Neyman, J. (1934), "On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection", Journal of the Royal Statistical Society 97(4), 558-625; JSTOR 2342192. Pages 579 and 580 were rendered as images with pdftoppm and read visually, because the equations are load-bearing.
Arguments
- y
per-stratum sample means ybar_h, one per stratum, or NULL. When supplied the stratified estimate of the population total sum_h N_h ybar_h is returned as total; it plays no part in the allocation, which by eq. (39) depends only on N_h and S_h.
- N_h
stratum sizes M_i, all positive.
- S_h
within-stratum standard deviations S_i, divisor M_i - 1.
- n
total number of units to allocate, m_0 of eq. (38).
Value
list: estimate, allocation, allocation_int, variance, variance_prop, A, B, C, total, n, method.
Details
p. 579, eq. (37), the variance of the estimate sum(M_i ubar_i) of the population total under stratified random sampling, is sigma^2 = sum_i { (M_i^2 / m_i) ((M_i - m_i) / (M_i - 1)) sigma_i^2 }.
p. 580 puts S_i^2 = M_i sigma_i^2 / (M_i - 1) and rewrites (37) as eq. (39), whose three terms the page names A, B and -C: A = ((M_0 - m_0) / m_0) sum_i M_i S_i^2; B = sum_i m_i ( M_i S_i / m_i - sum_j M_j S_j / m_0 )^2; C = (M_0 / m_0) sum_i M_i ( S_i - sum_j M_j S_j / M_0 )^2, with M_0 = sum_i M_i, m_0 = sum_i m_i and sigma^2 = A + B - C. Only B depends on the individual m_i. B is a sum of squares, so it is >= 0, and it vanishes exactly when m_i = m_0 M_i S_i / sum_j M_j S_j, which is Neyman's optimum allocation; there sigma^2 = A - C, the page's eq. (41). Under Bowley's proportional allocation m_i = m_0 M_i / M_0 instead, B = C identically and sigma^2 = A, the page's eq. (40).
ERRATUM in the original. Equation (40) on p. 580 is printed as "sigma^2 = ((M_0 - M_0) / m_0) sum(M_i S_i^2) = A". The second M_0 must be m_0: A is defined three lines above as the first term of (39), and as printed the leading factor is identically zero, which would make the variance under proportional allocation vanish for every population. This was read off the rendered page image, so it is a misprint in the 1934 typesetting and not a pdftotext artifact.
Argument convention. In eq. (37) sigma_i^2 is the within-stratum variance taken with divisor M_i; S_i^2 = M_i sigma_i^2 / (M_i - 1) is the same variance with divisor M_i - 1, which is the form survey practice reports. S_h below is S_i, so eq. (37) with sigma_i^2 eliminated is sigma^2 = sum_i M_i (M_i - m_i) S_i^2 / m_i, and that is what the variance element returns.