{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $x_1, \\dots, x_n \\in \\R^p$ be known regressor vectors, and let $\\Phi$ be the [[cumulative distribution function|0H8H7ZTHW2]] of the standard [[normal distribution|05AQE0QR0D]].\nThe *probit model* for responses $y_1, \\dots, y_n \\in \\set{0, 1}$ states three things.\nThe $y_i$ are independent, $y_i$ has the [[Bernoulli distribution|0386Y0Z7Y7]] with parameter $p_i$, and\n$$p_i = \\Phi(x_i^T \\beta)$$\nfor an unknown $\\beta \\in \\R^p$.\n\n## Remarks\n\nEquivalent latent variable form: let $y_i^* = x_i^T \\beta + \\varepsilon_i$ with independent $\\varepsilon_i \\sim N(0, 1)$, and $y_i = 1$ if $y_i^* \u003e 0$ and $y_i = 0$ otherwise.\nThen $P(y_i = 1) = P(\\varepsilon_i \u003e -x_i^T \\beta) = \\Phi(x_i^T \\beta)$, by the symmetry of the normal distribution.\nThe error variance is fixed at $1$ because $\\beta$ and the scale of $\\varepsilon_i$ cannot be identified separately from binary data.\n\nIt is the [[generalized linear model|K6C48KMRKC]] with Bernoulli responses and link $g = \\Phi^{-1}$, which is not the canonical link.\n\nExample: with $x_i = (1, t_i)$ and $\\beta = (-1, 0.5)$, at $t = 2$ one gets $p = \\Phi(0) = 1/2$.\n\nNon-example: the model $p_i = 1/(1 + e^{-x_i^T \\beta})$ uses the logistic function instead of $\\Phi$; it is [[logistic regression|JN1PJ5AJJS]], not the probit model.\nIts latent errors are logistic, with variance $\\pi^2/3$, so its coefficients are typically larger than probit coefficients by a factor of roughly $1.6$ to $1.8$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "XCR3V8SJ01",
    "kind": "definition",
    "msc": [
      "62J12",
      "62P20"
    ],
    "name": "Probit model",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Probit_model"
      }
    ],
    "tag": "GR4FN6XNWP",
    "uses": [
      "K6C48KMRKC",
      "0386Y0Z7Y7",
      "0H8H7ZTHW2",
      "05AQE0QR0D",
      "JN1PJ5AJJS"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/GR4FN6XNWP/",
  "links": [
    {
      "field": "defines",
      "target": "XCR3V8SJ01"
    },
    {
      "field": "uses",
      "target": "K6C48KMRKC"
    },
    {
      "field": "uses",
      "target": "0386Y0Z7Y7"
    },
    {
      "field": "uses",
      "target": "0H8H7ZTHW2"
    },
    {
      "field": "uses",
      "target": "05AQE0QR0D"
    },
    {
      "field": "uses",
      "target": "JN1PJ5AJJS"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GR4FN6XNWP.md",
  "msc": [
    "62J12",
    "62P20"
  ],
  "name": "Probit model",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GR4/GR4FN6XNWP.md",
  "status": null,
  "tag": "GR4FN6XNWP",
  "uri": "m2://GR4FN6XNWP",
  "url": "https://tamnd.github.io/math2/wiki/probit-model-definition/",
  "writing": {
    "tag": "GR4FN6XNWP",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}