Theory of Attributes- Basic concept and their applications:
Theory of Attributes- Basic concept and their applications:
ATTRIBUTES:
Literally, an attribute means a quality or characteristic.Theory of attributes deals with qualitative characteristics which not amenable to quantitative measurements are and hence need slightly different statistical treatment from that of the variables. Examples of attributes are drinking,smoking, blindness, health, honesty, etc. An attribute may be marked by its presence (possession) or absence (dispossession) in member of given population. It may be pointed out that the methods of statistical
analysis applicable to the study of variables can also be used to a great extent in the theory of attributes and vice-versa. For example, the presence or absence of an attribute may be regarded as changes in the values of a variable which can possess only two values, viz., 0 and 1.
NOTATIONS:
Suppose the population is divided into two classes according the to the presence or absence of a a single attribute. The positive class which denotes the presence of the attribute is generally written in capital Roman letters such as A, B, C, D, etc . and the negative class, denoting the absence of the attribute is written in corresponding small Greek letters such as ɑ, 𝛽,𝜸,δ, etc. For example, if A represents attributes of sickness and B represents blindness, than ɑ and ß represent the non-sickness (health) and sight respectively.
The two classes, viz., A (possession of the attribute) and ɑ (dispossession of the attribute) are said to be complementary classes and the attribute of ɑ used in the sense of not-A is often called the complementary attribute of A. Similarly 𝛽,𝜸,δ, are the complementary attributes of B, C, D respectively.
The combinations of attributes are denoted by grouping together the letters concerned,
e.g., AB is the combination of the attributes A and B. Thus for the attributes A (sickness) and B (smoking), AB would mean the simultaneous possession of sickness and smoking. Similarly Aß will represent sickness and non-smoking, ɑB non-sickness (health) and smoking, and ɑ𝛽 non-sickness and non-smoking.
If a third attribute be included to represent, say male, then ABC will stand for sick males who are smokers. Similar interpretations can be given to AB𝜸, A𝛽C, A𝛽𝜸, etc.
DICHOTOMY:
If the universe (population) is divided into two sub-classes or complementary classes and no more, with respect to each of the attributes A, B, C, etc., the division o classification is said to be 'dischotomous classification'. The classification is termed manifold if each class is further sub-divided.
CLASSES AND CLASS FREQUENCIES:
Different attributes in themselves are called different classes and the number or observations assigned to them are called class frequencies which are denoted by bracketing the class symbols. Thus (A) stands for the frequency of A and (AB) for the number of objects possessing the attribute AB.
Remark. Class frequencies of the type (A), (AB), (ABC), etc. are known as positive frequencies; ( ɑ), ( ɑ𝛽), ( ɑ𝛽𝜸), etc. are known as negative frequencies and (ɑ𝛽), (A𝛽C), (ɑ𝛽C), etc
are called the contrary frequencies.
Relationship between the class frequencies:
since αβ = -50, the ,given data is inconsistent.
Example 2 :Examine the consistency of the given data
N = 60 (A) = 51 (B) = 32 (AB) = 25
since all the frequencies are positive , it can be concluded that the given data are consistant.
INDEPENDENCE OF ATTRIBUTES:
Two attributes A and B are said to be independent it there exists no relationship of any kind between them. If A and B are independent, we would expect
(i) the same proportion of A's amongst B's as amongst 𝛽's and
(ii) the proportion of B's amongst A's is same as that amongst the ɑ's.
For example, if insanity and deafness are independent, the proportion of the insane people among deafs and non-deafs must be same.
Note: '' if the attribute A and B are independent, the proportion of AB's in the population is equal to the product of the proportions of A's and B's in the population.
ASSOCIATION OF ATTRIBUTES:
Two attributes A and B are said to be associated if they are not independent but are related in some way or other way. They are said to be
Positively associated if (AB) > (A) (B) / N
Negatively associated if (AB) < (A) (B) / N
In other words , two attributes A and B are Positively associated if δ >0 , Negatively associated if δ < 0 and are independent if δ =0
Remarks: Two attributes A and B are said to be comnpletely asosciated if A cannot occur without B, though B may occur without A and vice-versa. In other words for complete association either all A's are B's, i.e., (AB)=(A) or all B's are A's i.e., (AB) = B's (B) according as either A's or B' are in a minority. Similarly, complete dissociation means that no are A's are B's i.e., (AB) = 0 are no 𝞪's are 's 𝛃's, i.e., (𝞪) = 0 or more generally when either of these statements is true.
Yule's Coefficient of Association:
As a measure of the intensity of association between two attributes A and B, G. Udny Yule gave the coefficient of association Q, defined as follows:
Q = (AB) (𝞪𝛃) - (A𝛃) (𝞪B) / (AB) (𝞪𝛃) - (A𝛃) (𝞪B)
= Nδ / (𝞪B) (𝞪𝛃) + (A𝛃) (𝞪B)
If A and B independent, δ = 0 ⇒ Q = 0.
If A and B are completely associated, then
either (AB) = (A) ⇒ (A𝛃) = 0 or (AB) = (B) ⇒ (𝞪B) = 0
and in each case Q= + 1
If A and B are in complete dissociation, then
either (AB) = 0 or (𝞪𝛃) = 0 and we get Q = -1
Hence -1 ≤ Q ≤ 1
Applications of Attributes will be discussed tomorrow.
Keep Visiting
Keep Reading
Regards
A R Statistics
ATTRIBUTES:
Literally, an attribute means a quality or characteristic.Theory of attributes deals with qualitative characteristics which not amenable to quantitative measurements are and hence need slightly different statistical treatment from that of the variables. Examples of attributes are drinking,smoking, blindness, health, honesty, etc. An attribute may be marked by its presence (possession) or absence (dispossession) in member of given population. It may be pointed out that the methods of statistical
analysis applicable to the study of variables can also be used to a great extent in the theory of attributes and vice-versa. For example, the presence or absence of an attribute may be regarded as changes in the values of a variable which can possess only two values, viz., 0 and 1.
NOTATIONS:
Suppose the population is divided into two classes according the to the presence or absence of a a single attribute. The positive class which denotes the presence of the attribute is generally written in capital Roman letters such as A, B, C, D, etc . and the negative class, denoting the absence of the attribute is written in corresponding small Greek letters such as ɑ, 𝛽,𝜸,δ, etc. For example, if A represents attributes of sickness and B represents blindness, than ɑ and ß represent the non-sickness (health) and sight respectively.
The two classes, viz., A (possession of the attribute) and ɑ (dispossession of the attribute) are said to be complementary classes and the attribute of ɑ used in the sense of not-A is often called the complementary attribute of A. Similarly 𝛽,𝜸,δ, are the complementary attributes of B, C, D respectively.
The combinations of attributes are denoted by grouping together the letters concerned,
e.g., AB is the combination of the attributes A and B. Thus for the attributes A (sickness) and B (smoking), AB would mean the simultaneous possession of sickness and smoking. Similarly Aß will represent sickness and non-smoking, ɑB non-sickness (health) and smoking, and ɑ𝛽 non-sickness and non-smoking.
If a third attribute be included to represent, say male, then ABC will stand for sick males who are smokers. Similar interpretations can be given to AB𝜸, A𝛽C, A𝛽𝜸, etc.
DICHOTOMY:
If the universe (population) is divided into two sub-classes or complementary classes and no more, with respect to each of the attributes A, B, C, etc., the division o classification is said to be 'dischotomous classification'. The classification is termed manifold if each class is further sub-divided.
CLASSES AND CLASS FREQUENCIES:
Different attributes in themselves are called different classes and the number or observations assigned to them are called class frequencies which are denoted by bracketing the class symbols. Thus (A) stands for the frequency of A and (AB) for the number of objects possessing the attribute AB.
The number of observations or units belonging
to class is known as its frequency are denoted within bracket. Thus (A) stands for the frequency of
A and (AB) stands for the number objects possessing the attribute both A and B. The contingency table of order (2×2) for two attributes A and B can be displayed
as given below
A
|
α
|
Total
|
|
B
|
(A B)
|
(Α B)
|
(B)
|
β
|
(A β)
|
(α β)
|
(β)
|
Total
|
(A)
|
(Α)
|
Remark. Class frequencies of the type (A), (AB), (ABC), etc. are known as positive frequencies; ( ɑ), ( ɑ𝛽), ( ɑ𝛽𝜸), etc. are known as negative frequencies and (ɑ𝛽), (A𝛽C), (ɑ𝛽C), etc
are called the contrary frequencies.
Relationship between the class frequencies:
The frequency of a lower order class can always be expressed in
terms of the higher order class frequencies.
N = (A) +
(α) = (B) + (β)
(A)= (AB) + (Aβ)
(α)
= (αB) + (αβ)
(B)= (AB) + (αB)
(β)
= (Aβ) + (α β)
If
the number of attributes is n, then there will be 3n classes
and
we have 2n cell frequencies.
Consistency of the data:
In order to
find out whether the given data are consistent or not we have to
apply a very simple test. The test is to find out whether any or more of
the ultimate class-frequencies is negative or not. If none of the class
frequencies is negative we can safely calculate that the given data are consistent
(i.e the frequencies do not conflict in any way each other). On the other
hand, if any of the ultimate class frequencies
comes to be negative the given data are inconsistent.
Example 1: Test the consistency of the following data with the symbols having their usual meaning
N = 1000, (A) = 600, (B) = 500 and (AB) = 50
A
|
α
|
Total
|
|
B
|
50
|
450
|
500
|
β
|
550
|
-50
|
500
|
Total
|
600
|
400
|
1000
|
since αβ = -50, the ,given data is inconsistent.
Example 2 :Examine the consistency of the given data
N = 60 (A) = 51 (B) = 32 (AB) = 25
A
|
α
|
Total
|
|
B
|
25
|
7
|
32
|
β
|
26
|
2
|
28
|
Total
|
51
|
9
|
60
|
since all the frequencies are positive , it can be concluded that the given data are consistant.
INDEPENDENCE OF ATTRIBUTES:
Two attributes A and B are said to be independent it there exists no relationship of any kind between them. If A and B are independent, we would expect
(i) the same proportion of A's amongst B's as amongst 𝛽's and
(ii) the proportion of B's amongst A's is same as that amongst the ɑ's.
For example, if insanity and deafness are independent, the proportion of the insane people among deafs and non-deafs must be same.
Note: '' if the attribute A and B are independent, the proportion of AB's in the population is equal to the product of the proportions of A's and B's in the population.
ASSOCIATION OF ATTRIBUTES:
Two attributes A and B are said to be associated if they are not independent but are related in some way or other way. They are said to be
Positively associated if (AB) > (A) (B) / N
Negatively associated if (AB) < (A) (B) / N
In other words , two attributes A and B are Positively associated if δ >0 , Negatively associated if δ < 0 and are independent if δ =0
Remarks: Two attributes A and B are said to be comnpletely asosciated if A cannot occur without B, though B may occur without A and vice-versa. In other words for complete association either all A's are B's, i.e., (AB)=(A) or all B's are A's i.e., (AB) = B's (B) according as either A's or B' are in a minority. Similarly, complete dissociation means that no are A's are B's i.e., (AB) = 0 are no 𝞪's are 's 𝛃's, i.e., (𝞪) = 0 or more generally when either of these statements is true.
Yule's Coefficient of Association:
As a measure of the intensity of association between two attributes A and B, G. Udny Yule gave the coefficient of association Q, defined as follows:
Q = (AB) (𝞪𝛃) - (A𝛃) (𝞪B) / (AB) (𝞪𝛃) - (A𝛃) (𝞪B)
= Nδ / (𝞪B) (𝞪𝛃) + (A𝛃) (𝞪B)
If A and B independent, δ = 0 ⇒ Q = 0.
If A and B are completely associated, then
either (AB) = (A) ⇒ (A𝛃) = 0 or (AB) = (B) ⇒ (𝞪B) = 0
and in each case Q= + 1
If A and B are in complete dissociation, then
either (AB) = 0 or (𝞪𝛃) = 0 and we get Q = -1
Hence -1 ≤ Q ≤ 1
Applications of Attributes will be discussed tomorrow.
Keep Visiting
Keep Reading
Regards
A R Statistics
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