I want to know among what is the ratio of insured people with disease $x$ to uninsured people with disease $x$. Here are my numbers.
- Insured, disease $x = 3600$
- Uninsured, disease $x = 950$
- Insured, no disease $x = 100$
- Uninsured, no disease $x = 150$
I’m struggling to understand if I should use $3600:950$ or if I should use $(3600/3700):(950/1100)$. Thoughts?
Let:
$d_I=3600$ be the number of people with the disease among the insured;
$d_U=950$ be the number of people with the disease among the uninsured;
$\overline{d_I}=100$ be the number of people without the disease among the insured;
$\overline{d_U}=150$ be the number of people with the disease among the uninsured.
The ratio of insured people with disease $x$ to uninsured people with disease $x$ is: $$ \frac{d_I}{d_U} = \frac{3600}{950} $$ It is the odds of finding someone insured among the diseased.
Other odds that might be of interest are:
The odds of finding someone insured among the non-diseased $\frac{\overline{d_I}}{\overline{d_U}}$;
The odds of finding someone with the disease among the insured $\frac{d_I}{\overline{d_I}}$;
The odds of finding someone with the disease among the uninsured $\frac{d_U}{\overline{d_U}}$.
And finally, we could also compute the odds ratio $OR$:
$$ OR = \frac{\frac{d_I}{d_U}}{\frac{\overline{d_I}}{\overline{d_U}}} = \frac{\frac{d_I}{\overline{d_I}}}{\frac{d_U}{\overline{d_U}}} = \frac{d_I \cdot \overline{d_U}}{\overline{d_I} \cdot d_U} = \frac{3600 \times 150}{100 \times 950} \approx 5.68 $$
Since $OR > 1$, it means that the odds of having the disease (vs not having the disease) among the people with insurance is larger than the odds of having (vs not having the disease) the disease among the people without insurance.
Your final expression is the relative risk $RR$, a concept similar to the $OR$ but related with risk (probability) instead of odds.
The risk of finding someone with the disease among the insured $\frac{d_I}{d_I + \overline{d_I}}$;
The risk of finding someone with the disease among the uninsured $\frac{d_U}{d_U + \overline{d_U}}$;
The risk of finding someone insured among the diseased $\frac{d_I}{d_I + d_U}$;
The risk of finding someone insured among the non-diseased $\frac{\overline{d_I}}{\overline{d_I} + \overline{d_U}}$.
The relative risk of having the disease among the insured vs having the disease among the uninsured is then: $$ RR_d = \frac{\frac{d_I}{d_I + \overline{d_I}}}{\frac{d_U}{d_U + \overline{d_U}}} = \frac{\frac{3600}{3700}}{\frac{950}{1100}} \approx 1.13 $$
Note that this is different to the relative risk of being insured among the diseased vs being insured among the non-diseased : $$ RR_I = \frac{\frac{d_I}{d_I + d_U}}{\frac{\overline{d_I}}{\overline{d_I} + \overline{d_U}}} = \frac{\frac{3600}{4550}}{\frac{100}{250}} \approx 1.98 $$