**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**

**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**

**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**

**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**

**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**

**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**

**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 - 7: Contingency Table for 3 Attributes**

**Part - 8: Understanding the frequencies of 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**

**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**

**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**

**Part - 11: Solved Examples**

**Question: Find the missing frequencies? **

**Given:**

**(αB) = 500****(β) = 600****(α) = 800****(B) = 1000**

**Part - 12: Solved Examples**

**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**

**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**

**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**

**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**