Chapter two of the the Rasch book continues with the introduction of individual and item parameters. On my first reading of this, my powers of analysis were reduced to jelly because Rasch used the Greek letter ζ to represent ability. He also came up with the pronouncement.
where θvi is the probability of individual v making an error on item i. He introduces this expression by saying "Let us ... think of the probability of making a mistake in a test as a product of two factors (op. cit. page 16, his italics, but my emphasis, because he italicised the whole statement).
I mentioned earlier that, while at school, I studied Newtonian or macro-mechanics, which is classically deterministic. I also did econometric units at uni, which used linear algebra in essentially deterministic models of the economy. So had I been Rasch, I would have said: "Let us think of the probability of making a mistake as a function of two factors", and I might have expressed this algebraically as follows:
Only then, would I have added flesh to the model, by discussing what shape the relationship between the dependent variable y and the independent variables, x1 and x2 , might be. Rasch dispensed with such niceties. He just plunged in with a statement stating not only that the dependent variable is directly proportional to the first independent variable and inversely proportional to the second independent variable, but also that it is exactly equal to the quotient of the two variables (which in itself is a bit strange, because in the previous sentence he had talked about a product).
Of course the beauty of modelling, whether in balsa wood or in mathematics, is that the model designer gets to create the rules, and he can make the rules anything he pleases, as long as they are internally consistent with each other. If I set out to make a balsa wood aeroplane, I can choose the model of aeroplane, and the scale. But once I have set out the rules I have to stick to them (I cannot take some dimensions from the Spitfire and others from a Messerschmitt, and it would be silly to use a 1:50 scale for some components and a 1:40 scale for others), and my decisions will determine the shape of the finished model. Similarly, the rules for a probabilistic model can be quite arbitrary, and the model should work within it's own parameters if the modeller applies them consistently, but the detail of the outcomes will be shaped by the rules set out at the beginning.
So there is nothing intrinsically wrong with Rasch not explaining the exact shape of the relationship between the probability of an outcome and his person and item parameters. He said he was setting out a probabilistic model, and that he did. And his exposition of the model was much more detailed and erudite than I could ever manage. However, it is not the only model in the universe, nor is it necessarily the only model which can be reasonably be applied to individuals addressing items in a test.
I have mentioned in an earlier blog that I prefer a more concrete model, not only because it sits better with my more concrete brain, but also because it is easer to simulate with a few lines of code. Rasch himself, when introducing the concept of a probabilistic model talked of drawing coloured balls from a bag (op. cit. page 11). I prefer to extend that model to the definition of personal and item parameters in the context of a psychometric test. My conceptual stumbling block (and I have been thinking about this during the last few weeks) has been to define ability for the model without referring to item difficulty, and vice versa.
But I have decided I can legitimately force it, using a method analogous to partial differentiation in the calculus. In the calculus, if you want to describe the slope of a wavy surface on a 3d model, you can cheat a little, by holding one of the dimensions constant, and then applying the normal rules of differentiation to the resulting 2d curve. So I shall define ability as the probability of an individual submitting a correct answer in abstract. In an earlier blog I defined ability as the probability of an individual submitting a correct answer to an item of neutral difficulty, but that won't work in the model I want to propose. So I shall just define ability as an abstract quantity, without reference to items, simply as the probability of a correct answer being given by that individual. And I shall define difficulty as an abstract quantity, without reference to individuals, simply as the probability of an error being committed on that item.
Now I can model both individuals/people and test items as bags of coloured balls, or for consistency with my earlier blog, boxes of coloured beans. And I can model the person-item interaction as emptying the contents of one box into the other, and drawing a single bean from the resultant set. If red beans represent correct answers, and the box representing person j contains aj red beans out of a total of nj, then the ability of that person can be defined as aj/nj. Rasch used the letter v to refer to a specific individual, but I have used j, because it conforms with the notation used by my psychometric supervisor at UWA, and more common practice in contemporary journals. Rasch also defined his second parameter as difficulty, but I propose easiness as the item parameter. Then if the box representing item i contains ai red beans out of a total of ni, then the easiness of that item can be defined as ai/ni.
The probability, θij, of a successful outcome when person j interacts with item i can now be defined as:
|θij||=||(ai + aj)/(ni + nj)||(14)|
And unless you want ability and difficulty to be weighted unevenly, n should be equal for both boxes, so:
|θij||=||(ai + aj)/2n||(15)|
So in this model, the combined item-person probability is the mean of the individual probability and the item probability. And if you choose to express both individual probability and item probability as a percentage (or if you stipulate person and item boxes each containing 100 beads), the calculation is very simple.