Say we are reading about a recently completed trial investigating a new flu vaccine. At the beginning of the flu season, they randomly assigned 200,000 participants to be injected either with the experimental vaccine or a placebo (a saline solution only), winding up with 100,000 in each group. By the end of the flu season, of the participants who received the vaccine, 60% (60,000) developed the flu, whereas among those receiving the placebo, 80% (80,000) developed the flu. The results are summarized in the table below:
So, was the experimental vaccine a success? Would you recommend it to your patients?
One way to express this finding is in term of odds. We might ask, what were the odds of contracting the flu given the experimental vaccine and by comparison, what were the odds of coming down with the flu given the placebo.
Odds
The odds of an event such as getting the flu is expressed as a ratio of the number of times the event happened out of the total number of possible times the event could have happened divided by the number of times the event did not happen.
Note that this is distinct from probability, which is the ratio of the probability of an event occurring to the total number of chances of it occurring.
Looking at a visualization can make this easier to understand. In the flow chart below, you can see that the event (developing the flu) occurred 60,000 times out of the total of 100,000 possible times it could have occurred (because there are 100,000 people, the flu could in theory occur all 100,000 times). You can also see that the event did not occur 40,000 times. Therefore the odds of the flu given the flu vaccine is the ratio of 60,000 to 40,000.
Notice that odds are typically expressed in the # yes : # no format.
So you can conclude that for every 3 cases of the flu (yes-event), there are 2 cases
without the flu (no-event).
Let’s look at another example:
A box contains 5 shirts, 2 hats, and 9 pairs of socks
1. What are the odds of reaching blindly into the box and pulling out a hat?
Number of yes-event: 2 Number of no-event: 5 + 9 = 14
The odds would be the ratio of the number of yes-events (2) to the number of no-events (14). That’s 2 : 14, which can be simplified to 1 : 7.
Odds can be calculated from probabilities if they are available:
Therefore the odds of drawing a hat are 1:7.
2. What are the odds of reaching in the box and pulling out anything but a hat?
Number of yes-event: 5 + 9 Number of no-event: 2
Therefore the odds of drawing anything but a hat are 7:1.
What is the difference between Odds and Probability?
Probability is the likelihood of some event and is expressed as the proportion of times some event would occur out of some total number of chances for that event to occur.
For example, the probability of drawing a King from a single deck of playing cards is:
- Number of possible hits: 4 (4 Kings in a deck)
- Total number of chances: 52 (52 cards in a deck)
Thus the probability of drawing a King is .08 or 8%.
Whereas probability is a ratio of chances of hits to the total number of chances, the odds is the ratio of chances of hits to chances of misses.
- Number of possible hits: 4
- Number of possible misses: 48 (52 – 4)
Therefore the odds of drawing a King is 1:12
Odds Ratio
We calculated the odds of developing the flu given the experimental vaccine to be 3:2. So this tells us the odds of the event of interest (flu) given the experimental vaccine. But what we really want to know is how much better (or worse) the experimental treatment is compared to the control group. In other words, it would be useful to have a way of expressing odds of flu given the experimental vaccine relative to the odds of flu given the placebo. What we need is the odds ratio, often abbreviated OR.
Expressed more generally, the OR provides an index of the odds of something occurring (e.g., disease) given exposure to some condition (e.g., a treatment) relative to the risk of this event occurring without exposure.
If the odds of cancer given smoking is the same as the odds of cancer without smoking then OR = 1 and we could conclude that there is no greater risk of cancer (the event) given smoking (exposure) than the risk of cancer without the exposure. If however, OR = 4, that would mean that the odds of cancer given smoking would be 4 times higher compared to the odds without smoking.
OR = 1: conditions do not affect odds of outcome
OR > 1: conditions associated with increased odds of outcome OR < 1: conditions associated with decreased odds of outcome
OR can be visualized as follows:
NOTE: The presence of an OR > 1 does not automatically imply statistical significance. Confidence intervals and/or p-values are required to answer the question of significance.
Risk
A risk is the probability of something undesirable happening. Risk is identical in concept to probability except that risk implies a negative event (e.g., dying, developing cancer).
The flow figure above helps us to visualize risk. Among the 100,000 people (total # of possible cases) taking the experimental vaccine in which flu could possibly develop, 60,000 will fall into the group of individuals who develop a flu (# cases of event occurring).
So you can say that there is a 60% risk that one will develop a flu given the flu vaccine, or that the risk of developing a flu given the flu vaccine is 60%.
Let’s compare this with our calculation of the odds of coming down with the flu given the experimental vaccine:
Risk: 60,000 / 100,000
Odds: 60,000 / 40,000
Notice that they both express the number of times some event occurs (in this example, 6). But where they differ is in what is used as the comparison. Risk uses the total number of possible opportunities for the event to occur (i.e., 10), whereas odds uses the number of times the event did not occur as the comparison point.
Risk Ratio
The risk ratio is the ratio of the risk of an event (e.g., contracting the flu) occurring in one group (treatment) to the risk of the event occurring in a comparison group (placebo).
In the figure above you can see that the risk (ie. Probability) of contracting the flu given a flu vaccine is 60,000/100,000 = 0.60 = 60%, whereas the risk of contracting the flu given no vaccine is 80,000/100,000 = 0.80 = 80%. The risk ratio is therefore…
- A risk of 1 means that there is no difference in risk between the groups
- A risk > 1 means that there is a greater risk of the event in the treatmentgroup.
- A risk < 1 means that there is a lower risk of the event in the treatmentgroup.
Absolute Risk Difference (ARD)
Also known as…
- Absolute risk reduction (ARR)
- Risk difference (RD)
The absolute risk difference (ARD) simply the risk of some event given some treatment minus the risk of that event in a comparison group.
ARD = Risk in treatment group – Risk in control group
Say a new antibiotic is being tested in a clinical trial. It is found that out of 100 people in the treatment group, 10 are not cured, whereas out of 100 people in the control group 70 people are not cured.
So the risk of non-improvement in risk in treatment group = 10/100 = 1/10 Risk in control group = 70/100 = 7/10
Thus the absolute change in risk due to the treatment is 60%. That is, the antibiotic results in a 60% reduction in the risk of failure.
- The minus sign indicates a decrease in risk whereas a positive sign indicates an increase in risk.
A 0 indicates no difference in risk.
It can be helpful to multiply the risk figures by 100 to convert the risks into actual numbers. So out of 100 treatment patients, 10 will not be cured by the treatment. Out of 100 control patients, 70 will not be cured. Subtracting 70 from 10 results in -60. That means that out of a total of 100 patients 60 additional patients will be cured because of the treatment.
Relative Risk Difference (RRD)
Relative risk difference is the ratio of the risk of some event given exposure compared to the risk of that event without exposure. In the antibiotic example, it is the ratio of the risk of failure given the new antibiotic compared to the risk of failure without the new antibiotic.
.60/.80 = 75%
The problem with RRD is that it does not take into account the initial baseline risk of the outcome event. So let’s say for treatment A there is a 30% risk of a disease and for treatment B there is a 15% risk of disease. That’s a 50% relative risk reduction yet the absolute reduction in risk is only 15%!
Let’s take another example. Say for treatment A there is a 2% risk of disease whereas for treatment B there is a 1% risk of disease. Again the relative risk difference is 50%, but in this case, the absolute risk reduction is a measly 1%! The reason for such a big difference? In the second example, the risk of the disease is very small to begin with; that is, the baseline risk is small, only 2%. A change from 2% to 1% represents a 50% drop in risk yet it is very small in absolute terms.
Numbers Needed to Treat
The Number Needed to Treat (NNT) is the number of individuals that would need to be treated in order to prevent some event.
For example, if a headache drug is reported to have an NNT of 10, it means that 10 patients would need to be treated with the drug in order to prevent one additional headache.
NNT is the inverse of the Absolute Risk Difference measure.
where ARD = Comparison Event Rate – Treatment Event Rate. Always round up.
Say a new antibiotic is being tested in a clinical trial. It is found that out of 100 people in the treatment group, 10 are not cured, whereas out of 100 people in the control group 70 people are not cured.
Risk in treatment group = 10/100 = 1/10
Risk in control group = 70/100 = 7/10
Rounding up, this means that 2 patients need to be treatment for one to be cured. This would be considered very good. For some treatments, NNTs upwards of 100 are sometimes seen.
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