TL;DR - those not breastfed till 9 are 2.00x more likely it's weird than those breastfed till 9.

• those not breastfed till 9 is 34.00x more likely it's not weird than those breastfed till 9.

• it's weird is 11.67x more likely to be those breastfed till nine than it's not weird.

• it's not weird is 1.46x more likely to be those not breastfed till nine than it's weird.

• it's weird given those breastfed till nine is 5.00x more likely than it's weird given those not breastfed till 9.

• it's not weird given those not breastfed till 9 is 3.40x more likely than it's not weird given those breastfed till 9.

r/TheyDidTheMath

Let’s say the prevalence or prior probabilities for those breastfed till 9 is 9.09% (odds of 0.10x or chances of 100 for every 1100), and for those not breastfed till 9 is 90.91% (10.00x or 100 for every 110), whether or not it's weird.

In a world of those breastfed till 9, 75.00% (3.00x or 100 for every 133) is it's weird, let’s say, and 25.00% (0.33x or 100 for every 400) is it's not weird.

In a world of those not breastfed till 9, 15.00% (0.18x or 100 for every 667) is it's weird, let’s say, and 85.00% (5.67x or 100 for every 118) is it's not weird.

Thus, those breastfed till 9 is 5.00x as likely it's weird as those not breastfed till 9. Also, those breastfed till 9 are 0.29x as likely it's not weird as those not breastfed till 9. We know this as the Likelihood Ratio, Risk Ratio, or Bayes Factor.

The Relative Risk Increase is 400.00%, and the Absolute Risk Increase is 60.00% (1.50x or 100 for every 167). The prevalence of it's weird, or it's not weird, regardless of those breastfed till nine or those not breastfed till 9, is 20.45% (0.26x or 100 for every 489), and 79.55% (3.89x or 100 for every 126), respectively.

Therefore, which is more likely? In a world of it's weird, the posterior probability of those breastfed till 9 is 33.33% (0.50x or 100 for every 300), and those not breastfed till 9 is 66.67% (2.00x or 100 for every 150).

In a world of it's not weird, the posterior probability of those breastfed till 9 is 2.86% (0.03x or 100 for every 3500), and those not breastfed till 9 is 97.14% (34.00x or 100 for every 103).

That's an Attributable Risk or Risk Difference of 30.48% (0.44x or 100 for every 328). The Accuracy Rate (that is, 'true-positive' and 'true-negative') is 84.09% (5.29x or 100 for every 119), and the Inaccuracy Rate (that is, 'false-positive' and 'false-negative') is 15.91% (0.19x or 100 for every 629).

The probability of those breastfed till 9, and it's weird is 6.82% (0.07x or 100 for every 1467). The likelihood of those breastfed till 9, and it's not weird is 2.27% (0.02x or 100 for every 4400). The probability of those not breastfed till 9, and it's weird is 13.64% (0.16x or 100 for every 733). The likelihood of those not breastfed till 9, and it's not weird is 77.27% (3.40x or 100 for every 129).

Sensitivity analysis:

What would the prevalence or prior probabilities for those breastfed till 9, and those not breastfed till 9, whether or not it's weird, need to be such that in a world where those breastfed till nine given it's weird, and those not breastfed till nine given it's weird, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of those breastfed till nine would need to be 16.67% (0.20x or 100 for every 600), and those not breastfed till nine would need to be 83.33% (5.00x or 100 for every 120), all else being equal.

Similarly, what would the prevalence or prior probabilities for those breastfed till nine and those not breastfed till 9, whether or not it's not weird, need to be such that in a world where those breastfed till nine given it's not weird, and those not breastfed till nine given it's not weird, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of those breastfed till nine would need to be 77.27% (3.40x or 100 for every 129), and those not breastfed till nine would need to be 22.73% (0.29x or 100 for every 440), all else being equal.

What would the consequent probabilities or likelihoods for it's weird given those not breastfed till 9, and it's not weird given those not breastfed till 9, need to be such that in a world where those breastfed till nine given it's weird, and those not breastfed till nine given it's weird, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of it's weird given those not breastfed till nine would need to be 7.50% (0.08x or 100 for every 1333), and it's not weird given those not breastfed till nine would need to be 92.50% (12.33x or 100 for every 108), all else being equal.

Similarly, what would the consequent probabilities or likelihoods for it's weird given those not breastfed till 9, and it's not weird given those not breastfed till 9, need to be such that in a world where those breastfed till nine given it's not weird, and those not breastfed till nine given it's not weird, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of it's weird given those not breastfed till nine would need to be 97.50% (39.00x or 100 for every 103), and it's not weird given those not breastfed till nine would need to be 2.50% (0.03x or 100 for every 4000), all else being equal.

Using the Wald test, the relationship or association between those breastfed till nine or those not breastfed till 9, and it's weird is statistically significant (n=10,000). Odds Ratio (OR) = 17.00, p < .001, 95% Confidence Interval (CI) [14.48, 19.97]. We can say the same between those breastfed till nine or those not breastfed till 9, and it's not weird. OR = 0.06, p < .001, 95% CI [0.05, 0.07].

Bayes’ Rule: P(H|E) = P(H) x P(E|H) / P(H) x P(E|H) + P(H’) x P(E|H’), where:

H = those breastfed till 9

H’ = those not breastfed till 9

E = it's weird

E’ = it's not weird

Contingency Table: