remove redundant stuff

Author: drbenvincent

docs/source/knowledgebase/quasi_dags.ipynb

@@ -648,7 +648,7 @@
     "\n",
     "By conditioning on both $Y_0$ and $X$ we intercept backdoor paths 1 and 2, satisfying Pearl's back-door criterion. The treatment coefficient in the ANCOVA formula therefore identifies the causal effect of interest under the model's other assumptions (no measurement error, correct functional form, etc.).\n",
     "\n",
-    "Adjust for the baseline outcome $Y_0$ and any observed covariates that inform treatment assignment:  \n",
+    "If we can discount $U_1$ and $U_2$ then we can estimate the treatment effect $T \\to Y_1$ by running the linear regression:\n",
     "\n",
     "$$\n",
     "Y_1 = \\alpha + \\tau\\,T + \\gamma\\,Y_0 + \\mathbf{X}\\boldsymbol\\beta + \\varepsilon ,\n",
@@ -678,19 +678,16 @@
     "* $Y_0 \\to T$ Only mid-spenders qualify.  \n",
     "* $X \\to T$ Higher churn scores or certain regions raise the chance of receiving the email.  \n",
     "* $Y_0 \\to Y_1$ and $X \\to Y_1$ Past spend, tenure, and region predict future spend.    \n",
-    "* $T \\to Y_1$ The discount may increase (or cannibalise) post-campaign sales.\n",
-    "\n",
-    "If we can discount $U_1$ and $U_2$ then we can estimate the treatment effect $T \\to Y_1$ by running the linear regression:\n",
-    "\n",
-    "$$\n",
-    "Y_1 = \\alpha + \\tau T + \\gamma Y_0 + \\mathbf{X}\\boldsymbol\\beta + \\varepsilon\n",
-    "$$  \n",
-    "\n",
-    "where $\\tau$ is the average treatment effect (ATE).\n",
-    "\n",
-    "However, if there are unobserved confounders $U_1$ or $U_2$, then these will pose real threats to unbiased estimation of the treatment effect $T \\to Y_1$. In our example, $U_1$ could be a latent price sensitivity that affects both pre and post spend."
+    "* $T \\to Y_1$ The discount may increase (or cannibalise) post-campaign sales."
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "markdown",
    "metadata": {},