Theory of Attributes Assignment Help

Attributes definition

An attribute is a characteristic or property. Strong, for example, is an attribute of your body. Aside an attribute divides the universe into two classes, one possessing the attribute and another not possessing the attribute whereas the variable can divide the universe into any number of classes.

Association of Attribution

Association in statistical is different from the common meaning of association.

Methods used to measure the association of attributes refer to those techniques, which are used to measure the relationship between two such phenomena, whose size cannot be measured and where we can only find the presence or absence of an attribute.

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In the case of an association, we study the relationship between two attributes, which are not quantitatively measurable. For example, level of education and crime.

Types of Association

Two attributes are said to be associated if they are not independent but are related in some way or the other. There are three kinds of associations, which possibly occur between attributes.

• Positive association: In this, the presence of one attribute is accompanied by the presence of another attribute.

If (AB) > (A)(B)/N

This mean attributes A and B are positively associated.

• Negative association or disassociation: In this, the presence of one attribute ensures the absence of another attribute or vice versa.

If (AB) < (A)(B)/N

This mean attributes A and B are negatively associated.

• No association or independence: If two attributes are such that the presence or absence of one attribute has nothing to do with the absence or presence of another, they are said to independent or not associated

If (AB) = (A)(B)/N

This mean attributes A and B are independent.

Note:

1. Two attributes are said to be completely associated if one cannot occur without another. In other words, both are completely associated if all A’s are B’s i.e. (AB) = (A) or all B’s are A’s i.e. (AB) = (B), according to as for whether either A’s or B’s are in a minority.
2. Complete disassociation means that no A’s are B’s i.e. (AB) = 0 or no α’s are β’s i.e. (αβ) = 0.

The Symbols (AB)₀ and δ

Using following symbols to check types of association

(AB)₀ = (A)(B)/N, (αβ)₀ = (α)(β)/N

(αβ)₀ = (α)(β)/N, (Aβ)₀ = (A)(β)/N

δ = (AB) - (AB)₀ = (AB) - (A)(B)/N

If δ = 0, then (AB) = (A)(B)/N

• A and B are independent.

If δ > 0 then attributes A and B are positively associated and if δ < 0 then attributes A and B are negatively associated.

Note:

It is to be noted that if δ≠0 and its value is very small then it is possible that this association (either positive or negative) is just by chance and not really significant of any real association between the attributes. This difference is significant or not should be tested by the test statistic (Chi-square).

Example:

Question

Show whether A and B are independent, positively associated or negatively associated in each of the following cases:

1. N = 1000; (A) = 450; (B) = 600; (AB) = 340
2. (A) = 480; (AB) = 290; (α) = 585; (αβ) = 383

• N = 1000; (A) = 500; (B) = 400; (AB) = 200

Solution

1. (A)(B)/N = (450*600)/1000 = 270 = (AB)₀

Thus, (AB) = 340 > (A)(B)/N

Since (AB) > (AB)₀ hence they are positively associated

1. (B) = (AB) + (αβ) = 290 + 383 = 673

N = (A) + (α) = 480 + 585 = 1065

(A)(B)/N = (480*673)/1065 = 303.32 = (AB)₀

Thus, (AB) = 290 < 303.32 

Since (AB) < (AB)₀ hence they are negatively associated.

• (A)(B)/N = (500*400)/1000 = 200 = (AB)₀

Thus (AB) > (AB)₀ hence A and B are independent, i.e. δ = 0

Methods of measures of Association

Till far we have seen how to take a decision but it is not sufficient as we arealways interested in the extent of association so that we can measure the degree of association mathematically. In this, we will discuss the possibility of obtaining a coefficient of association, which can give some idea about the extent of association between two attributes. It would be easy for taking a decision if the coefficient of association is such that its value is 0 when two attributes are independent; +1 when they are perfectly associated and –1 when they are perfectly dissociated. In between –1 to +1 lie different levels of association.

Yule’s Coefficient of Association

Yule’s coefficient of association is named after its inventor G. Udny Yule. For two attributes A and B, the coefficient of association is given as

Q = (AB)(αβ) - (Aβ)(αB)/(AB)(αβ) + (Aβ)(αB))

Value of Q lies between –1 and +1.

• If Q = 1, A and B has perfect positive association. It can be verified that under perfect positive association

(AB) = (A) Þ (Aβ) = 0

(AB) = (B) Þ (αB) = 0

• If Q = –1, A and B possess perfect negative association. This leads to following relationship:

(AB) = 0 or (αβ) = 0

• If Q = 0, A and B are independent. Here, we have the following relation

(AB) (αβ) = (Aβ) (αB)

• Any value between –1 to +1 tells us the degree of relationship between two attributes A and B. Conventionally if Q > 0.5 the association between two attributes is considered to be of a high order and the value of Q less than 0.5 shows a low degree of association between two attributes.

Note:

Q is independent of the relative preposition of A’s or α’s in the data. This property of Q is useful when the prepositions are arbitrary.

Coefficient of Colligation

Another important coefficient of association given by Yule. It is defined as

It can be shown that

Q = = 2γ/1+γ²

The range of γ is from –1 to +1. It can be interpreted in the same manner as Q.

Example:

Question:

In a sample of 1000 children, 400 came from higher income group and rest from the lower income group. The number of delinquent children in these groups was 50 and 200 respectively. Calculate the coefficient of association between delinquency and income groups.

Solution:

Let A denotes the higher income group, then α would denote a lower income group. Let B denotes delinquent children then β would denote nondelinquent children. To get the frequencies of second order we form following nine square table

(α) = N – (A) = 1000 – 400 = 600

(B) = (AB) + (αB) = 50 + 200 = 250

(Aβ) = (A) – (AB) = 400 – 50 = 350

(αβ) = (B) – (AB) = 250 – 50 = 200

(αβ) = (α) – (αB) = 600 –200 = 400

Þ Q = (AB)(αβ) - (Aβ)(αB)/(AB)(αβ) + (Aβ)(αB)

= -0.55

Thus, there is a negative association between income and delinquency.