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this paper I'm reading is on MAOA and violent offenders. From what I understand, an odds ratio of 1.71 means that the violent offenders were almost 2x likely to exhibit the low activity MAOA compared to the controls? Also, because the CI doesn't contain 1, that means their results are statistically significant? Does the P-value represent the probability of the event occurring in the violent offending group?

The low-activity MAOA genotype was associated with violent offending in the crime cohort (odds ratio (OR) 1.71, P = 2.9 × 10 − 5; attributable risk 9%, 95% confidence interval (CI) 4–15%). This finding did not differ between males and females, and childhood maltreatment did not modify the risk (OR 1.62; Supplementary Tables 1a and 1b). The association was even stronger among extremely violent offenders (OR 2.66, P = 1.6 × 10 − 4, attributable fraction 16%, 95% CI 8–24%; Figure 1). In the cohort of homicide offenders (N = 96), the OR was slightly diluted (1.50, 95% CI 0.82– 2.72).

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Odds ratios are exactly that: ratios of odds. Odds are a bit strange to think about because they are not probabilities, and yet we are more used to thinking in probabilities. You cannot interpret an odds ratio as a multiplier to probability. For example, if you go from 30% to 60%, the odds go from 3/7 to 6/4, and the odds ratio is 3.5 (to get 3.5, you divide (6/4) by (3/7) ).

The confidence intervals there are actually for the attributable risk, which is defined as "Risk for Exposed - Risk for Unexposed", but yes, you can interpret that confidence interval [4% 15%] not including zero to mean that the difference is significant. If it was a confidence interval for the odds ratio then you are correct that you would look for whether the interval included 1 because a ratio of 1 means no effect.

p-values are another confusing topic. A p-value of 0.000029 means that if the actual effect was zero (that is, given the null hypothesis is true), you would expect to see a result as big or bigger than the result observed 0.0029% of the time. In other words, you'd be really unlikely to see results like these if there was no effect. This does not say anything about the probability of the event occurring in any group.

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To elaborate on Bryan's answer: how to interpret odds ratios.

First, think about what the odd is. Say, 13% of the individuals with the MAOA genotype are violent (incidence is 13%), whereas only 9% of individuals without the mutation are violent (incidence 9%). Then the odds for the MAOA genotype will be

$o_{MAOA}=\frac{0.13}{0.87}=0.15$

Basically, the odds will go from 0 to infinity, with $1$ meaning that both outcomes (violent / non violent) are equally likely.

We then look at the odds in the non-MAOA individuals:

$o_{ctrl}=\frac{0.09}{0.91}=0.01$

and finally, to calculate the OR,

$OR=\frac{o_{MAOA}}{o_{ctrl}}=\frac{0.15}{0.01}=15$

Another way of measuring this effect would be the slightly more intuitive Risk Ratio (RR) which is simply the ratio between the probability of violent behavior in the two genotypes:

$RR = \frac{0.13}{0.09}=1.44$

It is easy to see that this last value can be interpreted as "MAOA individuals are 1.44 more likely to be violent". With OR, such an easy interpretation is not possible.

For event probabilities close to 0 (like 1-2%), RR and OR are almost identical. The higher this probability, the more discrepant are OR and RR.

Bottom line, in most cases you cannot interpret $OR=2$ as a "twofold higher probability", unless the incidence is low (around 1%).

Here is an interesting paper on risk ratios and odds ratios.

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