Rather than treating each balloon as a unique “fixed effect” and estimating a difficulty specific to it, there may be other types of effects for which it is more effective and certainly more parsimonious to represent the difficulty as a composite, i.e., linear combination of more basic factors like size, distance, drafts. With estimates of the relevant effects in hand, we would have a good sense of the difficulty of any target we might face in the future. This is the idea behind Fischer’s (1973) Linear Logistic Test Model (LLTM), which dominates the Viennese school and has been almost totally absent in Chicago.
Rasch (1960) started with the Poisson model circa 1950 with his original problem in reading remediation, for seconds needed to read a passage or for errors made in the process. Andrich (1973) used it for errors in written essays. It could also be appropriate for points scored in almost any game. The Poisson can be viewed as a limiting case of the binomial (see Wright, 2003 and Andrich, 1988) where the probability of any particular error becomes small (i.e., bn-di large positively) enough that the di and the probabilities are all essentially equal.
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