3.5 Specific and crude rates
Rates are usually computed separately within categories of age and sex, because these affect almost every disease. A rate for a whole population, ignoring such categories, is a crude rate; a rate within a category is a specific rate (age-specific, sex-specific).
Proposition 3.5.1. If a population is divided into strata \(j = 1, \dots , J\) with person-time \(T_j\) and specific rates \(I_j\), its crude rate is \[ I_{\text {crude}} = \sum _j w_j I_j, \qquad w_j = \frac {T_j}{\sum _k T_k} , \] a weighted average of the specific rates with weights equal to the population’s own distribution of person-time across the strata.
Proof. The crude rate is \(\sum _j D_j/\sum _k T_k\), where \(D_j = I_j T_j\) is the number of cases in stratum \(j\). Substituting gives \(\sum _j I_j T_j/\sum _k T_k\). □
So two populations with identical specific rates can have very different crude rates if their age structures differ, and comparing crude rates compares age structures as much as risk.
Example 3.5.2. Two towns have the same death rates: \(2\) per \(1000\) per year under age \(50\) and \(20\) per \(1000\) at \(50\) and over. Town A has \(80{,}000\) people under \(50\) and \(20{,}000\) over; town B, a retirement town, has \(40{,}000\) under and \(60{,}000\) over. The deaths in a year are \[ \text {A}: 160 + 400 = 560, \qquad \text {B}: 80 + 1200 = 1280 , \] crude rates of \(5.6\) and \(12.8\) per \(1000\). Town B’s crude rate is more than twice town A’s although no person in it is at greater risk than the same person in A. Removing the effect of age structure from such comparisons is the subject of Section 8.
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