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I have the following odds ratios for a person in a given age group having a certain condition:

Ages 50 and above: $OR =1$

Ages 40-49: $OR = 1.67$

Ages 30-39: $OR = 2.43$

Ages 20-29: $OR = 3.36$

If I also know that $43.5\%$ of people aged 20-50 have this condition, then how do I determine what proportion in each age group have the condition?

Stanley
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1 Answers1

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Let $p_1, p_2, p_3, p_4$ be the proportions having the condition in age group "50 or above", "40-49", "30-39", and "20-29" respectively. By condition, you can set up the following system: \begin{align} & \frac{p_2}{1 - p_2} = 1.67\frac{p_1}{1 - p_1} \\ & \frac{p_3}{1 - p_3} = 2.43\frac{p_1}{1 - p_1} \\ & \frac{p_4}{1 - p_4} = 3.36\frac{p_1}{1 - p_1} \\ & p_1 + p_2 + p_3 + p_4 = 0.435 \end{align} Solve this system for $p_1, \ldots, p_4$ to get the result.

Zhanxiong
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  • I do follow, but shouldn't the last line be a weighted average of the $p_i$'s $= .435$? (weighted by the size of the age groups) – Stanley Jun 19 '15 at 23:31
  • No, just think the whole population has $N$ individuals, then there would be $N_1 = N \times (p_1 + \cdots + p_4)$ people having the condition, hence the proportion of having condition is given by $N_1/N = p_1 + \cdots + p_4$. – Zhanxiong Jun 19 '15 at 23:35