# 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:

    library(effectsize)

    odds_to_probs(13 / 4)

    > [1] 0.765

    # or
    odds_to_probs(3.25)

    > [1] 0.765

    # convert back
    probs_to_odds(0.764)

    > [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…

# References

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