Why Statisticians Enjoy Odds
Why Statisticians Enjoy Odds, Probability dominated your basic statistics class. Gamblers faced odds. However, it turns out that odds also play a significant influence in statistics.
Why not use probability instead of odds?. We can think of two important applications.
1. Logistic Regression Uses Odds
Logistic regression is the most helpful member of the linear model family, second only to linear regression.
It has been a strong and effective model for analyzing linkages between predictor variables and that outcome as well as for predicting the likelihood that an event would occur for decades.
Unfortunately, a linear model is poor at forecasting probabilities that do not rise linearly with predictor values.
A probability must be between 0 and 1 (nothing can happen with a greater than 100% likelihood). The chances are not as limited.
Odds can be any positive number; for example, a chance of 2/3 is equivalent to odds of 2/1. Instead, a linear model can be fitted if we utilize odds (actually, the log of odds, or logit).
It is simple to convert the chances of an outcome given a specific set of predictor values to a probability.
Another interpretation of the coefficient for a predictor in a linear regression model is the change in probabilities of an outcome per unit change in that predictor.
2. Retrospective Medical Studies Use Odds
The prospective study is a crucial research method in medicine. In prospective research, individuals are tracked over time to see if a disease develops.
Typically, we evaluate the risk (probability) of developing the disease in subjects with a certain characteristic or condition to the risk in those without that characteristic or condition.
We may compare the risk of heart disease among smokers and non-smokers, for instance.
The risk of heart disease for smokers divided by the risk of heart disease for non-smokers is the ratio that is generally used to express this comparison.
The “relative risk” ratio indicates how much more risk you are putting yourself at risk for by smoking.
Prospective studies, however, take a lot of time since you have to wait until the individuals have had enough time to develop the condition.
A different approach is to do a case-control retrospective research, in which you compare individuals who already have the condition to matched controls who do not and inquire whether they also have it.
Take a look at patients with heart disease and a group of matched controls, for instance, and inquire whether or not they smoke.
Since the study includes an equal number of heart disease patients and healthy participants, the likelihood of developing the disease cannot be calculated using this methodology.
Instead, these models determine the likelihood of the antecedent (smoking), not the result (heart disease).
For instance, we discover the likelihood that a patient with heart disease would smoke, not the likelihood that a smoker will develop heart disease.
The odds ratio for this risk factor can be calculated by dividing the probability of smoking by the odds of not smoking.
This design has the drawback of not providing information in terms that are most important to patients the increased likelihood of contracting a disease as a result of exposure to a risk factor.
However, it offers the benefit of being able to leverage already collected data and generate results right away.
Retrospective studies are therefore widely used because of their practicality and effectiveness.
As a result, odds ratios are well known to medical practitioners and statisticians who examine their data.
Gambling, probabilities, and behavioral research
Let’s look at betting or, more specifically, gambling.
Odds are a crucial component of betting because they serve as the medium via which gamblers and gambling establishments convert calculated probabilities into bets.
In betting, odds are expressed in terms of the return that a wager will produce.
You have a 3 to 1 chance of winning if you wager on 9 out of the 36 numbers on a 36-number roulette wheel.
This implies that if you win a $1 wager on a roulette wheel game, you will receive $3 (this is frequently referred to as a 2-1 payout, meaning that you will receive your initial $1 wager plus $2).
This amounts to a probability of 1/4 or 25% that one of your segments will be hit by the wheel.
The 3-to-1 odds are a reasonable wager if your play wins 25% of the time (getting you $3) and loses 75% of the time (getting you $1), implying that you will break even over the long run placing such a bet.
In order to generate unstoppable profits over the long term, gambling establishments create betting odds that are somewhat more favorable to themselves than a fair bet.
Or, to put it another way, the casino wants the probabilities implied by all of the outcomes’ odds to add up to more than 1.
In roulette, this is accomplished by adding a 0 (for Europe) or two (for the United States) to the 36 house-owned numbers.
If the roulette wheel falls on 0, no one wins, and the house keeps all of the wagers.
This house margin is small in open, competitive gambling games; if it were too large, another player could be able to provide better odds.
Beyond wagering on sporting events and casino games, betting has a huge potential impact.
In social psychology, hypothetical bets are a helpful experimental tool. This method was made popular by Daniel Kahneman through his study and book Thinking, Fast and Slow.
He offered the following option:
Scenario A: You’ve been given $1000. You must now choose between
●A 50% chance to win $1000
●Get $500 for sure
Scenario B: You’ve been given $2000. You must now choose between
●A 50% chance to lose $1000
●Lose $500 for sure
The selections in the two scenarios result in the same final levels of predicted wealth. However, almost everyone opts for the safe option in Scenario A and the wager in Scenario B.
This demonstrates the value of having a benchmark endowment of either $1000 or $2000. You can only make money from $1000, and that gain is secured by the sure bet of $500.
With a stake of $2000, you can only lose money; yet, there is a significant possibility you won’t.
The prospect theory developed by Kahneman and Tversky relies heavily on reference points.
Older theories that presupposed people to behave rationally to maximize predicted utility are enriched by prospect theory.
Kahneman’s wagers were fictitious wagers intended to illustrate academic notions; no real money was exchanged.
Political scientists have made an effort to go beyond this by designing marketplaces with tangible rewards.
The objective is to learn more about upcoming political events.
These “prediction markets” are believed to represent the total amount of information that is currently known about a subject (see The Wisdom of Crowds).
Similar to bid and offer prices in an economic market, bets in a prediction market have a purpose.
For a number of reasons, such political forecasting efforts have not been particularly successful.
One well-known attempt, the 2003 Policy Analysis Market of the U.S. Defense Department, was abandoned after it was suggested that a suitable topic would be upcoming terrorist strikes.
Many people felt that the American government should not support a market where those having inside information about upcoming attacks might make money from their information.
Betting enthusiasts believe it has enormous potential outside of the casino and racetrack. They are drawn to it because it can be used as a tool to clarify evaluation and judgment.
When opinions are qualified by words like “probably,” “rarely,” “a lot,” or something like that, people are frequently allowed to express their thoughts; but, continuing on to odds, probabilities, and bets is typically a step too far.