### Learning Objective:

What are characteristics in Theory of Attributes?

What are the types of Dichotomous Classification in Theory of Attributes?

What are Attribute Classes?

What are Attribute Frequencies?

What are Contingency Table?

Using Contingency Table.

Consistency of Data.

What is Independence of Attributes?

What is Association of Attributes?

What are methods to study Independence and Association of Attributes?

How to construct Contingency Table for 3 Attributes?

What is Coefficient of Colligation?

### Concepts

Measurable characteristics can be termed as Quantitative Characteristics.

Quantitative Characteristics are called as Variables.

Non-Measurable characteristics can be termed as Qualitative Characteristics.

Qualitative Characteristics can be called as Attributes.

Attributes types can be Positive or Negative.

Positive Attributes are denoted as "A", "B", "C"....

Negative Attributes are denoted as "α", "β", "γ"....

Attributes themselves represent a Group. Such groups can be termed as "Classes"

Positive Attributes can have Positive Frequencies and Negative Attributes can have Negative Frequencies.

Invalid classes can be "A" and "α"; "B" and "β"; "C" and "γ"

Relationship of Frequency can be studied in the Contingency Table.

Contingency Table can be for 2 OR 3 Attributes.

Contingency Table for 2 Attributes is also termed as 9 Square Table or 2 x 2 Table.

Any class frequencies cannot be greater than total population. Hence each class frequency must be less than or equal to "N"

Any class frequencies cannot be negative. Hence each class frequency must be greater than or equal to 0.

9 square table can be used to check consistency of data for 2 attributes.

Two attributes are said to be independent if there does not exist any kind of relation between them.

Two attributes are said to be associated if they are not independent i.e. they are related in some way or another.

Association of Attributes can be Positive or Negative.

When two attributes are present or absent together in the data then the attributes are positively associated.

When the presence of one attribute is associated with the absence of the other attribute then the attributes are negatively associated.

Yule computed another coefficient called coefficient of colligation (γ)

Yule's coefficient of colligation is generally restricted to 2 x 2 tables.

### Formulas

Proportion Method

Attributes are independent when (AB)/(B) = (Aβ)/(β)

Attributes are positively associated when (AB)/(B) > (Aβ)/(β)

Attributes are negatively associated when (AB)/(B) < (Aβ)/(β)

If A and B are independent then (A and β), (α and B) and (α and β) are also independent.

Comparison Method

Attributes are independent when (AB) = (A) x (B) / N

Attributes are positively associated when (AB) > (A) x (B) / N

Attributes are negatively associated when (AB) < (A) x (B) / N

Yule's Coefficient of Association

Q🇦🇧 = [(AB) (αβ) - (Aβ) (αB)] / [(AB) (αβ) + (Aβ) (αB)]

The value of Yule's Coefficient of Association lies between -1 and 1.

If the value of Yule's Coefficient is 0, the attributes are independent.

If the value of Yule's Coefficient lies between 0 and 1, the attributes are positively associated.

If the value of Yule's Coefficient lies between 0 and 0.5, the attributes are weakly positively associated.

If the value of Yule's Coefficient lies between 0.5 and 1, the attributes are strongly positively associated.

If the value of Yule's Coefficient lies between -1 and 0, the attributes are negatively associated.

If the value of Yule's Coefficient lies between 0 and -0.5, the attributes are weakly negatively associated.

If the value of Yule's Coefficient lies between -0.5 and -1, the attributes are strongly negatively associated.

If the value of Yule's Coefficient is 1, the attributes are completely associated.

If the value of Yule's Coefficient is -1, the attributes are completely disassociated.

### Part - 1: Introduction Theory of Attributes

### Concept Recap:

Measurable characteristics can be termed as Quantitative Characteristics.

Quantitative Characteristics are called as Variables.

Non-Measurable characteristics can be termed as Qualitative Characteristics.

Qualitative Characteristics can be called as Attributes.

### Part - 2: Dichotomous Classification and Types of Attributes

### Concept Recap:

Attributes types can be Positive or Negative.

Positive Attributes are denoted as "A", "B", "C"....

Negative Attributes are denoted as "α", "β", "γ"....

### Part - 3: Attributes Classes & Frequency and Contingency Table for 2

### Concept Recap:

Attributes can be Positive or Negative.

Positive Attributes can be represented by "A", "B", "C"....

Negative Attributes can be represented by "α", "β", "γ"....

Attributes themselves represent a Group. Such groups can be termed as "Classes"

Positive Attributes can have Positive Frequencies and Negative Attributes can have Negative Frequencies.

Invalid classes can be "A" and "α"; "B" and "β"; "C" and "γ"

Relationship of Frequency can be studied in the Contingency Table.

Contingency Table can be for 2 OR 3 Attributes.

Contingency Table for 2 Attributes is also termed as 9 Square Table or 2 x 2 Table.

### Part - 4: Contingency Table and Consistency of Data

### Concept Recap:

Any class frequencies cannot be greater than total population. Hence each class frequency must be less than or equal to "N"

Any class frequencies cannot be negative. Hence each class frequency must be greater than or equal to 0.

9 square table can be used to check consistency of data for 2 attributes.

### Part - 5: Independence and Association of Attributes

### Concept Recap:

Two attributes are said to be independent if there does not exist any kind of relation between them.

Two attributes are said to be associated if they are not independent i.e. they are related in some way or another.

Association of Attributes can be Positive or Negative.

When two attributes are present or absent together in the data then the attributes are positively associated.

When the presence of one attribute is associated with the absence of the other attribute then the attributes are negatively associated.

### Part - 6: Methods to study Independence and Association of Attributes

### Formulas to Study Independence and Association of Attributes:

Proportion Method

Attributes are independent when (AB)/(B) = (Aβ)/(β)

Attributes are positively associated when (AB)/(B) > (Aβ)/(β)

Attributes are negatively associated when (AB)/(B) < (Aβ)/(β)

If A and B are independent then (A and β), (α and B) and (α and β) are also independent.

Comparison Method

Attributes are independent when (AB) = (A) x (B) / N

Attributes are positively associated when (AB) > (A) x (B) / N

Attributes are negatively associated when (AB) < (A) x (B) / N

Yule's Coefficient of Association

Q🇦🇧 = [(AB) (αβ) - (Aβ) (αB)] / [(AB) (αβ) + (Aβ) (αB)]

The value of Yule's Coefficient of Association lies between -1 and 1.

If the value of Yule's Coefficient is 0, the attributes are independent.

If the value of Yule's Coefficient lies between 0 and 1, the attributes are positively associated.

If the value of Yule's Coefficient lies between 0 and 0.5, the attributes are weakly positively associated.

If the value of Yule's Coefficient lies between 0.5 and 1, the attributes are strongly positively associated.

If the value of Yule's Coefficient lies between -1 and 0, the attributes are negatively associated.

If the value of Yule's Coefficient lies between 0 and -0.5, the attributes are weakly negatively associated.

If the value of Yule's Coefficient lies between -0.5 and -1, the attributes are strongly negatively associated.

If the value of Yule's Coefficient is 1, the attributes are completely associated.

If the value of Yule's Coefficient is -1, the attributes are completely dissassociated.

### Part - 7: Contingency Table for 3 Attributes

### Part - 8: Understanding the frequencies of Contingency Table for 3 Attributes

### Part - 9: Obtaining missing frequencies using Contingency Table for 3 Attributes

### Question: Find the missing frequencies using contingency table?

### Given:

(ABC) = 26

(AβC) = 18

(ABγ) = 38

(Aβγ) = 12

(αBC) = 12

(αBC) = 12

(αβC) = 6

(αBγ) = 16

(αβγ) = 9

### Part - 10: Coefficient of colligation and Coefficient of contingency

### Concept Recap

Yule computed another coefficient called coefficient of colligation (γ)

Yule's coefficient of colligation is generally restricted to 2 x 2 tables.

### Part - 11: Solved Examples

### Question: Find the missing frequencies?

### Given:

(αB) = 500

(β) = 600

(α) = 800

(B) = 1000

### Part - 12: Solved Examples

### Question: Test for consistency of data when N = 1000, (A) = 150, (B) = 300 and (AB) = 200.

### Question: In a survey of 500 persons, 300 were married and 250 were successful executives. 198 successful executives were married. Is the data consistent?

### Part - 13: Solved Examples

### Question: State the type of association between the attributes when N = 126, (A) = 90, (B) = 56 and (AB) = 40.

### Question: State the type of association between the attributes when N = 180, (A) = 160, (β) = 130 and (αβ) = 6.

### Part - 14: Solved Examples

### Question: Find the Yule's coefficient of association when N = 35, (A) = 18, (AB) = 8 and (αβ) = 5.

### Question: Find the extent of association between intelligence of husband and the intelligence of wife when

Intelligent husbands with intelligent wives = 50

Intelligent husbands with dull wives = 200

Dull husbands with intelligent wives = 100

Dull husbands with dull wives = 300

### Part - 15: Solved Examples

### Question: Find the missing frequencies using contingency table?

### Given:

(ABC) = 210

(αBC) = 280

(ABγ) = 180

(αBγ) = 240

(AβC) = 250

(αβC) = 160

(Aβγ) = 360

(αβγ) = 32