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Logistic Regression Analysis: Understanding Odds and Probability

Probability and odds measure the same thing: the likelihood or propensity or possibility of a specific outcome.

People use the terms odds and probability interchangeably in casual usage, but that is unfortunate. It just creates confusion because they are not equivalent.

How Odds and Probability Differ

They measure the same thing on different scales. Imagine how confusing it would be if people used degrees Celsius and degrees Fahrenheit interchangeably. “It’s going to be 35 degrees today” could really make you dress the wrong way.

In measuring the likelihood of any outcome, we need to know two things: how many times something happened and how many times it could have happened, or equivalently, how many times it didn’t.

We call the outcome of interest a success, whether it’s a good outcome or not.

The other outcome is a failure. Each time one of the outcomes could occur is called a trial. Since each trial must end in success or failure, number of successes and number of failures adds up to total number of trials.

Probability is the number of successes compared to the total number of trials.

Odds are the number of successes compared to the number of  failures.

The Example

For example, to predict the likelihood of accidents at a particular intersection, we consider each car that goes through the intersection a trial. Each trial has one of two outcomes: accident or safe passage. If the outcome we’re most interested in modeling is an accident, that is a success (no matter how morbid it sounds).

Probability(success) = number of successes/total number of trials

Odds(success) = number of successes/number of failures

Odds are often written as:

Number of successes:1 failure

which is read as the number of successes for every 1 failure. But often the :1 is dropped.

I see a lot of researchers get stuck when learning logistic regression because they are not used to thinking of likelihood on an odds scale.

Equal odds are 1. 1 success for every 1 failure. 1:1

Equal probabilities are .5. 1 success for every 2 trials.

Odds can range from 0 to infinity. When odds are greater than 1, success is more likely than failure. When odds are less than 1, failure is more likely than success.

Probability can range from 0 to 1. When probability is greater than .5, success is more likely than failure. When probability is less than .5, failure is more likely than success.

The Calculations

In the last month, data from a particular intersection indicate that of the 1,354 cars that drove through it, 72 got into an accident.

72 Successes = Accident
1282 Failures = Safe Passage (1,354 – 72)

Failures = Total – Successes

Pr(Accident) = 72/1354 = .053
Pr(Safe Passage) = 1282/1354 = .947

Odds(Accident) = 72/1282 = .056
Odds(Safety) = 1282/72 = 17.87

Now get out your calculator, because you’ll see how these relate to each other.

Odds(Accident) = Pr(Accident)/Pr(Safety) = .053/.947

Why you Need to Understand Both Odds and Probability

So if we all find probability easier to understand and we’re more used to it, why do we ever need odds?

There are a few reasons.

One is that when probabilities get VERY close to 0 or 1, it’s actually easier to compare odds than it is probabilities.

Think about it. Probabilities of .0001 and .000001 don’t look all that different, intuitively.

But if you change them to odds 1 to 9,999 vs. 1 to 999,999, the difference in the order of magnitude is more intuitive.

Second, in logistic regression the only way to express the constant effect of a continuous predictor is with an odds ratio. Before you can understand or interpret an odds ratios, you need to understand an odds.

In many health-related fields, researchers really want to convert any odds to probabilities, and you’ll often see odds ratios converted to risk ratios. That works fine in a few situations, but there are just some situations where you can’t do it. So it’s best to be able to interpret both.

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