    Home Free Glossaries Free Dictionaries Post Your Translation Job! Free Articles Jobs for Translators  #### A Glossary of Statistics (in particular, re-randomisation statistics)

By Norman W. Marsh,

jw34[at]liverpool.ac.uk

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A

ALGORITHM(1)
A formal statement, clear complete and unambiguous, of how a certain process needs to be undertaken. Also see : ALGORITHM(2).

ALGORITHM(2)
An ALGORITHM(1) expressed in a PROGRAMMING LANGUAGE for a COMPUTER .

ALPHA
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 TYPES, P-VALUE.

ANSI
[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.

B

BERNOUILLI PROCESS
[()] 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.

BETA
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 : ERROR TYPES.

BINOMIAL DISTRIBUTION
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 rate parameters.

BINOMIAL TEST
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.

BOOTSTRAP
[()] 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.

BRANCH-AND-BOUND
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

'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.

CHI-SQUARED DISTRIBUTION
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.

CHI-SQUARED STATISTIC
[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, POISSON PROCESS.

COMPILER
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.

COMPUTER
An automatic data-processing device which is PROGRAMMABLE. Also see : COMPUTER PROGRAM, OBJECT CODE, PROGRAM.

COMPUTER PROGRAM
A specification of how to undertake a certain process, usually expressed via a PROGRAMMING LANGUAGE, for some chosen COMPUTER. Also see : PROGRAM.

CONFIDENCE INTERVAL
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 solvable.

CONTINUOUS DISTRIBUTION
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.

D

DECISION RULE
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).

DISCRETE DISTRIBUTION
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.

E

ERROR TYPES
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).

EXACT-STATS
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 : exact-stats@mailbase.ac.uk .

EXACT TEST(1)
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 DATA.

EXACT TEST(2)
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.

EXHAUSTIVE RE-RANDOMISATION
A series of samples from a RANDOMISATION SET which is known to generate every RANDOMISATION. In particular, sampling which generates every RANDOMISATION exactly once.

EXPERIMENTAL DESIGN
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.

EXTENDED PASCAL
See : PASCAL.

F

FACTORIAL
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.

FISHER TEST(1)
[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 TABLE.

FISHER TEST(2)
[()] 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.

FORTRAN
[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 and ISO.

FREEMAN-HALTON TEST
See FISHER TEST(2).

G

GOLD STANDARD(1)
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.

GOLD STANDARD(2)
The idea of a re-randomisation test as a standard of correctness by which to judge other tests which are not based upon principles of RE-RANDOMISATION.

I

INTERPRETER
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 COMPILER.

INTERVAL SCALE
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' TYPOLOGY.

ISO
[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.

L

LOGISTIC REGRESSION
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.

M

MANN-WHITNEY TEST
[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.

MEASUREMENT TYPE
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 unbounded.

MID-P
[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 of ALPHA.

MINIMAL-CHANGE SEQUENCE
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).

MONTE-CARLO TEST
[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 generator.

MULTINOMIAL DISTRIBUTION
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.

N

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 .

NOMINAL SCALE
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.

NON-PARAMETRIC TEST
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).

NORMAL DISTRIBUTION
[] 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.

NULL HYPOTHESIS
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.

O

OBJECT CODE
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.

ODDS RATIO
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.

ORDINAL SCALE
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.

OUTCOME VALUE
The value of the TEST STATISTIC for the data as initially observed, before any RE-RANDOMISATION..

P

P-VALUE
The ALPHA value arising from a statistical test. Also see : EXACT TEST(2)

PAS2C
One of a number of PROGRAMs for undertaking translations between STANDARD PROGRAMMING LANGUAGES.

PASCAL
[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.

PERMUTATION
This term has a distinct mathematical definition, but is also commonly used as a synonym for RE-RANDOMISATION.

PERMUTATION TEST
See : PERMUTATION, PITMAN PERMUTATION TEST(1), PITMAN PERMUTATION TEST(2).

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 TEST(2).

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)

POISSON DISTRIBUTION
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.

POISSON PROCESS
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.

POPULATION
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.

POWER
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.

PROGRAM
A sequence of instructions expressed in some PROGRAMMING LANGUAGE. Also see ALGORITHM(2).

PROGRAMMABLE
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.

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.

PSEUDO-RANDOM
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, MONTE-CARLO TEST.

R

RANDOM SAMPLE
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 SAMPLE.

RANDOMISATION(1)
Generation of whole or part of the RANDOMISATION SET. Also see : RANDOMISATION(3), RE-RANDOMISATION.

RANDOMISATION(2)
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.

RANDOMISATION(3)
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.

RANDOMISATION DISTRIBUTION
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 TEST.

RANDOMISATION SET
The collection of possible RE-RANDOMISATIONs of data within the constraints of the EXPERIMENTAL DESIGN. Also see : RANDOMISATION DISTRIBUTION.

RANDOMISATION TEST
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).

RANKED DATA
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 TEST.

RATIO SCALE
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.

RE-RANDOMISATION
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.

RE-RANDOMISATION STATISTICS
Also known as PERMUTATION or RANDOMISATION(1) statistics. These are the specific area of concern of this present glossary.

RELATIVE POWER
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 : BETA.

REPEATED-MEASURES
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.

REPLICATIONS
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.

REPRESENTATIVE
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.

RESAMPLING STATS
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.

RNG
Acronym for Random Number Generator. This is a process which uses a arithmetic algorithm to generate sequences of PSEUDO-RANDOM numbers. Also see : SEED.

S

SACROWICZ & COHEN CRITERION
[Sacrowicz & Cohen()] This is a TAIL DEFINITION POLICY which asserts that the ALPHA value should be

SAMPLE
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.

SAMPLE SIZE
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 DESIGN.

SCALE TYPE
See MEASUREMENT TYPE.

SEED
See PSEUDO-RANDOM.

SHIFT ALGORITHM
[()]. ALGORITHMs employing BRANCH-AND-BOUND methods for the PTIMAN PERMUTAION TEST(1) and the PITMAN PERMUTATION TEST(2).

SIGNIFICANCE
See : STATISTICAL SIGNIFICANCE.

SIZE
See ALPHA.

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.

STATISTIC
A number or code derived by a prior-defined consistent process of calculation, from a set of data. Also see : ALGORITHM(1), TEST STATISTIC.

STATISTICAL SIGNIFICANCE
See : ALPHA, NOMINAL ALPHA CRITERION LEVEL.

STEVENS' TYPOLOGY
[()] 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.

STRATIFIED
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.

T

TAIL
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 DEFINITION POLICY.

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.

TEST STATISTIC
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), OUTCOME VALUE.

TIED RANKS
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.

TIED VALUES
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.

TWO-WAY TABLE
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.

TYPE-1 ERROR
See : ALPHA.

TYPE-2 ERROR
See : BETA.

W

WILCOXON RANK-SUM TEST
See : WILCOXON TEST(1), WILCOXON TEST(2).

WILCOXON TEST(1)
[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).

WILCOXON 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 PERMUTATION TEST(1).

2-WAY TABLE
See : TWO-WAY TABLE.

2-BY-2 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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