In statistical terminology, "sufficient" means "all the information that this dataset has about ...". So, from a Rasch perspective, the "raw score" has all the information there is in the dataset about the "ability" of the respondent. This is also the Classical Test Theory perspective. But this is not the 2PL or 3PL perspective. They say that the "pattern of responses" has all the information there is in the dataset about the "ability" of the respondent.
1. Person n of ability B_{n} responds to test items (i) with difficulties (D_{i}) with scored (0,1) responses (X_{ni}). For the raw score R_{n} = ΣX_{ni} to be sufficient for estimating ability, indeed, for counting right answers to be useful at all, R_{n} must extract ability B_{n} out of (X_{ni}).
2. Bayes says that the probability of person n's response vector, given person ability B_{n} and items of difficulty (D_{i}), is the probability of the response vector given the raw score R_{n} multiplied by the probability of the raw score given the ability:
P{(X_{ni})B_{n},(D_{i})} = P{(X_{ni})R_{n},(D_{i})} * P{R_{n}B_{n},(D_{i})}
3. If P_{ni} = f(n,i) is the probability that n succeeds on i, then we can obtain the probability of person n's method of obtaining a score of R_{n}, and also all other methods of obtaining the same score.
4. R_{n} points unequivocally to ability B_{n} only if P{(X_{ni})R_{n}, (D_{i})}, the probability of the particular responses given the raw score, is entirely free from variation due to B_{n}. Thus the remainder must be locally independent of B_{n} so that it does not matter which particular (X_{ni}) are 1's and which are 0's provided that ΣX_{ni} = R_{n}. Thus the effect of B_{n} on the (X_{ni}) can be factored out of (3) and cancelled.
[This is saying that the set of response vectors are locally independent given the person abilities and the item difficulties. Georg Rasch called this property "specific objectivity".]
5. A sufficient factoring is obtained by asserting that
C_{ni} = B_{n} + D_{i},
P_{ni} = B_{n} / C_{ni}, and 1 P_{ni} = D_{i} / C_{ni}, then

so that
is seen to be free of B_{n}.
6. Thus a sufficient parameterization is
Reparameterizing,
produces the Rasch model, showing that this form of f(n,i) supports the use of R_{n} for estimating the ability of person n.
7. The necessity of this formulation can be demonstrated by dividing the numerator into the denominator in (3) to produce the following term in the denominator, which must not vary with Bn:
8. In T there are the same number of P_{ni} and P_{nj} terms in each product, because the raw score is invariant across the summation. The general condition for T not to vary with Bn, (apart from irrelevant specifics, such as when P_{ni} and P_{nj} are constants), is that
which takes us to step (2) of Model Necessary for a Thurstone Scale, RMT 2:1 p. 910, and thence to the Rasch model as the necessary f(n,i).
Benjamin D. Wright
Note: Since most of Classical Test Theory (CTT) regards the raw scores as sufficient for decisionmaking, CTT implicitly assumes that the Rasch model holds for their data.
Dichotomous Rasch Model derived from Counting Right Answers: Raw Scores as Sufficient Statistics. Wright BD. … Rasch Measurement Transactions, 1989, 3:2 p.62
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