{
  "slug": "length-biased-sampling",
  "term": "Length-biased sampling",
  "aliases": [
    "Length bias"
  ],
  "definition": "Length-biased sampling selects units with probability proportional to the duration being measured.",
  "note": "A cross-sectional sample of operating devices can overrepresent long lifetimes because those devices remain available for selection longer. Under stationary Poisson entry, a stable lifetime distribution and uniform selection among operating devices, the sampled full-lifetime density is t f(t) / E(T), for positive T with finite mean. This is a specific sampling model, not a property of every fleet snapshot. [chan-2012]\n\nThe mean full lifetime in that sample is E(T^2) / E(T) when the second moment is finite. The difference from the entrant mean is Var(T) / E(T), zero for identical lifetimes. It includes time before and after selection: it is neither current age nor remaining life. [wrigley-field-2022]\n\nSelection after survival to entry creates left-truncation; length bias is a special case. Right-censoring after selection is a separate observation limit and can complicate inference. Fleet growth, changing model mix and early retirement require a justified sampling model, not an automatic duration-weight correction. [chan-2012]",
  "status": "shipping",
  "advantages": [],
  "limitations": [],
  "related": [
    "left-truncation",
    "right-censoring",
    "survival-function"
  ],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "citations": [
    {
      "id": "chan-2012",
      "title": "Chan, Chen and Di (2012): Proportional mean residual life model for right-censored length-biased data, sections 1-2",
      "url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC3635658/",
      "accessed": "2026-11-05"
    },
    {
      "id": "wrigley-field-2022",
      "title": "Wrigley-Field and Feehan (2022): In a stationary population, the average lifespan of the living is a length-biased life expectancy, equations 5-6",
      "url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC8810607/",
      "accessed": "2026-11-05"
    },
    {
      "id": "hesela-age-audit",
      "title": "Hesela: Missing SMART ages and the limits of drive-age comparisons",
      "url": "https://hesela.com/analyses/backblaze-age-missing-data-audit/#length-biased-sampling",
      "accessed": "2026-11-05"
    }
  ],
  "provenance": [
    "Statistical sources establish a sampling mechanism, not an empirical estimate of bias in a storage fleet.",
    "The linked worked example uses fictional stationary arrivals and fixed lifetimes; it is not Backblaze data."
  ],
  "updated": "2026-11-05"
}