In a stratified survey, when does Neyman allocation beat proportional allocation?

## The allocation problem Stratification asks two separate questions. First, *what are the strata* — how do you partition the frame. Second, *how big is each stratum's sample* — allocation. The first is about which variables separate the outcome; the second is a pure optimisation, and it has clean answers. Write `N_h` for the population count in stratum `h`, `W_h = N_h / N` for its share, `S_h` for the standard deviation of the outcome *within* that stratum, and `n_h` for the number of units you sample there, with `sum of n_h = n` fixed. ## Proportional allocation Set `n_h = n * W_h`. Each stratum gets its population share of the sample. Two properties follow, and both matter operationally more than they look on paper. The design is **self-weighting**: every unit's inclusion probability is `n_h / N_h = n / N`, the same everywhere. So the plain unweighted sample mean is already the correct estimate of the population mean, and every analyst who touches the table afterwards gets the right answer without knowing the design existed. And it is **robust across outcomes**. A single survey usually measures many things — spend, satisfaction, feature usage. Proportional allocation is not optimal for any of them, but it is never badly wrong for any of them either. ## Neyman allocation Set `n_h` proportional to `N_h * S_h`. This is the allocation that minimises the variance of the stratified mean estimator subject to a fixed total `n`. The intuition is direct: sampling effort should go where two things coincide — a lot of population, and a lot of disagreement inside it. A stratum where everyone answers nearly the same number is cheap to pin down with a handful of interviews; a stratum whose answers span orders of magnitude needs many. A worked case. A customer base has 90,000 free accounts with a within-stratum spend standard deviation of about 5, and 10,000 enterprise accounts with a standard deviation of about 50. Total budget: 1,000 interviews. - Proportional: 900 free, 100 enterprise. - Neyman: the products `N_h * S_h` are `90,000 * 5 = 450,000` and `10,000 * 50 = 500,000`, summing to 950,000. So free gets `1000 * 450/950` which is about 474, and enterprise about 526. The enterprise tier is 10% of the population and receives more than half the sample. That is not a mistake — it is where the uncertainty about the population total lives. ## Optimal allocation with costs If a response costs `c_h` in stratum `h` and the constraint is a budget rather than a headcount, the variance-minimising rule becomes `n_h` proportional to `N_h * S_h / sqrt(c_h)`. Expensive-to-reach strata are sampled less than their variability alone would justify. Neyman allocation is exactly the special case where every stratum costs the same. ## What Neyman allocation costs you **You must guess `S_h` in advance.** The optimum depends on a quantity you are running the survey to learn. In practice you use a pilot, last year's survey, or an operational proxy such as the spread of billing records. A wrong guess costs precision, not correctness: as long as you use the true inclusion probabilities as weights, the estimator remains unbiased — you have simply spent interviews in the wrong places and landed somewhere between the optimum and proportional. **The sample is no longer self-weighting.** Inclusion probabilities now differ by stratum, so the plain average of the responses estimates the wrong thing, and every downstream calculation must apply weights. This is a real organisational cost, not a formality. **It is optimal for one variable at a time.** The `S_h` that minimises variance for spend is not the one that minimises it for satisfaction. Surveys measuring many outcomes usually settle for a compromise — an allocation near proportional, nudged toward the more variable strata. ## Domain estimates change the calculus Allocation rules above optimise the *overall* estimate. If you also need to report each stratum separately, precision inside a stratum depends on that stratum's own sample size, and a rule that hands a 200-account segment four interviews is useless. Practical designs therefore impose a floor: a minimum sample per reportable stratum, then allocate what is left by proportional or Neyman logic, then weight everything back together. ## What an interviewer is listening for That you can state both rules without hesitation; that you know Neyman needs `S_h`, not the stratum *mean*; that you catch the variance-versus-standard-deviation confusion (an allocation proportional to the *variance* is wrong — it is the standard deviation that enters); and that you volunteer the weighting obligation that a disproportionate allocation creates rather than being reminded of it.