Odds and Odds Ratios

For an event with a given probability value, $p$, the corresponding odds is a numerical value given by

$$odds \{ Event \} = \frac{p}{1-p} .$$

The concept of odds first arose in gambling. If you bet on the occurrence of an event (e.g. Horse A wins the race) and that event has a known probability, $p_A = Pr \{ A  wins \}$, then for a one dollar bet the fair return should equal the odds. For example, if horse A has a 50% chance of winning ($p_A = .5$), the odds are $odds \{ A \} = \frac{.5}{1-.5} = 1$, an equitable return on a one dollar bet is one dollar. If the horse has a 25% chance of winning, the odds are $\frac{.25}{1-.25} = 1/3$. For a one dollar bet, one should expect three dollars in return if the horse wins.

Odds values are a simple one-to-one conversion of probabilities. If presented with a probability, it may be converted to odds as above. If provided with odds, the conversion to probabilities follows the formula $p =\frac{\mathit{odds}} { 1 + \mathit{odds}}$. Thus if one knows that the odds for an event are 1 in 10, i.e. .10, the probability of the event is $\frac{.1}{1 + .1} = \frac{1} {11} \approx .091$. Note in this instance, the probability of .091 is nearly equal to the odds of .1. Whenever the odds are a small value, the odds be a reasonable approximation to the probability, and vice versa.

The odds ratio is a comparative measure of two odds relative to different events. For two probabilities, $p_A = Pr \{ event A occurs \}$ and $p_B = Pr \{ event B occurs \}$ the corresponding odds of $A$ occurring relative to $B$ occurring is

$$odds  ratio \{ A  vs.  B \} = \frac{odds \{ A \} } {odds \{ B \} } = \frac{ p_A / ( 1 - p_A ) }{p_B / (1- p_B )} .$$

The "practical" interpretation of an odds ratios is this. If confronted with a choice between wagers hinging on the occurrence of either A or of B, with potential winnings of either $X$ dollars for wager A or $Y$ dollars for wager $B$, the sound choice is to choose A if $odds  ratio \{ A  vs.  B \} > \frac{ Y }{ X }$.

In statistics, the odds ratio is also used as comparative measure of two probabilities $p_A$ and $p_B$ used as an alternative to the risk difference and the relative risk. Recall the latter are defined by:

Risk difference = $p_A - p_B$

Relative risk = $\frac{p_A}{p_B}$

While these measures are somewhat more intuitive and directly interpretable, the odds ratio has some "mathematical" advantages. Suppose one has conducts a comparative study to investigate the relationship of $p_A$ and $p_B$. Then the relevant events to consider are event $A$, which corresponds to a deleterious outcome if treated with drug $A$ and event $B$, corresponding to a deleterious outcome if treated with drug $B$. The standard tabulation of the resulting data would be the two way table of observed frequencies:

OUTCOME	Deleterious	Not Deleterious	Total
Drug
A	a	b	a+b
B	c	d	c+d
Total	a+c	b+d	n

The estimated odds ratio for $A$ relative to $B$ based on data as above is $\frac{a \times d }{b \times c }$. This formula is simpler than that for the risk difference $= \frac{a}{a+b} - \frac{c}{c+d}$ or the risk ratio $= \frac{a / (a + b) } { c / ( c + d ) }$. In a mathematical sense, the odds ratio is the most natural comparative measure, i.e. leads to the simplest (and aesthetically pleasing) algebra. In addition, when both probabilities being compared are small, the odds ratio will closely approximate the relative risk.