Independent t-test by hand in Python: with equal sample sizes and variance

t_test_1.ipynb

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    "# Independent two-sample t-test with equal sample sizes and variance\n",
    "> Detail: https://www.nlpinpython.com/article/article-detail/4/\n",
    "> Medium:https://xiaoouwang.medium.com/t-test-by-hand-in-python-d5513d1b55eb\n",
    "\n",
    "> Author: Xiaoou Wang. https://www.linkedin.com/in/xiaoou-wang "
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      "The mean of a is around 4 -> 3.9028591091939 \n",
      "The mean of b is around 0 -> -0.16958838211535887\n",
      "The variance of a is around 1 -> 1.418239361307898 \n",
      "The variance of b is around 1 -> 0.9982804180510149\n",
      "The degree of freedom is -> 18\n"
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   "source": [
    "## necessary packages\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "from numpy.random import seed\n",
    "\n",
    "## generate data\n",
    "# set the seed\n",
    "seed(1)\n",
    "N = 10\n",
    "# Gaussian distributed data with mean = 2 and var = 1\n",
    "# see https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.randn.html\n",
    "a = np.random.randn(N) + 4\n",
    "#Gaussian distributed data with with mean = 0 and var = 1\n",
    "b = np.random.randn(N)\n",
    "print(\"The mean of a is around 4 ->\",np.mean(a),\"\\nThe mean of b is around 0 ->\",np.mean(b))\n",
    "print(\"The variance of a is around 1 ->\",np.var(a),\"\\nThe variance of b is around 1 ->\",np.var(b))\n",
    "df = N*2 - 2\n",
    "print(\"The degree of freedom is ->\",df)"
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    "## Formula for variance\n",
    "\n",
    "$$s_N = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N \\left(x_i - \\bar{x}\\right)^2}.$$\n",
    "\n",
    "## Formula for corrected variance\n",
    "\n",
    "An unbiased estimator for the *variance* is given by applying [Bessel's correction](https://www.wikiwand.com/en/Bessel%27s_correction), using *N−1* instead of *N* to yield the *unbiased sample variance* denoted *s<sup>2</sup>*:\n",
    "$$s^2 = \\frac{1}{N - 1} \\sum_{i=1}^N \\left(x_i - \\bar{x}\\right)^2.$$\n",
    "\n",
    "## Formula for t statistic\n",
    "\n",
    "$$ t = \\frac{\\bar{X}_1 - \\bar{X}_2}{s_p \\sqrt\\frac{2}{n}} $$\n",
    "\n",
    "where\n",
    "$$ s_p = \\sqrt{\\frac{s_{X_1}^2+s_{X_2}^2}{2}}.$$"
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      "t hand calculated -> 7.859258455468933\n",
      "p hand calculated -> 3.152614480583793e-07\n",
      "t using scipy -> 7.8592584554689315\n",
      "p using scipy -> 3.152614479389834e-07\n"
     ]
    }
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   "source": [
    "# ddof=1 -> corercted variance\n",
    "var_a = a.var(ddof=1)\n",
    "var_b = b.var(ddof=1)\n",
    "sp = (var_a/2+var_b/2)**0.5\n",
    "\n",
    "## Calculate the t-statistics\n",
    "t = (a.mean() - b.mean())/(sp*np.sqrt(2/N))\n",
    "\n",
    "# p-value\n",
    "p = 1 - stats.t.cdf(t,df=df)\n",
    "\n",
    "print(\"t hand calculated ->\",t)\n",
    "print(\"p hand calculated ->\",2*p)\n",
    "\n",
    "## Cross Checking with the internal scipy function\n",
    "t2, p2 = stats.ttest_ind(a,b)\n",
    "print(\"t using scipy ->\",t2)\n",
    "print(\"p using scipy ->\",p2)"
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