Converting Between Probabilities, Odds (Ratios), and Risk Ratios

The effectsize package contains function to convert among indices of effect size. This can be useful for meta-analyses, or any comparison between different types of statistical analyses.

Converting Between p and Odds

Odds are the ratio between a probability and its complement:

\[ Odds = \frac{p}{1-p} \]

\[ p = \frac{Odds}{Odds + 1} \] Say your bookies gives you the odds of Doutelle to win the horse race at 13:4, what is the probability Doutelle’s will win?

Manually, we can compute \(\frac{13}{13+4}=0.765\). Or we can

Odds of 13:4 can be expressed as \((13/4):(4/4)=3.25:1\), which we can convert:


odds_to_probs(13 / 4)
> [1] 0.765
# or
> [1] 0.765
# convert back
> [1] 3.24

Will you take that bet?

Odds are not Odds Ratios

Note that in logistic regression, the non-intercept coefficients represent the (log) odds ratios, not the odds.

\[ OR = \frac{Odds_1}{Odds_2} = \frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}} \] The intercept, however, does represent the (log) odds, when all other variables are fixed at 0.

Converting Between Odds Ratios, Risk Ratios and Absolute Risk Reduction

Odds ratio, although popular, are not very intuitive in their interpretations. We don’t often think about the chances of catching a disease in terms of odds, instead we instead tend to think in terms of probability or some event - or the risk. Talking about risks we can also talk about the change in risk, either as a risk ratio (RR), or a(n absolute) risk reduction (ARR).

For example, if we find that for individual suffering from a migraine, for every bowl of brussels sprouts they eat, their odds of reducing the migraine increase by an \(OR = 3.5\) over a period of an hour. So, should people eat brussels sprouts to effectively reduce pain? Well, hard to say… Maybe if we look at RR we’ll get a clue.

We can convert between OR and RR for the following formula (Grant 2014):

\[ RR = \frac{OR}{(1 - p0 + (p0 \times OR))} \]

Where \(p0\) is the base-rate risk - the probability of the event without the intervention (e.g., what is the probability of the migraine subsiding within an hour without eating any brussels sprouts). If it the base-rate risk is, say, 85%, we get a RR of:

OR <- 3.5
baserate <- 0.85

(RR <- oddsratio_to_riskratio(OR, baserate))
> [1] 1.12

That is - for every bowl of brussels sprouts, we increase the chances of reducing the migraine by a mere 12%! Is if worth it? Depends on you affinity to brussels sprouts…

Similarly, we can look at ARR, which can be converted via

\[ ARR = RR \times p0 - p0 \]

riskratio_to_arr(RR, baserate)
> [1] 0.102

Or directly:

oddsratio_to_arr(OR, baserate)
> [1] 0.102

Note that the base-rate risk is crucial here. If instead of 85% it was only 4%, then the RR would be:

oddsratio_to_riskratio(OR, 0.04)
> [1] 3.18

That is - for every bowl of brussels sprouts, we increase the chances of reducing the migraine by a whopping 318%! Is if worth it? I guess that still depends on your affinity to brussels sprouts…


Grant, Robert L. 2014. “Converting an Odds Ratio to a Range of Plausible Relative Risks for Better Communication of Research Findings.” Bmj 348: f7450.