Rasch Response 9 and 10

Here again are more comments by John and my responses.

9. Dick: The odds of getting the extreme items correct changes exponentially; so we convert these percentages into a logarithmic expression and the resulting log scale is linear.

9. John: Whoa Nellie! I’m lost in a logarithmic fog. I don’t understand this well enough even to ask a question but let me try this. Are you saying something like this: we have 10,000 kids take my 10-question exam. Ignoring the success rates for the first 8 easiest problems, let’s jump to the two toughest. Let’s say on the # 2 hardest, only 100 kids get it right. The # 1 hardest is so much tougher that only 10 kids get it right. Is this sort of where you are headed on the log stuff?

Not really since the Rasch analysis figures out the difficult level of each item and the ability level of each student. Perhaps I over simplified the idea of a logarithmic scale. If we know something increases by a set amount each interval, like the distance we travel at a given rate of speed, then we know this is best thought of as an arithmetic function or graph. The graph will show this arithmetic relationship between distance and time as a straight line. But we know some things change differently over time so we need a different model to understand them. If something doubles each unit of time, then we have an exponential rate of change. We have a rate of change that we can model on the exponential series of 2. So it proceed 2 to the first power, 2 to the second power, and on. We know that fruit flies multiply exponentially so that the graph of number of flies per unit of time isn’t flat, it curves upward. We immediately see that we need to imagine an exponential rate of change and so use an exponential growth curve to model the actual rate of multiplication.

Charter schools in the nation do not increase arithmetically but also follow an algebraic curve, like an exponential curve. So we have phenomena that follow different kinds of mathematical relationships or models. The normal distribution has a different rate of change, rather complicated, but one which statisticians see as following certain mathematical patterns which they can formulate.

So to understand various phenomena, we attribute a mathematical relationships to them. Lots of phenomena follow exponential relationships particularly where growth is involved, and so if we want to show these exponential relationships on a graph, we could use an arithmetic scale for each axis but since the rate of change is exponential, why not convert the scale to a logarithmic scale since a logarithm is simply an exponent? So now the intervals on our scale shows arithmetic intervals that are the increase in the logarithm. Place values positions represent exponential values. Each one goes up by one factor or exponent of ten. Since using a log scale matches the exponential rate of change of a growth phenomenon, the graph appears as a flat line and we can better see changes in its basic growth rate. The log graph shows us much more clearly whether the rate of change, whether the exponential rate of change is constant or not. So since we changed our scale, our graph matches the mathematical relationship we are using to model the change we find in reality, and we have an interval scale for an exponential type of relationship.

Lots of things in the social sciences follow logarithmic relationships. In modeling testing data, algebraic relationships, not arithmetic ones, predominate. Anytime there is more than a simple addition of things, then we need mathematical models to match the algebraic relationships of multiple factors. The probabilities of correct responses to test items also is an algebraic relationship. The curve of the chances from everyone getting the item correct to no one getting it correct varies from 0 up to a midpoint and then back to 0. It is not a straight line going up to a peak and then abruptly angling back down. So we need an algebraic model for probability distributions. We transform the probability by representing the probabilities in terms of a logarithm. Now the arithmetic increase in the logarithm expresses the logarithmic change in the probabilities. The scale is transformed into a simple interval scale that accurately models the rate of change.

Fundamental to the Rasch model is this transformation of the probabilities of each item into its logarithmic representation in order to create an interval scale. It doesn’t mean each item’s value is an equal interval away from the next. It is not counting items we are doing. It means we create an underlying logarithmic scale and use it to represent the logarithmic value of the probability of each item. We create a log scale, or a scale of units called logits which we can then convert to something convenient like the state’s logit scale which they call a RIT scale.
10. Dick: Then we can figure the probability for each cell according to the percentages of a correct response in each cell according to the difficulty level of the item and ability of the student. We express this probability in terms of our logarithmic scale.

10. John: It would really help to use examples for this. Maybe you could play off my math exam illustration and show how it might work at this step.

To conclude, you might try thinking of each item as reprenting a certain difficulty level. A test is like using many individual single measure comparisons to the student so as to see if the student is above or below each item. We essentially have an ordered series of test items placed at various points along the interval scale, and we are attempting to find out where in that series the student fits. So he gets the easy ones below his ability level correct and the hard one above his level wrong. We know where he belongs on the scale from the items. The test items are like stick of different known lengths arranged in a series (although the spacing between them will vary depending on their actual lengths) and one by one we compare an object to each stick to see where in the series the student belongs. If the student is above a stick 4 1/2 feet and below a stick 4 7/8, then we know his length is 4 11/16 with a large error width since the student actually could be anywhere between 4 1/2 and 4 7/8.

Rasch modeled measurements are the only method that can currently be used to formulate tests that can provide an interval scale. They mark a huge advance over norm-referenced tests.