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   "source": [
    "### Example: Power calculations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eb55f8b5",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats\n",
    "import statsmodels.api as sm\n",
    "\n",
    "# import package for power calculations:\n",
    "import statsmodels.stats.power as smp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "557979a2",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We want to calculate the sample size needed in a new experiment\n",
    "\n",
    "# from previous experiments we have a good guess for the sample variation:\n",
    "sd = 1.62\n",
    "\n",
    "# we want to be able to detect a voltage drop of down to 0.5 volts\n",
    "delta = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "69b1ba56",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# we need some quantiles from the normal distribution (assuming the samples will be big enough for normal distribution assumption to apply)\n",
    "z_power = stats.norm.ppf(0.80, loc=0, scale = 1)\n",
    "z_signif = stats.norm.ppf(0.975, loc=0, scale = 1)\n",
    "\n",
    "print([z_power, z_signif])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9e727ac3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# use formula to calculate n_obs (needed number of observations):\n",
    "n_obs = (sd/delta*(z_power+z_signif))**2\n",
    "print(n_obs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a28cb940",
   "metadata": {},
   "source": [
    "So we need a sample of 21 observations "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c4486bde",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we can also use the python function TTestPower().solve_power (for one sample power calculations): \n",
    "print(smp.TTestPower().solve_power(effect_size=delta/sd, alpha=0.05, power=0.80))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87f57a34",
   "metadata": {},
   "source": [
    "i.e. 23 observations\n",
    "\n",
    "The Python function gives a larger estimate of needed sample size! The formula uses more assumptions than the Python version, so the result from the Python function is a little better/safer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d861fd86",
   "metadata": {},
   "outputs": [],
   "source": [
    "# What if we can only make 15 observations?\n",
    "n = 15\n",
    "\n",
    "# calculate the power:\n",
    "z_power_new = np.sqrt(n*(delta/sd)**2)-z_signif\n",
    "print(z_power_new)\n",
    "\n",
    "power = stats.norm.cdf(z_power_new)\n",
    "print(power)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0195ac82",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we can also use the python function TTestPower().solve_power (for one sample power calculations): \n",
    "print(smp.TTestPower().solve_power(effect_size=delta/sd, alpha=0.05, nobs=15))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39eff535",
   "metadata": {},
   "source": [
    "The result is almost the same (but the Python function is better/safer)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "958515f4",
   "metadata": {},
   "source": [
    "### Example: Power calculations, 2 samples"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae7c0f10",
   "metadata": {},
   "source": [
    "1)\n",
    "\n",
    "Find the sample size in a test where:\n",
    "\n",
    "power = 0.90 (beta = 0.10)\n",
    "\n",
    "n1 = n2 (k = 1)\n",
    "\n",
    "We want to be able to detect a difference of 0.4 (effect size is 0.4)\n",
    "\n",
    "using alpha = 0.05\n",
    "\n",
    "and assuming sigma = 1 in both populations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "60654430",
   "metadata": {},
   "outputs": [],
   "source": [
    "power = 0.90\n",
    "k = 1\n",
    "delta = 0.4\n",
    "sd = 1\n",
    "\n",
    "z_power = stats.norm.ppf(0.90, loc=0, scale = 1)\n",
    "z_signif = stats.norm.ppf(0.975, loc=0, scale = 1)\n",
    "\n",
    "n1 = (k+1) * (sd/delta*(z_signif+z_power))**2\n",
    "\n",
    "print(n1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "272daa02",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we can also use the python function TTestIndPower().solve_power (for one sample power calculations): \n",
    "print(smp.TTestIndPower().solve_power(effect_size=delta/sd, alpha=0.05, power=0.90, ratio=k))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d68a7d91",
   "metadata": {},
   "source": [
    "2)\n",
    "\n",
    "Find the power of an experiment where: \n",
    "\n",
    "n1 = n2 = 50 (k = 1)\n",
    "\n",
    "We want to be able to detect a difference of 20.4\n",
    "\n",
    "using alpha = 0.05\n",
    "\n",
    "and assuming sigma = 1 in both populations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1afa5306",
   "metadata": {},
   "outputs": [],
   "source": [
    "n1 = 50\n",
    "n2 = 50\n",
    "k = 1\n",
    "delta = 0.4\n",
    "sd = 1\n",
    "z_signif = stats.norm.ppf(0.975, loc=0, scale = 1)\n",
    "\n",
    "z = np.sqrt(n1/(k+1)*delta**2/sd**2) - z_signif\n",
    "power = stats.norm.cdf(z)\n",
    "\n",
    "print(power)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a702585d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we can also use the python function TTestIndPower().solve_power (for one sample power calculations): \n",
    "print(smp.TTestIndPower().solve_power(effect_size=delta/sd, alpha=0.05, nobs1=50, ratio=k))"
   ]
  }
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