IV. Doing the math (and a little algebra)

Estimates and Estimators: connecting model to data

The essential attributes of a Rasch model are sufficient statistics and separable parameters, which allow, but don’t guarantee, specific objectivity. Well, actually sufficient statistics come pretty close if they really are sufficient to capture all the relevant information in the data. We will come back to this in the discussion of what Rasch called control of the model and most of us call goodness of fit. The current topic is a demonstration, more intuitive than mathematical, of how to manipulate the model to estimate item difficulties.

The process begins with the basic Rasch model for how likely the person wins when one person takes one dichotomous item: . . . The Disappearing Beta Trick


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IIIf. Another Aspect, Reading Aloud

Truth emerges more readily from error than from confusion. Bacon

There is no such thing as measurement absolute; there is only measurement relative. Jeanette Winterson.

The Case of the Missing Person Parameters

Eliminating nuisance parameters and #SpecificObjectivity

It was a cold and snowy night when, while trying to make a living as a famous statistical consultant, Rasch was summoned to the isolated laboratory of a renowned reading specialist to analyze data related to the effect of extra instruction for poor readers. There may be better ways to make a statistician feel a valued and respected member of the team than to ask for an analysis of data collected years earlier but Rasch took it on (Rasch, 1977, p. 63.)

If we could measure, in the strictest sense, reading proficiency, measurements could be made before the intervention, after the intervention, and perhaps several points along the way. Then the analysis is no different, in principle, than if we were investigating the optimal blend of feed for finishing hogs or concentration of platinum for re-forming petroleum.

Continue reading . . .IIIf. Reading Aloud

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IIIe: A Spectrum of Math Proficiency and the Specter of Word Problems

In mathematics, one does not understand anything. You just get used to them. Johann Von Neumann

Defining mile posts along the way from counting your toes to doing calculus

The world has divided itself in two factions: those who think they don’t understand math and those who think they do. But we’re not talking about proving Fermat’s Last Theorem or correcting Stephan Hawking’s tensor algebra; we’re talking about counting, applying the four basic operators, and solving the dreaded word problems using basic algebra, geometry, and perhaps a little calculus. That just about covers the range from counting your toes to determining the spot in the outfield where a player should stand to catch a fly ball and should be good enough to get you through freshman math.

Continue reading . . . A Spectrum of Math Proficiency

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IIId. On Any Given Sunday

May the better team have better odds

Pair-wise comparisons and arbitrary labels

All of us have probably thought sometime during the football season that there must be a Rasch analysis in here somewhere. Every team in the National Football League plays a different selection of opponents but rankings are based, mostly[1], on a simple count of games won. We certainly know better than that. Here’s my answer.

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[1]There are more rules for resolving ties in the rankings but these are designed more to create excitement and sell tickets than to ensure that the best team wins.