Reading my first message again, I noticed that the table of critical values at 
the bottom of the page,
was not in the right format. I tried to rearrange it and hopefully this time it 
will appear in a better format.
Best wishes
Mojgan
-----Original Message-----
From:	mojgan naeeni [SMTP:[log in to unmask]]
Sent:	24 June 1999 00:19
To:	[log in to unmask]
Subject:	seeking reference for a weird nonparametric test.

Dear all,

I have accidentally come across the following nonparametric test, in the
appendix of an unpublished document, with an unknown author!
I have never seen this test before. If you can  recognise the test or the
author, would you please let me know the reference?

Many thanks,

Mojgan Naeeni

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Analysis of Rank Order Scores:

The following example considers the case of a comparison of 4  detergent
products (A, B, C, D) where the test involves 11 replicate washes using 2
article per wash, each wash being carried out on a separate day. A forced
choice scoring system is employed for the visual assessment for the cleaning
performance in which the product ranked best is given a score of 1 and the
product ranked worse a score of 4. 'No difference' is not allowed. The table
bellow illustrates the results obtained from assessment of the first day's
washes.

                                                 Product Rank Order
Day     Article   Assessor        A        B        C        D

 1           1            1                3        4        1         2
                           2                4        2         1         3

              2           1                3         4        2         1
                           2                3         4        1         2

                       Rank Sum     13        14      5          8

Note: The total rank sum over one ranking is ten, no ties being allowed.

Dividing by 4 gives the average rank sum per product per day over the two
article portions, that is,

A is 3.25, B is 3.5, C is 1.25 and D is 2.0

The same procedure of average ranking is continued until all 11 rankings are
obtained. Then add the average rank sums. For the present example intermediate
data results are not given , however the results obtained were as follows.

A is 38.5, B is 30.25, C is 13.75 and D is 27.5

Note in this case the sum has to be 11 times a total of 10, that is 110.
In order to proceed with the statistical analysis, first write the rank sum in
ascending order

i.e.  C is 13.75 D is 27.5 B is 30.25, A is 38.5

Confidence limits are established using critical values for differences between 
the rank sums. Table for the critical values are given below and depend upon:

n=Number of replicates(washes) per product
k=Number of products
m=Number of means over which a rank sum difference is being compared.

Application to the above data is as follows with n=11 and k=4:

The first comparisons is between two extremes i.e. C and A. The difference in
rank sum is 24.75. Since 4 means are involved in the rank sum comparison at
this stage m=4 and hence 95% critical value is 16. We can therefore conclude
that with 95% confidence C is superior to A.

The next comparison is D vs. A. Here m=3 as only 3 means are involved and the
critical value is 15. Since the rank sum difference is only 11 then the
difference is not significant. Since this difference is not significant then,
by definition B vs. A is also not significant.

The last comparison is C vs. D. The rank sum difference is 13.75 and m=2 as
only 2 means are being compared. The critical value from the table is 12 and
hence C is significantly different from D.

The overall conclusions from the study are therefore (a) that C is
significantly better than all the other test products and (b) there are no
significant difference between D, A and B.


Critical Values at 95% Confidence:

k        m       n=3    n=4   n=5    n=6    n=7   n=8    n=9    n=10  n=11

2        2         -        -         -        6         7       8        7 
      8       9
3        3         6       7        8        9         9      10      10     11 
     11
          2         5       6        7        7         8       8        9 
     9       10
4        4         9      10      11      12        13     14       15     15 
    16
          3         8       9       10      11        12     13      13     13 
     15
          2         7       8        9        9        10      11      11    12 
     12




k        m       n=12    n=13   n=14    n=15   n=16   n=17   n=18   n=19

2        2         8          9        10        9         10        9 
       10      11
3        3         12        12       13       13        14      14        15 
     15
          2         10        10       11       11        12      12        12 
     13
4        4         17        17       18       19        19      20        20 
     21
          3         15        16       17       17        18      18        19 
     19
          2         13        13       14       14        15      15        16 
     16




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