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    "# Polychoric Correlation using two-step and joint method\n",
    "\n",
    "Polychoric correlation in statsample requires installation of the [statsample-bivariate-extension](https://rubygems.org/gems/statsample-bivariate-extension) gem. This gem extends the Statsample::Bivariate class with useful algorithms for polychoric and tetrachoric correlation.\n",
    "\n",
    "Statsample will automatically detect presence of polychoric/tetrachoric algorithms so there is no need to explicitly require the gem.\n",
    "\n",
    "In this example we'll see how polychoric correlation can be performed using statsample."
   ]
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     "text": [
      "Analysis 2015-06-04 12:48:32 +0530\n",
      "= Statsample::Bivariate::Polychoric\n",
      "  == Polychoric with two-step\n",
      "    Contingence Table\n",
      "+-------+-----+-----+-----+-------+\n",
      "|       | Y=0 | Y=1 | Y=2 | Total |\n",
      "+-------+-----+-----+-----+-------+\n",
      "| X = 0 | 52  | 58  | 8   | 118   |\n",
      "| X = 1 | 18  | 15  | 7   | 40    |\n",
      "| X = 2 | 5   | 11  | 3   | 19    |\n",
      "+-------+-----+-----+-----+-------+\n",
      "| T     | 75  | 84  | 18  | 177   |\n",
      "+-------+-----+-----+-----+-------+\n",
      "\n",
      "    r: 0.1683\n",
      "    Thresholds\n",
      "+---------------+---------+\n",
      "|               |  Value  |\n",
      "+---------------+---------+\n",
      "| Threshold X 0 | 0.4307  |\n",
      "| Threshold X 1 | 1.2408  |\n",
      "| Threshold Y 0 | -0.1924 |\n",
      "| Threshold Y 1 | 1.2720  |\n",
      "+---------------+---------+\n",
      "\n",
      "    Iterations: 10\n",
      "    Test of bivariate normality: X^2 = 3.941, df = 3, p= 0.26795\n",
      "  == Polychoric with joint\n",
      "    Contingence Table\n",
      "+-------+-----+-----+-----+-------+\n",
      "|       | Y=0 | Y=1 | Y=2 | Total |\n",
      "+-------+-----+-----+-----+-------+\n",
      "| X = 0 | 52  | 58  | 8   | 118   |\n",
      "| X = 1 | 18  | 15  | 7   | 40    |\n",
      "| X = 2 | 5   | 11  | 3   | 19    |\n",
      "+-------+-----+-----+-----+-------+\n",
      "| T     | 75  | 84  | 18  | 177   |\n",
      "+-------+-----+-----+-----+-------+\n",
      "\n",
      "    r: 0.1682\n",
      "    Thresholds\n",
      "+---------------+---------+\n",
      "|               |  Value  |\n",
      "+---------------+---------+\n",
      "| Threshold X 0 | 0.4296  |\n",
      "| Threshold X 1 | 1.2411  |\n",
      "| Threshold Y 0 | -0.1936 |\n",
      "| Threshold Y 1 | 1.2731  |\n",
      "+---------------+---------+\n",
      "\n",
      "    Iterations: 1\n",
      "    Test of bivariate normality: X^2 = 3.940, df = 3, p= 0.26801\n",
      "  == Polychoric with polychoric series\n",
      "    Contingence Table\n",
      "+-------+------+------+------+-------+\n",
      "|       | Y=0  | Y=1  | Y=2  | Total |\n",
      "+-------+------+------+------+-------+\n",
      "| X = 0 | 52   | 58   | 8    | 118.0 |\n",
      "| X = 1 | 18   | 15   | 7    | 40.0  |\n",
      "| X = 2 | 5    | 11   | 3    | 19.0  |\n",
      "+-------+------+------+------+-------+\n",
      "| T     | 75.0 | 84.0 | 18.0 | 177.0 |\n",
      "+-------+------+------+------+-------+\n",
      "\n",
      "    r: 0.1701\n",
      "    Thresholds\n",
      "+---------------+---------+\n",
      "|               |  Value  |\n",
      "+---------------+---------+\n",
      "| Threshold X 0 | 0.4307  |\n",
      "| Threshold X 1 | 1.2408  |\n",
      "| Threshold Y 0 | -0.1924 |\n",
      "| Threshold Y 1 | 1.2720  |\n",
      "+---------------+---------+\n",
      "\n",
      "    Test of bivariate normality: X^2 = 3.941, df = 3, p= 0.26791\n",
      "\n"
     ]
    }
   ],
   "source": [
    "require 'statsample'\n",
    "\n",
    "Statsample::Analysis.store(Statsample::Bivariate::Polychoric) do \n",
    "  ct=Matrix[[rand(10)+50, rand(10)+50,  rand(10)+1],\n",
    "            [rand(20)+5,  rand(50)+4,   rand(10)+1],\n",
    "            [rand(8)+1,   rand(12)+1,   rand(10)+1]]\n",
    "\n",
    "  # Estimation of polychoric correlation using two-step (default)\n",
    "  poly=polychoric(ct, :name=>\"Polychoric with two-step\", :debug=>false)\n",
    "  summary poly\n",
    "\n",
    "  # Estimation of polychoric correlation using joint method (slow)\n",
    "  poly=polychoric(ct, :method=>:joint, :name=>\"Polychoric with joint\")\n",
    "  summary poly\n",
    "\n",
    "  # Uses polychoric series (not recomended)\n",
    "\n",
    "  poly=polychoric(ct, :method=>:polychoric_series, :name=>\"Polychoric with polychoric series\")\n",
    "  summary poly\n",
    "end\n",
    "\n",
    "Statsample::Analysis.run_batch"
   ]
  }
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