{
  "alt": "Line chart of the ratio of the pooled rate to the true rate against the expected number of deaths in a county-year. The line starts near zero, climbs steeply to a peak at a few expected deaths, then declines toward one and stays near one for larger expected counts. The second view shows the share of county-years published, falling to a minimum at a few expected deaths and rising to nearly all county-years as the expected count grows.",
  "caption": "The curve is below one when a county expects only a few deaths, because nearly every published county-year is a zero, crosses one near {fact:m_cross} expected deaths, and peaks where only {fact:m_pub_peak} of county-years are published and nearly all of them have a high count. It returns to within {fact:lam_tol} of one beyond {fact:m_flat} expected deaths. The second view plots the share of county-years that WONDER publishes. This is a calculation from the rule, not an estimate from data.",
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    {
      "lam": 21.3,
      "published": 99.769,
      "ratio": 1.0014
    },
    {
      "lam": 21.4,
      "published": 99.783,
      "ratio": 1.0013
    },
    {
      "lam": 21.5,
      "published": 99.796,
      "ratio": 1.0012
    },
    {
      "lam": 21.6,
      "published": 99.808,
      "ratio": 1.0012
    },
    {
      "lam": 21.7,
      "published": 99.819,
      "ratio": 1.0011
    },
    {
      "lam": 21.8,
      "published": 99.83,
      "ratio": 1.001
    },
    {
      "lam": 21.9,
      "published": 99.84,
      "ratio": 1.001
    },
    {
      "lam": 22,
      "published": 99.85,
      "ratio": 1.0009
    },
    {
      "lam": 22.1,
      "published": 99.859,
      "ratio": 1.0009
    },
    {
      "lam": 22.2,
      "published": 99.867,
      "ratio": 1.0008
    },
    {
      "lam": 22.3,
      "published": 99.875,
      "ratio": 1.0008
    },
    {
      "lam": 22.4,
      "published": 99.883,
      "ratio": 1.0007
    },
    {
      "lam": 22.5,
      "published": 99.89,
      "ratio": 1.0007
    },
    {
      "lam": 22.6,
      "published": 99.896,
      "ratio": 1.0006
    },
    {
      "lam": 22.7,
      "published": 99.903,
      "ratio": 1.0006
    },
    {
      "lam": 22.8,
      "published": 99.909,
      "ratio": 1.0006
    },
    {
      "lam": 22.9,
      "published": 99.914,
      "ratio": 1.0005
    },
    {
      "lam": 23,
      "published": 99.919,
      "ratio": 1.0005
    },
    {
      "lam": 23.1,
      "published": 99.924,
      "ratio": 1.0005
    },
    {
      "lam": 23.2,
      "published": 99.929,
      "ratio": 1.0005
    },
    {
      "lam": 23.3,
      "published": 99.933,
      "ratio": 1.0004
    },
    {
      "lam": 23.4,
      "published": 99.937,
      "ratio": 1.0004
    },
    {
      "lam": 23.5,
      "published": 99.941,
      "ratio": 1.0004
    },
    {
      "lam": 23.6,
      "published": 99.945,
      "ratio": 1.0004
    },
    {
      "lam": 23.7,
      "published": 99.948,
      "ratio": 1.0003
    },
    {
      "lam": 23.8,
      "published": 99.952,
      "ratio": 1.0003
    },
    {
      "lam": 23.9,
      "published": 99.955,
      "ratio": 1.0003
    },
    {
      "lam": 24,
      "published": 99.957,
      "ratio": 1.0003
    },
    {
      "lam": 24.1,
      "published": 99.96,
      "ratio": 1.0003
    },
    {
      "lam": 24.2,
      "published": 99.963,
      "ratio": 1.0002
    },
    {
      "lam": 24.3,
      "published": 99.965,
      "ratio": 1.0002
    },
    {
      "lam": 24.4,
      "published": 99.967,
      "ratio": 1.0002
    },
    {
      "lam": 24.5,
      "published": 99.969,
      "ratio": 1.0002
    },
    {
      "lam": 24.6,
      "published": 99.971,
      "ratio": 1.0002
    },
    {
      "lam": 24.7,
      "published": 99.973,
      "ratio": 1.0002
    },
    {
      "lam": 24.8,
      "published": 99.975,
      "ratio": 1.0002
    },
    {
      "lam": 24.9,
      "published": 99.976,
      "ratio": 1.0002
    },
    {
      "lam": 25,
      "published": 99.978,
      "ratio": 1.0001
    },
    {
      "lam": 25.1,
      "published": 99.979,
      "ratio": 1.0001
    },
    {
      "lam": 25.2,
      "published": 99.981,
      "ratio": 1.0001
    },
    {
      "lam": 25.3,
      "published": 99.982,
      "ratio": 1.0001
    },
    {
      "lam": 25.4,
      "published": 99.983,
      "ratio": 1.0001
    },
    {
      "lam": 25.5,
      "published": 99.984,
      "ratio": 1.0001
    },
    {
      "lam": 25.6,
      "published": 99.985,
      "ratio": 1.0001
    },
    {
      "lam": 25.7,
      "published": 99.986,
      "ratio": 1.0001
    },
    {
      "lam": 25.8,
      "published": 99.987,
      "ratio": 1.0001
    },
    {
      "lam": 25.9,
      "published": 99.988,
      "ratio": 1.0001
    },
    {
      "lam": 26,
      "published": 99.989,
      "ratio": 1.0001
    },
    {
      "lam": 26.1,
      "published": 99.989,
      "ratio": 1.0001
    },
    {
      "lam": 26.2,
      "published": 99.99,
      "ratio": 1.0001
    },
    {
      "lam": 26.3,
      "published": 99.991,
      "ratio": 1.0001
    },
    {
      "lam": 26.4,
      "published": 99.991,
      "ratio": 1.0001
    },
    {
      "lam": 26.5,
      "published": 99.992,
      "ratio": 1.0001
    },
    {
      "lam": 26.6,
      "published": 99.992,
      "ratio": 1.0001
    },
    {
      "lam": 26.7,
      "published": 99.993,
      "ratio": 1
    },
    {
      "lam": 26.8,
      "published": 99.993,
      "ratio": 1
    },
    {
      "lam": 26.9,
      "published": 99.994,
      "ratio": 1
    },
    {
      "lam": 27,
      "published": 99.994,
      "ratio": 1
    },
    {
      "lam": 27.1,
      "published": 99.995,
      "ratio": 1
    },
    {
      "lam": 27.2,
      "published": 99.995,
      "ratio": 1
    },
    {
      "lam": 27.3,
      "published": 99.995,
      "ratio": 1
    },
    {
      "lam": 27.4,
      "published": 99.996,
      "ratio": 1
    },
    {
      "lam": 27.5,
      "published": 99.996,
      "ratio": 1
    },
    {
      "lam": 27.6,
      "published": 99.996,
      "ratio": 1
    },
    {
      "lam": 27.7,
      "published": 99.996,
      "ratio": 1
    },
    {
      "lam": 27.8,
      "published": 99.997,
      "ratio": 1
    },
    {
      "lam": 27.9,
      "published": 99.997,
      "ratio": 1
    },
    {
      "lam": 28,
      "published": 99.997,
      "ratio": 1
    },
    {
      "lam": 28.1,
      "published": 99.997,
      "ratio": 1
    },
    {
      "lam": 28.2,
      "published": 99.997,
      "ratio": 1
    },
    {
      "lam": 28.3,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 28.4,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 28.5,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 28.6,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 28.7,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 28.8,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 28.9,
      "published": 99.998,
      "ratio": 1
    },
    {
      "lam": 29,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.1,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.2,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.3,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.4,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.5,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.6,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.7,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.8,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 29.9,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 30,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 30.1,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 30.2,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 30.3,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 30.4,
      "published": 99.999,
      "ratio": 1
    },
    {
      "lam": 30.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 30.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 30.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 30.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 30.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 31.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 32.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 33.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 34.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 35.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 36.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 37.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 38.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.1,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.2,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.3,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.4,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.5,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.6,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.7,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.8,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 39.9,
      "published": 100,
      "ratio": 1
    },
    {
      "lam": 40,
      "published": 100,
      "ratio": 1
    }
  ],
  "id": "bias_curve",
  "notes": [
    "A county-year with a given expected number of deaths is published when its Poisson count is zero or at least {fact:w_floor}. The ratio is the average death count in published county-years divided by the average count in all county-years, which is the ratio of the pooled rate to the true rate when counties share a population.",
    "The model assumes a constant rate in each county, independent years and Poisson counts. Real counts vary more than Poisson counts, so the curve gives direction and rough size, and it is not used to correct any estimate in this book."
  ],
  "source": "Calculation from the publication rule of the CDC WONDER county files, 2018-2024, applied to Poisson counts; full counts of the model, no sampling error, equal weight for every county-year",
  "spec": {
    "annotations": [
      {
        "label": "True rate",
        "type": "y-rule",
        "y": 1
      },
      {
        "label": "Peak",
        "type": "point",
        "x": 5.4,
        "y": 1.8274
      }
    ],
    "bandSeries": [],
    "colors": {},
    "endLabels": false,
    "x": "lam",
    "xDomain": [
      0,
      40
    ],
    "xLabel": "Expected deaths in a county-year",
    "xTicks": [
      0,
      5,
      10,
      15,
      20,
      25,
      30,
      35,
      40
    ],
    "y": "ratio",
    "yDomain": [
      0,
      2.2
    ],
    "yFormat": {
      "digits": 2
    },
    "yLabel": "Pooled rate over published county-years relative to the true rate (1 = unbiased)"
  },
  "subtitle": "Expected pooled rate over the county-years WONDER publishes (counts of zero and of at least 10) divided by the true rate, against the expected number of deaths in a county-year, Poisson counts and a constant rate",
  "title": "A rate pooled over published county-years is {fact:m_peak} times the true rate at {fact:m_peak_at} expected deaths a year, below it under {fact:m_cross} and within {fact:lam_tol} of it above {fact:m_flat}",
  "type": "line",
  "views": [
    {
      "id": "ratio",
      "label": "Bias of the pooled rate"
    },
    {
      "caption": "The share of county-years that WONDER publishes is smallest at a few expected deaths, where a county-year rarely has zero deaths and rarely reaches the limit, and rises to nearly all county-years as the expected count grows.",
      "id": "published",
      "label": "Share of county-years published",
      "spec": {
        "annotations": [
          {
            "label": "Peak of the bias",
            "type": "x-rule",
            "x": 5.4
          }
        ],
        "y": "published",
        "yDomain": [
          0,
          100
        ],
        "yLabel": "County-years published (%)"
      },
      "title": "WONDER publishes only {fact:m_pub_peak} of county-years at {fact:m_peak_at} expected deaths, and most county-years once a county expects more than {fact:m_flat}"
    }
  ],
  "width": "wide"
}