# Converting Between Probabilities, Odds (Ratios), and Risk Ratios" In effectsize: Indices of Effect Size

library(knitr)
options(knitr.kable.NA = "")
knitr::opts_chunk$set(comment = ">") options(digits = 3)  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) # or odds_to_probs(3.25) # convert back probs_to_odds(0.764)  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 $$RR = \frac{OR}{(1 - p0 + (p0 \times OR))}$$ 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))


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)


Or directly:

oddsratio_to_arr(OR, baserate)


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)


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...

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effectsize documentation built on Sept. 14, 2023, 5:07 p.m.