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help on ^gof^                                                    (STB-36: sg68)
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Goodness of fit statistics for multinomial distributions
--------------------------------------------------------

        ^gof^ obs exp [^if^ exp] [^in^ range] [^,^ 
              ^x2 cr g2 lr ft kl x2n la^mbda^(^numeric-list^) by(^varlist^)^ 
              ^df(^#^) li^st ^n(^#^) p^lot ^sc^ale ^sa^ving^(^filename^)^ ]

See help for @numlist@ (if installed) for a description of a numeric-list.


Description
-----------

^gof^ computes goodness-of-fit (gof) statistics between observed (^obs^) and 
expected (^exp^) values from a multinomial distribution. The statistics 
^CR(obs,exp,L)^ belong to the Cressie-Read family of discrepancy measures,

        ^CR(obs,exp,L) = 2 sum obs*((obs/exp)^^L-1) / (L*(L+1))^

with L(ambda) some real parameter, and summation is over all "cells" (catego-
ries). Well-known special cases are:

  L    description                      abbrev.          formula
------------------------------------------------------------------------------
 ^1.0^   Pearson's X2                       X2          sum (obs-exp) ^^2/exp
 ^2/3^   C & R's recommended statistic      CR
 ^0.0^   log-likelihood ratio (deviance)   LR=G2        2 sum obs log(obs/exp)
^-0.5^   Freeman-Tukey's statistic          FT      4 sum (sqrt(obs)-srqt(exp))^^2
^-1.0^   Kullbach-Leibner information       KL          2 sum exp log(exp/obs)
^-2.0^   Neyman's modified X2               X2N         sum (obs-exp)^^2/obs

If the estimate ^exp^ are efficient, the ^CR^ statistics are asymptotically 
first-order equivalent, and chi square distributed ^df^ degrees of freedom, with
df = #cells - #constraints.


Options to select ^L^
-------------------

^x2 cr g2 lr ft kl x2n^ specify that the respective tests are displayed.

^lambda(^numeric-lists^)^ specifies a range of powers L for the Cressie-Read
   statistics. We advise to restrict lambda to values between -2 and 2. See 
   below for an example of a numeric-list.

Specifying none of these options implies all of ^x2 cr lr ft kl x2n^, or, stated
differently, ^lambda(-2 -1 -.5 0 .667 1)^.


Other options
-------------

^by(^varlist^)^ specifies a list of variables on which to aggregate ^obs^ and ^exp^ 
    (join "cells" with the same values on varlist) before computing the 
    goodness-of-fit statistics.

^df^(#^)^ specifies the degrees of freedom used in the computations of chi square
    based approximate significance levels.

^list^ specifies that the table with statistic values is displayed. This option
    is effective only in combination with ^plot^. 

^n(^#^)^ specifies that the variables obs and exp are expressed as proportions
    with the total number of observations equal to #.

^plot^ specifies that a statistic-by-lambda plot is displayed. If ^df^ is speci-
    fied, horizontal lines at the 90%, 95%, and 99% critical values of the 
    (central) chi square distribution are shown. The usual output table is 
    suppressed in this case.

^saving(^filename^)^ specifies the name of a file for the statistic-by-lambda plot.

^scale^ specifies that the expected values may be scaled so that they sum to
    the number of observations. Otherwise obs and exp should have equal sums 
    (within a .001 multiplicative margin).


Examples
--------

    . ^gof obs exp^   
    . ^gof obs exp, df(3)^                           (specify degrees of freedom)
    . ^gof obs exp, df(3) lr x2 by(educate)^      (specify stats and aggregation)

    . ^gof obs exp, df(4) la(-2-2/0.25)^             (specify stats with numlist)
    . ^gof obs exp, df(5) l(-2-2/0.10) plot saving(gof)^             (plot stats)


Author
------

     Jeroen Weesie
     Utrecht University
     Netherlands
     weesie@@weesie.fsw.ruu.nl


References
----------

Cressie, N. A. C. & T. R. C. Read. 1984, Multinomial Goodness-of-Fit Tests, 
   Journal of the Royal Statistical Society, Series B, 46, 440-464.

Read, T. R. C & N. A. C. Cressie. 1988, Goodness-of-Fit Statistics for Discrete
   Multivariate Data, New York, Springer Verlag. 


Technical remarks
-----------------

For ^L^ << 0, the CR-statistic is particularly sensitive for observed values that
  are much smaller than the expected values.
For ^L^ >> 0, the CR-statistic is particularly sensitive for observed values that
  are much larger than the expected values.
Of course, obs and exp should be >= 0. Moreover, ^L^ < 0 requires that all 
  obs > 0, while ^L^ > 0 requires that all exp > 0.


Also see
--------

    STB:  STB-36 sg68
On-line:  help for @numlist@ (if installed)