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## Odds and Probabilities

• Odds and probabilities
• If there is a probability of something happening, then the odds can be considered the number of successes you expect to get for every failure on average. High odds correspond to high probabilities, low odds to low probabilities.
• To calculate the odds:

• To go from odds back to probabilities:

• While probabilities are bound to [0,1], odds are bound to [0,)
• Let's take the titanic example. What was the overall odds of survival on the Titanic. 38% of passengers survived the Titanic. Therefore:

Odds of survival

How can we say this in English: "For every one death on the Titanic, there was on average 0.61 survivors."
• Let's see how the odds change across the range of the probability.

 p O 0 0 0.01 .010 0.05 .052 0.10 .111 0.3333 0.5 0.40 0.6667 0.5 1 0.6 1.5 0.6667 2 0.75 3 0.9 9 0.95 19 0.999 999

• Odds ratio: the ratio of two odds. If one odds is twice as big as the other, then we say the odds ratio is 2. If it is half again as big, the odds ratio would be 1.5.

• Let's go back to the Titanic example. Let's look at the odds ratio between men and women on the Titanic. 19% of men survived while 73% of women survived. Let's look at this as a bar graph.

Now let's calculate the odds of survival for each group.

Odds for women

Odds for men

The odds of survival for women is 2.67 and the odds of survival for men is 0.24. The odds ratio is given by:

In English: "Women were eleven times as likely as men to survive the Titanic."

• Let's look at some of our previous examples, where we have an odds ratio of 2.

 0.05 .052 .1004 0.912 18.24 0.1 .1111 .2222 0.1818 18.18 0.3333 0.5 1 0.5 1.5 0.5 1 2 0.6667 1.333 0.75 3 6 0.857 1.14 0.9 9 18 0.947 1.052 0.95 19 38 0.974 1.025 0.999 999 1998 0.9995 1.0005

• The odds ratios are constant here, but not the ratios of the probabilities directly. A doubling of the odds when the odds is already high doesn't have nearly the effect on the probability of doubling the odds when the odds is low. The relationship is reversed if we halve the odds.
• Our interpretations of the parameters in the logistic regression model will be related to this odds ratio.

Next: Logistic Regression Up: Logistic Regression Previous: Logistic Regression   Contents
Aaron 2005-12-21