A Glossary of Statistics (in particular, re-randomisation statistics)
Norman W. Marsh,
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A formal statement, clear complete and unambiguous, of how
a certain process needs to be undertaken. Also see : ALGORITHM(2).
An ALGORITHM(1) expressed in a PROGRAMMING LANGUAGE
for a COMPUTER .
Also known as SIZE or TYPE-1 ERROR. This is
the probability that, according to some null hypothesis, a
statistical test will generate a false-positive error : affirming
a non-null pattern by chance. Conventional methodology for
statistical testing is, in advance of undertaking the test,
to set a NOMINAL ALPHA CRITERION LEVEL (often 0.05).
The outcome is classified as showing STATISTICAL SIGNIFICANCE
if the actual ALPHA (probability of the outcome
under the null hypothesis) is no greater than this NOMINAL
ALPHA CRITERION LEVEL (but see : TAIL DEFINITION
POLICIES). This reasoning is applicable for all types
of statistical testing, including RE-RANDOMISATION STATISTICS
which are the concern of this present glossary. Also
see : BETA, ERROR
[Initials/acronym for the American National Standards Institute]
This body publishes specifications for a number of STANDARD
PROGRAMMING LANGUAGES. The specifications are generally
arranged to concur with those of ISO.
[()] This is the simplest probability model - a single trial
between two possible outcomes such as a coin toss. The distribution
depends upon a single parameter,'p', representing the probability
attributed to one defined outcome out of the two possible
outcomes. Also see : BINOMIAL DISTRIBUTION, POISSON PROCESS.
Also known as TYPE-2 ERROR, BETA is the complement
to POWER : BETA = (1-POWER). This is the probability
that a statistical test will generate a false-negative error
: failing to assert a defined pattern of deviation from a
null pattern in circumstances where the defined pattern exists.
Conventional methodology for statistical testing is to set
in advance a NOMINAL ALPHA CRITERION LEVEL - the corresponding
level for BETA will depend upon the NOMINAL ALPHA CRITERION
LEVEL and upon further considerations including the strength
of the pattern in the data and the sample size. Interest is
generally in the RELATIVE POWER of different tests
rather than in an absolute value. It is questionable whether
the concept of BETA error is properly applicable without considering
the concept of sampling from a population, which is separate
from the concerns of this Glossary. Applicability of this
reasoning is also closely bound up with the choice of TEST
STATISTIC. Also see :
This is a special case of the MULTINOMIAL DISTRIBUTION where
the number of possible outcomes is 2. It is the distribution
of outcomes expected if a certain number of independent trials
are undertaken of a single BERNOUILLI PROCESS (e.g. multiple
tosses of a coin, or tosses of several coins with identical
characteristics). The distribution depends upon the single
parameter,'p', of the corresponding BERNOULLI PROCESS and
upon the number of trials, 'n'. An alternative characterisation
is as the outcome of two separate POISSON PROCESSEs with separate
This is a statistical test referring to a repeated binary
process such as would be expected to generate outcomes with
a BINOMIAL DISTRIBUTION. A value for the parameter 'p' is
hypothesised (null hypothesis) and the difference of the actual
value from this is assessed as a value of ALPHA. Also
see : EXACT BINOMIAL TEST.
[()] This is a form of RANDOMISATION TEST which is one of
the alternatives to EXHAUSTIVE RE-RANDOMISATION. The BOOTSTRAP
scheme involves generating subsets of the data on the basis
of random sampling with replacements as the data are sampled.
Such resampling provides that each datum is equally represented
in the randomisation scheme; however, the BOOTSTRAP procedure
has features which distinguish it from the procedure of a
MONTE-CARLO TEST. The distinguishing features of the BOOTSTRAP
procedure are concerned with over-sampling - there is no constraint
upon the number of times that a datum may be represented in
generating a single resampling subset; the size of the resampling
subsets may be fixed arbitrarily independently of the parameter
values of the EXPERIMENTAL DESIGN and may even exceed the
total number of data. The positive motive for BOOTSTRAP resampling
is the general relative ease of devising an appropriate resampling
ALGORITHM(1) when the EXPERIMENTAL DESIGN is novel
or complex. A negative aspect of the BOOTSTRAP is that the
form of the resampling distribution with prolonged resampling
converges to a form which depends not only upon the data and
the TEST STATISTIC, but also upon the BOOTSTRAP resampling
subset size - thus the resampling distribution should not
be expected to converge to the GOLD STANDARD(1) form of the
EXACT TEST as is the case for MONTE-CARLO resampling. An effective
necessity for the BOOTSTRAP procedure is a source of random
codes or an effective PSEUDO-RANDOM generator.
Exploration of a RANDOMISATION DISTRIBUTION in such a way
as to anticipate the effect of the next RANDOMISATION(3) relative
to the present RANDOMISATION(3). This allows selective search
of particular zones of a RANDOMISATION DISTRIBUTION; in the
context of a RANDOMISATION TEST such selective search may
be concerned with the TAIL of the RANDOMISATION DISTRIBUTION.
Also see : RANOMISATION TEST(1).
'C' [Named as one of a developmental sequence of theoretical programming
languages : 'A', 'B' (also the useful language BCPL)]. A PROGRAMMING
LANGUAGE of broad expressive power; thus suitable for both
numerical and general programming. 'C' is closely associated
with the construction of the ubiquitous computer operating
system 'unix'. COMPILERS for 'C' are supplied for virtually
all modern computers. 'C' is available as a STANDARD PROGRAMMING
LANGUAGE approved by ANSI and ISO.
Where expected frequencies are sufficiently high, hypothesised
distributions of counts may be approximated by a NORMAL DISTRIBUTION
rather than an exact BINOMIAL DISTRIBUTION. The corresponding
distribution of the CHI-SQUARED STATISTIC can be derived algebraically
- this is the CHI-SQUARED DISTRIBUTION which has been computed
and published historically as extensive printed tables. Use
of the tables is notably simple, as the CHI-SQUARED DISTRIBUTION
depends upon only one parameter, the DEGREES OF FREEDOM, defined
as one less than the number of categories.
[Named by E.S. Pearson ()?]. This is a long-established TEST
STATISTIC for measuring the extent to which a set of categorical
outcomes depart from a hypothesised set of probabilities.
It is calculated as a sum of terms over the available categories,
where each term is of the form : ((O-E)^2)/E ; 'O' represents
the observed frequency for the category and 'E' represents
the corresponding expected frequency based upon multiplying
the sample size by the hypothesised probability for the category
being considered (therefore 'E' will generally not be an integer
value). In situations where the number of categories is 2
an alternative procedure is to use an EXACT BINIOMIAL TEST.
Also see : CHI-SQUARED DISTRIBUTION, MULTINOMIAL DISTRIBUTION,
A PROGRAM supplied especially for a particular type of COMPUTER,
to enable the translation of code expressed in some PROGRAMMING
LANGUAGE into OBJECT CODE for that COMPUTER. A COMPILER
undertakes translation of the whole of the user's PROGRAM
to produce an OBJECT CODE version which is complete, undivided
and potentially permanent; this is in contrast to the action
of an INTERPRETER.
An automatic data-processing device which is PROGRAMMABLE.
Also see : COMPUTER PROGRAM, OBJECT CODE, PROGRAM.
A specification of how to undertake a certain process, usually
expressed via a PROGRAMMING LANGUAGE, for some chosen COMPUTER.
Also see : PROGRAM.
For a given RE-RANDOMISATION distribution, a family of related
distributions may be defined according to a range of hypothetical
values of the pattern which the TEST STATISTIC measures. For
instance, for the PITMAN PERMUTATION TEST(2) to test for a
scale shift between two groups, a related distribution may
be formed by shifting all the observations in one group by
a common amount, where this common shift is regarded as a
continuous variable. With finite numbers of data the number
of related distributions will be finite, and typically considerably
smaller than the number of points of the RANDOMISATION DISTRIBUTION.
The likelihood of the OUTCOME VALUE may be calculated for
each distribution in the family, and these likelihoods may
be then used to define a contiguous set of values which occupy
a certain proportion of the total unit weight of the likelihoods
integrated over all values of the TEST STATISTIC. The CONFIDENCE
INTERVAL is defined by the minimum and maximum values of the
range of values so defined. The proportion of the total weight
within the range of values is regarded as an ALPHA
probability that the value of the TEST STATISTIC lies within
this range. Generally the definition of a CONFIDENCE INTERVAL
cannot be unique without imposing further constraints. Approaches
to providing suitable constraints, such that a CONFIDENCE
INTERVAL will be unique, include defining the CONFIDENCE INTERVAL
: to include the whole of one TAIL of the distribution; or
to be centred in some sense upon the OUTCOME VALUE; or to
be centred between TAILS of equal weight. In the case of RE-RANDOMISATION
DISTRIBUTIONs, these are DISCRETE DISTRIBUTIONS so there will
generally be no range of values with weight corresponding
exactly to an arbitrary NOMINAL ALPHA CRITERION LEVEL,
and the problem of non-uniqueness is therefore not generally
A probability distribution of a continuous STATISTIC, based
upon an algebraic formula, such that for any possible value
of the cumulative probability there is an exact corresponding
value of the STATISTIC in question. Also see : DISCRETE DISTRIBUTION.
A rule for comparing the OUTCOME VALUE of ALPHA with
a NOMINAL ALPHA CRITERION LEVEL (such as 0.05). An
OUTCOME VALUE smaller (more extreme) than the NOMINAL
ALPHA CRITERION LEVEL leads to a decision of STATISTICAL
SIGNIFICANCE of the finding that the TEST STATISTIC has a
value other than its (null-) hypothesised value. Also see
: STATISTICAL SIGNIFICANCE, TAIL-DEFINITION POLICY.
DEGREES OF FREEDOM
An integer value measuring the extent to which an EXPERIMENTAL
DESIGN imposes constraints upon the pattern of the mean values
of data from various meaningful subsets of data. This value
is frequently referred to in the organisation of tables of
statistical distributions used in undertaking SIGNIFICANCE
TESTS. For simple one-way classifications the value of DEGREES
OF FREEDOM is defined as one less than the number of subsets.
DIFFERENCE OF MEANS
A TEST STATISTIC of intuitive appeal for measuring difference
in location between two samples with INTERVAL-SCALE data.
Employing this TEST STATISTIC in an EXACT TEST defines the
PITMAN PERMUTATION TESTs(1 or 2).
A probability distribution of some STATISTIC, based upon an
algebraic formula or upon re-randomisation or upon actual
data, in which the cumulative probability increases in non-infinitesmal
steps corresponding to non-infinitesmal weight associated
with possible values of the STATISTIC in question. This situation
is characteristic of RANDOMISATION DISTRIBUTIONs, and also
of TEST STATISTICs which are essentially discrete. Also see
: CONTINUOUS DISTRIBUTION.
See : ALPHA, BETA,
TYPE-1 ERROR, TYPE-2 ERROR.
EQUIVALENT TEST STATISTIC
Within a RANDOMISATION SET, it is possible that two different
STATISTICs may be inter-related in a manner which is provably
monotonic irrespective of the data. In such a situation a
RANDOMISATION TEST performed on either of these TEST STATISTICs
will necessarily have the same outcome in terms of ALPHA.
If one of the STATISTICs is of good descriptive validity whereas
the other is simpler to compute, then a RANDOMISATION TEST
upon the simpler STATISTIC may be used in place of a test
upon the descriptively more valid one, with corresponding
savings in amount of computation required. An example of such
EQUIVALENT TEST STATISTICs occurs for the situation of comparison
of levels of a single INTERVAL-SCALE variable between two
groups. In this situation, the descriptively valid statistic,
as defined for the PITMAN PERMUTATION TEST(1), is the difference
of means, but simpler EQUIVALENT TEST STATISTICS include the
mean for one designated group, or (most simply) the total
of scores in one designated group.
EXACT BINOMIAL TEST
A STATISTICAL TEST referring to the BINOMIAL DISTRIBUTION
in its exact algebraic form, rather than through continuous
approximations which are used especially where sample sizes
are substantial. Also see EXACT TEST(1).
This is the name of the academic initiative which produced
this present glossary. EXACT-STATS is a closed e-mail based
discussion group for the development and promulgation of the
ideas of re-randomisation statistics. The contact address
is : firstname.lastname@example.org .
The characteristic of a RE-RANDOMISATION TEST based upon EXHAUSTIVE
RE-RANDOMISATION, that the value of ALPHA will be fixed
irrespective of any random sampling of RANDOMISATIONS or upon
any distributional assumptions. Notable examples are the EXACT
BINOMIAL TEST, FISHER TEST(1), the PITMAN PERMUTATION TESTs(1
and 2), and various NON-PARAMETRIC TESTs based upon RANKED
A test which yields an ALPHA value which does not
depend upon the NOMINAL ALPHA CRITERION VALUE which
may have been set for ALPHA. This is in contrast to
the possible practice of producing only a yes/no decision
with regard to a NOMINAL ALPHA CRITERION VALUE. Note
that this reference to exactness is not (sic) the concern
of the EXACT-STATS initiative.
A series of samples from a RANDOMISATION SET which is known
to generate every RANDOMISATION. In particular, sampling which
generates every RANDOMISATION exactly once.
This term overtly refers to the planning of a process of data
collection. The term is also used to refer to the information
necessary to describe the interrelationships within a set
of data. Such a description involves considerations such as
number of cases, sampling methods, identification of variables
and their scale-types, identification of repeated measures
and replications. These considerations are essential to guide
the choice of TEST STATISTIC and the process of RE-RANDOMISATION.
Also see : DEGREES OF FREEDOM, REPEATED MEASURES,
REPLICATIONS, STRATIFIED, TWO-WAY TABLE.
See : PASCAL.
The FACTORIAL operator is applicable to a non-negative integer
quantity. It is notated as the postfixed symbol '!'. The resulting
value is the product of the increasing integer values from
1 up to the value of the argument quantity. For instance :
3! is 1x2x3 = 6. By convention 0! is taken as producing the
value 1. FACTORIAL values increase very rapidly wityh increase
in the argument value; this rapid growth is represented in
the similarly rapid growth in numbers of COMBINATIONS.
[Named after the statistician RA Fisher()]. This is an EXACT
TEST(1) to examine whether the pattern of counts in a 2x2
cross classification departs from expectations based upon
the marginal totals for the rows and columns. Such a test
is useful to examine difference in rate between two binomial
outcomes. The RANDOMISATION SET consists of those reassignments
of the units which produce tables with the same row- and column-
totals as the OUTCOME. The RANDOMISATION SET will thus consist
of a number of tables with different respective patterns of
counts; each such table will have a number of possible RANDOMISATIONS
which may be a very large number. For this test there are
several reasonable TEST STATISTICs, including : the count
in any one of the 4 cells, CHI-SQUARED(1), or the number of
RANDOMISATIONS for each 2x2 table with the given row- and
column- totals; these are EQUIVALENT TEST STATISTICS. The
calculation for the FISHER TEST(1) is relatively undemanding
computationally, making reference to the algebra of the hypergeometric
distribution, and the test was widely used before the appearance
of COMPUTERs. This test has historically been regarded as
superior to the use of CHI-SQUARED(2) where sample sizes are
small. Statistical tables have been published for the FISHER
TEST(1) for a number of small 2x2 tables defined in terms
of row- and column- totals. Also see FISHER TEST(2), TWO-WAY
[()] This is also known as the FREEMAN-HALTON TEST. It is
an extension of the logic of the FISHER TEST(1), for a 2-way
classification of counts where the extent of the cross-classification
may be greater than 2x2. The RANDOMISATION SET for an EXHAUSTIVE
RANDOMISATION TEST (EXACT TEST(1)) can be defined in the same
way as for the FISHER TEST(1). However, the various TEST STATISTICs
applicable when considering the FISHER TEST(1) will not all
be definable and will not clearly be EQUIVALENT TEST STATISTICs.
The TEST STATISTIC which is used is the number of RE-RANDOMISATIONS
for each table with the given row- and column- totals; this
TEST STATISTIC has the drawback of lacking any descriptive
significance in terms of the EXPERIMENTAL DESIGN.
[Name is an acronym : FORmula TRANslator]. A very long established
and widely implemented PROGRAMMING LANGUAGE, specialised substantially
for numerical applications. A number of STANDARD PROGRAMMING
LANGUAGE versions of FORTRAN have established at various dates
(e.g. FORTRAN IV, FORTRAN 90), approved as standard by ANSI
See FISHER TEST(2).
The GOLD STANDARD is the form of test which is most faithful
to the RANDOMISATION DISTRIBUTION, for a given TEST STATISTIC
and EXPERIMENTAL DESIGN. This involves EXHAUSTIVE RANDOMISATION.
Other RANDOMISATION TESTs may reasonably be judged by comparison
with this form. Also see : BOOTSTRAP, GOLD STANDARD(2), MONTE-CARLO.
The idea of a re-randomisation test as a standard of correctness
by which to judge other tests which are not based upon principles
A PROGRAM supplied especially for a particular type of COMPUTER,
to enable the translation of code expressed in some PROGRAMMING
LANGUAGE into OBJECT CODE for that type of COMPUTER. An INTERPRETER
undertakes translation of the user's PROGRAM in small functional
units (statements) to OBJECT CODE as the PROGRAM is used and
allows modification of the sequence of statements without
need to generate a full explicit OBJECT CODE version of the
PROGRAM; this is in contrast to the action of a COMPILER.
Use of an INTERPRETER is convenient and flexible for program
development; however, running a program produced in this way
generally requires more computational resource (particuarly
in terms of run time) than for the OBJECT CODE produced using
A characteristic of data such that the difference between
two values measured on the scale has the same substantive
meaning/significance irrespective of the common level of the
two values being compared. This implies that scores may meaningfully
be added or subtracted and that the mean is a representative
measure of central tendency. Such data are common in the domain
of physical sciences or engineering - e.g. lengths or weights.
Also see : MEASUREMENT TYPE, SCALE TYPES, STEVENS'
[Initials/acronym for the International Standards Organisation,
based in Geneva, Switzerland] This body publishes specifications
for a number of STANDARD PROGRAMMING LANGUAGES. The specifications
are arranged generally to concur with those of ANSI.
This relates to an EXPERIMENTAL DESIGN for predicting a binary
categorical (yes/no) outome on the basis of predictor variables
measured on INTERVAL SCALEs. For each of a set of values of
the predictor variables, the outcomes are regarded as representing
a BINOMIAL process, with the binomial parameter 'p' depending
upon the value of the predictor variable. The modelling accounts
for the logarithm of the ODDS RATIO as a linear function of
the predictor variable. Fitting is via a weighted least-squares
regression method. RANDOMISATION TESTS for this purpose have
been developed by Mehta & Patel.
[Devised by ()] This is a test of difference in location for
an EXPERIMENTAL DESIGN involving two samples with data measured
on an ORDINAL SCALE or better. The TEST STATISTIC is a measure
of ordinal precedence. For each possible pairing of an observation
in one group with an observation in the alternate group, the
pair is classified in one of three ways - according to whether
the difference is positive, zero or negative; the numbers
in these three categories are tallied over the RANDOMISATION
SET. The RANDOMISATION SET is the same as that for the PITMAN
PERMUTATION TEST(1). This test is generally recommended for
comparisons involving ORDINAL-SCALE data but is not confined
to this SCALE-TYPE. An equivalent formulation of the test,
based upon ranking the data and summing ranks within groups,
is the WILCOXON TEST(2). Also see : COMBINATIONS.
This is a distinction regarding the relationship between a
phenomenon being measured and the data as recorded. The main
distinctions are concerned with the meaningfulness of numerical
comparisons of data (NOMINAL SCALE versus ORDINAL SCALE versus
INTERVAL SCALE versus RATIO SCALE : this is known as STEVENS'
TYPOLOGY), whether the scale of the measurements (other than
NOMIMAL SCALE measurements) should be regarded as essentially
conituous or discrete, and whether the scale is bounded or
[Proposed by H.O Lancaster(), and further promoted by G.A.
Barnard] This is a TAIL DEFINITION POLICY that the ALPHA
value should be calculated as the sum of the proportion of
the TAIL for data strictly more extreme than the OUTCOME,
plus one half of the proportion of the DISTRIBUTION corresponding
to the exact OUTCOME value. This gives an unbiased estimate
Exploration of a RANDOMISATION DISTRIBUTION is such a sequence
that the successive RANDOMISATION(3)s differ is a simple way.
In the context of a RANODMISATION TEST this can mean that
the value of the TEST STATISTIC for a particular RANDOMISATION(3)
may be calculated by a simple adjustment to the value for
the preceding RANDOMISATION(3). Also see : RANDOMISATION(1).
[Named after the famous site of gambling casinos] A MONTE-CARLO
TEST involves generating a random subset of the RANDOMISATION
SET, sampled without replacement, and using the values of
the TEST STATISTIC to generate an estimate of the form of
the full RANDOMISATION DISTRIBUTION. This procedure is in
contrast to the BOOTSTRAP procedure in that the sampling of
the RANDOMISATION SET is without replacement. An advantage
of the MONTE-CARLO TEST over the BOOTSTRAP is that with successive
resamplings it converges to the GOLD STANDARD(1) form of the
EXACT TEST(1). An effective necessity for the MONTE-CARLO
procedure is a source of random codes or an effective PSEUDO-RANDOM
This is the distribution of outcomes expected if a certain
number of independent trials are undertaken of a several separate
BERNOUILLI PROCESSes, to determine a number of alternative
outcomes. A special case, where the number of outcomes is
2, is the BINOMIAL DISTRIBUTION. The distribution depends
upon the collection of parameter values of the corresponding
BERNOULLI PROCESSes and upon the number of trials, 'n'. An
alternative characterisation is as the outcome of a number
of separate POISSON PROCESSes with separate rate parameters.
Also see : TWO-WAY TABLEs.
NOMINAL ALPHA CRITERION LEVEL
A publicly agreed value for TYPE-1 ERROR, such that the outcome
of a statistical test is classified in terms of whether the
obtained value of ALPHA is extreme as compared with
this criterion level. The fine detail of the comparison involves
the TAIL DEFINITION POLICY. The outcome is classified as showing
STATISTICAL SIGNIFICANCE ('significant') if the outcome has
low ALPHA as compared with the NOMINAL ALPHA CRITERION
LEVEL, otherwise not ('non-significant'). The commonest
conventional values for the NOMINAL ALPHA CRITERION LEVEL
are 0.05 and 0.01 .
This is a type of MEASUREMENT SCALE with a limited number
of possible outcomes which cannot be placed in any order representing
the intrinsic properties of the measurements. Examples : Female
versus Male; the collection of languages in which an international
treaty is published.
A number of statistical tests were devised, mostly over the
period 1930-1960, with the specific objective of by-passing
assumptions about sampling from populations with data supposedly
conforming to theoretically modelled statistical distributions
wuch as the NORMAL DISTRIBUTION. Several of these tests were
explictly concerned with ORDINAL-SCALE data for which modelling
based upon continuous functions is clearly inappropriate.
These tests are implicitly RE-RANDOMISATION TESTS. Also see
: BINOMIAL TEST, MANN-WHITNEY TEST, WILCOXON TEST(1 and 2).
 The NORMAL DISTRIBUTION is a theoretical distribution applicable
for continuous INTERVAL-SCALE data. It is related mathematically
to the BINOMIAL and CHI-SQUARE(2) distributions and to several
named sampling distributions (including Student's t, Fisher's
F, Pearson's r); these sampling distributions are the characteristic
tools of parametric statisical infernece to which RE-RANDOMISATION
STATISTICS are an alternative.
In order to test whether a supposed interesting pattern exists
in a set of data, it is usual to propose a NULL HYPOTHESIS
that the pattern does not exist. It is the unexpectedness
of the degree of departure of the observed data, relative
to the pattern expected under the NULL HYPOTHESIS, which is
examined by the measure ALPHA. Reference to a NULL
HYPOTHESIS is common between RE-RANDOMISATION STATISTICS and
parametric statistics. Also see : BETA.
This is the code which a COMPUTER recognises and acts upon
as a direct consequence of its electromechanical construction.
Typically such code is highly abstract and unsuitable for
use in general use by human programmers. The OBJECT CODE to
specify a certain process is usually generated through use
of a COMPILER. Also see : PROGRAMMING LANGUAGE.
An alternative characterisation of the parameter 'p' for a
BINOMIAL PROCESS is the ratio of the incidences of the two
alternatives : p/(1-p) ; this quantity is termed the ODDS
RATIO; the value may range from zero to infinity. This relates
to a possible view of a BINOMIAL PROCESS as the combined activity
of two POISSON PROCESSes with a limit upon total count for
the two processes combined. Also see : LOGISITIC REGRESSION.
A MEASUREMENT TYPE for which the relative values of data are
defined solely in terms of being lesser, equa-to or greater
as compared with other data on the ORDINAL SCALE. These characteristics
may arise from categorical rating scales, or from converting
INTERVAL SCALE data to become RANKED DATA.
The value of the TEST STATISTIC for the data as initially
observed, before any RE-RANDOMISATION..
The ALPHA value arising from a statistical test. Also
see : EXACT TEST(2)
One of a number of PROGRAMs for undertaking translations
between STANDARD PROGRAMMING LANGUAGES.
[Named after the mathematician Blaise Pascal ( - )]. A PROGRAMMING
LANGUAGE designed for clarity of expression when published
in human-legible form, and for the teaching of programming.
PASCAL is to some extent specialised for numerical work. A
development is EXTENDED PASCAL. COMPILERS for PASCAL are widespread.
PASCAL and EXTENDED PASCAL are each represented as STANDARD
PROGRAMMING LANGUAGEs approved by ANSI and ISO.
This term has a distinct mathematical definition, but is also
commonly used as a synonym for RE-RANDOMISATION.
See : PERMUTATION, PITMAN PERMUTATION TEST(1), PITMAN PERMUTATION
PITMAN PERMUTATION TEST(1)
[Named after the statistician E.J. Pitman who described this
test, and the PITMAN PERMUTATION TEST(2), in 1937; this is
one of the earliest instances of an EXACT TEST(1)] An EXACT
RE-RANDOMISATION TEST in which the TEST STATISTIC is the DIFFERENCE
OF MEANS of two samples of univariate INTERVAL-SCALE data.
. Also see : EQUIVALENT TEST STATISTIC, PITMAN PERMUTATION
PITMAN PERMUTATION TEST(2)
An EXACT RE-RANDOMISATION TEST in which the TEST STATISTIC
is the MEAN DIFFERENCE of a single sample of univariate data
measured under two circumstances as REPEATED MEASURES. Also
see : PITMAN PERMUTATION TEST(1)
The distribution of number of events in a given time, arising
from a POISSON PROCESS. This differs from the BINOMIAL DISTRIBUTION
in that there is no upper limit, corresponding to the parameter
'n' of a BINOMIAL PROCESS, to the number of events which may
occur. Also see : ODDS RATIO.
A process whereby events occur independently in some continuum
(in many applications, time), such that the overall density
(rate) is statistically constant but that it is impossible
to improve any prediction of the position (time) of the next
event by reference to the detail of any number of preceding
observations. The corresponding distribution of intervals
between events is an exponential distribution. The conventional
example of a POISSON PROCESSES is concerned with occurence
of radioactive emissions in a substantial sample of radioactive
with a half-life very much longer than the total observation
period. Also see : POISSON DISTRIBUTION.
A definable set of individual units to which the findings
from statistical examination of a SAMPLE subset are intended
to be applied. The POPULATION will generally much outnumber
the SAMPLE. In RE-RANDOMISATION STATISTICs the process of
applying inferences based upon the SAMPLE to the POPULATION
is essentially informal. Also see : REPRESENTATIVE.
This is the probability that a statistical test will detect
a defined pattern in data and declare the extent of the pattern
as showing STATISTICAL SIGNIFICANCE. POWER is related
to TYPE-2 ERROR by the simple formula : POWER = (1-BETA)
; the motive for this re-definition is so that an increase
in value for POWER shall represent improvement of performance
of a STATISTICAL TEST. For more detail, see : BETA.
A sequence of instructions expressed in some PROGRAMMING LANGUAGE.
Also see ALGORITHM(2).
The characteristic of a COMPUTER which enables it to be used
to undertake a variety of different processes on different
occasions. Also see : ALGORITHM(2), PROGRAM, PROGRAMMING
LANGUAGE, STANDARD PROGRAMMING LANGUAGE.
A formal code for expressing to a COMPUTER how a certain process
should be undertaken. The translation from the code of the
PROGRAMMING LANGUAGE to the OBJECT CODE of the appropriate
COMPUTER is itself undertaken by a PROGRAM for that COMPUTER;
the translation program may take the form of either a COMPILER
of an INTERPRETER. Also see : ALGORITHM(1), ALGORITHM(2),
PROGRAM. STANDARD PROGRAMMING LANGUAGES.
A source of data which is effectively unpredictable although
generated by a determinate process. Successive PSEUDO-RANDOM
data are produced by a fixed calculation process acting upon
preceding data from the PSEUDO-RANDOM sequence. To start the
sequence it is necessary to decide arbitrarily upon a first
datum, which is termed the SEED value. Also see : BOOTSTRAP,
A SAMPLE drawn from a POPULATION in such a way that every
individual of the POPULATION has an equal chance of appearing
in the SAMPLE. This ensures that the SAMPLE is REPRESENTATIVE,
and provides the necessary basis for virtually all forms of
inference from SAMPLE to POPULATION, including the informal
inference which is characteristic of RE-RANDOMISATION statistics.
PSEUDO-RANDOM procedures can be useful in defining a RANDOM
Generation of whole or part of the RANDOMISATION SET. Also
see : RANDOMISATION(3), RE-RANDOMISATION.
The process of arranging for data-collection, in accordance
with the EXPERIMENTAL DESIGN, such that there should be no
foreseeable possibilty of any systematic relationship between
the data and any measureable characteristic of the procedure
by which the data was sampled. This is usually arranged by
assigning experimental units to groups, and REPEATED MEASURES
to experimental units, on a strictly random basis.
One of the arrangements making up the RANDOMISATION SET. These
arranegments will be encountered in the act of RANDOMISATION(1).
Also see : BRANCH AND BOUND, MINIMAL-CHANGE SEQUENCE.
A collection of values of the TEST STATISTIC obtained by undertaking
a number of RE-RANDOMISATIONS of the actual data within the
RANDOMISATION SET. ALso see : CONFIDENCE INTERVAL, RANDOMISATION
The collection of possible RE-RANDOMISATIONs of data within
the constraints of the EXPERIMENTAL DESIGN. Also see : RANDOMISATION
The rationale of a RANDOMISATION TEST involves exploring RE-RANDOMISATIONs
of the actual data to form the RANDOMISATION DISTRIBUTION
of values of the TEST STATISTIC. The OUTCOME VALUE value of
the TEST STATISTIC is judged in terms of its relative position
within the RE-RANDOMISATION DISTRIBUTION. If the OUTCOME VALUE
is near to one extreme of the RE-RANDOMISATION DISTRIBUTION
then it may be judged that it is in the extreme TAIL of the
distribution, with reference to a NOMINAL ALPHA CRITERION
VALUE, and thus judged to show STATISTICAL SIGNIFICANCE. Also
see : EXACT TEST(1).
This refers to the practice of taking a set of N data, to
be regarded as ORDINAL-SCALE, amd replacing each datum by
its rank (1 .. N) within the set. Also see : WILCOXON RANK-SUM
This is a type of MEASUREMENT SCALE for which it is meaningful
to reason in terms of differences in scores (see INTERVAL
SCALE) and also in terms of ratios of scores. Such a scale
will have a zero point which is meaningful in the sense that
it indicates complete absence of the property which the scale
measures. The RATIO SCALE may be either unipolar (negative
values not meaningful) or bipolar (both positive and negative
values meaningful), and either continuous or discrete.
The process of generating alternative arrangements of given
data which would be consistent with the EXPERIMENTAL DESIGN.
Also see : BOOTSTRAP, EXACT TEST(2), EXHAUSTIVE RE-RANDOMISATION,
MONTE-CARLO, RE-RANDOMISATION STATISTICS.
Also known as PERMUTATION or RANDOMISATION(1) statistics.
These are the specific area of concern of this present glossary.
A comparison of two or more statistical tests, for the same
EXPERIMENTAL DESIGN, SAMPLE SIZE, and NOMINAL ALPHA CRITERION
VALUE, in terms of the respective values of POWER. Also see
This is a feature of an EXPERIMENTAL DESIGN whereby several
observations measured on a common scale refer to the same
sampling unit. Identification of the relation of the individual
observations to the EXPERIMENTAL DESIGN is crucial to this
definition. Examples : the measurement of water level at a
particular site on several systematically-defined occasions;
measurement of reaction-time of an individual using right
hand and left hand separately. Also see : INDEPENDENT
GROUPS, REPLICATIONS, STRATIFIED.
This is a feature of an EXPERIMENTAL DESIGN whereby observations
on an experimental unit are repeated under the same conditions.
Identification of the position of a particular observation
within the sequence of replications is irrelevant. Also see
: REPEATED MEASURES, STRATIFIED.
Patterns in a SAMPLE of units may reasonably be attributed
to the POPULATION from which the SAMPLE is drawn, only if
the SAMPLE is REPRESENTATIVE. In practical terms, to ensure
that a SAMPLE is REPRESENTATIVE almost always means ensuring
that it is a RANDOM SAMPLE.
This is the name of an educational initiative involving the
use of a PROGRAMMING LANGUAGE, in the form of an INTERPRETER,
allowing the user to specify MONTE-CARLO RESAMPLING of a set
of data and accumulation of the RANDOMISATION DISTRIBUTION
of a defined TEST STATISTIC.
Acronym for Random Number Generator. This is a process which
uses a arithmetic algorithm to generate sequences of PSEUDO-RANDOM
numbers. Also see : SEED.
SACROWICZ & COHEN CRITERION
[Sacrowicz & Cohen()] This is a TAIL DEFINITION POLICY
which asserts that the ALPHA value should be
A set of individual units, drawn from some definable POPULATION
of units, and generally a small proportion of the POPULATION,
to be used for a statistical examination of which the findings
are intended to be applied to the POPULATION. It is essential
for such inference that the SAMPLE should be REPRESENTATIVE.
In RE-RANDOMISATION STATISTICS the process of applying inferences
based upon the SAMPLE to the POPULATION is essentially informal.
The number of experimental units on which observations are
considered. This may be less than the number of observations
in a data-set, due to the possible multipying effects of multiple
variables and/or REPEATED MEASURES within the EXPERIMENTAL
See MEASUREMENT TYPE.
[()]. ALGORITHMs employing BRANCH-AND-BOUND methods for the
PTIMAN PERMUTAION TEST(1) and the PITMAN PERMUTATION TEST(2).
See : STATISTICAL SIGNIFICANCE.
STANDARD PROGRAMMING LANGUAGE
A PROGRAMMING LANGUAGE which has a publicly agreed common
form across several different types of COMPUTER. Such standardisation
allows a PROGRAM to be transported conveniently between the
different types of COMPUTER and is thus suitable for communicating
general ideas about programming. Some STANDARD PROGRAMMING
LANGUAGES relevant to the present context are : FORTRAN, PASCAL,
'C'. There are a number of widely available programs for translating
SOURCE PROGRAMS from one STANDARD PROGRAMMING LANGUAGE to
another - e.g. the program PAS2C which translates source code
from PASCAL to 'C'. Also see : ALGORITHM(2), ANSI, ISO.
A number or code derived by a prior-defined consistent process
of calculation, from a set of data. Also see : ALGORITHM(1),
See : ALPHA, NOMINAL ALPHA CRITERION LEVEL.
[()] This is widely-observed scheme of distinctions between
types of MEASUREMENT SCALEs according to the meaningfulness
of arithmetic which may be performed upon data values. The
types are : NOMINAL SCALE versus ORDINAL SCALE versus INTERVAL
SCALE versus RATIO SCALE.
This is a feature of an EXPERIMENTAL DESIGN whereby a scheme
of observations is repeated entirely using further sets (strata)
of experimental units, with each such further set distinguished
by a level of a categorical variable which is distinct from
any categorical variables used to define the EXPERIMNATL DESIGN
within a single set (stratum). The data from the various strata
are regarded as distinct. This situation occurs when attempting
to make inferences based upon the results of several similar
independent experiments. Also see : REPEATED MEASURES, REPLICATIONS.
An area at the extreme of a RANDOMISATION DISTRIBUTION,
where the degree of extremity is sufficient to be notable
judged against some NOMINAL ALPHA CRITERION VALUE.
Also see : BRANCH-AND BOUND, RE-RANDOMISATION TEST, TAIL
TAIL DEFINITION POLICY
This is a defined method for dividing a DISCRETE DISTRIBUTION
into a TAIL area and a body area. The scope for
differing policies arises due to the non-infinitesmal amount
of probability measure which may be associated with the ACTUAL
OUTOME value. The conventional policy, based upon considerations
of simplicity and of conservatism in terms of ALPHA,
is to include the whole of the weight of outcomes equal to
the ACTUAL OUTCOME as part of the TAIL. Also see MID-P,
SACROWICZ & COHEN.
A STATISTIC measuring the strength of the pattern which a
statistical test undertakes to detect. In the context of RE-RANDOMISATION
TESTS one is concerned with the distribution of the values
of the TEST STATISTIC over the RANDOMISATION SET. An example
of a TEST STATISTIC is the DIFFERENCE OF MEANS as employed
in the PITMAN PERMUTATION TEST. Also see : EXACT TEST(1),
In a NONPARAMETRIC TEST involving RANKED DATA, if two data
have TIED VALUES then they will deserve to receive the same
rank value. It is generally agreed that this should be the
average of the ranks which would have been assigned if the
values had been discernably unequal. Thus, the ranks assigned
to a set of 6 data, with ties present might emerge as sets
such as : 1,3,3,3,5,6 or 1,2,3.5,3.5,5,6. The possibility
of TIED RANKS leads to elaborations in the otherwise-standard
tasks of computing or tabulating RANDOMISATION DISTRIBUTIONS
where data are replaced by ranks.
Where data are represented by ranks, TIED VALUES lead to TIED
RANKS. Whether or not data are rep[resnted by ranks, for any
TEST STATISTIC the occurrence of TIED VALUES will increase
the extent to which a RANDOMISATION DISTRIBUTION will be a
DISCRETE DISTRIBUTION rather than a CONTINUOUS DISTRIBUTION.
A representation of suitable data in a table organised as
rows and columns, such that the rows represent one scheme
of alternatives covering the whole of the the data represented,
the columns represent a further scheme of alternatives covering
the whole of the data represented, and the entries in the
TWO-WAY TABLE are the counts of numbers of observations conforming
to the respective cells of the two-way classification.
See : ALPHA.
See : BETA.
WILCOXON RANK-SUM TEST
See : WILCOXON TEST(1), WILCOXON TEST(2).
[Named after the statistician F, Wilcoxon ()] This test applies
to an EXPERIMENTAL DESIGN involving two REPEATED MEASURE observations
on a common set of experimental units, which need be only
ORDINAL-SCALE. The purpose is to measure shift in scale location
between the two levels of the REPEATED MEASURE distinction.
The TEST STATISTIC is derived from the set of differences
between the two levels of the REPEATED MEASURE distinction
- one difference score for each observational unit. The procedure
is somewhat variable between authors, although the variants
each correspond to valid well-sized EXACT TEST(1)s. Wilcoxon's
original procedure commences by discarding entirely the observations
from any experimental units for which the data values are
equal at each level of the REPEATED MEASURE comparison. Thus
or otherwise, the next step is RANKING the differences, providing
a rank for each retained experimental unit; the ranks are
according to the absolute values of the differences. The ranks
are summed separately into two or three categories : negative
differences; zero differences (if any); positive differences.
The TEST STATISTIC is the smaller of the outer categories,
plus an adjustment for the middle (zero-difference) category.
Also see : PITMAN PERMUTATION TEST(2).
[Named after the statistician F, Wilcoxon ()] This is a test
for an EXPERIMENTAL DESIGN involving two INDEPENDENT GROUPS
of experimental units, where data need be only ORDINAL-SCALE.
The purpose is to measure shift in scale location between
the two groups. The TEST STATISTIC is the sum, for a nominated
group, of the ranks of the data for the groups combined. This
test has an EQUIVALENT TEST STATISTIC to that for the MANN-WHITNEY
TEST, so the two tests must always agree. Also see : PITMAN
See : TWO-WAY TABLE.
This is a TWO-WAY TABLE where the numbers of levels
of the row- and column-classifications are each 2. If the
row- and column- classifications each divide the observational
units into subsets, then it is likely that it will be useful
to analyse the data using the FISHER TEST(1)
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