What does direct standardisation of a mortality rate to a fixed age mix let you compare?

## Crude rates carry the population's own shape A **crude rate** is total events divided by total population — deaths per 1,000 people, say. Written out over strata it is a weighted average: ``` crude = sum over strata i of w_i * r_i ``` where `r_i` is the stratum-specific rate (deaths per 1,000 among people aged 65-74, for instance) and `w_i` is that stratum's share of the population. The `w_i` are a property of the population, not of its health. So a crude rate answers "how many people died here", which is the right question for planning hospital beds, and a very poor question for "is it more dangerous here". ## A worked reversal | | share young | rate young | share old | rate old | crude | |---|---|---|---|---|---| | Country A | 0.60 | 2 / 1,000 | 0.40 | 20 / 1,000 | 9.2 / 1,000 | | Country B | 0.85 | 3 / 1,000 | 0.15 | 25 / 1,000 | 6.3 / 1,000 | Country A's crude mortality (9.2) is far higher than Country B's (6.3). Yet A's age-specific rates are **lower in both bands**: 2 against 3 among the young, 20 against 25 among the old. A simply has a much older population, and old age carries a high rate everywhere. Standardise both to a 50/50 age split: ``` A: 0.5*2 + 0.5*20 = 11.0 per 1,000 B: 0.5*3 + 0.5*25 = 14.0 per 1,000 ``` The comparison reverses, and now it means what people thought the crude rate meant. ## The mechanics of direct standardisation Pick one **standard population** — a published reference age distribution, or the pooled population of the groups you are comparing. Then for each group compute ``` standardised rate = sum over strata of w_std_i * r_i(group) ``` Every group is scored with the same weights and its own rates. Two consequences follow directly: - The difference between two standardised rates is driven **only** by differences in stratum-specific rates. - The number is counterfactual. It is what the group's death rate would be if it had the standard population's age structure. Nobody experienced it, so it must never appear in a sentence about how many people died, or in a capacity plan. ## The choice of standard matters Because the standardised rate is a weighted average of the group's rates, changing the weights changes the number. Usually it only rescales, and the ordering is stable. But if the two groups' age-specific rates **cross** — one better among the young, worse among the old — the ordering itself can flip with the standard chosen. Suppose C has rates 1 (young) and 30 (old), and D has 3 and 20. Against a young standard of 90/10: ``` C: 0.9*1 + 0.1*30 = 3.9 D: 0.9*3 + 0.1*20 = 4.7 C looks better ``` Against an old standard of 20/80: ``` C: 0.2*1 + 0.8*30 = 24.2 D: 0.2*3 + 0.8*20 = 16.6 D looks better ``` Same data, opposite conclusion. This is why the standard must be fixed **before** the comparison, documented, and kept identical across every group and every period in the report. Choosing it afterwards is a way of choosing the answer. ## Indirect standardisation When a group's stratum-specific rates are unstable — small strata, few events — direct standardisation amplifies that noise, because it multiplies a shaky rate by a large standard weight. **Indirect** standardisation reverses the roles: apply a set of standard stratum-specific rates to the group's own population structure to get expected events, then form ``` SMR = observed events / expected events ``` A value above 1 means more events than the standard rates would predict for a population of that shape. The trade-off is that SMRs computed against the same standard are not strictly comparable **with each other**, because each uses its own population's weights. ## Limits worth stating - Standardisation neutralises exactly the variable you stratified on. An age-standardised rate says nothing about differences in sex composition, income, altitude or reporting completeness. - You can standardise on several variables by cross-classifying strata (age by sex), but cells thin out quickly and the stratum rates become unstable — the practical ceiling arrives fast. - The stratum boundaries are a choice. Very wide age bands leave composition differences inside a band, which the method cannot remove. - If the two populations barely overlap in some stratum — one has almost nobody over 80 — its standardised rate rests on a rate estimated from a handful of people. ## What interviewers are checking That you can write the crude rate as a weighted average and see the weights as the culprit; that you know the standardised figure is a comparison device rather than a real rate; that you would fix the standard population in advance; and that you can name indirect standardisation and when it is preferred.