# Independence of Attributes

Let us consider certain examples before we discuss the independence in a formal manner.

Example:

Consider the following sample data on the liking of males and females for fish.

 Gender Males Females Total Like Fish $80$ $80$ $160$ Do not like Fish $20$ $20$ $40$ Total $100$ $100$ $200$

Discussion: There are 100 males out of which 80 like fish and out of 100 females 80 like fish. Males and females have the same liking for fish. We say that there is independence between gender and liking or disliking of fish. Another way of saying the same thing is that there is no relation between the gender and liking for fish.

Example:

Consider the following sample data on smoking by adult males and females:

 Gender Males Females Total Smokers $20$ $1$ $21$ Non-Smokers $80$ $99$ $179$ Total $100$ $100$ $200$

There are 100 males out of which 20 are smokers and out of 100 females there is only one smoker. It means that the smokers in males are 20 times more than the smokers among females. Males have a strong relation or association with smoking. Thus males and smoking are strongly associated. We say that there is positive association between males and smoking. There are 99 females who are non-smokers as compared to 80 male-non smokers. Thus females are inclined towards non-smoking. The association between females and non-smoking is also of positive type. There is only 1 female smoker as compared to 20 male smokers. Thus there is negative association between females and smoking and there is also negative association between males and non - smoking. Thus in a certain contingency table, when there is positive association between two attributes, then in the same table there exists the negative association between some other pairs of attributes. If there is positive association between $A$ and $B$, then $\alpha$ and $\beta$ are also positively associated. In this case there is negative association between $A$ and $\beta$, and between $\alpha$ and $B$.

The data may be written as

 Males $A$ Females $\alpha$ Total Non-Smokers, $B$ $\left( {AB} \right) = 80$ $\left( {\alpha B} \right) = 99$ $179$ Smokers, $\beta$ $\left( {A\beta } \right) = 20$ $\left( {\alpha \beta } \right) = 1$ $21$ Total $100$ $100$ $200$

In this table 80 is less than 99 and 20 is greater than 1 (or 1 is less than 20). There is negative association between $A$ and $B$ and between $\alpha$ and $\beta$. There is positive association between $A$ and $\beta$ and between $\alpha$ and $B$. If the attributes in the one diagonal have positive association, then the attributes in the other diagonal have negative association.