Everitt B.S. The Cambridge Dictionary of Statistics

Подождите немного. Документ загружается.

exogeneity assumption is violated,

ﬁxed-effects models

with ﬁxed cluster-speciﬁceffectsα

are sometimes preferred. Multilevel models are sometimes called mixed models because they

contain both ﬁxed effects β

; β

; ...; β

and a random effects 

More complex linear multilevel models also include at least one random coefﬁcient 

¼ β

þ β

1ij

þ β

2ij

þ ...þ β

qij

þ 

1ij

þ 

¼ðβ

þ 

Þþðβ

þ 

Þx

1ij

þ β

2ij

þ ...þ β

qij

þ 

where the effect of the covariate X

1ij

(which varies within clusters) vary between clusters

with random slope β þ 

. The random effects 

and 

have a bivariate normal distribu-

tion with zero means.

Multilevel generalized linear models or generalized linear mixed models are multilevel

models where random effects are introduced in the linear predictor of

generalized linear

models

. In addition to linear models for continuous responses, such models include, for

instance, logistic random effects models for dichotomous, ordinal and nominal responses

and log-linear random effects models for counts.

Multilevel models can also be speciﬁed for higher-level data where units are nested in

clusters which are nested in superclusters. An example of such a design would be measure-

ment occasions nested in subjects who are nested in communities. Other terms sometimes used

for multilevel models include mixed models. hierarchical models, random effects models and

random coefﬁencnt models.[Multilevel Analysis, 1999, T. A. B. Snijders, Sage, London]

M ultimodal distribution: A probability distribution or frequency distribution with several modes.

Multimodality is often taken as an indication that the observed distribution results from the

mixing of the distributions of relatively distinct groups of observations. An example of a

distribution with four modes is shown in Fig. 96.

Multinomialcoefficient: The number of ways that k distinguishable items can be distributed into

n containers so that there are k

items in the ith container. Given by

! k

Fig. 96 A frequency distribution with four modes.

289

Multinomial distribution: A generalization of the

binomial distribution

to situations in which r

outcomes can occur on each of n trials, where r>2. Speciﬁcally the distribution is given by

Pðn

; n

; ...; n

Þ¼

! n

p

where n

is the number of trials with outcome i, and p

is the probability of outcome i

occurring on a particular trial. The expected value of n

is np

and its variance is np

ð1 p

Þ.

The covariance of n

and n

is  np

. [STD Chapter 26.]

Multinomial logit model: See multinomial logistic regression.

Multinomial logistic regression: A form of

logistic regression

for use when the categorical

response variable has more than two unordered categories. If we let k be the number of

categories of the response variable, Y, then the model used (sometimes called the multi-

nomial logit model) is the following;

ðxÞ¼

expðx

s¼1

expðx

; r ¼ 1; ...; k

where p

ðxÞ¼PrðY ¼ r jxÞ; r ¼ 1; ...; k, x is a vector of explanatory variables and β

is a

vector of regression coefﬁcients for category r. Because it is only possible to investigate the

effect of x upon the ‘preference’ of a response category compared to other categories, not all

the parameter vectors are identiﬁable so it is necessary to choose a reference category,

commonly category k and to take β

¼ 0. See also proportional odds model.[Applied

Logistic Regression, 2nd edn, 2000, D. W. Hosmer and S. Lemeshow, Wiley, New York.]

Multinormal distribution: Synonym for multivariate normal distribution.

Multiphasic screening: A process in which tests in

screening studies

may be performed in

combination. For example, in cancer screening, two or more anatomic sites may be screened

for cancer by tests applied to an individual during a single screening session. [American

Journal of Public Health, 1964, 54, 741–50.]

Mu lt ipl e compa r iso n tests: Procedures for detailed examination of the differences between a set

of means, usually after a general hypothesis that they are all equal has been rejected. No

single technique is best in all situations and a major distinction between techniques is how

they control the possible inﬂation of the type I error. See also Bonferroni correction,

Duncan’s multiple range test, Scheffé’s test and Dunnett’s test.[Biostatistics: A

Methodology for the Health Sciences , 2nd edition, 2004, G. Van Belle, L. D. Fisher,

P. J. Heagerty and T. S. Lumley, Wiley, New York.]

Multiplecorrelationcoefficient: The correlation between the observed values of the dependent

variable in a

multiple regression

, and the values predicted by the estimated regression

equation. Often used as an indicator of how useful the explanatory variables are in predicting

the response. The square of the multiple correlation coefﬁcient gives the proportion of

variance of the response variable that is accounted for by the explanatory variables. [SMR

Chapter 12.]

Multipleendpoints: A term used to describe the variety of outcome measures used in many

clinical

trials

. Typically there are multiple ways to measure treatment success, for example, length of

patient survival, percentage of patients surviving for two years, or percentage of patients

experiencing tumour regression. The aim in using a variety of such measures is to gain better

overall knowledge of the differences between the treatments being compared. The danger

290

with such an approach is that the performance of multiple signiﬁcance tests incurs an

increased risk of a

false positive result

. See also Bonferroni correction.[Statistics in

Medicine, 14, 1995, 1163–76.]

Multiple-frame surveys: Surveys that refer to two or more

sampling frames

that can cover a target

population. For example, in a study of AIDS a general population frame might be used in

addition to drug treatment centres and hospitals. Information from the samples is then

combined to estimate population quantities. Such surveys are very useful for sampling

rare or hard-to-reach populations and can result in considerable cost savings over a single

frame design with comparable precision. See also respondent-driven sampling and snow-

ball sampling.[Journal of the American Statistical Association, 2006, 10, 1019–1030.]

Multiple imputation: A

Monte Carlo method

in which

missing values

in a data set are replaced by

m>1 simulated versions, where m is typically small (say 3–10). Each of the simulated

complete datasets is analysed by the method appropriate to the investigation at hand, and the

results are later combined to produce estimates, conﬁdence intervals etc. The imputations are

created by a Bayesian approach which requires speciﬁcation of a parametric model for the

complete data and, if necessary, a model for the mechanism by which data become missing.

Also required is a

prior distribution

for the unknown model parameters.

Bayes’ theorem

used to simulate m independent samples form the conditional distribution of the missing

values given the observed values. In most cases special computation techniques such as

Markov chain Monte Carlo methods

will be needed. See also SOLAS.[The Analysis of

Incomplete Multivariate Data, 1997, J. Schafer, CRC/Chapman and Hall.]

Multiple indicator multiple cause model (MIMIC): A

structural equation model

in which

there are multiple indicators and multiple causes of each

latent variable

.[Modelling

Covariance and Latent Variables using EQS, 1993, G. Dunn, B. Everitt and A. Pickles,

Chapman and Hall/CRC Press, London.]

Multiple linear regression: Synonym for multiple regression.

Multiple regression: A term usually applied to models in which a continuous response variable, y,is

regressed on a number of explanatory variables, x

; x

; ...; x

. Explicitly the model ﬁtted is

E ð yjx

; ...; x

Þ¼β

þ β

þþβ

where E denotes expected value. By introducing a vector y

¼½y

; y

; ...; y

 and an

n ðq þ 1Þ matrix X given by

X ¼

1 x

 x

1 x

 x

1 x

 x

the model for n observations can be written as

y ¼ Xβ þ 

where 

¼½

;

; ...;

 contains the residual error terms and β

¼½β

; β

; ...; β

.

Least squares estimation

of the parameters involves the following set of equations

β ¼ðX

XÞ

1

The regression coefﬁcients, β

; β

; ...; β

give the change in the response variable corre-

sponding to a unit change in the appropriate explanatory variable, conditional on the other

variables remaining constant. Signiﬁcance tests of whether the coefﬁcients take the value

291

zero can be derived on the assumption that for a given set of values of the explanatory

variables, y has a normal distribution with constant variance. See also regression diagnos-

tics, selection methods in regression and beta coefﬁcient. [ARA Chapter 3.]

Multiple time response data: Data arising in studies of episodic illness, such as bladder cancer

and epileptic seizures. In the former, for example, individual patients may suffer multiple

bladder tumours at observed times, t



M ultiple time series: Observations taken simultaneously on two or more

time series

. Also known

as multivariate time series and vector time series. Environmental examples include readings

of lead concentrations at several sites at 5 minute intervals, air temperature readings taken at

hourly intervals at a ﬁxed height above sea level at several locations and monthly ozone

levels at several recording stations. If the m series are represented as X

¼½x

1;t

; x

2;t

; ...; x

m;t



for t ¼ 1; 2; ...; n then the cross-covariance function, γ

ðkÞis covðx

i;t

; x

j; tk

Þand the cross-

correlation function, 

ðkÞ is deﬁned as



ðkÞ¼

ðkÞ

½γ

ð0Þγ

ð0Þ

The matrix rðkÞ with ijth entry equal to 

ðkÞ can be used to deﬁne the spectral density

matrix, fðωÞ,deﬁned as

fðωÞ¼

l¼1

expð2piωlÞrðkÞ0:5  ω  0:5

where i

¼1. The ijth element of this matrix is the cross-spectral density. The real and

imaginary parts of this matrix deﬁne the co-spectrum and quadrature spectrum. Multivariate

generalizations of

autoregressive moving average models

are available for modelling such a

collection of time series. See also coherence. [TMS Chapter 8.]

Multiplication rule for probabilities: For events A and B that are independent, the probability

that both occur is the product of the separate probabilities, i.e.

PrðA and BÞ¼PrðAÞ PrðBÞ

where Pr denotes probability. Can be extended to k independent events, B

; B

; ...; B

Prð\

i¼1

Þ¼

i¼1

PrðB

[KA1 Chapter 8.]

Multiplicative intensity model: A generalization of the

Cox’s proportional hazards model

.[The

Annals of Statistics, 1978, 6, 701–26.]

Multiplicative model: A model in which the combined effect of a number of factors when applied

together is the product of their separate effects. See also additive model.

MULT ISCALE : Software for

multidimensional scaling

set in an inferential framework. [J. O. Ramsay,

Department of Psychology, McGill University, Steward Biological Sciences Building, 1205

Dr. Penﬁeld Avenue, Montreal, QC, Canada H34 1B1.]

Mu lt istage sam pli ng: Synonym for cluster sampling.

Mu lt istate models: Models that arise in the context of the study of

survival times

. The experience

of a patient in such a study can be represented as a process that involves two (or more) states.

In the simplest situation at the point of entry to the study, the patient is in a state that

292

corresponds to being alive. Patients then transfer from this ‘live’ state to the ‘dead’ state at

some rate measured by the

hazard function

at a given time. More complex models will

involve more states. For example, a three-state model might have patients alive and tumour

free, patients alive and tumour present and the ‘dead’ state. See also Markov illness–death

model.[Statistics in Medicine, 1988, 7, 819–42.]

M u lt itaper spectra lesti mators: A very powerful class of procedures for estimating the

spectral

density

of a

time series

, which use the average of several direct spectral estimators. [Signal

Processing, 1997, 58, 327–32.]

Mu ltit rait^ mult imethod model ( MTMM ) : A form of

conﬁrmatory factor analysis

model in

which different methods of measurement are used to measure each latent variable. Allows

the variance of each measure to be decomposed into trait variance, method variance and

measurement error variance. [Modelling Covariances and Latent Variables using EQS,

1993, G. Dunn, B. Everitt and A. Pickles, Chapman and Hall/CRC Press, London.]

Multivariate adaptive regression splines (MARS): A method of ﬂexible

nonparametric

regression modelling

that works well with moderate sample sizes and with more than two

explanatory variables. [Journal of Computational and Graphical Statistics, 1997, 6,74–91.]

M ultivariate analysis: A generic term for the many methods of analysis important in investigating

multivariate data. Examples include

cluster analysis

principal components analysis

and

factor analysis

. [MV1 and MV2.]

Multivariate analysis of variance: A procedure for testing the equality of the mean vectors of

more than two populations for a multivariate response variable. The technique is directly

analogous to the

analysis of variance

of univariate data except that the groups are compared on

q response variables simultaneously. In the univariate case, F-tests are used to assess the

hypotheses of interest. In the multivariate case, however, no single test statistic can be

constructed that is optimal in all situations. The most widely used of the available test statistics

is Wilk’slambda(L) which is based on three matrices W (the within groups matrix of sums of

squares and products), T (the total matrix of sums of squares and cross-products)andB (the

between groups matrix of sums of squares and cross-products), deﬁned as follows:

T ¼

i¼1

j¼1

ðx





xÞðx





xÞ

W ¼

i¼1

j¼1

ðx





Þðx





B ¼

i¼1







xÞð







xÞ

where x

; i ¼ 1; ...; g; j ¼ 1; ...; n

represent the jth multivariate observation in the ith

group, g is the number of groups and n

is the number of observations in the i th group. The

mean vector of the ith group is represented by



and the mean vector of all the observations



x. These matrices satisfy the equation

T ¼ W þB

Wilk’s lambda is given by the ratio of the determinants of W and T, i.e.

L ¼

jWj

jTj

jWj

jW þ Bj

The statistic, L, can be transformed to give an F-test to assess the null hypothesis of the

equality of the population mean vectors. In addition to L a number of other test statistics are

available

293

Roy’s largest root criterion: the largest

eigenvalue

of BW

1

The Hotelling –Lawley trace: the sum of the eigenvalues of BW

1

The Pillai–Bartlett trace: the sum of the eigenvalues of BT

1

It has been found that the differences in power between the various test statistics are

generally quite small and so in most situations which one is chosen will not greatly affect

conclusions. [MV1 Chapter 6.]

Multivariate Bartlett test: A test for the equality of a number (k) of

variance-covariance

matrices

. The test statistic used is M given by

M ¼



...jS



jSj



where 

¼ n

 1 where n

is the sample size of the ith sample, S

is the covariance matrix of

the ith sample,  ¼

i¼1



and S is the pooled sample covariance matrix given by

S ¼

i¼1



i¼1



The statistic u ¼2ð1  c

Þlog M is under the null hypothesis of the equality of the

population covariance matrices, approximately distributed as chi-squared with

ðk  1Þpðp þ 1Þ degrees of freedom where p is the number of variables and

i¼1



ð1=

i¼1



þ 3p  1

6ðp þ1Þðk  1Þ



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

B. Vidakovic, Wiley, New York.]

Mu lt i va r i at e Cauch y di str ibution: See Student’s t-distribution.

M ultivariate counting process: A

stochastic process

with k components counting the occur-

rences, as time passes, of k different types of events, none of which can occur simulta-

neously. [Statistical Analysis of Counting Processes, 1982, M. Jacobsen, Springer-Verlag,

New York.]

Multivariate data: Data for which each observation consists of values for more than one random

variable. For example, measurements on blood pressure, temperature and heart rate for a

number of subjects. Such data are usually displayed in the form of a data matrix, i.e.

X ¼

 x



 x

where n is the number of subjects, q the number of variables and x

the observation on

variable j for subject i. [MV1 Chapter 1.]

Multivariate distribution: The simultaneous probability distribution of a set of random variables.

See also multivariate normal distribution and Dirichlet distribution. [KA1 Chapter 15.]

M u ltivar iate growth data: Data arising in studies investigating the relationships in the growth of

several organs of an organism and how these relationships evolve. Such data enable

biologists to examine growth gradients within an organism and use these as an aid to

understanding its form, function and biological niche as well as the role of evolution in

bringing it to its present form. [Psychometrika, 2004, 69,65–79.]

294

Multivariate hypergeometric distribution: A probability distribution associated with

sam-

pling without replacement

from a population of ﬁnite size. The population consists of r

elements of type i, i ¼ 1; 2; ...; k with

i¼1

¼ N . Then the probability of ﬁnding x

elements of the ith type when a random sample of size n is drawn ðn ¼

Þ is

Prðx

; x

; ...; x

Þ¼







When k=2 this reduces to the

hypergeometric distribution

. The distribution is important in

conditional inference for 2  k

contingency tables

.[Statistical Analysis of Categorial Data,

1999, C. J. Lloyd, Wiley, New York.]

M ultivariate kurtosis measure: See Mardia’s multivariate normality test.

M ultivariate normal distribution: The probability distribution of a set of variables

¼½x

; x

; ...; x

 given by

f ðx

; x

; ...; x

Þ¼ð2pÞ

q=2

jSj



exp 

ðx  mÞ

1

ðx  mÞ



where m is the mean vector of the variables and S is their

variance–covariance matrix

. This

distribution is assumed by multivariate analysis procedures such as

multivariate analysis of

variance

. See also bivariate normal distribution. [KA1 Chapter 15.]

M ultivariate power exponential distribution: A multivariate probability distribution, f(x)

given by

f ðxÞ¼



ﬃﬃﬃﬃﬃﬃ

jSj

G 1 þ

2β



1þ

2β

exp 

½ðx Þ0S

1

ðx Þ



The distribution has mean vector µ and

variance-covariance matrix

given by

nþ2

2β



2β



When β ¼ 1; f ðxÞ becomes the

multivariate normal distribution

. In general f ðxÞ is an

elliptically contoured distribution in which β determines the

kurtosis

. For β

1 the distri-

bution has heavier tails than the normal distribution. [Nonlinear Models in Medical

Statistics, 2001, J. K. Lindsey, Oxford University Press, Oxford.]

M u lt iva riate prob it analysis: A method for assessing the effect of explanatory variables on a set

of two or more correlated binary response variables. See also probit analysis.[Biometrics,

1981, 37, 541–51.]

M u lt iva r iate skew -norma l d ist ri buti o n: See skew-normal distribution.

Multivariate Student’s t-distribution: See Student’s t-distribution.

M ultivariate time series: Synonym for multiple time series.

Mu lt i v a r i at e ZIP model ( M ZIP): A multivariate version of

zero-inﬂated Poisson regression

useful in monitoring manufacturing processes when several types of defect are possible.

[Technometrics, 1999, 41,29–38.]

Mutation distance: A distance measure for two amino acid sequences, deﬁned as the minimal

number of nucleotides that would need to be altered in order for the gene of one sequence to

295

code for the other. [Branching Processes with Biological Applications, 1975, P. Jagers,

Wiley, New York.]

M utual ly exclusive events: Events that cannot occur jointly.

MYCIN: An

expert system

developed at Stanford University to assist physicians in the diagnosis and

treatment of infectious diseases. The system was never used in practice, not because of any

weakness in performance, but because of ethical and legal problems related to the use of

computers in medicine – if it gives the wrong diagnosis who would you sue? [Rule-Based

Expert Systems, 1985, B. G. Buchanan and E. H. Shortliffe, Addison-Wesley, Reading,

MA.]

MZIP: Abbreviation for multivariate ZIP model.

296

N ada raya- Watson esti mat or: An estimator used to estimate regression functions based on data

; y

. The estimator produces an estimate of Y at any requested value of X (not only the

ones in the data). A

kernel function

is used to assign weights to be put on values in the data

near X in estimating Y.[Nonparametric Functional Data Analysis: Theory and Practice,

2006, F. F. Erraty and P. V. Ieu, Springer, New York.]

NAG: Numerical Algorithms Group producing many useful subroutines relevant to statistics. [NAG

Ltd., Wilkinson House, Jordan Hill Road, Oxford, OX2 8DR; NAG Inc., 1431 Opus Place,

Suite 220, Downers Grove, Illinois 60515-1362, USA; www.nag.co.uk]

Naor’s distribution: A discrete probability distribution that arises from the following model;

Suppose an urn contains n balls of which one is red and the remainder are white.

Sampling with replacement of a white ball (if drawn) by a red ball continues until a

red ball is drawn. Then the probability distribution of the required number of draws,

Y,is

PrðY ¼ yÞ¼

ðn 1Þ!y

ðn yÞ!n

After n − 1 draws the urn contains only red balls and so no more than n draws are required.

[Univariate Discrete Distributions, 2005, N. L. Johnson, A. W. Kemp and S. Kotz, Wiley,

New York.]

National lotteries: Games of chance held to raise money for particular causes. The ﬁrst held in the

UK took place in 1569 principally to raise money for the repair of the Cinque Ports. There

were 400 000 tickets or lots, with prizes in the form of plate, tapestries and money.

Nowadays lotteries are held in many countries with proceeds either used to augment the

exchequer or to fund good causes. The current UK version began in November 1994 and

consists of selecting six numbers from 49 for a one pound stake. The winning numbers are

drawn ‘at random’ using one of a number of ‘balls-in-drum’ type of machine. [Chance

Rules, 2nd edn, 2008, B. S. Everitt, Springer-Verlag, New York.]

Natural-pairing: See paired samples.

N earest-nei ghbour cl ust eri ng: Synonym for single linkage clustering.

N earest-nei ghbour methods I: Methods of

discriminant analysis

based on studying the

training

set

subjects most similar to the subject to be classiﬁed. Classiﬁcation might then be decided

according to a simple majority verdict among those most similar or ‘nearest’ training set

subjects, i.e. a subject would be assigned to the group to which the majority of the

‘neighbours’ belonged. Simple nearest-neighbour methods just consider the most similar

neighbour. More general methods consider the k nearest neighbours, where k>1.

[Discrimination and Classiﬁcation, 1981, D. J. Hand, Wiley, Chichester.]

297

Nearest -neig hbour methods II: Methods used in the preliminary investigation of

spatial data

assess depatures from

complete spatial randomness

. For example, histograms of distances

between nearest neighbours are often useful. [Spatial Processes: Models and Applications,

1981, A. D. Cliff and K. Ord, Penn, London.]

Necessarily empty cells: Synonym for structural zeros.

Negative binomial distribution: The probability distribution of the number of failures, X, before

the kth success in a sequence of

Bernoulli trials

where the probability of success at each trial

is p and the probability of failure is q =1−p. The distribution is given by

PrðX ¼ xÞ¼

k þ x  1



0  x

The mean, variance,

skewness

and

kurtosis

of the distribution are as follows:

mean ¼ kq=p

variance ¼ kq=p

skewness ¼ð1 þ qÞðkqÞ



kurtosis ¼ 3 þð1 þ4p þ p

Þ=ðkpÞ

Often used to model

overdispersion

in count data. [STD Chapter 28.]

Negative exponential distr ibution: Synonym for exponential distribution.

Negative hypergeometric distribution: In

sampling without replacement

from a population

consisting of r elements of one kind and N − r of another, if two elements corresponding to

that selected are replaced each time, then the probability of ﬁnding x elements of the ﬁrst

kind in a random sample of n elements is given by

PrðxÞ¼

r þ x 1



N r þ n  x  1

n  x



N þn  1



The mean of the distribution is Nr/N and the variance is (nr/N)(1−r/N)(N+n)/(N+1).

Corresponds to a

beta binomial distribution

with integral parameter values. [KA1

Chapter 5.]

N egat ive mult i no m i na l distr i buti on: A generalization of the

negative binomial distribution

which r > 2 outcomes are possible on each trial and sampling is continued until m outcomes

of a particular type are obtained. [Biometrika, 1949, 36,47–58.]

Negative predictive val ue: The probability that a person having a negative result on a

diagnostic

test

does not have the disease. See also positive predictive value. [SMR Chapter 14.]

Negative skewness: See skewness.

Negative study: A study that does not yield a statistically signiﬁ cant result.

Negative synergism: See synergism.

N ei gh bourh ood cont ro ls: Synonym for community controls.

Ne lder ^ M ead si mplex algo rithm: Type of simplex algorithm.

Nelson ^ Aalen estimator: A nonparametric estimator of the

cumulative hazard function

from

censored

survival data

. Essentially a

method of moments

estimator. [Biometrics , 1996, 52,

1034–41.]

298