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Theory
of Attributes-
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Basic concept and
their applications. (part 1)
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Introduction.
Literally, an attribute means a quality or
characteristic. Theory of attributes deals with qualitative characteristics
which are not amenable to quantitative measurements.Examples of attributes are
drinking, smoking, blindness, health, honesty, etc.
Data
might be such that it may not be possible for an investigator to measure their
magnitude. In such cases, the observer can only study the presence or absence
of a particular quality in a group of individuals. Examples of such phenomena
are blindness, insanity, deaf-mutism, sickness, honesty, extravagance etc. In
such cases. an observer cannot measure the magnitude of the data ; for example,
he cannot measure the extent of blindness or honesty in quantitative form. All
he can do is to count the number of persons who are blind or who are honest. He
has to take decision on the basis of some standard definition of the term in
question. Such data in which the quantitative measurement of the magnitude is
not possible and in which only the presence or absence of an attribute can be
studied are called statistics of attributes.
CLASSIFICATION OF DATA
In the analysis of statistics relating to
attributes, the first thing is the classification of data. Here, data are classified
on the basis of presence or absence of particular attributes. If only one
attribute, say, blindness, is being studied the population would be divided in
two classes-one consisting of those people in whom this attribute is present
and the other consisting of those in whom this attribute is not present. Thus,
one class would be of the “blinds” and the other of “non-blinds.” If more than
one attribute are taken into account, the number of classes would be more than
two. If, for example, the attribute of deafness is also studied, there would be
a number of classes in which the universe/population would be divided; There
would be “blinds”, “non-blinds”, “deafs”, “non:deafs”, “blind and not deafs”,
“blind and deafs” “not-blinds and deafs” and “not-blinds and not-deafs”.
Classification by dichotomy : If only one attribute is being studied, the
universe is divided in to two parts one in which the attribute is present and
the other in which it is not present. These classes are mutually exclusive.
Such a classification where the universe is divided in two parts is called
"Classification by. Dichotomy". In actual analysis, usually there are
more than two classes in which the universe is divided and such classification
is called manifold classification.
NOTATION AND TERMINOLOGY
For the sake of convenience, in analysis,
it is necessary to use certain symbols to represent different Classes and their
frequencies. Usually, capital letters A, B and C etc., are used' to denote the
presence or attributes and the Greek letters, α, β and ϒ etc. are used to
denote absence of these attributes respectively. Thus if A represents the
attribute of blindness, α would represent absence of blindness, if B represents
deafness,β would represent absence of deafness'and if C represents insanity, ϒ
would represent absence of insanity. The number of units possessing a
particular attribute represented by A would be termed belonging to Class A, and
similarly, those in whom this attribute is absent would be termed belonging to
Class α.
If two attributes are being studied,
their combination can be represented by the combination of the letters
representing the two attributes. Thus, if blindness is represented by A and
deafness by B, then AB would represent blindness and deafness ; Aβ would
represent blindness and absence of deafness
αB would represent absence of blindness and presence of deafness ; and
αβ would represent absence of blindness and absence of deafness. '
Class Frequencies
The number of units in different classes
are called "class frequencies. ” Thus, if the number of blind and deaf people
is 20, the frequency of class AB is 20. Class frequencies are denoted by
enclosing the class symbols by brackets. Thus, (AB) would represent the
frequency of the class AB.
Number of Classes lf there is one attribute represented by A, the total number of
classes is 3 (if the total or ‘N‘ is also taken as a class), they would be A, α
and N. If the number of attributes is two, represented by A and B. the total
number of classes (including N) would be 9. They would be N, A, B, α, β, AB Aβ,
αB and αβ.
If the
number of attributes is three,the total number of classes (including N) would
be 27. The total number of classes is always equal to
where n stands for the
number of attributes. Thus, when there are three attributes, the total number
of classes would be
or 27 ; if the number of attributes is 4, the
total number of classes would be
or 81.
Order of Classes . The various
classes and their frequencies are demarcated on the basis of an order.
Thus, N is a class of Zero order
Similarly, the frequencies of these classes
are also known as frequencies of the Zero, First or Second order. If there are
only two attributes under study, then the Second order classes and frequencies
are called the classes or the fequencies of the ultimate order. Since these are
the last set of classes and frequencies, the number of classes of the ultimate
order is equal to
where n stands for the number of attributes. Thus, in case of 2
attributes, the number of the classes of the ultimate order would be
or 4 and in case of three
attributes
or 8
and so on.
Positive and Negative Classes The classes which represent the presence of an
attribute or attributes are called positive classes. The classes which
represent the absence of an attribute or attributes are called negative
classes. The classes in which one attribute is present and the other absent are
called pairs of contraries. Thus :
N,A,
B and AB are positive classes
and β
are negative classes
Aβ and
αB are pairs of contraries
RELATIONSHIP BETWEEN CLASSES OF VARIOUS ORDERS
ln classifying statistical data
according to attributes, the following simple rule should be kept in mind. Any
class frequency can always be expressed in terms of class frequencies of higher
order: Thus, the frequencies of first order can be expressed in terms of the
frequencies of the second order which in turn can be expressed in terms of
frequencies of the third order and so on. On the basis of this rule we can set
up various types of relationships between the frequencies of different orders.
If there is one attribute only, represented by A, the frequency of the universe
or N can be divided into two classes (A) and (α). Thus,
N=(A)+(α)
Now, if one more attribute B is taken into
account the first order classes, i.e., A and α can each be divided into two classes-one
in which attribute B is present and the other in which it is not present.
Thus
The following examples would clarify the above
rules.
Example
I : Given the following ultimate class frequencies, find the frequencies of the
positive and negative classes and the total number of observations :
(AB) = 250 (Aβ) = 120
(αB) = 200 (αβ) = 70
Solution
‘ N =
(A) + (α) = (AB) + (A β) + (αB) + (αβ) = 250 + 120 + 200 + 70 = 640
Next
in Part ----2
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