# [Rasch] Likelihood

Svend Kreiner svkr at sund.ku.dk
Sun Mar 15 20:27:00 EST 2015

```Patrick,

"I am looking the proportion of people who took the exam in the first quarter vs. the fourth quarter and given the greater propensity to fail in the fourth quarter, it is X times more likely that you will fail the exam if you wait until the fourth quarter to take the exam".

The measure you are looking for is the so-called odds-ratio comparing the odds for failure in the fourth quarter to the odds for failure in the first quarter rather than the probabilities.

it is very easy to calculate

4413   430
92     43

and the odds-ratio is

(4413x43)/(430x92) = 4.8

This is the measure that epidemiologists use when they compare risk in different groups.

I was a little surprised to see this question here, because it has nothing to do with Rasch models, but there are two results that connects it to what we usually discuss here.

The first is that there are two old papers showing that the odds-ratio (and functions of the odds-ratio) is the only measure of association in 2 x 2 tables that does not depend on the way persons are sampled.

The first is                         Edwards, AWF (1963): The Measure of association in a 2 x 2 table. J.R. Statist. Soc. A, 126, 109-114
The second is                   Altham, P (1970) The measurement of association of rows and columns for an r x s contingency table.  J.R. Statist. Soc. B, 32, 63-73

Since objective comparisons of items (according to Rasch) requires that comparisons of items must not depend on how persons are sampled, toy might say that (functions of) the odds-ratio is the only objective comparisons of risks.

The second is that you can calculate the odds-ratio by a logistic regression analysis with failure as dependent and quarter as the independent variable. The result is a beta parameter describing the effect of  quarter on failure,
that can be transformed to an odds-ratio statistic by odds-ratio = exp(beta).

The beta parameter is another objective measure comparing failure in the two quarters. Since the definition of logits are given by logit=ln(odds), you can say 1) that the beta is the logit difference between the risk in the fourth quarter and the risk in the first quarter,
and 2) that the beta statistic is a measure of the different risks on exactly the same scale as the scale we use for item and person parameters in the Rasch model. If you are an experienced Rasch modeller and therefore understand and can interpret logit-differences,
you may prefer this measure to the odds-ratio, but you have to remember, that the rest of the world do not understand logits and therefore prefer odds-ratios.

Svend

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