Friedman's Analysis of Variance Test (and Kendall's Coefficient of Concordance)
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         ^friedman^ varlist {^in^ range] [^if^ exp]

^friedman^ estimates Friedman's nonparametric two-way analysis of variance
^and^ Kendall's Coefficient of Concordance (a descriptive measure of the
agreement between k sets of rankings).  The two tests are equivalent and one
p-value is given for both.  Note the the value of Kendall's statistic must be
between 0 and 1 and therefore may be easier to interpret.

Missing values are not allowed.  I have tried to trap this, but may not find
all cases--if you ever obtain a negative result for the test statistics, then
you have at least one missing value.

For Kendall's coefficient, the variables are the rankers or judges or tests,
while the cases are the things being judged or ranked.  For example, one
common use is to compare the rankings of students on each of several tests;
the following is taken from Gibbons, JD (1985), Nonparametric Statistical
Inference, 2nd edition, NY:  Marcel Dekker, Inc., pp. 326-327.

                          S t u d e n t
         --------------------------------------------
 T E S T      1    2    3    4    5    6    7    8
-----------------------------------------------------
 1            90   60   45   48   58   72   25   85
 2            62   81   92   76   70   75   95   72
 3            60   91   85   81   90   76   93   80

If you enter this as three variables, then you can just type in
    ^friedman varlist^
and all is fine.  If you enter the data as 8 variables, one per student, then
use the ^xpose, c^ command first.  You can then enter the ^friedman^ command,
as above.  Note that if you need to transpose the data you should first
eliminate all variables that you will not be analyzing.  Otherwise you may get
a surprise.

The above data set is included as gibbons2.dta--as 8 variables.  I also
include Gibbons' example Friedman data set from p. 325 of her book as
gibbons1.dta.  This one also needs to be transposed prior to testing.  This
code is written expecting the raters to be the variables.  This is because the
genrank program ranks variables across cases, not the other way around.  Note
that in other programs you would do things differently--thus, in SYSTAT enter
this data set as 8 variables to obtain the same results as below.

The results from the above data are:
         Friedman =   2.8889
         Kendall =   0.1376
         p-value =   0.8951

Kendall's coefficient of concordance ranges from 0 to 1, with 0 meaning no
agreement across raters (judges).  The null hypothesis (Friedman) is that the
treatments are equal; the null hypothesis (Kendall) is that there is no
agreement between rankings or test results.

I do not produce the sums of ranks for each case, though other software does.
To add it to your output, add the line ^li sum_rs^ near the end (I recommend
either just before or just after the current display lines).

The p-value is an approximation, though it appears to be pretty good as long
as there are at least 8 cases and 3 variables.

WARNING: do not add "compress" or in any other way try to change the code so
that the new variables are bytes rather than floats--your answer will be
WRONG!  (Note that all new variables are dropped at the end of the ado file
anyway.)