Added notebook on Equivalence tests

minireference · Dec 28, 2023 · 9031ac7 · 9031ac7
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diff --git a/examples/one_sample_t-test.ipynb b/examples/one_sample_t-test.ipynb
@@ -5,7 +5,7 @@
    "id": "229b9ce5-9ba8-406a-bb1a-5979d1e2a64f",
    "metadata": {},
    "source": [
-    "### One-sample $z$-test for the mean\n",
+    "### One-sample t-test for the mean\n",
     "\n",
     "The goal of the one-sample $t$-test to check if the mean $\\mu$ of an unknown population $X \\sim \\mathcal{N}(\\mu, \\sigma_0)$,\n",
     "equals the mean $\\mu_0$ of a theoretical distribution $X_0 \\sim \\mathcal{N}(\\mu_0, \\sigma_0)$.\n",
@@ -99,18 +99,30 @@
    "outputs": [],
    "source": []
   },
+  {
+   "cell_type": "markdown",
+   "id": "7674af35-fd9d-4608-970a-b74c27ca3475",
+   "metadata": {},
+   "source": [
+    "#### Estimates\n",
+    "\n",
+    "Calculate the sample mean $\\overline{\\mathbf{x}} = \\Mean(\\mathbf{x})$\n",
+    "and the sample standard deviation $s_{\\mathbf{x}} = \\Std(\\mathbf{x})$.\n",
+    "We can then calculate the estimated standard error of the mean\n",
+    "$\\stderrhat{\\overline{\\mathbf{x}}} =  \\frac{ s_{\\mathbf{x}} }{ \\sqrt{n} }$."
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "ab880b56-b783-4c32-b2cd-8cb66b59f766",
    "metadata": {},
    "source": [
     "#### Test statistic\n",
     "\n",
-    "Compute $t = \\frac{\\overline{\\mathbf{x}} - \\mu_0}{s/\\sqrt{n}}$,\n",
-    "where $\\overline{\\mathbf{x}}$ is the sample mean,\n",
-    "$\\mu_0$ is the theoretical population mean,\n",
-    "$n$ is the sample size.\n",
-    "and $s$ is the sample standard deviation, TODO UPDATE ME WITH NEW NOTATION\n"
+    "Compute $t = \\frac{\\overline{\\mathbf{x}} - \\mu_0}{ \\stderrhat{\\overline{\\mathbf{x}}} }$ from\n",
+    "the sample mean $\\overline{\\mathbf{x}}$,\n",
+    "the theoretical population mean $\\mu_0$,\n",
+    "the estimated standard error $\\stderrhat{\\overline{\\mathbf{x}}}$."
    ]
   },
   {
@@ -128,7 +140,7 @@
    "id": "15cf13c7-78bf-495a-baf8-2312c6210c2a",
    "metadata": {},
    "source": [
-    "#### Calculating $p$-values"
+    "#### P-value calculation"
    ]
   },
   {
@@ -261,7 +273,7 @@
    "id": "d563fb81-7b86-41e1-9a6a-c781d7cde415",
    "metadata": {},
    "source": [
-    "Let's generate a sample `xBs` of size $n=20$ from the random variable $X = \\texttt{rvXA}$."
+    "Let's generate a sample `xAs` of size $n=20$ from the random variable $X = \\texttt{rvXA}$."
    ]
   },
   {
@@ -341,7 +353,7 @@
     {
      "data": {
       "text/plain": [
-       "(5.413149619618617, 3.1862282356088544e-05)"
+       "(5.413149619618617, 3.186228235607658e-05)"
       ]
      },
      "execution_count": 8,
@@ -357,8 +369,7 @@
     "sehat = std / np.sqrt(n)\n",
     "obst = (obsmean - muX0) / sehat\n",
     "rvT0 = tdist(df=n-1)\n",
-    "from stats_helpers import tailprobs\n",
-    "pvalue = tailprobs(rvT0, obst, alt=\"two-sided\")\n",
+    "pvalue = rvT0.cdf(-obst) + 1-rvT0.cdf(obst)\n",
     "obst, pvalue"
    ]
   },
@@ -436,7 +447,7 @@
    "id": "869366ba-2513-4501-ae31-f98ee4dcb134",
    "metadata": {},
    "source": [
-    "Let's generate a sample `xs` of size $n=20$ from the random variable $X = \\texttt{rvX}$,\n",
+    "Let's generate a sample `xBs` of size $n=20$ from the random variable $X = \\texttt{rvX}$,\n",
     "which has the same distribution as the theoretical distribution we expect under the null hypothesis."
    ]
   },

diff --git a/examples/one_sample_z-test.ipynb b/examples/one_sample_z-test.ipynb
@@ -5,7 +5,7 @@
    "id": "229b9ce5-9ba8-406a-bb1a-5979d1e2a64f",
    "metadata": {},
    "source": [
-    "### One-sample $z$-test for the mean\n",
+    "### One-sample z-test for the mean\n",
     "\n",
     "The goal of the one-sample $z$-test to check if the mean $\\mu$ of an unknown population $X \\sim \\mathcal{N}(\\mu, \\sigma_0)$,\n",
     "equals the mean $\\mu_0$ of a theoretical distribution $X_0 \\sim \\mathcal{N}(\\mu_0, \\sigma_0)$.\n",
@@ -129,7 +129,7 @@
    "id": "15cf13c7-78bf-495a-baf8-2312c6210c2a",
    "metadata": {},
    "source": [
-    "#### Calculating $p$-values"
+    "#### P-value calculation"
    ]
   },
   {
@@ -260,7 +260,7 @@
    "id": "d563fb81-7b86-41e1-9a6a-c781d7cde415",
    "metadata": {},
    "source": [
-    "Let's generate a sample `xBs` of size $n=20$ from the random variable $X = \\texttt{rvXA}$."
+    "Let's generate a sample `xAs` of size $n=20$ from the random variable $X = \\texttt{rvXA}$."
    ]
   },
   {
@@ -333,29 +333,28 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 20,
    "id": "b80ab201-a6c9-4325-85c4-15dc290bea12",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "(3.1180664906014357, 0.0018204172963375855)"
+       "(3.1180664906014357, 0.0018204172963375287)"
       ]
      },
-     "execution_count": 8,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "mean = np.mean(xAs)\n",
+    "obsmean = np.mean(xAs)\n",
     "se = sigmaX0 / np.sqrt(n)\n",
-    "obsz = (mean - muX0) / se\n",
+    "obsz = (obsmean - muX0) / se\n",
     "rvZ = norm(0,1)\n",
-    "from stats_helpers import tailprobs\n",
-    "pval = tailprobs(rvZ, obsz, alt=\"two-sided\")\n",
-    "obsz, pval"
+    "pvalue = rvZ.cdf(-obsz) + 1-rvZ.cdf(obsz)\n",
+    "obsz, pvalue"
    ]
   },
   {
@@ -369,7 +368,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 21,
    "id": "d76a7c12-eab2-4f44-8040-3d170b6f16d2",
    "metadata": {},
    "outputs": [
@@ -379,7 +378,7 @@
        "0.0018204172963375855"
       ]
      },
-     "execution_count": 9,
+     "execution_count": 21,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -432,7 +431,7 @@
    "id": "869366ba-2513-4501-ae31-f98ee4dcb134",
    "metadata": {},
    "source": [
-    "Let's generate a sample `xs` of size $n=20$ from the random variable $X = \\texttt{rvX}$,\n",
+    "Let's generate a sample `xBs` of size $n=20$ from the random variable $X = \\texttt{rvX}$,\n",
     "which has the same distribution as the theoretical distribution we expect under the null hypothesis."
    ]
   },
@@ -566,7 +565,7 @@
    "source": [
     "#### Related\n",
     "\n",
-    "- Use the one-sample $t$-test if you want to use the sample standard deviation\n",
+    "- Use the [one-sample $t$-test](./one_sample_t-test.ipynb) if you want to use the sample standard deviation\n",
     "- Identical calculations to the one-saple $z$-test for proportions"
    ]
   },