SFWR ENG 3S03: Software Testing - McMaster …...SFWR ENG 3S03: Software Testing Dr. Ridha Khedri...

SFWR ENG 3S03:Software Testing

Dr. R. Khedri

Outline

Preliminaries

TheRepresentationalTheory ofMeasurement

Measurement andModels

MeasurementScales and ScaleTypes

StatisticalOperations onMeasures

(Slide 1 of 64)

SFWR ENG 3S03: Software Testing

Dr. Ridha Khedri

Department of Computing and Software, McMaster UniversityCanada L8S 4L7, Hamilton, Ontario

Acknowledgments: Material based on [FP97, Chapter 2] and [Zus97, Chapter 4]

Dr. R. Khedri SFWR ENG 3S03: Software Testing


Dr. R. Khedri

Outline

Preliminaries





(Slide 2 of 64)

1 Preliminaries2 The Representational Theory of Measurement

Empirical relationsThe Rules of the MappingThe Representation Condition of MeasurementDirect Product StructureExamples of Specific Measures used in SoftwareEngineering

3 Measurement and ModelsDefining AttributesDirect and indirect measurementMeasurement for Prediction

4 Measurement Scales and Scale TypesNominal ScaleOrdinal ScaleInterval ScaleRatio ScaleAdmissible Transformation (revisited)

5 Statistical Operations on Measures



Dr. R. Khedri

Outline

Preliminaries





(Slide 3 of 64)

The Basics of MeasurementåPreliminaries

We use measurement every day, to understand, controland improve what we do and how we do it

We examine measurement in more depth

In our daily life, to measure we use both tools andprinciples that we now take for granted

These sophisticated measuring devices and techniqueshave been developed over time, based on the growth ofunderstanding of the attributes we are measuring



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As we understood more about an attributes and therelationships between them, we develop

a framework for describing them

tools for measuring them

Unfortunately, we have no deep understanding ofsoftware attributes

Many questions that are relatively easy to answer fornon-software entities are difficult for software



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Consider the following questions:

Does a count of the number of ”bugs” found in asystem during integration testing measure the qualityof the system?

For instance, is it meaningful to talk about doubling adesign’s quality? If not, how do we compare twodifferent designs?

Is it sensible to compute average productivity for agroup of developers, or the average quality of a set ofmodules?

How can we measure how quality such as security orprivacy?



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To answer these questions, we must establish thebasics of a theory of measurement

We begin by examining formal measurement theory

We see how the concepts of measurement theory applyto software

This theory tells us

how to measure

how to analyze and depict data

how to tie the results back to our original questions



Dr. R. Khedri

Outline

Preliminaries


Empirical relations

The Rules of theMapping

The RepresentationCondition ofMeasurement

Direct ProductStructure

Examples of SpecificMeasures used inSoftware Engineering




(Slide 7 of 64)

The Basics of MeasurementåThe Representational Theory of Measurement

In any measurement activity, there are rules to befollowed

help us to be consistent in our measurement

provide us with a basis for interpreting data

laying the groundwork for developing and reasoningabout all kinds of measurement

This rule-based approach is common in many sciences(theory)



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Empirical relations








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Relationship between theory and measurement

We can use rules about measurement to codify ourinitial understanding, and then expand our horizons aswe analyze our software

However, there are several theories for field

Euclidean and non-EuclideanPsychology Theories: provide a set of guiding principlesand concepts that describe and explain humandevelopmentClassical mechanics and quantum mechanics are thetwo major sub-fields of mechanics

Depending on the theory chosen, there are also severaltheories of measurement



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åEmpirical relations

The representational theory of measurement formalizesour intuition about the studied entities

The data we obtain as measures should representattributes of the entities

Manipulation of the data should preserve relationshipsthat we observe among the entities

Thus, our intuition is the starting point for allmeasurement



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Empirical relations








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We tend to understand things by comparing them, notby assigning numbers to them

We observe that certain people are taller than otherswithout actually measuring them

Our observation reflects a set of rules that we areimposing on the set of people

We form pairs of people and define a binary relation”taller than” on them

Given any two people, x and y , we can compare x andy using ”taller than”

”taller than” is an empirical relation for height



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A (binary) empirical relation is one for which there is areasonable consensus about which pairs are in therelation

We can define more than one empirical relation on thesame set (e.g., ”much taller than”, “almost the samehight”)

Empirical relations NEED NOT be binary

We can think of these relations as mappings from theempirical, real world to a formal mathematical world



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For example: Any measure of height should assign ahigher number to Jack than to Jill

HOWEVER, the representation condition ofmeasurement needs to preserve intuition andobservation

Example

Suppose we are evaluating the four best-sellingwordprocessing programs: A, B, C, and D. We ask 100independent computer users to rank these programsaccording to their functionality, and the results are shown inthe following table. Each cell of the table represents thepercentage of respondents who preferred the row’s programto the column’s program.



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Table: More functionality

A B C D

A – 80 10 80

B 20 – 5 50

C 90 95 – 96

D 20 50 4 –

”greater functionality than”def“ cell (x, y) exceeds 60%

Table: More User-friendly

A B C D

A – 45 50 44

B 55 – 52 50

C 50 48 – 51

D 54 50 49 –

”user-friendliness” relation: no real consensus



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Table: Historical advances in temperature measurement

ă 2000 BC Rankings ”hotter than”1600 AD First thermometer measuring ”hotter than”1720 AD Fahrenheit scale1742 AD Celsius scale1854 AD Absolute zero (Kelvin scale)

We can use relatively unsophisticated relationships thatrequire no measuring tools

With more accumulated knowledge, we may build moresophisticated ways and tools



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Empirical relations








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Definition (Measurement)

A measurement is a mapping from the empirical world tothe formal, relational world. Consequently, a measure is avalue or a symbol assigned to an entity by this mapping inorder to characterize an attribute.

Sometimes, the empirical relations for an attribute arenot yet agreed upon (e.g., personal preference, nocommon understanding)

They enable us to establish the basis for empiricalrelations, so that formal measurement may be possiblein the future



Dr. R. Khedri

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Preliminaries


Empirical relations








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åThe Rules of the Mapping

mapping

Real World

Mathematical World



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A measure must specify the domain and range as well as therule for performing the mapping.

Homeland Security Advisory System was a color-codedterrorism threat advisory scale

(red, orange, yellow, blue, green) = (severe, high,significant, general, low) risk

(Severe, High, Elevated, Guarded, Low)

Why not p4, 3, 2, 1, 0q, p7777, 777, 77, 7, 0qor (¨, , ª, «, g)?



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A bijective mapping does not exist in the area of(software) measurement

Why? Measurements values can be identical fordifferent objects

There are four types of mappings:Injective, but not surjective (Not a measurementmapping)

Not injective, but surjective (Could be a measurementmapping)

Bijective (Not a measurement mapping)

Neither Injective nor surjective (The most of themeasurement mappings)



Dr. R. Khedri

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Preliminaries


Empirical relations








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Dr. R. Khedri

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Preliminaries


Empirical relations








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Dr. R. Khedri

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Preliminaries


Empirical relations








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åThe Representation Condition of Measurement

Each relation in the empirical relational systemcorresponds via the measurement to an element in acarrier set in the mathematical world (could a set ofnumbers)

We want to have preservation of operation andrelations

This rule is called the representation condition



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Empirical relations








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The mapping that we call a measure is sometimescalled a representation or homomorphism

We use the notion of representation to define validity:any measure that satisfies the representation conditionis a valid measure

How can we build more sophisticated measuresfrom simple ones?



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Empirical relations








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åDirect Product Structure

Definition (Direct Product)

Let tMiuI “ `

Mi , tfiuf PF , tR

iuRPR˘

| i P I(

, be anI -indexed family of L-structures. The direct product ΠIMi

of the family is defined as follows:

The support set if ΠIMi (i.e., the Cartesian Product ofMi )

Operations on the product are defined componentwise

Given R P R, the relation RΠ on ΠIMi is defined as

follows:

px1, ¨ ¨ ¨ , xmq P RΠ

ðñ @pi | i P I : px1piq, ¨ ¨ ¨ , xmpiqq P Ri q,

where m is the arity mpRq of R andpx1, ¨ ¨ ¨ , xmq P pΠIAi q

m.Dr. R. Khedri SFWR ENG 3S03: Software Testing


Dr. R. Khedri

Outline

Preliminaries


Empirical relations








(Slide 25 of 64)


åDirect Product Structure

Clearly, ΠIMi “`

ΠIMi , tfΠuf PF , tR

ΠuRPR˘

as it isdefined has the same language as L as each of thestructures in the family tMiuI .

The set I can be empty: the empty product ΠH has asupport with one element e.

RH “ tpe, ¨ ¨ ¨ , equ

If @pi , j | i , j P I : Mi “ N “Mj q, thenΠIMi “ N |I | denoted N I .

N I def“ ΠIMi is called I -direct power of the

L-structure N .



Dr. R. Khedri

Outline

Preliminaries


Empirical relations








(Slide 26 of 64)


åExamples of Specific Measures used in Software Engineering



Dr. R. Khedri

Outline

Preliminaries



Defining Attributes

Direct and indirectmeasurement

Measurement forPrediction



(Slide 27 of 64)

The Basics of MeasurementåMeasurement and Models

Software Engineers use several types of models (e.g.,cost-estimation models, quality models,capability-maturity models)

A model is an abstraction of reality, allowing us to stripaway detail and view an entity or concept from aparticular perspective

Cost models permit us to examine only those projectaspects that contribute to the project’s final cost

Models come in many different forms (e.g., equations,mappings, or diagrams)



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Defining Attributes





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As far as a model is concerned, the world can bedivided into three parts:

1 Things whose effects are neglected

2 Things that affect the model but whose behavior themodel is not designed to study

3 Things the model is designed to study the behavior of

When we use a model to draw conclusion, we make adeductive process:

If the assumptions are true, the conclusionmust be true



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Defining Attributes





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Example

To measure length of programs using lines of code, we needa model of a program. The model would specify how aprogram differs from a subroutine, whether or not to treatseparate statements on the same line as distinct lines ofcode, whether or not to count comment lines, whether ornot to count data declarations, and so on. The model wouldalso tell us what to do when we have programs written in acombination of different languages. It might distinguishdelivered operational programs from those underdevelopment, and it would tell us how to handle situationswhere different versions run on different platforms.



Dr. R. Khedri

Outline

Preliminaries



Defining Attributes





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åDefining Attributes

When measuring, there is always a danger that wefocus too much on the formal, mathematical system,and not enough on the empirical one

We should give careful thought to the relationshipsamong entities and their attributes in the real world



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Preliminaries



Defining Attributes





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Example

Our intuition tells us that the “complexity” of aprogram can affect the time it takes to code it, test it,and fix it

We suspect that “complexity” can help us tounderstand when a module is prone to contain faults

However, there are few researchers who have builtmodels of exactly what it means for a module to becomplex

Many software developers define program complexity asthe cyclomatic number proposed by McCabe

This number, based on a graph-theoretic concept,counts the number of linearly independent pathsthrough a program



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Defining Attributes





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Dr. R. Khedri

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Preliminaries



Defining Attributes





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On the basis of empirical research, McCabe claimedthat modules with high values of η were those mostlikely to be fault-prone and unmaintainable

He proposed a threshold value of 10 for each moduleAny module with η greater than 10 should beredesigned to reduce η

Limitations:The cyclomatic number presents only a partial view ofcomplexity

There are many programs that have a large number ofdecisions but are easy to understand, code andmaintain (ηpgq “ d ` 1)

A more complete model of complexity is needed



Dr. R. Khedri

Outline

Preliminaries



Defining Attributes





(Slide 34 of 64)


åDirect and indirect measurement

Definition (Direct measurement)

Direct measurement of an attribute of an entity involves noother attribute or entity.

length of a physical object can be measured withoutreference to any other object or attribute

Density of a physical object can be measured onlyindirectly in terms of mass and volume

Model

density “mass

volume



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Preliminaries



Defining Attributes





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The following direct measures are commonly used insoftware engineering:

Length of source code (measured by lines of code)

Duration of testing process (measured by elapsed timein hours)

Number of defects discovered during the testingprocess (measured by counting defects)

Time a programmer spends on a project (measured bymonths worked)



Dr. R. Khedri

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Preliminaries



Defining Attributes





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Dr. R. Khedri

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Preliminaries



Defining Attributes





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Direct measurement to assess a product [PFP94]



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Defining Attributes





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Indirect measurement to assess a product [PFP94]



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Preliminaries



Defining Attributes





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åMeasurement for Prediction

When wemeasure, we usually mean that we wish toassess some entity that already exists

In many circumstances, we would like to predict anattribute of some entity

The distinction between measurement for assessmentand prediction is not always clear-cut

In general, measurement for prediction always requiressome kind of mathematical model



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Preliminaries



Defining Attributes





(Slide 40 of 64)


åMeasurement for Prediction

Example (Reliability model)

A well known reliability model is based on an exponentialdistribution for the time to the ith failure of the product.This distribution is described by the formulaF ptq “ 1´ e´pN´i`1qat , where

t is time

N represents the number of faults initially residing inthe program

a represents the overall rate of occurrence of failures

There are many ways that the model parameters N anda can be estimated (e.g., Maximum LikelihoodEstimation)



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale

AdmissibleTransformation(revisited)


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The Basics of MeasurementåMeasurement Scales and Scale Types

There are differences among the different kind ofmappings

The differences among the mappings can restrict thekind of analysis we can do

We discuss the notion of a measurement scale and thenwe use the scale to understand which analyses areappropriate

Measurement scale = measurement mapping (M) +the empirical and numerical relation systems (i.e.,dom , ran )



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



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There are three important questions concerningrepresentations and scales:

How do we determine when one numerical (or,symbolic) relation system is preferable to another?

How do we know if a particular empirical relationsystem has a representation in a given numericalrelation system? (This question is about therepresentation problem)

What do we do when we have several different possiblerepresentations (and hence many scales) in the samenumerical relation system? (This question is about theuniqueness problem)



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Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



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In general, there are many different representations fora given empirical relation system pS , tRi | i P I uq,where I is a set of indices

Higher is the size of I , the fewer are the representations

pS , tRi | i P I uq is richer than pS , tQi | i P I uq iff

@pi | i P I : Qi Ď Ri q

The richer the empirical relation system, the morerestrictive the set of representations, and so the moresophisticated the scale of measurement



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



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We classify measurement scales as one of five majortypes: Nominal richer than Ordinal richer than Intervalricher than ratio richer than Absolute

There are other scales that can be defined (e.g.,logarithmic scale)

What is a scale?Example:

We may measure the length of physical objects byusing a mapping from length to inchesThere are equally acceptable measures in feet, meters,furlongs, miles, etc.All of the acceptable measures are very closely related(we can convert one to another)

A Scale is defined by a homomorphism



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 45 of 64)


What is a scale type?It defined by the notion ofadmissible transformation

A mapping from one acceptable measure to another iscalled an admissible transformation (also calledrescaling)

Scales and scale types lead directly to the notion ofmeaningfulness

Meaningfulness

A statement with measurement values is meaningful iff itstruth or falsity value is invariant to admissibletransformations.



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 46 of 64)


Another widely discussed notion in the area of softwaremeasurement is wholeness

Wholeness

The whole is equally or more big than the sum of the parts.

pP, ˝q be a structure of flowgraphs with theconcatenation operation ˝

Let P,Q P P the set of flowgraphs.

Let µ be a software measure (e.g., ν linearlyindependent paths in the graph)

The requirement of wholeness property translates intorequiring µpP ˝ Qq ě µpPq ` µpQq



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 47 of 64)


åNominal Scale

Definition (Nominal Scale)

Let pP,«q be an empirical relational system, where P is anon-empty countable set and where « is an equivalencerelation on P. Let pR,“q be numerical mathematicalstructure with R its carrier set and “ is its identity relation.Let µ : P ÝÑ R be a real value function. The system`

pP,«q, pR,“q, µ˘

is a nominal scale iff

@pp, q | p, q P P : p « q ðñ µppq “ µpqq q

Meaningful Statistical Operations

The admissible transformation is only a one-to-onetransformation



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 48 of 64)


åOrdinal Scale

Definition (Ordinal Scale)

Let pP,Áq be an empirical relational system, where P is anon-empty countable set and where Á is an empiricalrelation describing ranking properties on P. Let pR,ďq benumerical mathematical structure with R its carrier set andď is its partial order. Let µ : P ÝÑ R be a real valuefunction. The system

`

pP,Áq, pR,ěq, µ˘

is an ordinal scaleiff

1 @pp, q | p, q P P : p Á q ðñ µppq ě µpqq q

2 @pp, q, r | p, q, r P P : p Á q ^ q Á r ùñ p Á r q(Transitivity)

3 @pp, q | p, q P P : p Á q _ q Á p q (Completeness)



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 49 of 64)


åInterval Scale

Interval Scale

The empirical conditions behind the interval scale arenot intuitive in the area of software measurementbecause they consider empirical distances

If we consider the empirical relation equally or moredifficult to maintain, then we have to considerdistances of maintainability

Among others, ratio scale measures can be transformedto interval measures by admissible transformations, ifusers require that



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



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åInterval Scale

For this reason, the interval scale in the area ofsoftware measure is obsolete

In physics, examples are the transformations fromKelvin to Celsius or Fahrenheit

It is the movement of the zero point (In physicsdistances are well known)

But, the definition of distances objects of softwareengineering is more difficult



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



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åInterval Scale

An interval scale can be defined by an algebraicdifference structure

The concept of an algebraic difference structure isa set of objects

and a 2-arity relation on Aˆ A

pa, bq Á pc , dqdef“

The difference between a and b

is ě to the difference between c and d

Example

pbeer,wineq Á pcoffee, teaqdef“

my preference to beer over wine is

equal or greater than my preference to coffee over tea



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 52 of 64)


åInterval Scale

pS ,ěq is a weak order:

@px , y | x , y P S : x ě y _ y ě x q (completeness orconnectness)@px , y , z | x , y , z P S : x ě y ^ y ě z ùñ px ě zq q (transitivity)

A sequence pa1, a2, ¨ ¨ ¨ , ai , ¨ ¨ ¨ q is bounded if thereexists a real number M such that@pi | i is an index : |ai | ď M q



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 53 of 64)


åInterval Scale

Definition (Algebraic Difference Structure)

Let pAˆA,Áq be an algebraic difference structure iff, for alla, b, c , a1, b1, and c 1 P A and all sequencesa1, a2, ¨ ¨ ¨ , ai , ¨ ¨ ¨ P A, the following five axioms satisfied:

1 pAˆ A,Áq is a weak order

2 pa, bq Á pc , dq ùñ pd , cq Á pb, aq

3 pa, bq Á pa1, b1q ^ pb, cq Á pb1, c 1q ùñ pa, cq Á pa1, c 1q

4 pa, bq Á pc , dq ^ pc , dq Á pa, aq ùñ

Dpd 1, d2 | d 1, d2 P A : pa, d 1q Á pc , dq ^ pc , dq Á pd2, bq q

5 If a sequence a1, a2, ¨ ¨ ¨ , ai , ¨ ¨ ¨ P A is strictly boundedsequence, then it is finite.



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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



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åInterval Scale

Definition (Interval Scale)

An Interval Structure is the system`

pAˆ A,Áq, pRˆ R,ěq, µ˘

, where

1 pAˆ A,Áq is an algebraic difference structure

2 µ be a real valued function on A such that@pa, b, c , d | a, b, c , d P A : pa, bq Á pc , dq ðñµpaq ´ µpbq ě µpcq ´ µpdq q

3 If another real value function g satisfies property (2),then there exist real value numbers α, β ą 0, such thatgpxq “ αµpxq ` β holds

Le last condition can be written as followsDpg |

@pa, b, c , d | a, b, c , d P A : pa, bq Á pc , dq ðñ gpaq ´ gpbq ě gpcq ´ gpdq q :Dpα, β | α, β ą 0 : gpxq “ αµpxq ` β for all x P A q q



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 55 of 64)


åInterval Scale

The axioms of the interval scale show us that no singleelements of a set A are used, but pairs of elements of A

These pairs are treated as differences (intervals) of theelements of A

The relation Á is here no ranking order of elements ofA, but an order of differences (intervals) of A

Difference structures can be applied, if we canformulate differences or intervals empirically

Axiom 2 of the difference structure shows that the setAˆ A contains positive pa, bq and negative pb, aqdifferences



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 56 of 64)


åRatio Scale

Definition (Ratio Scale)

Let pP,Á, ˝q be an empirical relational system, where P is anon-empty countable set, Á is an empirical relationdescribing ranking properties on P, and ˝ is a binaryoperation on P. Let pR,ě,`q be numerical mathematicalstructure with R its carrier set, ě is its partial order, and `is the addition on R. Let µ : P ÝÑ R be a real valuefunction. The system

`

pP,Á, ˝q, pR,ě,`q, µ˘

is a ratioscale iff

1 @pp, q | p, q P P : p Á q ðñ µppq ě µpqq q

2 @pp, q | p, q P P : µpp ˝ qq “ µppq ` µpqq q



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 57 of 64)


åRatio Scale

Theorem

Let`

pP,Á, ˝q, pR,ě,`q, µ˘

be a structure such that itsrelational part is an ordinal scale. A real value functionν : P ÝÑ R satisfies (1) and (2) of the above definition iff

Dpα | α P R ^ α “ 0 : @pp | p P P : νppq “ αµppq q q

νppq “ αµppq is the admissible transformation for ratioscale

An additive measure assume an extensive additivestructure



Dr. R. Khedri

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Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 58 of 64)


åAdmissible Transformation (revisited)Definition (Admissible Transformation)

Let pA,B, µq be a scale, where A (resp. B) is the carrier set(or underlying set) of A (resp. B). A mapping g : A ÝÑ Ris an admissible transformation iff pA,B, gq is also a scale.

Example

pA,B, µq def“

`

pP,Á, ˝q, pR,ě,`q, LOC˘

, where LOCis function that take a program and returns the numberof lines of code

The question is whether a measure LOC’ exists that

pA,B, µq def“

`

pP,Á, ˝q, pR,ě,`q, LOC’˘

,

where LOC’ppq “ LOCppq1000 , for p P P ????

Yes. gpxq “ 11000µpxq. LOC’ is denoted KLOC



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 59 of 64)


åAdmissible Transformation (revisited)

Table: Scale classification using admissible transformation

Scale Admissible Transformation gNominal Any one-to-oneOrdinal g is a strictly increasing monotonic

functionInterval gpxq “ aµpxq`b “ a1x`b, a, a1 ą 0Ratio gpxq “ aµpxq “ a1x , a, a1 ą 0

Scales are ”defined” by a homomorphism

Scale types are ”defined” by admissible transformations



Dr. R. Khedri

Outline

Preliminaries




Nominal Scale

Ordinal Scale

Interval Scale

Ratio Scale



(Slide 60 of 64)


åAdmissible Transformation (revisited)

Metrical

Nominal

Ordinal

Interval

Ratio

n−affine

n−linear

n−Euclidean

Figure: General Classification of Scales



Dr. R. Khedri

Outline

Preliminaries





(Slide 61 of 64)

The Basics of MeasurementåStatistical Operations on Measures

Table: List of what can be computed on a scale

You can compute ... Nominal Ordinal Interval RatioFrequency distribution Yes Yes Yes YesMedian and percentiles NO Yes Yes YesAdd or Substruct NO NO Yes YesMean, standard devia-tion, standard error ofthe mean

NO NO Yes Yes

Ratio, or coefficient ofvariation

NO NO NO Yes



Dr. R. Khedri

Outline

Preliminaries





(Slide 62 of 64)

The Basics of MeasurementåStatistical Operations on Measures

Figure: Summary of measurement scales and statistics relevant toeach



Dr. R. Khedri

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Preliminaries





(Slide 63 of 64)References I

Norman E. Fenton and Shari Lawrence Pfleeger,Software metrics: A rigorous and practical approach,second ed., PWS Publishing Company, 1997.

S.L. Pfleeger, N.E. Fenton, and S. Page, Evaluatingsoftware engineering standards, IEEE Computer 27(1994), no. 9, 71–79.

Horst Zuse, A framework of software measurement,Walter de Gruyter, 1997.



Dr. R. Khedri

Outline

Preliminaries





(Slide 64 of 64)


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