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					                   SOEN 384
      Management, Measurement and Quality
                    Control
                 http://users.encs.concordia.ca/~s384_2/




                             Lecture 15:

     Software Engineering Management (SEM) :
      Representational Theory of Measurement
•

•Fenton’s book


                             SOEN384-F10-L15:
                          representational theory of       1
                                measurement
         Agenda
   Review
   Representational theory of
    measurement
     Scale types and meaningful
      statistics
        Nominal

        ordinal

   Next?
                     SOEN384-F10-L15:
                  representational theory of
                        measurement            2
Evolution of an information need into a
          measurement plan




               SOEN384-F10-L15:
            representational theory of
                  measurement            3
Plan Measurement activity




             SOEN384-F10-L15:
          representational theory of
                measurement            4
GQM
      Review: measurement information model
      (ISO/IEC 15939:2007 standard)




                                               Select &
                     SOEN384-F10-L15:           Specify
                  representational theory of
                        measurement
                                               Measures
                                                     6
Measurable Concepts and
   Related Questions




          SOEN384-F10-L15:
       representational theory of
             measurement            7
Select and Specify Measures




Are we measuring correctly the
specified measurable concepts?
                      SOEN384-F10-L15:
                   representational theory of
                         measurement            8
         Agenda
   Review
   Representational theory of
    measurement
     Scale types and meaningful
      statistics
        Nominal

        ordinal

   Next?
                     SOEN384-F10-L15:
                  representational theory of
                        measurement            9
Representational theory of
measurement
     Empirical                               Numerical
    Relational    Measurement Procedure      Relational
    Structure E                              Structure N

   Measurement is a mapping of
    empirical (observed) objects to
    numerical (mathematical) objects using
    a valid and reliable mechanism, called
    a homomorphism

                   soen337-W09-L11: review                 10
          Representational Theory of
          Measurement in Examples
   Tom is tall.                       185cm
   John is taller than Tom. 195cm >185cm
   Software A is reliable.      300 bugs/10 KLOC
   Software B is more reliable. 150 bugs /1 KLOC
•to measure is to assign numbers to the entities Tom,
Software A, etc. according to clearly defined rules;
•“cm”, “bug”, “KLOC” are measurement units
•Here, “tall”, “taller than”, “reliable”, “more reliable” are
relations.                 SOEN384-F10-L15:
                        representational theory of
                              measurement                 11
      Representational Problem
                    Measurement is a
                    mapping of
                    empirical
                    (observed) objects             Have to be
                    to numerical                   meaningful!
                    (mathematical)
http://www.micha    objects using a
elbach.de/ot/ang_
hering/index.html
                    valid and reliable
                    mechanism, called
                    a homomorphism




                         SOEN384-F10-L15:
                      representational theory of
                            measurement                          13
               Meaningfulness of
          Mathematical Manipulations
   Example:
       Assume there are four football players with
        the numbers 2, 4 , 6, 8 on their shirts.
       We now calculate the arithmetic mean
                        AR=(2+4+6+8)/4 = 5
       Can we say that the player with number 5
        represents the average performance of the
        four players?
       Answer: no because the numbers are only
        identification of the players …


                         SOEN384-F10-L15:
                      representational theory of
                            measurement               15
      Meaningfulness of the analysis of
           measurement data
   Example 2:
      Given two distances D1=10km and

       D2=5km:
      Can we say that the distance D1 is twice

       as long as the distance D2? (D1=D2*2)
      Will this statement be meaningful if the

       length is measured in meters or yards?


                     SOEN384-F10-L15:
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                        measurement               16
        Meaningfulness and Scale Types of
               measurement data

   Five Major Types:
    Nominal, Ordinal, Interval, Ratio, Absolute
       Shown in increasing level of richness
        (the relations of each are contained in
        the next, when applicable)



                        SOEN384-F10-L15:
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                           measurement            17
    Nominal Scale Type
    Defines categories of attributes of
    objects (entities)
   places each entity in a particular
    category, based on the value of its
    attribute.
    We do not know which categories are
    better (no preference)
                   SOEN384-F10-L15:
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                      measurement            18
Nominal Scale Type:
Example 2.6, Fenton
   Entity: software failure
   Attribute: criticality
   Categories of software failures
    according to their criticality:
       R1: delayed response
       R2: incorrect output
       R3: data loss
                     SOEN384-F10-L15:
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                        measurement            19
   Example 2.6:
Measurement Method




       SOEN384-F10-L15:
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          measurement            20
     Nominal Scale Type Properties
   Structure-Preserving Mapping
      Given two empirical entities a, b:




   Examples of Permissible Statistics & Statistical
    Tests:
       Frequency
       Mode
       Nonparametric tests which do not depend on preference or ranking




                                 SOEN384-F10-L15:
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                                    measurement                      21
Nominal Scale Type: Key Points
   Classification
      Partitions the set of empirical entities into categories
       (also called equivalence classes) with respect to
       a certain attribute
      Equivalence operation:

      No knowledge about relationships among categories

   Empirical Requirements for the partitioning:
       Empirical Classes are jointly exhaustive
          ALL CATEGORIES TOGETHER SHOULD COVER ALL POSSIBLE

           CATEGORIES FOR THE ATTRIBUTE
       Empirical Classes are mutually exclusive
          A SIBJECT CAN BE CLASSIGIED INTO ONE AND ONLY ONE

           CATEGORY
   Numerical Requirements:
       Math. Operation: =
                          soen337-W09-L10: a different number
        Each category is represented by theory (2)               22
             Back to the 1st example:
   …
       Assume there are four football players with
        the numbers 2, 4 , 6, 8 on their shirts.
       Scale type?
       We now calculate the arithmetic mean
                      AR=(2+4+6+8)/4 = 5
       Is AR a meaningful statistical data?
       Answer: no because the arithmetic mean
        is not a meaningful mathematical operation
        on the nominal scale.

                         SOEN384-F10-L15:
                      representational theory of
                            measurement               23
               Example 2.6:
          from Nominal to Ordinal
   Next: add a new binary relation x more
    critical than y :
       Each data loss failure (R3) is more critical than
        incorrect output (R2) and delayed response
        failure (R1)
       Each incorrect failure (R2) is more critical than
        delayed response failure (R1)
   The measurement method shall be revised to
    account for this new binary relations
                        SOEN384-F10-L15:
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                           measurement                  24
        Example 2.6:
Revised Measurement Method




           SOEN384-F10-L15:
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              measurement            25
        Ordinal Scale Type
   Useful to augment the nominal scale with
    information about an ordering of the
    categories (such as preferences).
    The numbers represent ranking only, so
    addition, subtraction, and other arithmetic
    operations have no meaning. Why?
       No knowledge about the magnitude of the
        differences between categories of entities
                            SOEN384-F10-L15:
                         representational theory of
                               measurement            26
        Ordinal Scale Type Properties

       Structure-Preserving Mapping:
       preserves equivalence
       Given two empirical entities a, b:




   Math. Operations: =, <, >
       The numbers represent ranking only, so addition,
        subtraction, and other arithmetic operations have no
        meaning.             SOEN384-F10-L15:
                            representational theory of
                                  measurement                  27
         Ordinal Scale Type: Meaningful
            mathematical operations


   Examples of Permissible Statistics &
    Statistical tests:
       All nominal scale statistics
       Median
       Rank order statistics (f.e, Spearman’s correlation
        coefficient)
       Non-parametric


                              SOEN384-F10-L15:
                           representational theory of
                                 measurement                 28
    Exercises on Nominal and
      Ordinal Scale Types
   Determine the scale type of:
       Gender
       Marital status
       Distance
       Intelligence Score
       “Yes-No” voting system
       Blood types (A,B,AB,O)
       Letter grade system

                  SOEN384-F10-L15:
               representational theory of
                     measurement            29
        Ordinal Scale Type
   Useful to augment the nominal scale with
    information about an ordering of the
    categories.
    The numbers represent ranking only, so
    addition, subtraction, and other arithmetic
    operations have no meaning. Why?
       No knowledge about the magnitude of the
        differences between categories of entities

                        soen337-W09-L10: theory (2)   30
        Ordinal Scale Type Properties

       Structure-Preserving Mapping:
       preserves equivalence
       Given two empirical entities a, b:




   Math. Operations: =, <, >
       The numbers represent ranking only, so addition,
        subtraction, and other arithmetic operations have no
        meaning.
                           SOEN337-W08-L7-L8: Theory           31
    Ordinal Scale Type (4)

   Examples of Permissible Statistics & Statistical
    tests:
      All nominal scale statistics

      Median

      Rank order statistics (f.e, Spearman’s correlation

       coefficient)
      Non-parametric




                        soen337-W09-L10: theory (2)         32
    Exercises on Nominal and
      Ordinal Scale Types
   Determine the scale type of:
       Gender
       Marital status
       Distance
       Intelligence Score
       “Yes-No” voting system
       Blood types (A,B,AB,O)
       Letter grade system



              soen337-W09-L9: theory (1)   33
 Qualitative & Quantitative Aspects of
     Software Measurement Data

Qualitative + Body of Knowledge
          = Quantitative




             soen337-W09-L11: review   34
  Select and Specify Measures:
           Key Points




Are we measuring correctly the
specified measurable concepts?
    Yes, if we apply the
 representational theory of
        measurement   SOEN384-F10-L15:
                       representational theory of
                             measurement            35
 Representational
    theory of
  measurement



    Meaningful
statistical analysis
                          SOEN384-F10-L15:
                       representational theory of
                             measurement            36
          Software Measurement Theory:
                   Key Points

   Representational theory of measurement
    [Fenton 1998 ]
       rigorous framework for determining when a
        proposed measure really does characterize the
        attribute it is supposed to.
        rules for determining the scale types of
        measures, and hence to determine what
        statistical analysis is meaningful.
                           SOEN384-F10-L15:
                        representational theory of
                              measurement            37
Questions?
   …




         soen337-W09-L9: theory (1)   38
Next?
   More scale types:
       Interval, Ratio, Absolute




                  soen337-W09-L9: theory (1)   39

				
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