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Appendix 3: Significance of a
Chi-Square Statistic

       For 30 or fewer degrees of freedom, an exact series expansion is used; otherwise
       the Peizer-Pratt approximation is used.


Notation
       The following notation is used in this appendix:

           X                Value of the chi-square statistic
           k                Degrees of freedom
           Q                Significance level (right-tail probability)



Computation
       •       If X ≤ 0 or k < 1,


               Q =1


       •       If k = 1 ,


               Q = 2Q N     e Xj
               where Q N    e X j is the standard normal one-tailed significance probability.
       •       For k ≤ 30 , an exact series expansion is used (Abramowitz and Stegun, 1965,
               eqs. 26.4.4 and 26.4.5)




                                                555
556 Appendix 3



                   R
                   |2Q d X i + R 2 expF − X I        k odd
                   |
                   |
                            N
                                  π   H2K
                 Q=S
                   | −X
                   |expF I × a1 + Rf
                   | H2K
                   T                                 k even


                 where


                   Ra f
                   |     k −1 2   r −1 2
                   | ∑ 1⋅ 3Ka2r − 1f
                   |
                            X
                                            k odd
                          r =1
                   |
                 R=S
                   |a f
                   |     k −2 2   r
                   | ∑ 2 ⋅ 4XK2r
                   |                        k even
                   T      r =1



            •    If k > 30 , the Peizer-Pratt approximation is used (Peizer and Pratt, 1968, eq
                 2.24a).

            •    If X ≥ 150 ,


                 Q=0

                 otherwise


                 Q = QN Z  bg
                 where
                                                                     Appendix 3 557



             R
             |FG −F 1 + 0.08 I IJ d 2k − 2 i
             |H H 3 k K K                                      if X = k − 1
             |
             |
           Z=S
             |
             |F d k − 1 logF k − 1I + X − k − 1 I
             |GH a f H X K a f JK                      a f
                                                    X − k −1   if X ≠ k − 1
             |
             T
           where


           d = X − k + 2 3 − 0.08 k


       •   If Z < 0 ,


           Q = 1 − QN Z bg

References
       Peizer and Pratt (1968)