libs/math/doc/html/math_toolkit/stat_tut/weg/nccs_eg/nccs_power_eg.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Tables of the power function of the chi2 test.</title>
   5 <link rel="stylesheet" href="../../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../../../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../nccs_eg.html" title="Non Central Chi Squared Example">
   9 <link rel="prev" href="../nccs_eg.html" title="Non Central Chi Squared Example">
  10 <link rel="next" href="../error_eg.html" title="Error Handling Example">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="../nccs_eg.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../nccs_eg.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../error_eg.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.nccs_eg.nccs_power_eg"></a><a class="link" href="nccs_power_eg.html" title="Tables of the power function of the chi2 test.">Tables
  28           of the power function of the chi<sup>2</sup> test.</a>
  29 </h5></div></div></div>
  30 <p>
  31             This example computes a table of the power of the &#967;<sup>2</sup>
  32 test at the 5% significance
  33             level, for various degrees of freedom and non-centrality parameters.
  34             The table is deliberately the same as Table 6 from "The Non-Central
  35             &#967;<sup>2</sup> and F-Distributions and their applications.", P. B. Patnaik, Biometrika,
  36             Vol. 36, No. 1/2 (June 1949), 202-232.
  37           </p>
  38 <p>
  39             First we need some includes to access the non-central chi squared distribution
  40             (and some basic std output of course).
  41           </p>
  42 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">non_central_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  43 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chi_squared</span><span class="special">;</span>
  44 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">non_central_chi_squared</span><span class="special">;</span>
  45
  46 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  47 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
  48 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
  49
  50 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
  51 <span class="special">{</span>
  52 </pre>
  53 <p>
  54             Create a table of the power of the &#967;<sup>2</sup> test at 5% significance level, start
  55             with a table header:
  56           </p>
  57 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"[table\n[[[nu]]"</span><span class="special">;</span>
  58 <span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">lam</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">&lt;=</span> <span class="number">20</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
  59 <span class="special">{</span>
  60    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"[[lambda]="</span> <span class="special">&lt;&lt;</span> <span class="identifier">lam</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span><span class="special">;</span>
  61 <span class="special">}</span>
  62 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"]\n"</span><span class="special">;</span>
  63 </pre>
  64 <p>
  65             (Note: the enclosing [] brackets are to format as a table in Boost.Quickbook).
  66           </p>
  67 <p>
  68             Enumerate the rows and columns and print the power of the test for each
  69             table cell:
  70           </p>
  71 <pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">n</span> <span class="special">&lt;=</span> <span class="number">20</span><span class="special">;</span> <span class="special">++</span><span class="identifier">n</span><span class="special">)</span>
  72 <span class="special">{</span>
  73    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"[["</span> <span class="special">&lt;&lt;</span> <span class="identifier">n</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span><span class="special">;</span>
  74    <span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">lam</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">&lt;=</span> <span class="number">20</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
  75    <span class="special">{</span>
  76 </pre>
  77 <p>
  78             Calculate the &#967;<sup>2</sup> statistic for a 5% significance:
  79           </p>
  80 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">cs</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">chi_squared</span><span class="special">(</span><span class="identifier">n</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span>
  81 </pre>
  82 <p>
  83             The power of the test is given by the complement of the CDF of the non-central
  84             &#967;<sup>2</sup> distribution:
  85           </p>
  86 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">lam</span><span class="special">),</span> <span class="identifier">cs</span><span class="special">));</span>
  87 </pre>
  88 <p>
  89             Then output the cell value:
  90           </p>
  91 <pre class="programlisting">         <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"["</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">beta</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span><span class="special">;</span>
  92       <span class="special">}</span>
  93       <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  94    <span class="special">}</span>
  95    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  96 <span class="special">}</span>
  97 </pre>
  98 <p>
  99             The output from this program is a table in Boost.Quickbook format as
 100             shown below.
 101           </p>
 102 <p>
 103             We can interpret this as follows - for example if &#957;=10 and &#955;=10 then the
 104             power of the test is 0.542 - so we have only a 54% chance of correctly
 105             detecting that our null hypothesis is false, and a 46% chance of incurring
 106             a type II error (failing to reject the null hypothesis when it is in
 107             fact false):
 108           </p>
 109 <div class="informaltable"><table class="table">
 110 <colgroup>
 111 <col>
 112 <col>
 113 <col>
 114 <col>
 115 <col>
 116 <col>
 117 <col>
 118 <col>
 119 <col>
 120 <col>
 121 <col>
 122 </colgroup>
 123 <thead><tr>
 124 <th>
 125                     <p>
 126                       &#957;
 127                     </p>
 128                   </th>
 129 <th>
 130                     <p>
 131                       &#955;=2
 132                     </p>
 133                   </th>
 134 <th>
 135                     <p>
 136                       &#955;=4
 137                     </p>
 138                   </th>
 139 <th>
 140                     <p>
 141                       &#955;=6
 142                     </p>
 143                   </th>
 144 <th>
 145                     <p>
 146                       &#955;=8
 147                     </p>
 148                   </th>
 149 <th>
 150                     <p>
 151                       &#955;=10
 152                     </p>
 153                   </th>
 154 <th>
 155                     <p>
 156                       &#955;=12
 157                     </p>
 158                   </th>
 159 <th>
 160                     <p>
 161                       &#955;=14
 162                     </p>
 163                   </th>
 164 <th>
 165                     <p>
 166                       &#955;=16
 167                     </p>
 168                   </th>
 169 <th>
 170                     <p>
 171                       &#955;=18
 172                     </p>
 173                   </th>
 174 <th>
 175                     <p>
 176                       &#955;=20
 177                     </p>
 178                   </th>
 179 </tr></thead>
 180 <tbody>
 181 <tr>
 182 <td>
 183                     <p>
 184                       2
 185                     </p>
 186                   </td>
 187 <td>
 188                     <p>
 189                       0.226
 190                     </p>
 191                   </td>
 192 <td>
 193                     <p>
 194                       0.415
 195                     </p>
 196                   </td>
 197 <td>
 198                     <p>
 199                       0.584
 200                     </p>
 201                   </td>
 202 <td>
 203                     <p>
 204                       0.718
 205                     </p>
 206                   </td>
 207 <td>
 208                     <p>
 209                       0.815
 210                     </p>
 211                   </td>
 212 <td>
 213                     <p>
 214                       0.883
 215                     </p>
 216                   </td>
 217 <td>
 218                     <p>
 219                       0.928
 220                     </p>
 221                   </td>
 222 <td>
 223                     <p>
 224                       0.957
 225                     </p>
 226                   </td>
 227 <td>
 228                     <p>
 229                       0.974
 230                     </p>
 231                   </td>
 232 <td>
 233                     <p>
 234                       0.985
 235                     </p>
 236                   </td>
 237 </tr>
 238 <tr>
 239 <td>
 240                     <p>
 241                       3
 242                     </p>
 243                   </td>
 244 <td>
 245                     <p>
 246                       0.192
 247                     </p>
 248                   </td>
 249 <td>
 250                     <p>
 251                       0.359
 252                     </p>
 253                   </td>
 254 <td>
 255                     <p>
 256                       0.518
 257                     </p>
 258                   </td>
 259 <td>
 260                     <p>
 261                       0.654
 262                     </p>
 263                   </td>
 264 <td>
 265                     <p>
 266                       0.761
 267                     </p>
 268                   </td>
 269 <td>
 270                     <p>
 271                       0.84
 272                     </p>
 273                   </td>
 274 <td>
 275                     <p>
 276                       0.896
 277                     </p>
 278                   </td>
 279 <td>
 280                     <p>
 281                       0.934
 282                     </p>
 283                   </td>
 284 <td>
 285                     <p>
 286                       0.959
 287                     </p>
 288                   </td>
 289 <td>
 290                     <p>
 291                       0.975
 292                     </p>
 293                   </td>
 294 </tr>
 295 <tr>
 296 <td>
 297                     <p>
 298                       4
 299                     </p>
 300                   </td>
 301 <td>
 302                     <p>
 303                       0.171
 304                     </p>
 305                   </td>
 306 <td>
 307                     <p>
 308                       0.32
 309                     </p>
 310                   </td>
 311 <td>
 312                     <p>
 313                       0.47
 314                     </p>
 315                   </td>
 316 <td>
 317                     <p>
 318                       0.605
 319                     </p>
 320                   </td>
 321 <td>
 322                     <p>
 323                       0.716
 324                     </p>
 325                   </td>
 326 <td>
 327                     <p>
 328                       0.802
 329                     </p>
 330                   </td>
 331 <td>
 332                     <p>
 333                       0.866
 334                     </p>
 335                   </td>
 336 <td>
 337                     <p>
 338                       0.912
 339                     </p>
 340                   </td>
 341 <td>
 342                     <p>
 343                       0.943
 344                     </p>
 345                   </td>
 346 <td>
 347                     <p>
 348                       0.964
 349                     </p>
 350                   </td>
 351 </tr>
 352 <tr>
 353 <td>
 354                     <p>
 355                       5
 356                     </p>
 357                   </td>
 358 <td>
 359                     <p>
 360                       0.157
 361                     </p>
 362                   </td>
 363 <td>
 364                     <p>
 365                       0.292
 366                     </p>
 367                   </td>
 368 <td>
 369                     <p>
 370                       0.433
 371                     </p>
 372                   </td>
 373 <td>
 374                     <p>
 375                       0.564
 376                     </p>
 377                   </td>
 378 <td>
 379                     <p>
 380                       0.677
 381                     </p>
 382                   </td>
 383 <td>
 384                     <p>
 385                       0.769
 386                     </p>
 387                   </td>
 388 <td>
 389                     <p>
 390                       0.839
 391                     </p>
 392                   </td>
 393 <td>
 394                     <p>
 395                       0.89
 396                     </p>
 397                   </td>
 398 <td>
 399                     <p>
 400                       0.927
 401                     </p>
 402                   </td>
 403 <td>
 404                     <p>
 405                       0.952
 406                     </p>
 407                   </td>
 408 </tr>
 409 <tr>
 410 <td>
 411                     <p>
 412                       6
 413                     </p>
 414                   </td>
 415 <td>
 416                     <p>
 417                       0.146
 418                     </p>
 419                   </td>
 420 <td>
 421                     <p>
 422                       0.27
 423                     </p>
 424                   </td>
 425 <td>
 426                     <p>
 427                       0.403
 428                     </p>
 429                   </td>
 430 <td>
 431                     <p>
 432                       0.531
 433                     </p>
 434                   </td>
 435 <td>
 436                     <p>
 437                       0.644
 438                     </p>
 439                   </td>
 440 <td>
 441                     <p>
 442                       0.738
 443                     </p>
 444                   </td>
 445 <td>
 446                     <p>
 447                       0.813
 448                     </p>
 449                   </td>
 450 <td>
 451                     <p>
 452                       0.869
 453                     </p>
 454                   </td>
 455 <td>
 456                     <p>
 457                       0.911
 458                     </p>
 459                   </td>
 460 <td>
 461                     <p>
 462                       0.94
 463                     </p>
 464                   </td>
 465 </tr>
 466 <tr>
 467 <td>
 468                     <p>
 469                       7
 470                     </p>
 471                   </td>
 472 <td>
 473                     <p>
 474                       0.138
 475                     </p>
 476                   </td>
 477 <td>
 478                     <p>
 479                       0.252
 480                     </p>
 481                   </td>
 482 <td>
 483                     <p>
 484                       0.378
 485                     </p>
 486                   </td>
 487 <td>
 488                     <p>
 489                       0.502
 490                     </p>
 491                   </td>
 492 <td>
 493                     <p>
 494                       0.614
 495                     </p>
 496                   </td>
 497 <td>
 498                     <p>
 499                       0.71
 500                     </p>
 501                   </td>
 502 <td>
 503                     <p>
 504                       0.788
 505                     </p>
 506                   </td>
 507 <td>
 508                     <p>
 509                       0.849
 510                     </p>
 511                   </td>
 512 <td>
 513                     <p>
 514                       0.895
 515                     </p>
 516                   </td>
 517 <td>
 518                     <p>
 519                       0.928
 520                     </p>
 521                   </td>
 522 </tr>
 523 <tr>
 524 <td>
 525                     <p>
 526                       8
 527                     </p>
 528                   </td>
 529 <td>
 530                     <p>
 531                       0.131
 532                     </p>
 533                   </td>
 534 <td>
 535                     <p>
 536                       0.238
 537                     </p>
 538                   </td>
 539 <td>
 540                     <p>
 541                       0.357
 542                     </p>
 543                   </td>
 544 <td>
 545                     <p>
 546                       0.477
 547                     </p>
 548                   </td>
 549 <td>
 550                     <p>
 551                       0.588
 552                     </p>
 553                   </td>
 554 <td>
 555                     <p>
 556                       0.685
 557                     </p>
 558                   </td>
 559 <td>
 560                     <p>
 561                       0.765
 562                     </p>
 563                   </td>
 564 <td>
 565                     <p>
 566                       0.829
 567                     </p>
 568                   </td>
 569 <td>
 570                     <p>
 571                       0.879
 572                     </p>
 573                   </td>
 574 <td>
 575                     <p>
 576                       0.915
 577                     </p>
 578                   </td>
 579 </tr>
 580 <tr>
 581 <td>
 582                     <p>
 583                       9
 584                     </p>
 585                   </td>
 586 <td>
 587                     <p>
 588                       0.125
 589                     </p>
 590                   </td>
 591 <td>
 592                     <p>
 593                       0.225
 594                     </p>
 595                   </td>
 596 <td>
 597                     <p>
 598                       0.339
 599                     </p>
 600                   </td>
 601 <td>
 602                     <p>
 603                       0.454
 604                     </p>
 605                   </td>
 606 <td>
 607                     <p>
 608                       0.564
 609                     </p>
 610                   </td>
 611 <td>
 612                     <p>
 613                       0.661
 614                     </p>
 615                   </td>
 616 <td>
 617                     <p>
 618                       0.744
 619                     </p>
 620                   </td>
 621 <td>
 622                     <p>
 623                       0.811
 624                     </p>
 625                   </td>
 626 <td>
 627                     <p>
 628                       0.863
 629                     </p>
 630                   </td>
 631 <td>
 632                     <p>
 633                       0.903
 634                     </p>
 635                   </td>
 636 </tr>
 637 <tr>
 638 <td>
 639                     <p>
 640                       10
 641                     </p>
 642                   </td>
 643 <td>
 644                     <p>
 645                       0.121
 646                     </p>
 647                   </td>
 648 <td>
 649                     <p>
 650                       0.215
 651                     </p>
 652                   </td>
 653 <td>
 654                     <p>
 655                       0.323
 656                     </p>
 657                   </td>
 658 <td>
 659                     <p>
 660                       0.435
 661                     </p>
 662                   </td>
 663 <td>
 664                     <p>
 665                       0.542
 666                     </p>
 667                   </td>
 668 <td>
 669                     <p>
 670                       0.64
 671                     </p>
 672                   </td>
 673 <td>
 674                     <p>
 675                       0.723
 676                     </p>
 677                   </td>
 678 <td>
 679                     <p>
 680                       0.793
 681                     </p>
 682                   </td>
 683 <td>
 684                     <p>
 685                       0.848
 686                     </p>
 687                   </td>
 688 <td>
 689                     <p>
 690                       0.891
 691                     </p>
 692                   </td>
 693 </tr>
 694 <tr>
 695 <td>
 696                     <p>
 697                       11
 698                     </p>
 699                   </td>
 700 <td>
 701                     <p>
 702                       0.117
 703                     </p>
 704                   </td>
 705 <td>
 706                     <p>
 707                       0.206
 708                     </p>
 709                   </td>
 710 <td>
 711                     <p>
 712                       0.309
 713                     </p>
 714                   </td>
 715 <td>
 716                     <p>
 717                       0.417
 718                     </p>
 719                   </td>
 720 <td>
 721                     <p>
 722                       0.523
 723                     </p>
 724                   </td>
 725 <td>
 726                     <p>
 727                       0.62
 728                     </p>
 729                   </td>
 730 <td>
 731                     <p>
 732                       0.704
 733                     </p>
 734                   </td>
 735 <td>
 736                     <p>
 737                       0.775
 738                     </p>
 739                   </td>
 740 <td>
 741                     <p>
 742                       0.833
 743                     </p>
 744                   </td>
 745 <td>
 746                     <p>
 747                       0.878
 748                     </p>
 749                   </td>
 750 </tr>
 751 <tr>
 752 <td>
 753                     <p>
 754                       12
 755                     </p>
 756                   </td>
 757 <td>
 758                     <p>
 759                       0.113
 760                     </p>
 761                   </td>
 762 <td>
 763                     <p>
 764                       0.198
 765                     </p>
 766                   </td>
 767 <td>
 768                     <p>
 769                       0.297
 770                     </p>
 771                   </td>
 772 <td>
 773                     <p>
 774                       0.402
 775                     </p>
 776                   </td>
 777 <td>
 778                     <p>
 779                       0.505
 780                     </p>
 781                   </td>
 782 <td>
 783                     <p>
 784                       0.601
 785                     </p>
 786                   </td>
 787 <td>
 788                     <p>
 789                       0.686
 790                     </p>
 791                   </td>
 792 <td>
 793                     <p>
 794                       0.759
 795                     </p>
 796                   </td>
 797 <td>
 798                     <p>
 799                       0.818
 800                     </p>
 801                   </td>
 802 <td>
 803                     <p>
 804                       0.866
 805                     </p>
 806                   </td>
 807 </tr>
 808 <tr>
 809 <td>
 810                     <p>
 811                       13
 812                     </p>
 813                   </td>
 814 <td>
 815                     <p>
 816                       0.11
 817                     </p>
 818                   </td>
 819 <td>
 820                     <p>
 821                       0.191
 822                     </p>
 823                   </td>
 824 <td>
 825                     <p>
 826                       0.286
 827                     </p>
 828                   </td>
 829 <td>
 830                     <p>
 831                       0.387
 832                     </p>
 833                   </td>
 834 <td>
 835                     <p>
 836                       0.488
 837                     </p>
 838                   </td>
 839 <td>
 840                     <p>
 841                       0.584
 842                     </p>
 843                   </td>
 844 <td>
 845                     <p>
 846                       0.669
 847                     </p>
 848                   </td>
 849 <td>
 850                     <p>
 851                       0.743
 852                     </p>
 853                   </td>
 854 <td>
 855                     <p>
 856                       0.804
 857                     </p>
 858                   </td>
 859 <td>
 860                     <p>
 861                       0.854
 862                     </p>
 863                   </td>
 864 </tr>
 865 <tr>
 866 <td>
 867                     <p>
 868                       14
 869                     </p>
 870                   </td>
 871 <td>
 872                     <p>
 873                       0.108
 874                     </p>
 875                   </td>
 876 <td>
 877                     <p>
 878                       0.185
 879                     </p>
 880                   </td>
 881 <td>
 882                     <p>
 883                       0.276
 884                     </p>
 885                   </td>
 886 <td>
 887                     <p>
 888                       0.374
 889                     </p>
 890                   </td>
 891 <td>
 892                     <p>
 893                       0.473
 894                     </p>
 895                   </td>
 896 <td>
 897                     <p>
 898                       0.567
 899                     </p>
 900                   </td>
 901 <td>
 902                     <p>
 903                       0.653
 904                     </p>
 905                   </td>
 906 <td>
 907                     <p>
 908                       0.728
 909                     </p>
 910                   </td>
 911 <td>
 912                     <p>
 913                       0.791
 914                     </p>
 915                   </td>
 916 <td>
 917                     <p>
 918                       0.842
 919                     </p>
 920                   </td>
 921 </tr>
 922 <tr>
 923 <td>
 924                     <p>
 925                       15
 926                     </p>
 927                   </td>
 928 <td>
 929                     <p>
 930                       0.105
 931                     </p>
 932                   </td>
 933 <td>
 934                     <p>
 935                       0.179
 936                     </p>
 937                   </td>
 938 <td>
 939                     <p>
 940                       0.267
 941                     </p>
 942                   </td>
 943 <td>
 944                     <p>
 945                       0.362
 946                     </p>
 947                   </td>
 948 <td>
 949                     <p>
 950                       0.459
 951                     </p>
 952                   </td>
 953 <td>
 954                     <p>
 955                       0.552
 956                     </p>
 957                   </td>
 958 <td>
 959                     <p>
 960                       0.638
 961                     </p>
 962                   </td>
 963 <td>
 964                     <p>
 965                       0.713
 966                     </p>
 967                   </td>
 968 <td>
 969                     <p>
 970                       0.777
 971                     </p>
 972                   </td>
 973 <td>
 974                     <p>
 975                       0.83
 976                     </p>
 977                   </td>
 978 </tr>
 979 <tr>
 980 <td>
 981                     <p>
 982                       16
 983                     </p>
 984                   </td>
 985 <td>
 986                     <p>
 987                       0.103
 988                     </p>
 989                   </td>
 990 <td>
 991                     <p>
 992                       0.174
 993                     </p>
 994                   </td>
 995 <td>
 996                     <p>
 997                       0.259
 998                     </p>
 999                   </td>
1000 <td>
1001                     <p>
1002                       0.351
1003                     </p>
1004                   </td>
1005 <td>
1006                     <p>
1007                       0.446
1008                     </p>
1009                   </td>
1010 <td>
1011                     <p>
1012                       0.538
1013                     </p>
1014                   </td>
1015 <td>
1016                     <p>
1017                       0.623
1018                     </p>
1019                   </td>
1020 <td>
1021                     <p>
1022                       0.699
1023                     </p>
1024                   </td>
1025 <td>
1026                     <p>
1027                       0.764
1028                     </p>
1029                   </td>
1030 <td>
1031                     <p>
1032                       0.819
1033                     </p>
1034                   </td>
1035 </tr>
1036 <tr>
1037 <td>
1038                     <p>
1039                       17
1040                     </p>
1041                   </td>
1042 <td>
1043                     <p>
1044                       0.101
1045                     </p>
1046                   </td>
1047 <td>
1048                     <p>
1049                       0.169
1050                     </p>
1051                   </td>
1052 <td>
1053                     <p>
1054                       0.251
1055                     </p>
1056                   </td>
1057 <td>
1058                     <p>
1059                       0.341
1060                     </p>
1061                   </td>
1062 <td>
1063                     <p>
1064                       0.434
1065                     </p>
1066                   </td>
1067 <td>
1068                     <p>
1069                       0.525
1070                     </p>
1071                   </td>
1072 <td>
1073                     <p>
1074                       0.609
1075                     </p>
1076                   </td>
1077 <td>
1078                     <p>
1079                       0.686
1080                     </p>
1081                   </td>
1082 <td>
1083                     <p>
1084                       0.752
1085                     </p>
1086                   </td>
1087 <td>
1088                     <p>
1089                       0.807
1090                     </p>
1091                   </td>
1092 </tr>
1093 <tr>
1094 <td>
1095                     <p>
1096                       18
1097                     </p>
1098                   </td>
1099 <td>
1100                     <p>
1101                       0.0992
1102                     </p>
1103                   </td>
1104 <td>
1105                     <p>
1106                       0.165
1107                     </p>
1108                   </td>
1109 <td>
1110                     <p>
1111                       0.244
1112                     </p>
1113                   </td>
1114 <td>
1115                     <p>
1116                       0.332
1117                     </p>
1118                   </td>
1119 <td>
1120                     <p>
1121                       0.423
1122                     </p>
1123                   </td>
1124 <td>
1125                     <p>
1126                       0.512
1127                     </p>
1128                   </td>
1129 <td>
1130                     <p>
1131                       0.596
1132                     </p>
1133                   </td>
1134 <td>
1135                     <p>
1136                       0.673
1137                     </p>
1138                   </td>
1139 <td>
1140                     <p>
1141                       0.74
1142                     </p>
1143                   </td>
1144 <td>
1145                     <p>
1146                       0.796
1147                     </p>
1148                   </td>
1149 </tr>
1150 <tr>
1151 <td>
1152                     <p>
1153                       19
1154                     </p>
1155                   </td>
1156 <td>
1157                     <p>
1158                       0.0976
1159                     </p>
1160                   </td>
1161 <td>
1162                     <p>
1163                       0.161
1164                     </p>
1165                   </td>
1166 <td>
1167                     <p>
1168                       0.238
1169                     </p>
1170                   </td>
1171 <td>
1172                     <p>
1173                       0.323
1174                     </p>
1175                   </td>
1176 <td>
1177                     <p>
1178                       0.412
1179                     </p>
1180                   </td>
1181 <td>
1182                     <p>
1183                       0.5
1184                     </p>
1185                   </td>
1186 <td>
1187                     <p>
1188                       0.584
1189                     </p>
1190                   </td>
1191 <td>
1192                     <p>
1193                       0.66
1194                     </p>
1195                   </td>
1196 <td>
1197                     <p>
1198                       0.728
1199                     </p>
1200                   </td>
1201 <td>
1202                     <p>
1203                       0.786
1204                     </p>
1205                   </td>
1206 </tr>
1207 <tr>
1208 <td>
1209                     <p>
1210                       20
1211                     </p>
1212                   </td>
1213 <td>
1214                     <p>
1215                       0.0961
1216                     </p>
1217                   </td>
1218 <td>
1219                     <p>
1220                       0.158
1221                     </p>
1222                   </td>
1223 <td>
1224                     <p>
1225                       0.232
1226                     </p>
1227                   </td>
1228 <td>
1229                     <p>
1230                       0.315
1231                     </p>
1232                   </td>
1233 <td>
1234                     <p>
1235                       0.402
1236                     </p>
1237                   </td>
1238 <td>
1239                     <p>
1240                       0.489
1241                     </p>
1242                   </td>
1243 <td>
1244                     <p>
1245                       0.572
1246                     </p>
1247                   </td>
1248 <td>
1249                     <p>
1250                       0.648
1251                     </p>
1252                   </td>
1253 <td>
1254                     <p>
1255                       0.716
1256                     </p>
1257                   </td>
1258 <td>
1259                     <p>
1260                       0.775
1261                     </p>
1262                   </td>
1263 </tr>
1264 </tbody>
1265 </table></div>
1266 <p>
1267             See <a href="../../../../../../example/nc_chi_sq_example.cpp" target="_top">nc_chi_sq_example.cpp</a>
1268             for the full C++ source code.
1269           </p>
1270 </div>
1271 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1272 <td align="left"></td>
1273 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
1274       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
1275       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
1276       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
1277       Daryle Walker and Xiaogang Zhang<p>
1278         Distributed under the Boost Software License, Version 1.0. (See accompanying
1279         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
1280       </p>
1281 </div></td>
1282 </tr></table>
1283 <hr>
1284 <div class="spirit-nav">
1285 <a accesskey="p" href="../nccs_eg.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../nccs_eg.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../error_eg.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
1286 </div>
1287 </body>
1288 </html>