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

Source:`vignettes/convert_p_OR_RR.Rmd`

`convert_p_OR_RR.Rmd`

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…