Week 3 Supplemental: The Odds...

...Never tell me them

Stat 305 Notes. Week 3 Supplemental Page 1 / 23

OddsOdds are a lot like probability, but are calculated differently.

Probability of event =

Times event occurs

-------------------------

Times anything occurs

Stat 305 Notes. Week 3 Supplemental Page 2 / 23

Example:

The probability of rolling a “4” on a six-sided die is:

Pr( Rolling a 4) = One face / Six faces in total = 1/6.

Odds is calculated as

Odds of event =

Times event occurs / Times event DOESN’T occur

In odds, “ / ” should be read “to”.

In probability, “ / ” is read “in”.

Example: The ODDS of rolling a 4 on a six-sided die are:

Odds( Rolling a 4 ) = One face / Five faces = 1/5, 1 to 5.

Probability:

Odds:

The formula for odds, can be computed from probability.

Sometimes odds of doing something are interpreted as:

We use odds when we’re interested in comparing how often

an event happens to its opposite, or its complement.

The odds of something happening is how many times more likely the event happen is to something not happening.

Another example: If the probability of something 0.6,

Then the odds of the effect are 0.6 / 0.4 = 3/2 or 1.5

That means the event is 1.5 times as likely as it not happening.

The formula for getting odds from probability P is:

Other equivalent interpretations:

Odds = Pr( Event) / Pr ( Not Event )

Odds = Pr( Success) / Pr ( Failure )

Special case 1: If something is impossible, it has probability

zero.

An impossible event has odds zero as well.

So an impossible event happening is zero times as likely as it not happening.

Example 2: If an event has 0.5 probability, it has odds one.

This means if an event has probability 0.5, it’s just as likely that the event will happen as it not happening.

(Flipping a coin as heads has probability 0.5. This event is just as likely as flipping and getting a tails.)

Special Case 2: If something is certain to happen, it has

probability one. This event has odds infinity. (1 / 0)*

A certain event is infinitely more likely to occur than it not happening.

These examples also provide the limits of odds. Odds are always between 0 and infinity, and never negative.

Some other example values

Probability Odds0 00.1 0.11110.25 0.33330.5 10.66667 20.8 50.9999 99991 + Infinity

for interest: You can get the probability from odds by using

… it’s handy to check your work.

Now we can handle a multitude of problems!

Odds Ratio

As the name suggests, odds ratio is the ratio of odds under twodifferent conditions.

Example: If the odds of having lung cancer by age 70 is 0.15 if you smoke tobacco, and 0.008 if you don’t smoke anything, then the odds ratio of getting cancer for smoking vs. non-smoking is…

Odds1 / Odds2 = 0.150 / 0.008 = 18.75

We can also compute the odds ratio by hand in cases where there is only one explanatory variable, and it is also categorical.

Consider a sample of 20 heart attack patients, in which we know...

- Whether they had a second heart attack within a year (response variable, categorical, 2 levels)

- Whether they attended traditional anger management therapy after their first heart attack (explanatory, categorical, 2 levels)We can describe the relationship between these two variables as a crosstabulation, or crosstab for short.

Anger Management Therapy

2nd Heart Attack None Traditional

No (0) 4 6

Yes (1) 7 3

This means, for example, that 7 of the 20 patients had a secondheart attack and did not receive anger management therapy

Anger Management Therapy

2nd Heart Attack None Traditional

No (0) 4 6

Yes (1) 7 3

We can estimate the odds of a 2nd attack under each condition:

Odds of 2nd attack with no therapy: Pr(event) / Pr(not event),estimated by # with 2nd attacks / # without 2nd attacks

= 7 / 4= 1.75

Anger Management Therapy

2nd Heart Attack None Traditional

No (0) 4 6

Yes (1) 7 3

Likewise, we estimateOdds of 2nd attack WITH therapy: 3 / 6 = 0.5

Then we estimate the odds ratio:Odds without therapy / Odds with therapy = 1.75 / 0.5 = 3.5

So the odds of second heart attack are 3.5 times as high without anger management therapy.

That estimate of the true odds ratio comes from a sample, so itis only a statistic .

We only had 20 patients in the sample, so that statistic is going to come with a LOT of uncertainty as well.

Some consequences of the standard error of log odds:

1. The standard error gets smaller as the sample (n1 + n2 + n3 + n4) gets larger.

In other words: We become more confident in our results as we collect more information.

2. A very small group (low n) can make the standard error large, no matter how big the other groups are.

In other words: Our results are only as good as our smallest group (important for rare diseases or cases!!!)

Lots of new examples? No problem!