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{\plain {\fs20 \fs32 Probabilities for matching
 things}{\fs20 \par \par Mathcad 6.0 plus
 worksheet to describe calculation of the
 probabilities of getting a certain number
 of matched cases in the "categorical mismatched
 cases" paradigm for determining whether
 there is statistically significant information
 being contained in case descriptions.  
This can be viewed using Mathbrowser and
 is mounted on the WWW as:\par    http://www.sghms.ac.uk/mhs/psychotherapy/mch/dichot1.mcd\par 
}{\i \fs20 It is mounted as copyleft.  I.e.
 I, Chris Evans, claim copyright, however,
 I am very happy for anyone to duplicate
 it however they like provided they do not
 change it in any way (other than tinkering
 with any of the parameters you wish, which
 is the full glory of the Mathcad/Mathbrowser
 capability).  Try modelling other mismatchings
 and other scores as you go through.  If
 you want to change it or tell me about 
errors contact me at:\par Psychotherapy 
Section, St. George's Hospital Medical School,
 Cranmer Terrace,\par London SW17 0RE Britain
        Email: C.Evans@sghms.ac.u}{\fs20 
k\par \par N.B. The maths of this can be
 used in other scenarios, this worksheet
 was actually prepared to deterimine the
 statistical significances for matching 
"birth stories" of patients against their
 own Cognitive Analytic Therapy (C.A.T.)
 self-descriptions which had been elicited
 independently of the birth stories.  Each
 judge was given 16 birth stories and for
 each was given the self-description of 
the patient and a randomly selected self-description
 of another patient.  The possible matching
 score clearly ranges between zero and 16.\par 
\par The probability of achieving a perfect
 16 out of 16 match (or n out of n for the
 more general case) is easy.  Assuming that
 the null model is of random matching and
 taking a one-tailed approach of discounting
 the probability of significantly }{\fs20 
\i bad }{\fs20 matching the probability 
must be .5 (random matching) raised to the
 power n (here .5^16=0.00001526).\par \par 
The probabilities of lesser matches (0 <=
 r <= n) form a classical statistical distribution:
 the Binomial (see e.g. Evans (no relation!),
 Hastings & Peacock (1993)).  In this distribution
 each score r has probability equal to the
 number of different ways of achieving that
 score multiplied by the probability of 
r correct (p^r) and the probability of (n-r)
 incorrect ((1-p)^(n-r) = p^(n-r) since 
p = .5 = (1-p)).\par \par The number of 
ways getting that score is n!/r!(n-r)! where
 the "!" sign indicates a "factorial":\par 
  n! = 1.2.3. .... (n-3).(n-2).(n-1).n\par 
where 0! = 1 by convention, 1! = 1, 2! =
 2, 3! = 6, 4! = 24 etc.\par \par O.K. now
 for the Mathcad formulae}}
}
.EQN 80 0 3 0
{0:comb}NAME({0:r}NAME,{0:n}NAME):{0:if}NAME({0:n}NAME<{0:r}NAME,0,(({0:n}NAME)!)/(({0:r}NAME)!*(({0:n}NAME-{0:r}NAME))!))
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{0:factorial}NAME(2)={0}?_n_u_l_l_
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{0:factorial}NAME(3)={0}?_n_u_l_l_
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Cg a7.125000,17.000000,18
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{\plain {\fs20 Checking!}}
}
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{0:factorial}NAME(0)={0}?_n_u_l_l_
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{0:factorial}NAME(1)={0}?_n_u_l_l_
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{\plain {\fs20 More checking!}}
}
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{\plain {\fs20 Now lets consider the possibility
 (achieved by the researcher herself in 
the study described) of correctly matching
 all 16 birth stories to their self-descriptions}}
}
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{\plain {\fs20 Now to define the binomial
 probability}}
}
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{\plain {\fs20 That looks pretty statistically
 significant!}}
}
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{\plain {\fs20 The probability of a lower
 score can be calculated in exactly the 
same way:}}
}
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{\plain {\fs20 However, we're really interested
 in the probability of doing as well, or
 better, than this so we need a cumulative
 probability for that score and any higher
 score.  Another definition will do this:}}
}
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{\plain Let's check that works for the silly
 option of zero or more correct matches}
}
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{\plain \par }
}
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{\plain {Grand!}}
}
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{\plain Now let's do some other silly checks:}
}
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{\plain {\fs20 In the experiment the researcher
 achieved the perfect 16 out of 16 and the
 independent rater achieved 12 of 16:}}
}
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{\plain {\fs20 Still statistically significant}}
}
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{\plain {\fs20 Let's get Mathcad to tabulate
 all the possible scores for n=1 to n=20:}}
}
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{0:r}NAME:0;20
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{\plain {\fs20 As a final thought: consider
 matching against a number of mismatched
 cases i.e. only one true match but two 
distractors.  In this case the pertinent
 p for the null model is .33 rather than
 .5}}
}
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{\plain {\fs20 This shows the dichotomous
 case\par The z axis (coming R to L toward
 you)\par shows the number of cases to be
 matched\par The x axis (back to forwardL
 to R) shows the score and \par the y axis
 shows the probability of scoring \par more
 than or equal to that score }}
}
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{\plain {\fs20 This shows the trichotomous
 case, i.e. where there are two possible
 mismatches for each match. You can see 
how the probabilities fall away faster as
 the scores increase}}
}