Selection for significance.

selection_for_significance.ipynb

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    "# Quantile function under selection for significance"
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    "Assume we conduct a study with an unknown effect $\\theta$, and let\n",
    "$Z$ be an estimator of $\\theta$ with known mean, distributed according\n",
    "to the usual $N\\left(\\theta,n^{-1}\\right)$. Under no selection for\n",
    "significance, $\\sqrt{n}Z\\sim\\varphi\\left(\\sqrt{n}\\theta\\right)$,\n",
    "while under selection for significance (with a one-sided test), $\\sqrt{n}Z\\sim\\varphi\\left(\\sqrt{n}\\theta\\right)\\mid_{\\geq c_{\\alpha}}$,\n",
    "where $c_{\\alpha}$ is some critical value and ``$|_{\\geq c_{\\alpha}}$''\n",
    "denotes a truncated distribution. An example is $c_{\\alpha}=\\Phi\\left(1-\\alpha\\right)$\n",
    "for $\\alpha=0.05$, which yields the critical value for a test for\n",
    "a postive effect at the $0.05$ level. Anyhow, the quantile function\n",
    "of the truncated $Z$ is given by\n",
    "\n",
    "$$\n",
    "Q\\left(p;n,\\theta,\\alpha\\right)=\\theta+\\frac{1}{\\sqrt{n}}\\left(p\\left(1-\\Phi\\left(\\sqrt{n}\\theta-c_{\\alpha}\\right)\\right)+\\Phi\\left(c_{\\alpha}\\right)\\right).\n",
    "$$\n",
    "This is the case since the quantile function of a normal with mean\n",
    "$\\sqrt{n}\\theta$ is $\\sqrt{n}\\theta+\\Phi^{-1}\\left(p\\right)$, while\n",
    "the quantile function of a lower truncated distribution at $c_{\\alpha}$\n",
    "is given by the solution to $p=\\int_{c_{\\alpha}}^{q}\\frac{f\\left(x\\right)}{1-F\\left(c_{\\alpha}\\right)}dx,$\n",
    "so that \n",
    "$$\n",
    "p\\left(1-F\\left(c_{\\alpha}\\right)\\right)=\\int_{-\\infty}^{q}f\\left(x\\right)dx-\\int_{\\infty}^{c_{\\alpha}}f\\left(x\\right)dx.\n",
    "$$\n",
    "Rearrange to obtain $p\\left(1-F\\left(c_{\\alpha}\\right)\\right)+F\\left(c_{\\alpha}\\right)=\\int_{-\\infty}^{q}f\\left(x\\right)dx,$\n",
    "which is the defining equation of the function $\\sqrt{n}\\theta+\\Phi^{-1}\\left(p\\left(1-F\\left(c_{\\alpha}\\right)\\right)+F\\left(c_{\\alpha}\\right)\\right)$.\n"
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