What is the difference between rate, risk and odds?

Oftentimes, you will ready a study that says, treatment A reduces the rate of something by X%, or the treatment reduces the risk by Y%, or reduces the odds by Z%.  These are basically the same thing, right?

That is not entirely correct.  While all these concepts summarize the frequency or likelihood of something occuring in some way, their exact definition varies.  Below, I provide a list of commonly use measures and what their official definition is.

• Probability: (# events that occured in a period)/(number of entities eligible).  This number ranges between 0 and 1.  For instance, if I want to know the probability or risk that I get a cold this winter, I will want to count the number of people who lived in my city over that winter and got a cold during the winter and divide that by the total number of people living in my city that winter.
• Rate.  (# events that occurred in a period)/(total time period experience by all subjects).  Now consider the case where different people live in my cite different amounts of time.  I could measure the total number of flu cases divided by the total number of days the eligilbe people lived in my city.  Another example would be a health plan that measures the rate at which people are hospitalized.  People often enroll and disenroll in health plans so for a given year, the health plan would count up all the hospitalizations in the year and divide by the total months enrolled of all these people.
• Relative risk. (probability of outcome in the exposed)/(probability of the outcome in the unexposed).  Let’s go back to my flu example.  Let’s say that kids spread the flu easily.  Assume one did a study where 40% of adults with young children got the flu, but only 20% of adults without young children got the flu.  In this case, the relative risk is 2 (i.e., 40%/20% =2).  A relative risk number can vary between 0 and infinity.
• Odds: (probability of an outcome)/(1-probability of an outcome). In my flue example, the odds of getting the flue for parents are 40%/(1-40%) = 40%/60% = 2/3, or 0.67:1.  For, adults without young children, the odds are 20%/(1-20%) = 20%/80% = 1/4 or 0.25:1.
• Odds ratio.  (Odds of exposed group)/(odds in unexposed group). Back to the flu example, the odds ratio would be (2/3)/(1/4) = (8/12)/(3/12) = 8/3 = 2.67.
• Risk difference. Risk in exposed group – risk in unexposed group.   For the flu example, this would be 40% – 20% = 20%.  The parents with young kids have a 20 percentage point higher chance of getting the flu.
• Mean.  Number of events/number of observations.

In short, these measures aim to capture similar but not identical concepts.  Relative risk and odds ratios are comparative of two groups, whereas probability, rate, odds and measure the frequency of occurrence within a specific group.