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\u0627\u0637\u0631\u0627\u0641 \u06a9\u06cc \u0633\u06cc\u062f\u06be\u06cc \u0644\u06a9\u06cc\u0631 \u0633\u06d2 \u0644\u06af \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u06cc\u06c1 \u062f\u06cc\u06a9\u06be\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u06c1 \u0622\u06cc\u0627 \u0622\u067e \u06a9\u0627 \u062c\u0648\u0627\u0628 \u062f\u0631\u0633\u062a \u06c1\u06d2\u060c \u06a9\u0626\u06cc \u0628\u06cc\u0646\u0688\u0648\u062a\u06be\u0633 \u067e\u0631 \u0644\u06a9\u06cc\u0631\u06cc \u0627\u0648\u0631 \u0686\u0648\u06a9\u0648\u0631 \u062a\u0635\u0631\u06cc\u062d\u0627\u062a \u06a9\u0648 \u0622\u0632\u0645\u0627\u0626\u06cc\u06ba\u06d4<\/p>\n<p>\u06c1\u0645 \u067e\u0648\u0631\u06d2 \u0645\u0636\u0645\u0648\u0646 \u0645\u06cc\u06ba \u062a\u06cc\u0632 RDDs \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u06a9\u06cc\u0648\u0646\u06a9\u06c1 \u0645\u062e\u062a\u0635 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0627 \u0627\u06cc\u06a9 \u0641\u06cc\u0635\u0644\u06c1 \u06a9\u0646 \u0641\u0639\u0644 \u06c1\u06d2 (0.85 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u06c1\u0645\u06cc\u0634\u06c1 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u060c \u0627\u0648\u0631 0.85 \u0633\u06d2 \u0627\u0648\u067e\u0631 \u06c1\u0645\u06cc\u0634\u06c1 \u0633\u0633\u062a\u0627 \u06c1\u0648\u062a\u0627 \u06c1\u06d2)\u06d4 \u062c\u0628 \u062d\u062f\u06cc\u06ba \u0627\u0633\u0679\u0627\u06a9\u0633\u0679\u06a9 \u06c1\u0648\u062a\u06cc \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u062a\u0639\u0645\u06cc\u0644 \u062c\u0632\u0648\u06cc \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u060c \u062a\u0648 \u0688\u06cc\u0632\u0627\u0626\u0646 \u0627\u06cc\u06a9 \u0645\u0628\u06c1\u0645 RDD \u06c1\u06d2 \u0627\u0648\u0631 \u0627\u0633 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u06cc\u06a9 \u0627\u0646\u0633\u0679\u0631\u0648\u0645\u06cc\u0646\u0679\u0644 \u0645\u062a\u063a\u06cc\u0631 \u0641\u0631\u06cc\u0645 \u0648\u0631\u06a9 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2 \u062c\u0633\u06d2 \u0627\u0633 \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u0644\u0627\u06af\u0648 \u06a9\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u06d4 <code>rdrobust<\/code> \u0627\u0632\u06af\u0631 \u067e\u06cc\u06a9\u062c\u06d4<\/p>\n<h2 id=\"heading-prerequisites\">\u0634\u0631\u0637\u06cc\u06ba<\/h2>\n<p>\u0622\u067e \u06a9\u0648 Python 3.11 \u06cc\u0627 \u0627\u0633 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1\u060c \u067e\u0627\u0646\u0688\u0648\u06ba \u0627\u0648\u0631 \u0634\u0645\u0627\u0631\u06cc\u0627\u062a\u06cc \u0645\u0627\u0688\u0644\u0632 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0633\u06a9\u0648\u0646\u060c \u0627\u0648\u0631 \u0644\u06a9\u06cc\u0631\u06cc \u0631\u062c\u0639\u062a \u0627\u0648\u0631 \u062a\u0639\u0627\u0645\u0644 \u06a9\u06cc \u0634\u0631\u0627\u0626\u0637 \u06a9\u06d2 \u0628\u0627\u0631\u06d2 \u0645\u06cc\u06ba \u0642\u0637\u0639\u06cc \u0639\u0644\u0645 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u06af\u06cc\u06d4<\/p>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0645\u06cc\u06ba \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06c1\u0648\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u067e\u06cc\u06a9\u062c\u0632 \u06a9\u0648 \u0627\u0646\u0633\u0679\u0627\u0644 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-shell\">pip install numpy pandas statsmodels matplotlib scipy\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u062a\u0634\u062e\u06cc\u0635\u06cc \u062a\u0635\u0648\u0631 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0686\u0627\u0631 \u0645\u0639\u06cc\u0627\u0631\u06cc \u0633\u0627\u0626\u0646\u0633\u06cc Python \u0644\u0627\u0626\u0628\u0631\u06cc\u0631\u06cc\u0627\u06ba \u0627\u0648\u0631 matplotlib\u06d4 \u0627\u0633 \u06a9\u06d2 \u0628\u0627\u0631\u06d2 \u0645\u06cc\u06ba \u063a\u06cc\u0631 \u0645\u0644\u06a9\u06cc \u06a9\u0648\u0626\u06cc \u0686\u06cc\u0632 \u0646\u06c1\u06cc\u06ba \u06c1\u06d2\u06d4<\/p>\n<p>\u0633\u0627\u062a\u06be\u06cc \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u06cc\u06ba \u0627\u0648\u0631 \u0627\u06cc\u06a9 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0628\u0646\u0627\u0626\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-shell\">git clone https:\/\/github.com\/RudrenduPaul\/product-experimentation-causal-inference-genai-llm.git\ncd product-experimentation-causal-inference-genai-llm\npython data\/generate_data.py --seed 42 --n-users 50000 --out data\/synthetic_llm_logs.csv\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u0688\u06cc\u0679\u0627 \u062c\u0646\u0631\u06cc\u0679\u0631 \u06c1\u06d2\u06d4 <code>query_confidence<\/code> \u0628\u06cc\u0679\u0627 (5,2) \u06a9\u06cc \u062a\u0642\u0633\u06cc\u0645 \u06a9\u0627 \u0633\u06a9\u0648\u0631 \u0631\u0648\u0679\u0646\u06af \u06a9\u0627 \u0627\u0635\u0648\u0644 \u06c1\u06d2 (<code>routed_to_premium = query_confidence < 0.85<\/code>)\u060c +6% \u067e\u0648\u0627\u0626\u0646\u0679 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u0627\u062b\u0631 <code>task_completed<\/code>. \u06c1\u0631 \u0628\u0627\u0631 \u0627\u06cc\u06a9 \u06c1\u06cc \u0628\u06cc\u062c\u060c \u0627\u06cc\u06a9 \u06c1\u06cc \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679\u06d4<\/p>\n<h2 id=\"heading-setting-up-the-working-example\">\u0648\u0631\u06a9\u0646\u06af \u0645\u062b\u0627\u0644 \u0633\u06cc\u0679 \u0627\u067e<\/h2>\n<p>\u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679 \u0627\u06cc\u06a9 SaaS \u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u06cc \u0646\u0642\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0627\u0633\u06a9\u0648\u0631 \u06a9\u06cc \u0628\u0646\u06cc\u0627\u062f \u067e\u0631 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0645\u0627\u0688\u0644 \u0627\u0648\u0631 \u0627\u06cc\u06a9 \u0633\u0633\u062a\u06d2 \u0645\u0627\u0688\u0644 \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0648 \u0631\u0648\u0679 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u062d\u062f 0.85 \u06c1\u06d2 \u0627\u0648\u0631 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u06a9\u0627 \u0627\u0635\u0644 \u06a9\u0627\u0631\u0622\u0645\u062f \u0627\u062b\u0631 \u06a9\u0627\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u067e\u0631 +6 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679\u0633 \u06c1\u06d2\u06d4 \u0627\u0628 \u062c\u0628 \u06a9\u06c1 \u0622\u067e \u06a9\u0648 \u062d\u0642\u06cc\u0642\u062a \u0645\u0639\u0644\u0648\u0645 \u06c1\u0648 \u06af\u0626\u06cc \u06c1\u06d2\u060c \u0622\u067e \u0627\u0633 \u0628\u0627\u062a \u06a9\u0648 \u06cc\u0642\u06cc\u0646\u06cc \u0628\u0646\u0627 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba \u06a9\u06c1 \u0622\u067e \u06a9\u0627 RDD \u0627\u0633\u06d2 \u0628\u062d\u0627\u0644 \u06a9\u0631 \u0644\u06d2\u06d4<\/p>\n<p>\u0627\u067e\u0646\u0627 \u0688\u06cc\u0679\u0627 \u0644\u0648\u0688 \u06a9\u0631\u06cc\u06ba \u0627\u0648\u0631 \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u062a\u062c\u0632\u06cc\u0627\u062a \u06a9\u0648 \u062f\u0631\u06cc\u0627\u0641\u062a \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport pandas as pd\nimport statsmodels.formula.api as smf\n\ndf = pd.read_csv(\"data\/synthetic_llm_logs.csv\")\nprint(f\"Loaded {len(df):,} rows, {df.shape[1]} columns\")\n\nprint(\"nRouting breakdown:\")\ncounts = df.routed_to_premium.value_counts().to_dict()\nprint(f\"  Premium-routed (confidence < 0.85):  {counts.get(1, 0):,}\")\nprint(f\"  Cheap-routed   (confidence >= 0.85): {counts.get(0, 0):,}\")\n\nprint(\"nQuery confidence distribution:\")\nprint(df.query_confidence.describe().round(3))\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-python\">Loaded 50,000 rows, 16 columns\n\nRouting breakdown:\n  Premium-routed (confidence < 0.85):  38,874\n  Cheap-routed   (confidence >= 0.85): 11,126\n\nQuery confidence distribution:\ncount    50000.000\nmean         0.715\nstd          0.159\nmin          0.078\n25%          0.611\n50%          0.736\n75%          0.838\nmax          0.998\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u062a\u0642\u0631\u06cc\u0628\u0627\u064b 78% \u0633\u0648\u0627\u0644\u0627\u062a 0.85 \u06a9\u0679 \u0622\u0641 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u0622\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u062d\u0627\u0635\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0628\u06cc\u0679\u0627 (5,2) \u06a9\u06cc \u062a\u0642\u0633\u06cc\u0645 0.736 \u06a9\u06d2 \u0645\u06cc\u0688\u06cc\u0646 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u0648\u067e\u0631\u06cc \u0633\u0631\u06d2 \u06a9\u06cc \u0637\u0631\u0641 \u0645\u062a\u0648\u062c\u06c1 \u06c1\u06d2\u060c \u0632\u06cc\u0627\u062f\u06c1 \u062a\u0631 \u0645\u0627\u0633\u0632 \u0627\u0628 \u0628\u06be\u06cc 0.85 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0628\u0627\u0642\u06cc 22% \u0633\u0648\u0627\u0644\u0627\u062a \u06c1\u06cc\u06ba \u062c\u0646 \u06a9\u06d2 \u0628\u0627\u0631\u06d2 \u0645\u06cc\u06ba \u0645\u0627\u0688\u0644 \u067e\u06c1\u0644\u06d2 \u06c1\u06cc \u067e\u0631\u0627\u0639\u062a\u0645\u0627\u062f \u06c1\u06d2 \u0627\u0648\u0631 \u0633\u0633\u062a\u06d2 \u0645\u0627\u0688\u0644 \u067e\u0631 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u0631\u062c\u0639\u062a \u06a9\u0627 \u062a\u062c\u0632\u06cc\u06c1 \u06a9\u0631\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2\u060c \u06cc\u06c1\u0627\u06ba \u0627\u0646 \u0633\u0627\u062f\u06c1 \u0645\u0648\u0627\u0632\u0646\u0648\u06ba \u067e\u0631 \u0627\u06cc\u06a9 \u0646\u0638\u0631 \u06c1\u06d2 \u062c\u0648 \u06c1\u0631 \u067e\u0631\u0648\u0688\u06a9\u0679 \u0679\u06cc\u0645 \u0686\u0644\u0627\u0646\u0627 \u0686\u0627\u06c1\u06d2 \u06af\u06cc:<\/p>\n<pre><code class=\"language-python\">naive = (\n    df[df.routed_to_premium == 1].task_completed.mean()\n    - df[df.routed_to_premium == 0].task_completed.mean()\n)\nprint(f\"Naive premium-vs-cheap effect: {naive:+.4f}  (ground truth = +0.06)\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-python\">Naive premium-vs-cheap effect: +0.0632  (ground truth = +0.06)\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u0633\u0627\u062f\u06c1 \u062a\u062e\u0645\u06cc\u0646\u06c1 +0.0632 \u06c1\u06d2\u060c \u062c\u0648 \u0645\u0634\u06a9\u0648\u06a9 \u0637\u0648\u0631 \u067e\u0631 \u0633\u0686\u0627\u0626\u06cc \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u06c1\u06d2\u06d4 \u06cc\u06c1 \u0627\u06cc\u06a9 \u0627\u062a\u0641\u0627\u0642 \u06c1\u06d2 \u06a9\u06c1 \u0627\u0633 \u0645\u062e\u0635\u0648\u0635 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0627\u0648\u0631 \u06a9\u0645 \u0642\u06cc\u0645\u062a \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646 \u0648\u0627\u062d\u062f \u0627\u0644\u062c\u06be\u0646\u06d2 \u0648\u0627\u0644\u0627 \u0639\u0646\u0635\u0631 \u06c1\u06d2\u06d4 <code>query_confidence<\/code> \u06cc\u06c1 \u0648\u06c1\u06cc \u06c1\u06d2 \u062c\u0648 \u06cc\u06c1 \u06c1\u06d2 \u0627\u0648\u0631 \u0646\u062a\u0627\u0626\u062c \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0639\u0644\u0627\u0648\u06c1 \u0648\u0634\u0648\u0633\u0646\u06cc\u06cc\u062a\u0627 \u067e\u0631 \u0645\u0646\u062d\u0635\u0631 \u0646\u06c1\u06cc\u06ba \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u067e\u06cc\u062f\u0627\u0648\u0627\u0631 \u0645\u06cc\u06ba\u060c \u0622\u067e \u06a9\u0648 \u0634\u0627\u0630 \u0648 \u0646\u0627\u062f\u0631 \u06c1\u06cc \u0627\u0633 \u0642\u0633\u0645 \u06a9\u06cc \u0642\u0633\u0645\u062a \u0645\u0644\u062a\u06cc \u06c1\u06d2\u06d4 \u0635\u0627\u0631\u0641 \u06a9\u06cc \u0645\u06c1\u0627\u0631\u062a\u060c \u062a\u06cc\u0632 \u0631\u0641\u062a\u0627\u0631 \u0627\u0638\u06c1\u0627\u0631\u060c \u0648\u0642\u062a\u060c \u0627\u0648\u0631 12 \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0633\u0648\u0627\u0644 \u06a9\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u0633\u0628 \u06a9\u0627 \u062a\u0639\u0644\u0642 \u0627\u0639\u062a\u0645\u0627\u062f \u0627\u0648\u0631 \u062a\u06a9\u0645\u06cc\u0644 \u0633\u06d2 \u06c1\u06d2\u06d4<\/p>\n<p>\u062d\u0642\u06cc\u0642\u06cc \u0646\u0638\u0627\u0645\u0648\u06ba \u06a9\u06d2 \u0633\u0627\u062f\u06c1 \u0645\u0648\u0627\u0632\u0646\u06c1 \u062f\u0648\u0646\u0648\u06ba \u0633\u0645\u062a\u0648\u06ba \u0645\u06cc\u06ba 50% \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u062e\u062a\u0644\u0641 \u06c1\u0648 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 RDD \u0634\u0646\u0627\u062e\u062a \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0645\u062e\u0641\u06cc \u0627\u0644\u062c\u06be\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0639\u0648\u0627\u0645\u0644 \u06a9\u06cc \u0639\u062f\u0645 \u0645\u0648\u062c\u0648\u062f\u06af\u06cc \u067e\u0631 \u0645\u0646\u062d\u0635\u0631 \u0646\u06c1\u06cc\u06ba \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-step-1-a-sharp-rdd-with-local-linear-regression\">\u0645\u0631\u062d\u0644\u06c1 1: \u0645\u0642\u0627\u0645\u06cc \u0644\u06a9\u06cc\u0631\u06cc \u0631\u062c\u0639\u062a \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u062a\u06cc\u0632 \u0622\u0631 \u0688\u06cc \u0688\u06cc<\/h3>\n<p>\u0628\u0646\u06cc\u0627\u062f\u06cc \u062a\u06cc\u0632 \u0622\u0631 \u0688\u06cc \u0688\u06cc \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0644\u06af\u0627\u0646\u06d2 \u0648\u0627\u0644\u0627 \u0645\u0642\u0627\u0645\u06cc \u0644\u06a9\u06cc\u0631\u06cc \u0631\u062c\u0639\u062a \u06c1\u06d2\u06d4 \u06c1\u0645 \u0627\u067e\u0646\u06d2 \u0622\u067e \u06a9\u0648 \u0627\u0646 \u0635\u0627\u0631\u0641\u06cc\u0646 \u062a\u06a9 \u0645\u062d\u062f\u0648\u062f \u0631\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba \u062c\u0646 \u06a9\u0627 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0679 \u0622\u0641 \u06a9\u06cc \u0628\u06cc\u0646\u0688\u0648\u062a\u06be \u06a9\u06d2 \u0627\u0646\u062f\u0631 \u06c1\u06d2\u060c \u06c1\u0631 \u0637\u0631\u0641 \u0627\u0644\u06af \u0627\u0644\u06af \u0644\u06a9\u06cc\u0631\u06cc \u0688\u06be\u0644\u0648\u0627\u0646 \u0641\u0679 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0627\u0648\u0631 0.85 \u067e\u0631 \u0686\u06be\u0644\u0627\u0646\u06af \u067e\u0691\u06be\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-python\">\u06a9\u0679 \u0622\u0641 = 0.85 bw = 0.10 \u0642\u0631\u0628\u062a = df[(df.query_confidence > cutoff - bw)\n          & (df.query_confidence < cutoff + bw)].copy() \u06a9\u06d2 \u0642\u0631\u06cc\u0628[\"below_cutoff\"] = \u0642\u0631\u06cc\u0628 (near.query_confidence < cutoff).astype(int)[\"rc\"] = Near.query_confidence - cutoff rdd_model = smf.ols( \"task_completed ~ below_cutoff + rc + below_cutoff:rc\", data=near, .fit(cov_type=\"HC3\") \u0627\u062b\u0631 = rdd_model.params[\"below_cutoff\"]\n\u067e\u0631\u0646\u0679 \u06a9\u0631\u06cc\u06ba['below_cutoff']:.4f}\") \u067e\u0631\u0646\u0679(f\"p-values: {rdd_model.pvalues['below_cutoff']:.4f}\") \u067e\u0631\u0646\u0679 (f\"N \u0635\u0627\u0631\u0641\u06cc\u0646 [0.75, 0.95):     {len(near):,}\")\n<\/code><\/pre>\n<p><strong>Expected output:<\/strong><\/p>\n<pre><code class=\"language-python\">RDD effect at cutoff (LATE): +0.0548\nStd error (HC3):             0.0131\np-value:                     0.0000\nN users in [0.75, 0.95):     21,689\n<\/code><\/pre>\n<p><strong>Here's what's happening:<\/strong> the model fits separate intercepts and slopes on each side of 0.85 (<code>below_cutoff<\/code> is the side indicator, <code>rc<\/code> is confidence centered at the cutoff). The coefficient on <code>below_cutoff<\/code> reads off the vertical jump at the threshold, which is the local average treatment effect (LATE) for queries with confidence near 0.85. You get +0.0548, within sampling noise of the +0.06 ground truth.<\/p>\n<p>Three notes on the specification. First, <code>task_completed<\/code> is binary, so this is a linear probability model. For RDD with a binary outcome at the cutoff, the linear probability model is standard practice because local linearity is the identifying assumption either way. Logit at the cutoff is an alternative if you need bounded predictions globally.<\/p>\n<p>Second, the standard errors are used <code>cov_type=\"HC3\"<\/code> to relax the homoskedasticity assumption, which is almost always wrong for binary outcomes.<\/p>\n<p>Third, the dataset has one query per user with no within-user clustering, so cluster-robust standard errors aren't needed here. In a setting with multiple queries per user, you'd cluster on <code>user_id<\/code>.<\/p>\n<p>The next diagnostic to look at is the confidence distribution near the cutoff. Figure 2 shows what 50,000 queries look like in the bandwidth window:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/umang.pk\/wp-content\/uploads\/2026\/05\/1778427135_376_\u0631\u062c\u0639\u062a-\u06a9\u06d2-\u0648\u0642\u0641\u06d2-\u06a9\u06d2-\u0633\u0627\u062a\u06be-\u067e\u0631\u0648\u0688\u06a9\u0679-\u06a9\u0627-\u062a\u062c\u0631\u0628\u06c1-LLM-\u0627\u0639\u062a\u0645\u0627\u062f.png\" alt=\"9ecb8a4c-6eac-4732-95ae-2a5981917f54\" style=\"display:block;margin:0 auto\" width=\"600\" height=\"400\" loading=\"lazy\" title=\"\"><\/p>\n<p><em>Figure 2. Real distribution from the 50,000-user synthetic dataset. Unlike the schematic in Figure 1, this shows the actual query density by confidence score, with the routing threshold annotated. The bottom panel counts how many queries land in each 2-percentage-point bin near the cutoff (2,461 \/ 2,481 \/ 2,335 \/ 2,229 \/ 2,048 across the 0.80\u20130.90 range). The roughly uniform spread is the visual signal that no manipulation is concentrating users on one side of the threshold.<\/em><\/p>\n<h3 id=\"heading-step-2-try-different-bandwidths\">Step 2: Try Different Bandwidths<\/h3>\n<p>Bandwidth choice matters. Too narrow and you have too few observations, so the confidence interval blows up. Too wide and you're extrapolating into regions where the linear specification is no longer a reasonable local approximation.<\/p>\n<p>The honest move is to try multiple bandwidths and report whether the estimate holds.<\/p>\n<pre><code class=\"language-python\">results = []\nBW \u06a9\u06d2 \u0644\u06cc\u06d2 [0.05, 0.10, 0.15, 0.20]:sub=df[(df.query_confidence > cutoff - bw)\n             & (df.query_confidence < cutoff + bw)].copy() \u0630\u06cc\u0644\u06cc[\"below_cutoff\"] = (sub.query_confidence <cutoff).astype(int) \u0630\u06cc\u0644\u06cc[\"rc\"] = sub.query_confidence - cutoff m = smf.ols( \"task_completed ~ below_cutoff + rc + below_cutoff:rc\", data=sub, .fit(cov_type=\"HC3\") results.append({ \"bandwidth\": bw, \"n\": len(sub. m), \"effect:\"[\"below_cutoff\"]\"se\": m.bse[\"below_cutoff\"]\"p\": mp \u0642\u062f\u0631[\"below_cutoff\"]}) \u067e\u0631\u0646\u0679(pd.DataFrame(\u0646\u062a\u0627\u0626\u062c)) \u0631\u0627\u0624\u0646\u0688(4).to_string(index=False))\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-python\"> bandwidth      n  effect     se       p\n      0.05  11554  0.0635  0.0183  0.0005\n      0.10  21689  0.0548  0.0131  0.0000\n      0.15  29137  0.0618  0.0112  0.0000\n      0.20  34074  0.0614  0.0107  0.0000\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06cc\u06c1 \u06a9\u0679 \u0622\u0641 \u06a9\u06d2 \u0627\u0631\u062f \u06af\u0631\u062f \u00b10.05 \u0633\u06d2 \u00b10.20 \u062a\u06a9 \u06a9\u06cc \u0686\u0627\u0631 \u0628\u06cc\u0646\u0688\u0648\u062a\u06be\u0633 \u067e\u06cc\u0634 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c \u06c1\u0631 \u0627\u06cc\u06a9 \u062f\u0648\u0628\u0627\u0631\u06c1 \u0627\u0633\u06cc RDD \u0648\u0636\u0627\u062d\u062a\u0648\u06ba \u0633\u06d2 \u0645\u0634\u0631\u0648\u0637 \u06c1\u06d2\u06d4 \u062a\u062e\u0645\u06cc\u0646\u0648\u06ba \u06a9\u06cc \u062d\u062f +0.0548 \u0633\u06d2 +0.0635 \u062a\u06a9 \u06c1\u06d2\u060c \u06cc\u06c1 \u0633\u0628 \u06a9\u0686\u06be +0.06 \u0632\u0645\u06cc\u0646\u06cc \u0633\u0686\u0627\u0626\u06cc \u06a9\u06cc \u0637\u0631\u062d \u06c1\u06d2\u060c \u062c\u0633 \u0645\u06cc\u06ba \u0645\u0639\u06cc\u0627\u0631\u06cc \u063a\u0644\u0637\u06cc\u0627\u06ba \u06a9\u0645 \u06c1\u0648\u062a\u06cc \u062c\u0627\u062a\u06cc \u06c1\u06cc\u06ba \u062c\u06cc\u0633\u06d2 \u062c\u06cc\u0633\u06d2 \u0628\u06cc\u0646\u0688\u0648\u062a\u06be \u06a9\u0645 \u06c1\u0648\u062a\u06cc \u062c\u0627\u062a\u06cc \u06c1\u06d2 \u0648\u0633\u06cc\u0639 \u062a\u0631 \u06c1\u0648\u062a\u06cc \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u062a\u0645\u0627\u0645 p-values \u200b\u200b0.05 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0622\u06cc\u0627 \u062a\u062e\u0645\u06cc\u0646\u06c1 \"\u0645\u0633\u062a\u062d\u06a9\u0645\" \u06c1\u06d2 \u0627\u0633 \u06a9\u0627 \u0627\u0646\u062d\u0635\u0627\u0631 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u06d2 \u0627\u0631\u062f \u06af\u0631\u062f \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0648\u0642\u0641\u06d2 \u067e\u0631 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0648 \u0645\u0631\u062d\u0644\u06c1 5 \u0645\u06cc\u06ba \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e\u0646\u06af \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u062a\u06cc\u0627\u0631 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-step-3-checking-for-manipulation-at-the-threshold\">\u0645\u0631\u062d\u0644\u06c1 3: \u062f\u06c1\u0644\u06cc\u0632 \u067e\u0631 \u06c1\u06cc\u0631\u0627 \u067e\u06be\u06cc\u0631\u06cc \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u06cc\u06ba\u06d4<\/h3>\n<p>RDDs \u0635\u0631\u0641 \u0627\u0633 \u0648\u0642\u062a \u062f\u0631\u0633\u062a \u06c1\u06cc\u06ba \u062c\u0628 \u0635\u0627\u0631\u0641 \u06a9\u0679 \u0622\u0641 \u06a9\u06d2 \u0627\u0631\u062f \u06af\u0631\u062f \u0627\u06cc\u06af\u0632\u06cc\u06a9\u06cc\u0648\u0634\u0646 \u0645\u062a\u063a\u06cc\u0631\u0627\u062a \u06a9\u0648 \u062f\u0631\u0633\u062a \u0637\u0631\u06cc\u0642\u06d2 \u0633\u06d2 \u062c\u0648\u0691 \u0646\u06c1\u06cc\u06ba \u0633\u06a9\u062a\u0627\u06d4 \u0627\u06af\u0631 \u0635\u0627\u0631\u0641 (\u06cc\u0627 \u0633\u0633\u0679\u0645) \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u067e\u0631 \u0645\u062c\u0628\u0648\u0631 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0627\u0633\u06a9\u0648\u0631 \u06a9\u0648 0.85 \u0633\u06d2 \u0628\u0627\u0644\u06a9\u0644 \u0646\u06cc\u0686\u06d2 \u0627\u06cc\u0688\u062c\u0633\u0679 \u06a9\u0631 \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u060c \u062a\u0648 \u06cc\u06c1 \u06a9\u0679 \u0622\u0641 \u067e\u0631 \u06a9\u062b\u0627\u0641\u062a \u0645\u06cc\u06ba \u0627\u0636\u0627\u0641\u06d2 \u06a9\u0627 \u0633\u0628\u0628 \u0628\u0646\u06d2 \u06af\u0627 \u0627\u0648\u0631 RDD \u062a\u062e\u0645\u06cc\u0646\u0648\u06ba \u06a9\u0648 \u0622\u0644\u0648\u062f\u06c1 \u06a9\u0631 \u062f\u06d2 \u06af\u0627\u06d4<\/p>\n<p>\u0645\u0639\u06cc\u0627\u0631\u06cc \u062a\u0634\u062e\u06cc\u0635\u06cc \u0645\u06cc\u06a9\u06a9\u0631\u06cc \u06a9\u062b\u0627\u0641\u062a \u0679\u06cc\u0633\u0679 \u06c1\u06d2\u060c \u062c\u0648 \u06cc\u06c1 \u062c\u0627\u0646\u0686\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0622\u06cc\u0627 \u0686\u0644\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0645\u062a\u063a\u06cc\u0631 \u06a9\u06cc \u062a\u0642\u0633\u06cc\u0645 \u0645\u06cc\u06ba \u06a9\u0679 \u0622\u0641 \u067e\u0631 \u062a\u06cc\u0632 \u0686\u06be\u0644\u0627\u0646\u06af \u06c1\u06d2\u06d4 \u0633\u0627\u062f\u06c1 \u0648\u0631\u0698\u0646: 0.85 \u06a9\u06d2 \u0627\u0631\u062f \u06af\u0631\u062f \u0688\u06cc\u0679\u0627 \u06a9\u0648 \u0633\u062e\u062a\u06cc \u0633\u06d2 \u062f\u0631\u062c\u06c1 \u0628\u0646\u062f\u06cc \u06a9\u0631\u06cc\u06ba \u0627\u0648\u0631 \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba \u06a9\u06c1 \u0622\u06cc\u0627 \u062f\u0648\u0646\u0648\u06ba \u0627\u0637\u0631\u0627\u0641 \u06a9\u06cc \u06af\u0646\u062a\u06cc \u0627\u06cc\u06a9 \u062c\u06cc\u0633\u06cc \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">\u067e\u0631\u0646\u0679 (\"2% \u067e\u0648\u0627\u0626\u0646\u0679 \u0631\u06cc\u0646\u062c \u0645\u06cc\u06ba \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06cc \u062a\u0639\u062f\u0627\u062f \u062a\u0642\u0631\u06cc\u0628\u0627\u064b 0.85 \u06c1\u06d2:\") \u06a9\u06d2 \u0644\u06cc\u06d2 [0.80, 0.82, 0.84, 0.86, 0.88]: hi = lo + 0.02 cnt = ((df.query_confidence >= lo) & (df.query_confidence < hi)).sum() print(f\"  [{lo:.2f}, {hi:.2f}):  n = {cnt:,}\")\n<\/code><\/pre>\n<p><strong>Expected output:<\/strong><\/p>\n<pre><code class=\"language-python\">User counts in 2-percentage-point bins around 0.85:\n  [0.80, 0.82):  n = 2,461\n  [0.82, 0.84):  n = 2,481\n  [0.84, 0.86):  n = 2,335\n  [0.86, 0.88):  n = 2,229\n  [0.88, 0.90):  n = 2,048\n<\/code><\/pre>\n<p><strong>Here's what's happening:<\/strong> counts trend gently downward across the bandwidth because Beta(5,2) places more mass at higher confidence levels, and the density tapers as it approaches 1.0. There's no spike or dip at the 0.84\u20130.86 bin that straddles the cutoff. The 433-user spread across all five bins is consistent with smooth tapering of the underlying density.<\/p>\n<p>That's the pattern you want when manipulation is absent. For a more rigorous test, the <code>rddensity<\/code> Python package implements the formal McCrary procedure with bias-corrected standard errors.<\/p>\n<p>What manipulation looks like when it's real: a spike in users at confidences just barely below 0.85 (they're being nudged into premium routing) and a dip just above. If you see that pattern, the RDD estimate overstates the causal effect because the users right below 0.85 differ in motivation from those right above. They cared enough to manipulate the score, and they'd have shown different outcomes even under random routing.<\/p>\n<h3 id=\"heading-step-4-quadratic-specification-as-a-robustness-check\">Step 4: Quadratic Specification as a Robustness Check<\/h3>\n<p>If the true relationship between confidence and task completion isn't exactly linear, a local linear RDD can mistake the curvature for a jump. The standard robustness check allows quadratic terms on both sides of the cutoff and tests whether the estimate holds.<\/p>\n<pre><code class=\"language-python\">near = df[(df.query_confidence > cutoff - 0.10)\n         & (df.query_confidence < cutoff + 0.10)].copy() \u06a9\u06d2 \u0642\u0631\u06cc\u0628[\"below_cutoff\"] = \u0642\u0631\u06cc\u0628 (near.query_confidence < cutoff).astype(int)[\"rc\"] = Near.query_confidence - \u06a9\u0679 \u0622\u0641 \u06a9\u06d2 \u0642\u0631\u06cc\u0628[\"rc2\"] = Near.rc ** 2 rdd_quad = smf.ols( \"task_completed ~ below_cutoff + rc + below_cutoff:rc\" \" + rc2 + below_cutoff:rc2\", data=near, .fit(cov_type=\"HC3\") print(f\"Linear RDD (bw=0. 0.1 \u0627\u062b\u0631): <0=0.1 0.0001\") \u067e\u0631\u0646\u0679 (f\"\u06a9\u0648\u0627\u0688\u0631\u0627\u0679\u06a9 RDD (bw=0.10): \u0627\u062b\u0631 = \" f\"{rdd_quad.params['below_cutoff']:+.4f}\u060c \"f\"p = {rdd_quad.pvalues['below_cutoff']:.4f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Linear RDD    (bw=0.10):  effect = +0.0548, p < 0.0001\nQuadratic RDD (bw=0.10):  effect = +0.0569, p = 0.0036\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u0686\u0648\u06a9\u0648\u0631 \u062a\u0635\u0631\u06cc\u062d \u0627\u06cc\u06a9 \u0645\u0631\u0628\u0639 \u0627\u0635\u0637\u0644\u0627\u062d \u0627\u0648\u0631 \u06a9\u0679 \u0622\u0641 \u06a9\u06cc \u0646\u0645\u0627\u0626\u0646\u062f\u06af\u06cc \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u0633 \u06a9\u0627 \u062a\u0639\u0627\u0645\u0644 \u062c\u0648\u0691\u062a\u06cc \u06c1\u06d2 \u062a\u0627\u06a9\u06c1 \u0631\u0634\u062a\u06c1 \u062f\u0648\u0646\u0648\u06ba \u0627\u0637\u0631\u0627\u0641 \u0645\u06cc\u06ba \u0645\u062e\u062a\u0644\u0641 \u0637\u0631\u06cc\u0642\u06d2 \u0633\u06d2 \u0645\u0646\u062d\u0646\u06cc \u06c1\u0648\u06d4 \u06a9\u06c1 <code>below_cutoff<\/code> \u06af\u062a\u0627\u0646\u06a9 \u0627\u0628 \u0628\u06be\u06cc \u0632\u06cc\u0627\u062f\u06c1 \u0644\u0686\u06a9\u062f\u0627\u0631 \u062a\u0635\u0631\u06cc\u062d \u06a9\u06d2 \u0645\u0637\u0627\u0628\u0642 \u062d\u062f \u0645\u06cc\u06ba \u0686\u06be\u0644\u0627\u0646\u06af \u06a9\u0648 \u067e\u06a9\u0691\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u062f\u0648\u0646\u0648\u06ba \u0627\u0646\u062f\u0627\u0632\u0648\u06ba \u0645\u06cc\u06ba 0.0022 \u06a9\u0627 \u0641\u0631\u0642 \u06c1\u06d2\u060c \u062f\u0648\u0646\u0648\u06ba \u0632\u0645\u06cc\u0646\u06cc \u0633\u0686\u0627\u0626\u06cc \u06a9\u06d2 \u0642\u0631\u06cc\u0628 +0.06 \u06c1\u06cc\u06ba\u060c \u0627\u0648\u0631 \u062f\u0648\u0646\u0648\u06ba p <0.01 \u067e\u0631 \u0627\u06c1\u0645 \u06c1\u06cc\u06ba\u06d4 \u0645\u0627\u0688\u0644 \u06a9\u0648 \u0645\u0648\u0691\u0646\u06d2 \u0633\u06d2 \u062c\u0648\u0627\u0628 \u0646\u06c1\u06cc\u06ba \u0628\u062f\u0644\u062a\u0627\u06d4<\/p>\n<p>\u0627\u06af\u0631 \u0644\u06a9\u06cc\u0631\u06cc \u0627\u0648\u0631 \u0686\u0648\u06a9\u0648\u0631 \u062a\u0635\u0631\u06cc\u062d\u0627\u062a \u0628\u0638\u0627\u06c1\u0631 \u0645\u0645\u0627\u062b\u0644 \u0646\u06c1\u06cc\u06ba \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0622\u067e \u06a9\u06d2 \u067e\u0627\u0633 \u0627\u06cc\u06a9 \u062d\u0642\u06cc\u0642\u06cc \u0633\u06af\u0646\u0644 \u06c1\u06d2\u06d4 \u0686\u06be\u0648\u0679\u06d2 \u0646\u0645\u0648\u0646\u0648\u06ba \u06a9\u06d2 \u0644\u06cc\u06d2 (\u06c1\u0632\u0627\u0631\u0648\u06ba \u062a\u0646\u06af \u0628\u06cc\u0646\u0688\u0648\u0688\u062a\u06be \u067e\u0631)\u060c \u062f\u0648\u0633\u0631\u06d2 \u0622\u0631\u0688\u0631 \u0648\u0627\u0644\u06d2 \u0648\u0631\u0698\u0646 \u06a9\u06cc \u0637\u0627\u0642\u062a \u062e\u062a\u0645 \u06c1\u0648 \u0633\u06a9\u062a\u06cc \u06c1\u06d2 \u06a9\u06cc\u0648\u0646\u06a9\u06c1 \u0686\u0627\u0631 \u0627\u0636\u0627\u0641\u06cc \u067e\u06cc\u0631\u0627\u0645\u06cc\u0679\u0631\u0632 \u06a9\u06cc \u0634\u0646\u0627\u062e\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u0688\u06cc\u0679\u0627 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u0645\u0639\u06cc\u0627\u0631\u06cc \u0627\u0642\u062f\u0627\u0645 \u0628\u06cc\u0646\u0688\u0648\u062a\u06be \u06a9\u0648 \u0648\u0633\u06cc\u0639 \u06a9\u0631\u0646\u0627 \u0627\u0648\u0631 \u062f\u0648\u0646\u0648\u06ba \u062a\u0635\u0631\u06cc\u062d\u0627\u062a \u06a9\u0648 \u062f\u0648\u0628\u0627\u0631\u06c1 \u06a9\u0631\u0646\u0627 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0627\u0628 \u0628\u06be\u06cc \u0648\u0633\u06cc\u0639 \u0628\u06cc\u0646\u0688\u0648\u0688\u062a\u06be \u0645\u06cc\u06ba \u06a9\u0648\u0626\u06cc \u062a\u0636\u0627\u062f \u06c1\u06d2 \u062a\u0648\u060c \u0644\u06a9\u06cc\u0631\u06cc \u062a\u062e\u0645\u06cc\u0646\u06c1 \u063a\u0644\u0637 \u06c1\u06d2 \u0627\u0648\u0631 \u062f\u0648\u0646\u0648\u06ba \u0646\u0645\u0628\u0631\u0648\u06ba \u06a9\u06cc \u0627\u0637\u0644\u0627\u0639 \u062f\u06cc \u062c\u0627\u0646\u06cc \u0686\u0627\u06c1\u06cc\u06d2\u06d4<\/p>\n<h3 id=\"heading-step-5-bootstrap-confidence-intervals\">\u0645\u0631\u062d\u0644\u06c1 5: \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0627 \u0648\u0642\u0641\u06c1<\/h3>\n<p>\u0627\u0633 \u0645\u0636\u0645\u0648\u0646 \u0645\u06cc\u06ba \u062a\u0645\u0627\u0645 \u0646\u0642\u0637\u06c1 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0645\u062d\u062f\u0648\u062f \u0646\u0645\u0648\u0646\u0648\u06ba \u0633\u06d2 \u0648\u0627\u062d\u062f \u0646\u0645\u0628\u0631 \u06c1\u06cc\u06ba\u06d4 \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e\u0633 \u0627\u0633 \u0628\u0627\u062a \u06a9\u0627 \u0627\u0646\u062f\u0627\u0632\u06c1 \u0644\u06af\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u06a9\u06c1 \u062f\u0648\u0628\u0627\u0631\u06c1 \u0646\u0645\u0648\u0646\u06d2 \u0644\u06cc\u0646\u06d2 \u067e\u0631 \u0648\u06c1 \u0646\u0645\u0628\u0631 \u06a9\u062a\u0646\u06d2 \u0634\u0641\u0679 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062c\u0648 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0627 \u0648\u0642\u0641\u06c1 \u0628\u06cc\u0627\u0646 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">def bootstrap_ci(df, cutoff, bw, quadratic=False, n_reps=500, seed=7):\n    rng = np.random.default_rng(seed)\n    near = df[(df.query_confidence > cutoff - bw)\n              & (df.query_confidence < cutoff + bw)].copy()\n    near[\"below_cutoff\"] = (near.query_confidence < cutoff).astype(int)\n    near[\"rc\"] = near.query_confidence - cutoff\n    if quadratic:\n        near[\"rc2\"] = near.rc ** 2\n        formula = (\"task_completed ~ below_cutoff + rc + below_cutoff:rc\"\n                   \" + rc2 + below_cutoff:rc2\")\n    else:\n        formula = \"task_completed ~ below_cutoff + rc + below_cutoff:rc\"\n\n    n = len(near)\n    estimates = np.empty(n_reps)\n    for i in range(n_reps):\n        sample = near.iloc[rng.integers(0, n, size=n)]\n        m = smf.ols(formula, data=sample).fit()\n        estimates[i] = m.params[\"below_cutoff\"]\n    return (np.percentile(estimates, 2.5), np.percentile(estimates, 97.5))\n\n\nprint(\"Linear RDD (bw=0.10):\")\nlo, hi = bootstrap_ci(df, cutoff, bw=0.10)\nprint(f\"  effect = +0.0548   95% CI: [{lo:+.4f}, {hi:+.4f}]\")\n\nprint(\"nBandwidth sensitivity:\")\nfor bw, eff in [(0.05, 0.0635), (0.10, 0.0548), (0.15, 0.0618), (0.20, 0.0614)]:\n    lo, hi = bootstrap_ci(df, cutoff, bw=bw)\n    print(f\"  bw = {bw:.2f}   effect = {eff:+.4f}   \"\n          f\"95% CI: [{lo:+.4f}, {hi:+.4f}]\")\n\nprint(\"nQuadratic RDD (bw=0.10):\")\nlo, hi = bootstrap_ci(df, cutoff, bw=0.10, quadratic=True)\nprint(f\"  effect = +0.0569   95% CI: [{lo:+.4f}, {hi:+.4f}]\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Linear RDD (bw=0.10):\n  effect = +0.0548   95% CI: [+0.0278, +0.0817]\n\nBandwidth sensitivity:\n  bw = 0.05   effect = +0.0635   95% CI: [+0.0244, +0.0986]\n  bw = 0.10   effect = +0.0548   95% CI: [+0.0278, +0.0817]\n  bw = 0.15   effect = +0.0618   95% CI: [+0.0381, +0.0823]\n  bw = 0.20   effect = +0.0614   95% CI: [+0.0420, +0.0808]\n\nQuadratic RDD (bw=0.10):\n  effect = +0.0569   95% CI: [+0.0205, +0.0959]\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0628\u06cc\u0646\u0688\u0648\u0688\u062a\u06be \u06a9\u06d2 \u0645\u062d\u062f\u0648\u062f \u0688\u06cc\u0679\u0627 \u06a9\u0648 500 \u0628\u0627\u0631 \u0645\u062a\u0628\u0627\u062f\u0644 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\u0627\u0646\u062c\u06cc\u0646\u0626\u0631\u0646\u06af \u06a9\u06d2 \u06a9\u0627\u0645\u060c \u0627\u0639\u062a\u0645\u0627\u062f \u0628\u0691\u06be\u0627\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0679\u0631\u06a9\u0633) RDD \u06a9\u0648 \u062a\u0648\u0691 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u06c1\u0631 \u0628\u0627\u0631 \u062a\u06cc\u0646 \u0642\u062f\u0645\u06cc \u06a9\u062b\u0627\u0641\u062a \u06a9\u06cc \u062c\u0627\u0646\u0686 \u06a9\u0631\u06cc\u06ba\u06d4 \u0627\u06af\u0631 \u06c1\u0645\u06cc\u06ba \u06c1\u06cc\u0631\u0627 \u067e\u06be\u06cc\u0631\u06cc \u06a9\u0627 \u067e\u062a\u06c1 \u0686\u0644\u062a\u0627 \u06c1\u06d2\u060c \u062a\u0648 \u06c1\u0645 \u06cc\u0627 \u062a\u0648 \u0627\u06cc\u06a9 \u0645\u0628\u06c1\u0645 RDD \u067e\u0631 \u0633\u0648\u0626\u0686 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u062c\u0648 \u0645\u0645\u06a9\u0646\u06c1 \u0637\u0648\u0631 \u067e\u0631 \u062d\u062f\u0648\u06ba \u06a9\u0648 \u0633\u0646\u0628\u06be\u0627\u0644\u062a\u0627 \u06c1\u06d2\u060c \u06cc\u0627 \u06c1\u0645 \u0646\u0642\u0637\u06c1 \u0646\u0638\u0631 \u06a9\u0648 \u062a\u0631\u06a9 \u06a9\u0631 \u062f\u06cc\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p><strong>\u062f\u0648\u0633\u0631\u06cc \u067e\u0627\u0644\u06cc\u0633\u06cc\u0627\u06ba \u0628\u06be\u06cc \u0627\u0633\u06cc \u0628\u0646\u06cc\u0627\u062f \u067e\u0631 \u0644\u0627\u06af\u0648 \u06c1\u0648\u062a\u06cc \u06c1\u06cc\u06ba\u06d4<\/strong> (\u0645\u0641\u0631\u0648\u0636\u06c1 2 \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc)\u06d4 \u0627\u06af\u0631 \u0622\u067e \u06a9\u06d2 \u067e\u0631\u0648\u0688\u06a9\u0679 \u0645\u06cc\u06ba 0.85 \u0645\u06cc\u06ba \u0627\u0636\u0627\u0641\u06cc \u0642\u0648\u0627\u0639\u062f \u0641\u0639\u0627\u0644 \u06c1\u06cc\u06ba (\u0639\u0644\u06cc\u062d\u062f\u06c1 UI \u06c1\u06cc\u0646\u0688\u0644\u0646\u06af\u060c \u0645\u062e\u062a\u0644\u0641 \u0644\u0627\u06af\u0646\u06af \u0644\u06cc\u0648\u0644\u0632\u060c \u0645\u062e\u062a\u0644\u0641 \u062f\u0648\u0628\u0627\u0631\u06c1 \u06a9\u0648\u0634\u0634 \u06a9\u06cc \u067e\u0627\u0644\u06cc\u0633\u06cc\u0627\u06ba)\u060c RDD \u062f\u06cc\u06af\u0631 \u067e\u0627\u0644\u06cc\u0633\u06cc \u0627\u062b\u0631\u0627\u062a \u0633\u06d2 \u0631\u0648\u0679\u0646\u06af \u0627\u062b\u0631\u0627\u062a \u06a9\u0648 \u0627\u0644\u06af \u0646\u06c1\u06cc\u06ba \u06a9\u0631 \u0633\u06a9\u06d2 \u06af\u0627\u06d4 \u0627\u0646 \u062a\u0645\u0627\u0645 \u0627\u0646\u062f\u0631\u0627\u062c\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u067e\u0648\u0631\u06cc \u0631\u0648\u0644 \u0628\u06a9 \u06a9\u0627 \u0622\u0688\u0679 \u06a9\u0631\u06cc\u06ba \u062c\u0648 \u062d\u062f \u06a9\u0627 \u0627\u0634\u062a\u0631\u0627\u06a9 \u06a9\u0631\u062a\u06cc \u06c1\u06cc\u06ba\u06d4<\/p>\n<p><strong>\u062f\u06c1\u0644\u06cc\u0632 \u0645\u06cc\u06ba \u0634\u0648\u0631 \u06cc\u0627 \u0627\u0648\u0648\u0631 \u0631\u0627\u0626\u0688 \u06c1\u06d2\u06d4<\/strong> (\u06cc\u06c1 \u0633\u0627\u062e\u062a\u06cc \u0645\u0639\u0646\u0648\u06ba \u0645\u06cc\u06ba \u0645\u0641\u0631\u0648\u0636\u06c1 1 \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc \u06a9\u0631\u062a\u0627 \u06c1\u06d2)\u06d4 \u06c1\u0648\u0633\u06a9\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0631\u0648\u0679\u0646\u06af 0.85 \u067e\u0631 \u0633\u062e\u062a\u06cc \u0633\u06d2 \u0637\u06d2 \u0634\u062f\u06c1 \u0646\u06c1 \u06c1\u0648\u06d4 \u0628\u06d2 \u062a\u0631\u062a\u06cc\u0628 \u06af\u06be\u0645\u0628\u06cc\u0631 \u06c1\u0648 \u0633\u06a9\u062a\u06cc \u06c1\u06d2 \u0627\u0648\u0631 \u0628\u0639\u0636 \u0635\u0648\u0631\u062a\u0648\u06ba \u0645\u06cc\u06ba \u062f\u0648\u0633\u0631\u0627 \u0627\u0635\u0648\u0644 \u0628\u0646\u06cc\u0627\u062f\u06cc \u0627\u0635\u0648\u0644 \u06a9\u0648 \u0627\u0648\u0648\u0631 \u0631\u0627\u0626\u06cc\u0688 \u06a9\u0631 \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u062c\u0628 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0645\u0627\u0688\u0644\u0632 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0645\u062e\u062a\u0635 \u06a9\u0631\u0646\u0627 \u0627\u06cc\u06a9 \u0645\u062a\u0639\u06cc\u0646 \u062e\u0635\u0648\u0635\u06cc\u062a \u0646\u06c1\u06cc\u06ba \u06c1\u06d2\u06d4 <code>query_confidence<\/code>\u0627\u06cc\u0633\u06d2 \u0645\u0628\u06c1\u0645 RDDs \u06c1\u06cc\u06ba \u062c\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u06cc\u06a9 \u0627\u0646\u0633\u0679\u0631\u0648\u0645\u06cc\u0646\u0679\u0644 \u0645\u062a\u063a\u06cc\u0631 \u0641\u0631\u06cc\u0645 \u0648\u0631\u06a9 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4 \u06a9\u06c1 <code>rdrobust<\/code> \u067e\u06cc\u06a9\u06cc\u062c\u0646\u06af \u0645\u06cc\u06ba \u062a\u06cc\u0632 \u0627\u0648\u0631 \u0645\u0628\u06c1\u0645 \u0688\u06cc\u0632\u0627\u0626\u0646 \u062f\u0648\u0646\u0648\u06ba \u0634\u0627\u0645\u0644 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p><strong>\u06af\u06be\u0645\u0627\u0624 \u0686\u06be\u0644\u0627\u0646\u06af \u06a9\u06d2 \u0637\u0648\u0631 \u067e\u0631 \u0628\u06be\u06cc\u0633 \u0645\u06cc\u06ba<\/strong> (\u0644\u06a9\u06cc\u0631\u06cc \u0642\u0631\u0628\u062a \u06a9\u0648 \u062a\u0648\u0691\u0646\u0627 \u062c\u0648 \u06a9\u0679 \u0622\u0641 \u067e\u0631 \u0634\u0646\u0627\u062e\u062a \u06a9\u06cc \u062d\u0645\u0627\u06cc\u062a \u06a9\u0631\u062a\u0627 \u06c1\u06d2) \u062a\u06cc\u0632 \u0622\u0631 \u0688\u06cc \u0688\u06cc \u0641\u0631\u0636 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0644\u06a9\u06cc\u0631\u06cc \u0627\u06cc\u06a9 \u0645\u0639\u0642\u0648\u0644 \u0645\u0642\u0627\u0645\u06cc \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0628\u0646\u06cc\u0627\u062f\u06cc \u0646\u062a\u06cc\u062c\u06c1 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0627 \u0631\u0634\u062a\u06c1 \u0645\u0636\u0628\u0648\u0637\u06cc \u0633\u06d2 \u0645\u0691\u0627 \u06c1\u0648\u0627 \u06c1\u06d2\u060c \u062a\u0648 \u0627\u06cc\u06a9 \u0644\u06a9\u06cc\u0631\u06cc \u062a\u0635\u0631\u06cc\u062d \u0648\u06a9\u0631 \u06a9\u0648 \u0686\u06be\u0644\u0627\u0646\u06af \u0644\u06af\u0627\u0646\u06d2 \u06a9\u06cc \u063a\u0644\u0637\u06cc \u06a9\u0631 \u0633\u06a9\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u0645\u0631\u062d\u0644\u06c1 4 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u062b\u0627\u0646\u0648\u06cc \u0645\u0636\u0628\u0648\u0637\u06cc \u0679\u06cc\u0633\u0679 \u0645\u0639\u06cc\u0627\u0631\u06cc \u062a\u0634\u062e\u06cc\u0635 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0644\u06a9\u06cc\u0631\u06cc \u0627\u0648\u0631 \u0686\u0648\u06a9\u0648\u0631 \u0645\u0645\u0627\u062b\u0644 \u0646\u06c1\u06cc\u06ba \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0628\u06cc\u0646\u0688\u0648\u062a\u06be \u06a9\u0648 \u0686\u0648\u0691\u0627 \u06a9\u0631\u06cc\u06ba \u0627\u0648\u0631 \u062f\u0648\u0646\u0648\u06ba \u06a9\u0648 \u062f\u0648\u0628\u0627\u0631\u06c1 \u0686\u0644\u0627\u0626\u06cc\u06ba\u06d4<\/p>\n<p><strong>extrapolation \u062a\u0639\u0635\u0628<\/strong> (\u062a\u0633\u0644\u0633\u0644 \u06a9\u06d2 \u0645\u0633\u0627\u0626\u0644\u060c \u062a\u0634\u06a9\u06cc\u0644 \u0646\u0648) RDD \u06a9\u06d2 \u062a\u062e\u0645\u06cc\u0646\u06d2 \u0633\u062e\u062a\u06cc \u0633\u06d2 \u06a9\u0679 \u0622\u0641 \u062a\u06a9 \u0645\u062d\u062f\u0648\u062f \u06c1\u06cc\u06ba\u06d4 0.85 \u067e\u0631 +0.06 \u06a9\u0627 \u0627\u062b\u0631 \u06c1\u0645\u06cc\u06ba \u0627\u0633 \u0628\u0627\u0631\u06d2 \u0645\u06cc\u06ba \u06a9\u0686\u06be \u0646\u06c1\u06cc\u06ba \u0628\u062a\u0627\u062a\u0627 \u06a9\u06c1 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af 0.30 \u06cc\u0627 0.99 \u06a9\u06d2 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u06cc\u0627 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u06af\u0631 \u0622\u067e \u0645\u062c\u0645\u0648\u0639\u06cc \u0627\u0648\u0633\u0637 \u0627\u062b\u0631 \u0686\u0627\u06c1\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0622\u067e \u06a9\u0648 \u062f\u0648\u0633\u0631\u06cc \u062a\u06a9\u0646\u06cc\u06a9\u0648\u06ba \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u06d2\u060c \u062c\u06cc\u0633\u06d2 \u06a9\u06c1 \u0631\u062c\u062d\u0627\u0646 \u06a9\u06d2 \u0637\u0631\u06cc\u0642\u06d2\u060c \u0627\u0644\u062c\u06be\u0627\u0624 \u0627\u06cc\u0688\u062c\u0633\u0679\u0645\u0646\u0679 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0631\u062c\u0639\u062a\u060c \u06cc\u0627 \u062d\u0642\u06cc\u0642\u06cc \u062a\u062c\u0631\u0628\u0627\u062a\u06d4<\/p>\n<h2 id=\"heading-what-to-do-next\">\u0622\u06af\u06d2 \u06a9\u06cc\u0627 \u06a9\u0631\u0646\u0627 \u06c1\u06d2\u06d4<\/h2>\n<p>RDD \u0627\u06cc\u06a9 \u0645\u0648\u0632\u0648\u06ba \u0679\u0648\u0644 \u06c1\u06d2 \u062c\u0628 AI \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u06a9\u0648 \u0645\u0633\u0644\u0633\u0644 \u0633\u06a9\u0648\u0631 \u0627\u0648\u0631 \u062a\u06cc\u0632 \u062d\u062f \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u06af\u0631 \u0641\u06cc\u0686\u0631 \u06a9\u0648 \u0635\u0627\u0631\u0641 \u06a9\u06d2 \u0632\u06cc\u0631 \u06a9\u0646\u0679\u0631\u0648\u0644 \u0679\u0648\u06af\u0644 \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u060c \u062a\u0648 \u067e\u0631\u067e\u06cc\u0646\u0633\u0679\u06cc \u0633\u06a9\u0648\u0631 \u06a9\u0627 \u0637\u0631\u06cc\u0642\u06c1 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u0646\u0627\u0633\u0628 \u06c1\u06d2\u06d4 \u062c\u0628 \u06a9\u0627\u0645 \u06a9\u06cc \u062c\u06af\u06c1 \u067e\u0631 \u0645\u0631\u062d\u0644\u06c1 \u0648\u0627\u0631 \u0631\u0648\u0644 \u0622\u0624\u0679 \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2 \u062a\u0648 \u0627\u062e\u062a\u0644\u0627\u0641\u0627\u062a \u0645\u06cc\u06ba \u0641\u0631\u0642 \u0627\u0633 \u06a9\u0648 \u0633\u0646\u0628\u06be\u0627\u0644\u062a\u0627 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u06a9\u0648\u0626\u06cc \u0686\u06cc\u0632 \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a \u0642\u0627\u0628\u0644 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0646\u06c1\u06cc\u06ba \u06c1\u06d2 \u0644\u06cc\u06a9\u0646 \u0627\u0633 \u06a9\u0648 \u0642\u0648\u0627\u0639\u062f \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2 \u062c\u0633 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0628\u06d2 \u062a\u0631\u062a\u06cc\u0628 \u062c\u0632\u0648 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u060c \u062a\u0648 \u0622\u0644\u06c1 \u0645\u062a\u063a\u06cc\u0631 \u0635\u062d\u06cc\u062d \u0627\u0646\u062a\u062e\u0627\u0628 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u067e\u06cc\u062f\u0627\u0648\u0627\u0631 RDD \u062a\u062c\u0632\u06cc\u06c1 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba: <code>rdrobust<\/code> \u0627\u0632\u06af\u0631 \u067e\u06cc\u06a9\u062c\u06d4 \u0628\u06cc\u0646\u0688\u0648\u0688\u062a\u06be \u06a9\u0627 \u0628\u06c1\u062a\u0631\u06cc\u0646 \u0627\u0646\u062a\u062e\u0627\u0628 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 (Calonico, Cattaneo, and Titiunik 2014)\u060c \u062a\u0639\u0635\u0628 \u0633\u06d2 \u062f\u0631\u0633\u062a \u0634\u062f\u06c1 \u0645\u0639\u06cc\u0627\u0631\u06cc \u062e\u0627\u0645\u06cc\u0627\u06ba\u060c \u0627\u0648\u0631 \u0628\u0644\u0679 \u0627\u0646 \u067e\u0644\u0627\u0679\u0646\u06af \u06cc\u0648\u0679\u06cc\u0644\u06cc\u0679\u06cc\u0632\u06d4 \u0633\u0627\u062a\u06be\u06cc <code>rddensity<\/code> \u067e\u06cc\u06a9\u062c \u0645\u06cc\u06a9\u06a9\u0631\u06cc \u06a9\u062b\u0627\u0641\u062a \u0679\u06cc\u0633\u0679 \u06a9\u0648 \u0646\u0627\u0641\u0630 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0633\u06d2 \u063a\u06cc\u0631 \u0631\u0633\u0645\u06cc \u0637\u0648\u0631 \u067e\u0631 \u0645\u0631\u062d\u0644\u06c1 3 \u0645\u06cc\u06ba \u062f\u06cc\u06a9\u06be\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u06a9\u0627 \u0627\u0628\u062a\u062f\u0627\u0626\u06cc \u0648\u0631\u0698\u0646 \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u06a9\u0648 \u0638\u0627\u06c1\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 rd-packages \u0627\u0633\u0679\u06cc\u06a9 \u062c\u0627\u0626\u0632\u06c1 \u0644\u06cc\u0646\u06d2 \u0648\u0627\u0644\u0648\u06ba \u06a9\u0648 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u06cc\u06a9 \u0686\u06cc\u0632 \u062c\u0648 \u0644\u06cc\u0679 \u0646\u06c1\u06cc\u06ba \u06a9\u0631\u062a\u06cc \u06c1\u06d2 \u0648\u06c1 \u0622\u067e \u06a9\u0648 \u0627\u0646 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u062b\u0631 \u0628\u062a\u0627\u062a\u06cc \u06c1\u06d2 \u062c\u0648 \u06a9\u0679 \u0622\u0641 \u0633\u06d2 \u062f\u0648\u0631 \u06c1\u06cc\u06ba\u06d4 \u0627\u06af\u0631 \u0622\u067e \u06a9\u06cc \u067e\u0627\u0626\u067e \u0644\u0627\u0626\u0646 \u0645\u06cc\u06ba \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u06a9\u0648 \u0628\u0631\u0642\u0631\u0627\u0631 \u0631\u06a9\u06be\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 0.85 \u0633\u06d2 +0.06 \u0644\u06cc\u0679 \u06a9\u0627\u0641\u06cc \u06c1\u06d2\u060c \u062a\u0648 \u0622\u067e \u06a9\u0627 \u06a9\u0627\u0645 \u06c1\u0648 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0622\u067e \u06a9\u0648 \u06cc\u06c1 \u062c\u0627\u0646\u0646\u06d2 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u06d2 \u06a9\u06c1 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0627\u0646 \u0622\u0633\u0627\u0646 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u06cc\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0622\u067e \u0641\u06cc \u0627\u0644\u062d\u0627\u0644 \u0633\u0633\u062a\u06d2 \u0645\u06cc\u06ba \u0628\u06be\u06cc\u062c\u062a\u06d2 \u06c1\u06cc\u06ba (\u06cc\u0627 \u0646\u06cc\u0686\u06d2 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u0633\u0628 \u0633\u06d2 \u0645\u0634\u06a9\u0644 \u0633\u0648\u0627\u0644\u0627\u062a)\u060c \u0627\u06af\u0644\u0627 \u0645\u0631\u062d\u0644\u06c1 \u0627\u0633 \u0639\u0644\u0627\u0642\u06d2 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0686\u06be\u0648\u0679\u0627 \u0628\u06d2 \u062a\u0631\u062a\u06cc\u0628 \u0631\u0648\u0644 \u0622\u0624\u0679 \u06c1\u06d2\u060c \u062c\u0648 RDD \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u06d2 \u062e\u0644\u0627\u0641 \u06a9\u06cc\u0644\u06cc\u0628\u0631\u06cc\u0634\u0646 \u0686\u06cc\u06a9 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u0633\u06a9\u0648\u0631 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0628\u063a\u06cc\u0631 \u062b\u0628\u0648\u062a \u06a9\u06d2 \u062f\u06cc\u0631 \u06a9\u0648 \u0639\u0627\u0645 \u0646\u06c1 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0633\u0627\u062a\u06be\u06cc \u0646\u0648\u0679 \u0628\u06a9 \u06cc\u06c1\u0627\u06ba GitHub \u067e\u0631 \u06c1\u06d2\u06d4 \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u06cc\u06ba\u060c \u0627\u06cc\u06a9 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0628\u0646\u0627\u0626\u06cc\u06ba\u060c \u0627\u0648\u0631 \u0627\u0633\u06d2 \u0686\u0644\u0627\u0626\u06cc\u06ba\u06d4 <code>rdd_demo.ipynb<\/code> \u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0633\u06d2 \u062a\u0645\u0627\u0645 \u06a9\u0648\u0688 \u0628\u0644\u0627\u06a9\u0633 \u06a9\u0648 \u062f\u0648\u0628\u0627\u0631\u06c1 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>\u062a\u06be\u0631\u06cc\u0634\u0648\u0644\u0688 \u0631\u0648\u0679\u0646\u06af \u067e\u0631\u0648\u0688\u06a9\u0634\u0646 LLM \u0633\u0633\u0679\u0645\u0632 \u0645\u06cc\u06ba \u0633\u0628 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0639\u0627\u0645 \u0646\u0645\u0648\u0646\u0648\u06ba \u0645\u06cc\u06ba \u0633\u06d2 \u0627\u06cc\u06a9 \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0627\u0633\u0679\u06cc\u06a9 \u0645\u06cc\u06ba \u06c1\u0631 \u0627\u0639\u062a\u0645\u0627\u062f \u067e\u0631 \u0645\u0628\u0646\u06cc \u0631\u0648\u0679\u0646\u06af \u06a9\u0627 \u0641\u06cc\u0635\u0644\u06c1 \u0645\u0645\u06a9\u0646\u06c1 RDD \u06c1\u06d2\u06d4 \u062a\u062c\u0632\u06cc\u06c1 \u0686\u0644\u0627\u0626\u06cc\u06ba\u06d4<\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>LLM \u067e\u0631 \u0645\u0628\u0646\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u0648\u062c\u06c1 \u06a9\u0627 \u0627\u0646\u062f\u0627\u0632\u06c1 \u0627\u06cc\u06a9 \u0633\u0648\u0627\u0644 \u0633\u06d2 \u0634\u0631\u0648\u0639 \u06c1\u0648\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0627\u06cc\u0688\u06cc\u0679\u0631\u0632 \u06a9\u0633\u06cc \u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u0648 \u0644\u0627\u0646\u0686 \u06a9\u0631\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2 \u067e\u0648\u0686\u06be\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06a9\u06cc\u0627 \u062a\u0628\u062f\u06cc\u0644\u06cc \u0646\u06d2 \u062d\u0642\u06cc\u0642\u062a \u0645\u06cc\u06ba \u0645\u06cc\u0679\u0631\u06a9 \u06a9\u0648 \u0645\u0646\u062a\u0642\u0644 \u06a9\u06cc\u0627\u060c \u06cc\u0627 \u06a9\u06cc\u0627 \u0645\u06cc\u0679\u0631\u06a9 \u0635\u0631\u0641 \u0645\u0646\u062a\u0642\u0644 \u06c1\u0648\u0627\u061f \u0641\u0631\u0636 \u06a9\u0631\u06cc\u06ba \u06a9\u06c1 \u0622\u067e \u06a9\u06cc \u0679\u06cc\u0645 \u0646\u06d2 \u0627\u06cc\u06a9 \u0631\u0648\u0679\u0646\u06af \u067e\u0631\u062a \u0628\u0646\u0627\u0626\u06cc \u06c1\u06d2 \u062c\u0648 \u0622\u0646\u06d2 [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":23759,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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