Log odds

Odds (odds of success): It is defined as the chances of success divided by the chances of failure. Say, there is a 90% chance that winning a wager implies that the ‘odds are in our favour’ as the winning odds are 90% while the losing odds are just 10%. It is also known defined as odds ratio as it is in the form of a ratio. So the odd ratio of the above discussed example would be:

Formally, the odds ratio of an event A is defined as the probability that A occurs upon the probability that A does not occur (i.e. complement of A). (As shown by equation given below)

Odds, not the same as probability

It is important to note that odds of an event occurring is not the same as its probability. Say the odds of my basketball team winning the tournament is 1 to 5. 

The odds of my team winning =  but,
The probability of my team winning = 

However, mathematically the end result remains the same for both as in case of probability, the denominator gets canceled out. (As shown below)

Problem with odds

The problem with odds ratio is due to the presence of asymmetricity created between the odds of winning and the odds of losing. Let us try to understand this with the help of an example. 72 countries participate in the Commonwealth Games. 

Considering every country has an equal chance of winning, the odds of the Indian team winning a gold are against, 1 to 71 or 
But assuming an idealistic scenario where the odds of Indian team winning the gold are in favour, 71 to 1 or 

Fig 1: Domain for favourable vs against odds

As observed in Fig 1, when the odds are against our favour, the value always tends to lie between 0 and 1 which is a very small value. However, if the odds are in our favour, the value can lie anywhere in between 1 and infinity, which can be a very big value! To solve this problem, the concept of Log odds came into picture.

Log odds: It is the logarithm of the odds ratio. (As shown by the equation given below)

As per the above-mentioned example, 

The log of odds of the Indian team winning a gold are against, 1 to 71 = 

The log of odds of Indian team winning the gold are in favour, 71 to 1 = 

Fig 2: Log Odds

As observed in Fig. 2, taking the log of the odds ratio brings about a certain symmetricity in the results, making it easier to interpret and use in various statistics. 

INTERESTING NOTE: The log odds of a certain event gives a normal distribution when plotted on a histogram! This is what makes log odds so useful. 

Real-life example 

Log odds are used in medical research for predicting the likelihood of symptoms the subject may get based on his/her previous symptoms. Consider the following example. A medical research is done on 1000 test subjects showing symptoms of fever and cough/cold at random. The objective is to find out the likelihood that people having cough/cold have a much more chance of getting a fever or not. (Data given below in Fig 3.)

Fig 3: Sample research data

Solution: 

The odds that a person having cough/cold, also has fever 

The odds that a person not having cough/cold, also has fever 

∴ The objective odds 

and , to provide symmetricity to the various results that can be obtained from the research. 

This shows that the likelihood that a person having cough/cold also has fever is 21 times more likely than that of a person not having cough/cold. The log odds or odds ratio is very similar to the R-squared test as it tells the relationship between two factors. So, it can be said that the higher the odds value, the more related the two factors tend to be. This is the power of log odds/odds ratio.