Nobody Wins 102% of the Time.

Home field in baseball is worth one number: +0.14. That single constant buys a coin-flip club 3.5 percentage points and a dominant club only 1.2. Percentages cannot hold a constant like that. Log-odds can, and that is the whole reason the log is there.

The Sports Page · The Professor · answering a question we were asked

The question, asked plainly: why do statisticians take the log of the odds instead of just using the odds? And what is an odds ratio actually for?

The answer: because two advantages have to be added together, and probabilities refuse to be added.

Why that is surprising. Most people assume you can just add a few percentage points for home field. You cannot, and the reason is not fussiness — do it and you will eventually claim a club wins 104% of the time.

What you get: the ability to read the phrase “odds ratio” correctly, which most reporting does not.

The problem our own bracket handed us

We built a postseason bracket this week. To fill it in we had to combine two separate things for every series: how good each club is, and who is at home.

Try that in percentages. Home clubs in baseball win about 53.5% of the time, so home field looks like it is worth three and a half points. Hand those points to a club that already wins 90% of the time and you get 93.5%. Hand them to one that wins 99% and you get 102.5%.

The trouble is not that the last number is silly. The trouble is that the first one was wrong too, and it did not announce itself.

First move: stop using probability, use odds

Odds are just the probability of a thing against the probability of no thing.

A 75% chance is odds of 3 to 1, because 75 divided by 25 is 3. A 50% chance is 1 to 1. A 90% chance is 9 to 1.

This helps, because odds have no ceiling. They run from zero to infinity, so nothing can overflow. Advantages now multiply rather than add: being twice as likely to win is a clean statement about odds.

But odds are still awkward, and lopsided in a way that will bite you. Doubling your odds from 1 to 2 is not the mirror image of halving them from 1 to 0.5, even though those are the same size of change. One has room to run and the other is squeezed against zero.

Second move: take the log

Take the logarithm of the odds and both problems disappear at once.

Multiplying becomes adding. Zero means a coin flip. The scale runs from minus infinity to plus infinity and is symmetric — a step up is the exact mirror of the same step down. Nothing can escape, because when you convert back, every number on the line lands somewhere between 0 and 1.

Now home field is one constant, forever. Home clubs win 53.5%, so playing at home rather than away is worth 0.14 in log-odds. Add it. Convert back. Done.

Look at what that single number does. To a club at 50% it is worth 3.5 points. To a good club at 70%, 2.9. To a dominant one at 90%, only 1.2.

That is not the constant misbehaving; that is the constant behaving. It is much harder to improve on 90% than on 50%, and the curve knows it. In percentages you would need a different home-field number for every club. In log-odds you need one.

So what is an odds ratio?

Exactly what it says: one club’s odds divided by another’s. It is the natural way to answer “how much does this thing help?”

Use our own data from a fortnight ago. Of the college football teams that won their opening game, 599 finished at .500 or better and 244 did not — odds of 2.46. Of the teams that lost the opener, 205 finished at .500 or better and 249 did not — odds of 0.82.

Divide one by the other and the odds ratio is 2.98. Winning the opener multiplies your odds of a winning season by three.

And because we did the arithmetic in logs, an interval comes almost free. The log of 2.98 is 1.09, its standard error is 0.12, and converting the ends back gives 2.35 to 3.78. Try building that interval on the raw ratio and it comes out lopsided and wrong, because ratios are not symmetric. Logs are.

The mistake nearly everyone makes with it

Here is the part to carry around, because you will meet it this month in a health story.

An odds ratio of 2.98 does not mean the teams were three times as likely to have a winning season.

Their actual chances were 71% against 45%. That is a ratio of 1.57 — barely more than half of 2.98. The odds ratio overstated the thing people think it measures by 89%.

This is not a rounding quibble. Odds ratios and risk ratios agree only when the outcome is rare. When the outcome is common, as a winning season is, the odds ratio runs away and every headline that reads it as “three times likelier” is badly wrong.

So when a study reports an odds ratio, ask one follow-up: how common was the outcome? If it happened to a third of people, the odds ratio is not a risk. If it happened to one in a thousand, the two are near enough the same.

What to take home: logs are where you add things up

You will now recognize this machinery everywhere, because almost everything runs on it.

Every time a study says it “controlled for” age and income and smoking, it added those effects together in log-odds. Credit scores work this way. So do medical risk calculators, spam filters and our bracket. The reason is always the one above: it is the only scale where separate advantages simply add and nothing can leave the interval.

And when the answer comes back out, remember it has been converted. The log-odds is where the work happens; the percentage is the translation. A constant effect in one is never a constant effect in the other — which is why the same home field is worth three points to a coin-flip club and one point to a great one.