The Odds ratio

• The odds ratio is a very useful device for the analysis of categorical data.
  • It measures association
  • and underlies the maths behind loglinear models and logistic regression.

• What are odds?
  • The odds of outcome 1 versus outcome 2 are the probability (or frequency) of outcome 1 divided by the probability (or frequency) of outcome 2.

  • Contrast this with probability or proportions: one category's probability is its frequency divided by the total frequency:

    
    

    $$O{ij} = \frac{f_i}{f_j}$$
    
    
    
    
    $$\pi_i = \frac{f_i}{f_+}$$
    
    

• Where there are only two outcomes (e.g., winning and losing):
  
  $$O_{ij} = \frac{f_i}{f_+ - f_i} \equiv \frac{p_i}{1 - p_i}$$
  
  

• A horse given odds of 5:2 has 2 chances in 7 ($= 5 + 2$) of winning ( $p = 2 \div 7 = 0.286$).

• Odds measure the frequency or probability of one outcome relative to another.

• Odds ratios involve comparing the odds of a pair of outcomes on one variable, for a pair of categories of a second variable: $OR_{ij,kl} = O_{i,kl}/O_{j,kl}$

  	Owners	Private	Total
  		renters
  Cons	1977	211	2188
  Lab	2353	378	2731
  Total	4330	589	4919

• The overall odds of owning versus renting are $4330 \div 589 = 7.35$. That is, there are far more owners.

• However, separately for Conservative and Labour voters:

  Conservative	$1977 \div 211 = 9.37$
  Labour	$2353 \div 378 = 6.22$
  Ratio	$9.37 \div 6.22 = 1.51$

• The Odds-Ratio for conservative versus labour voters being owners versus private renters is 1.51. That is, even though labour voters are predominantly house-owners, conservative voters are much more likely to be house-owners.

• A $2\times2$ table under independence will have an odds ratio of 1, whatever the marginals.

• If we know the odds ratio and the marginals for a $2\times2$ table we can calculate all the cell sizes.

• For bigger 2-D tables, we can reconstruct all the cell counts from the marginals and a set of odds ratios that involve all the cells ( $(r-1)\times(c-1)$ is sufficient).

• The odds ratios are therefore equivalent to the structure of association.

• Loglinear models essentially define a pattern of odds ratios, apply the marginals to them, and compare the resulting table with the observed table, in pretty much the same way we apply the Pearson $X^2$ test for association. The big difference is the pattern we define can be much more complicated than independence.