with categories

1  2  3 c, are modelled as linear functions of the explanatory variables via the

logistic transformation

, that is

log

Prðy  kÞ

1 Prðy  kÞ

¼ τ

þ β

þþβ

where τ

is a category speciﬁ c parameter and β

; ...; β

are regression c oefﬁcients. The

name proportional-odds arises since the

odds ratio

of having a score of k or less for two

different sets of values of the explanatory variables does not depend on k.[GLM

Chapter 5.]

344

Proportionate mortality ratio (P M R): The number of deaths from a speciﬁc cause in a speciﬁc

period of time per 100 deaths from all causes in the same time period. The ratio is used to

approximate the

standardized mortality ratio

when death data are available but the popula-

tion at risk is not known. See also mortality odds ratio.[Journal of Chronic Diseases, 1978,

31,15–22.]

Prospective study: Studies in which individuals are followed-up over a period of time. A common

example of this type of investigation is where samples of individuals exposed and not

exposed to a possible risk factor for a particular disease, are followed forward in time to

determine what happens to them with respect to the illness under investigation. At the end of

a suitable time period a comparison of the incidence of the disease amongst the exposed and

non-exposed is made. A classic example of such a study is that undertaken among British

doctors in the 1950s, to investigate the relationship between smoking and death from lung

cancer. All

clinical trials

are prospective. See also retrospective study, cohort study and

longitudinal study. [SMR Chapter 10.]

Protective efficacy of a vaccine: The proportion of cases of disease prevented by the vaccine,

usually estimated as

PE ¼ðARU  ARVÞ=ARU

where ARV and ARU are the

attack rates

of the disease under study among the vaccinated

and unvaccinated cohorts, respectively. For example, if the rate of the disease is 100 per 10

000 in a non-vaccinated group but only 30 per 10 000 in a comparable vaccinated group, the

PE is 70%. Essentially equivalent to

attributable risk

Protocol: A formal document outlining the proposed procedures for carrying out a

clinical trial

. The

main features of the document are study objectives, patient selection criteria, treatment

schedules, methods of patient evaluation, trial design, procedures for dealing with

protocol

violations

and plans for statistical analysis. [Clinical Trials in Psychiatry, 2003, B. S. Everitt

and S. Wessley, Oxford University Press, Oxford.]

Protocol violations: Patients who either deliberately or accidentally have not followed one or other

aspect of the protocol for carrying out a

clinical trial

. For example, they may not have taken

their prescribed medication. Such patients are said to show

non-compliance

Protopathic bias: A type of bias (also know as reverse-causality) that is a consequence of the

differential misclassiﬁcation of exposure related to the timing of occurrence. Occurs when a

change in exposure taking place in the time period following disease occurrence is incorrectly

thought to precede disease occurrence. For example, a ﬁnding that alcohol has a protective

effect for clinical gallstone disease might be explained by a reduction in alcohol use because of

symptoms related to gallstone disease. [Biological Psychiatry, 1997, 41,257–8.]

Proximity matrix: A general term for either a

similarity matrix

or a

dissimilarity matrix

. In general

such matrices are symmetric, but

asymmetric proximity matrices

do occur in some situa-

tions. [MV1 Chapter 5.]

Pruning algorithms: Algorithms used i n the design of

artiﬁcial neural networks

for selecting the

right-sized network. This is important since the use of the right-sized network leads to an

improvement in performance. [IEEE Transactions in Neural Networks, 1993, 4,740–7.]

Pseudo-in verse: Synonym for Moore–Penrose inverse.

Pseudo-li kel ihood: A term that has many meanings in statistics but the common theme is to signal

that such a ‘likelihood’ differs from a standard likelihood. For instance, the term has been used

in the context of inference based on complex survey data with probability weights, multi-stage

345

estimation where estimates from previous stages are plugged in at later stages, and for

approximations based on linearization for generalized linear mixed models.

Pseudorando m numbers: A sequence of numbers generated by a speciﬁc computer

algorithm

which satisfy particular statistical tests of randomness. So although not random, the numbers

appear so. See also congruential methods. [KA1 Chapter 9.]

Pseud o R

: An index sometimes used in assessing the ﬁt of speciﬁc types of models particularly

logistic regression and those used for modelling survival times. It is deﬁned as

1 

ln Lðpresent modelÞ

ln Lðmodel without covariatesÞ

where L represents

likelihood

. Expresses the relative decrease of the

log-likelihood

of the

present model as opposed to the model without covariates. [Sociological Methods and

Research, 2002, 31,27–74.]

Psyc hom et r ics: The study of the measurements of psychological characteristics such as abilities,

aptitudes, achievement, personality traits and knowledge.

Pub l icatio n b ias: The possible bias in published accounts of, for example,

clinical trials

, produced

by editors of journals being more likely to accept a paper if a statistically signiﬁcant effect

has been demonstrated. See also funnel plot . [SMR Chapter 8.]

P-value : The probability of the observed data (or data showing a more extreme departure from the

null

hypothesis

) when the null hypothesis is true. See also misinterpretation of P-values,

signiﬁcance test and signiﬁcance level. [SMR Chapter 8.]

Pyke, R onald (1 932^20 05): Pyke was born in Hamilton, Ontario and graduated with honours from

McMaster University in 1953. In 1965 he completed a Ph.D. in mathematics at the same

university, the title of his thesis being ‘On one-sided distribution free statistics’. On completing

his Ph.D., Pyke spent two years in Stanford and then moved to the University of Washington

where he remained until he retired in 1998. Pyke made important contributions to the

Kolmogorov–Smirnov statistics and goodness of ﬁt statistics

, and branching processes, and

was editor of the Annals of Probability during the ﬁrst four years of its existence; he was also

President of the Institute of Mathematical Statistics in 1986–7. Pyke died on 22 October 2005.

Pyramids: Generalizations of hierarchical classiﬁcations in which each of the constituent classes

comprises an interval of an ordering of the set of objects, and two classes may have some

but not all their objects in common. [Classiﬁcation, 2nd edition, 1999, A. D. Gordon,

Chapman and Hall/CRC Press, London.]

346

Q^Q plot: Abbreviation for quantile–quantile plot.

Q-tech nique: The class of data analysis methods that look for relationships between the individuals

in a set of

multivariate data

. Includes

cluster analysis

and

multidimensional scaling

although the term is most commonly used for a type of

factor analysis

applied to an

n × n matrix of ‘correlations’ between individuals rather than between variables. See

also R-techniques.[A User’s Guide to Principal Components, 2nd edition, 2003,

J. K. Jackson, Wiley, New York.]

Quadrant sampling: A sampling procedure used with

spatial data

in which sample areas (the

quadrants) are taken and the number of objects or events of interest occurring in each

recorded. See also distance sampling and line-intersect sampling.[Quarterly Review of

Biology, 1992, 67, 254.]

Quadratic discriminant function: See discriminant analysis.

Quadratic function: A function of the form

Q ¼

i¼1

j¼1

important in

multivariate analysis

and

analysis of variance

Quadratic loss function: See decision theory.

Quad ratu re: See numerical integration.

Quad ratu re spectrum: See multiple time series.

Quality-adjusted survivalanalysis: A methodology for evaluating the effects of treatment on

survival which allows consideration of quality of life as well as quantity of life. For

example, a highly toxic treatment with many side effects may delay disease recurrence and

increase survival relative to a less toxic treatment. In this situation, the trade-off between

the negative quality-of-life impact and the positive quantity-of-life impact of the more

toxic therapy should be evaluated when determining which treatment is most likely to

beneﬁt a patient. The method proceeds by deﬁning a quality function that assigns a ‘score’

to a patient which is a composite measure of both quality and quantity of life. In general

the quality function assigns a small value to a short life with poor quality and a high value

to a long life with good quality. The assigned scores are then used to calculate

quality-

adjusted survival times

for analysis. [Statistics in Medicine, 1993, 12, 975–88.]

Quality-adjusted survival times: The weighted sum of different time episodes making up a

patient’s

survival time

, with the weights reﬂecting the quality of life of each period.

[Statistics in Medicine, 1993, 12, 975–88.]

Quality assurance: Any procedure or method for collecting, processing or analysing data that is

aimed at maintaining or improving the reliability or validity of the data.

347

Quality control angle chart: A modiﬁed version of the

cusum

, derived by representing each

value taken in order by a line of some unit length in a direction proportional to its value. The

line for each successive value starts at the end of the preceeding line. A change in direction

on the chart indicates a change in the mean value. [Statistics in Medicine, 1987, 6, 425–40.]

Quality control procedures: Statistical procedures designed to ensure that the precision and

accuracy of, for example, a laboratory test, are maintained within acceptable limits. The

simplest such procedure involves a chart (usually called a control chart) with three hori-

zontal lines, one drawn at the target level of the relevant

control statistic

, and the others,

called action lines, drawn at some pre-speciﬁed distance above and below the target

level. The process is judged to be at an acceptable quality level as long as the observed

control statistic lies between the two lines and to be at a rejectable quality level if either

of these lines are crossed. [Modern Methods of Quality Control and Improvement, 2nd

edition, 2001, H. M. Wadsworth, K. S. Stephen and A. B. Godfrey, Wiley, New York.]

Quality function: See quality-adjusted survival analysis.

Quangle: Synonym for quality control angle chart.

Quantalassay: An experiment in which groups of subjects are exposed to different doses of, usually,

a drug, to which a particular number respond. Data from such assays are often analysed

using the

probit transformation

, and interest generally centres on estimating the

median

effective dose

lethal dose 50

.[Analysis of Quantal Response Data, 1992, B. J. T. Morgan,

Chapman and Hall/CRC Press, London.]

Quantal variable: Synonym for binary variable.

Quantil e ^ quantil e (Q ^ Q) plot: See probability plot.

Quantile regression: An extension of classical least squares from the estimation of conditional

mean models to the estimation of a variety of models for several conditional quantile

functions. An example of such a model is

least absolute deviation regression

. Quantile

regression is often capable of providing a more complete statistical analysis of the stochastic

relationship between random variables. [Journal of Economic Perspectives, 2001, 15,

143–56.]

Quantiles: Divisions of a probability distribution or frequency distribution into equal, ordered

subgroups, for example, quartiles or percentiles. [SMR Chapter 7.]

Quantit model: A three-parameter non-linear

logistic regression model

.[Biometrics, 1977, 33,

175–86.]

Quartiles : The values that divide a frequency distribution or probability distribution into four equal

parts. [SMR Chapter 7.]

Quasi-experiment: A term used for studies that resemble experiments but are weak on some of

the characteristics, particularly that allocation of subjects to groups is not under the

investigator’s control. For example, if interest centred on the health effects of a natural

disaster, those who experience the disaster can be compared with those who do not, but

subjects cannot be deliberately assigned (randomly or not) to the two groups. See also

prospective studies, experimental design and clinical trials.[Experimental and Quasi-

experimental Designs for Research, 1963, D. T. Campbell and J. C. Stanley, Houghton

Mifﬂin, Boston.]

348