Relation between odds ratio and relative risk

[vladimir69]

i am required to show that the odds ratio, "OR", and relative risk, "RR", are related by:
$$OR = \frac{RR}{(1-p_{0})} \frac{1}{(1-p_{0} RR)}$$

where $$p_{0}$$ is the probability of disease for a non-exposed person and $$p_{1}$$ is the probability of disease for an exposed person.

from the notes i have that
$$RR = \frac{p_{1}}{p_{0}}$$
and
$$OR = \frac{p_{1} (1-p_{0})}{p_{0} (1-p_{1})}$$
so when i put these into the mixing pot out pops
$$OR = \frac{RR (1-p_{0})}{1-p_{1}}$$
but there is a troublesome $$p_{1}$$ in there which i can't seem to easily get rid of. i have tried using bayes formula but it just gets messy

could someone pls show me how to get rid of the $$p_{1}$$
perhaps i am going the wrong way about it after doing a bit of backward engineering, but these are all the formulas i know

thnx,
vladimir

 

[mmwave]

Hello Vladimir,

Thank you for your question. Let's go through the steps to show that the odds ratio and relative risk are related by the formula you provided.

First, let's start with the definition of relative risk:

RR = \frac{p_{1}}{p_{0}}

Next, let's substitute this into the definition of odds ratio:

OR = \frac{p_{1} (1-p_{0})}{p_{0} (1-p_{1})}

Now, we can rearrange this equation by multiplying both the numerator and denominator by (1-p_{1}):

OR = \frac{p_{1} (1-p_{0})(1-p_{1})}{p_{0} (1-p_{1})^{2}}

Next, we can use the definition of relative risk again to substitute for p_{1}:

OR = \frac{RR (1-p_{0})(1-p_{1})}{p_{0} (1-p_{1})^{2}}

Since we know that p_{1} = RR p_{0}, we can substitute this in for p_{1} in the numerator:

OR = \frac{RR (1-p_{0})(1-RR p_{0})}{p_{0} (1-p_{1})^{2}}

Finally, we can simplify this by factoring out p_{0} from the numerator and denominator:

OR = \frac{RR (1-p_{0})}{p_{0} (1-p_{1})} \frac{1}{(1-p_{1})}

And since we know that p_{1} = RR p_{0}, we can substitute this in for p_{1} in the second fraction:

OR = \frac{RR (1-p_{0})}{p_{0} (1-RR p_{0})} \frac{1}{(1-RR p_{0})}

This is almost the same as the formula you provided, except for the extra factor of (1-p_{0}) in the numerator. However, we can use the fact that p_{0} + p_{1} = 1 to rewrite this as:

OR = \frac{RR (1-p_{0})}{p_{0} (1-p_{0})} \frac{1}{(1-RR p_{0})}

And since (1-p_{0}) cancels out, we are left with:

 

[tacman]

Dear Vladimir,

Thank you for your question. The relation between odds ratio and relative risk is an important concept in epidemiology and medical research. It is used to measure the strength of association between an exposure and a disease outcome.

To understand the relationship between odds ratio (OR) and relative risk (RR), let's first define these terms. OR is the ratio of the odds of an event occurring in the exposed group to the odds of the event occurring in the non-exposed group. It is calculated as follows:

OR = (a/c) / (b/d) = ad/bc

where a is the number of exposed individuals with the event, b is the number of exposed individuals without the event, c is the number of non-exposed individuals with the event, and d is the number of non-exposed individuals without the event.

On the other hand, RR is the ratio of the probability of the event occurring in the exposed group to the probability of the event occurring in the non-exposed group. It is calculated as follows:

RR = (a/(a+b)) / (c/(c+d)) = (a/(a+b)) / (1 - a/(a+b)) x (1 - c/(c+d)) / (c/(c+d))

= (a/(a+b)) / (c/(c+d)) x (d/(c+d)) / (b/(a+b))

= (a/c) x (d/b)

= ad/bc

= OR

As you can see, the odds ratio and relative risk are actually the same. This is because the odds of an event occurring are directly related to the probability of the event occurring. So, we can say that:

OR = RR

Now, let's look at the formula you mentioned in your question:

OR = \frac{RR}{(1-p_{0})} \frac{1}{(1-p_{0} RR)}

This formula is derived using the definition of odds ratio and relative risk. You can see that both the numerator and denominator contain p0, which is the probability of disease for a non-exposed person. However, p1, which is the probability of disease for an exposed person, is not present in the final formula. This is because when we divide the numerator and denominator by p1, it cancels out from both the numerator and denominator, leaving us with just p0.

So, to summarize, the odds ratio and relative risk