Independence

• Structure: Is there association?
  • Does the distribution of one variable differ across the categories of the other variable?
  • Examine row or column percentages:
    • For instance, retirement is the status of 19% of women but only 16% of men,
    • for self employment the corresponding figures are 3.5% and 12.3%.
    • That is, the distribution of employment status clearly differs by sex (and vice versa).

• Association is a non-directional:
  • row or column percentages equally useful.

• If there is no association the variables are said to be independent. In a table showing independence, the percentage distributions within any row (or column) will be the same.

  Two unassociated variables
  Raw numbers
  	A	B	Total
  I	10	40	50
  II	14	56	70
  Total	24	96	120
  Row percents
  	A	B	Total
  I	20%	80%	50
  II	20%	80%	70
  Total	24	96	120
  Column percents
  	A	B	Total
  I	41.67%	41.67%	50
  II	88.33%	88.33%	70
  Total	24	96	120

• Given a set of marginals (i.e., row and column totals) we can calculate the expected values under the assumption of independence according to the following formula:

  
  

  $$E_{ij}=\frac{n_{i+}n_{+j}}{n_{++}}$$