Learning Java

On page 18 of the text , Rasch has defined the misreadings in two tests as a vi1 and a vi2 , and the sum of the two as a v . He has also equated the observed number of misreadings in test i with λ vi , the expected number. And from equation 20 in my previous blog: λ vi = τ i /ζ v (20) So if: a vi = λ vi and: a v = a vi1 + a vi2 then: λ v = (τ i1 + τ vi2 )/ζ v (24) Rasch calls this the "additivity of impediments" ( op. cit. page 16). Leaving aside the fact that additivity is not listed in the (MS) dictionary, I can't see the point in this nomenclature just yet. He also rearranges the equation, and suddenly replaces his bold equality with an approximation: ζ v ≈ ( τ i1 + τ vi2 )/a v (25) I shall not discuss the remaining text on that page, because it relates solely to reading tests, and...

I have decided that I really don't like the Rasch personal and item parameters, because they seem counter intuitive, and I don't like his use of the Poisson Distribution, because the assumptions required by it are unrealistic, certainly if you broaden the discussion beyond a very easy reading test. But I have resolved to continue anyway. I've been thinking about this for years, and at last I've bought the book , so I don't have to worry about library or other loan returns. I can take as long as I like, and I have resolved to understand every equation in the book, if only to disagree with it, or at least with it's use. I mentioned at the end of my last blog that Rasch refers to the "probability of misreading" ( op. cit. page 16) in a test, but he is a bit vague as to whether this a single error or several. The vagueness continues into the next section ( op. cit. page 17), when he refers back to his equation 2.1, which is my equation 12: ...

Returning to the original Rasch definition of person and item parameters, given by: θ vi = δ i /ζ v (12) Rasch puts flesh on them by defining a test, test zero, for which δ is unity. In this case: θ v 0 = 1/ζ v (16) He then says: "the ability of the person is the reciprocal of the probability of misreading in the reference test". Let's not think too hard about the reference test for a minute, but say the probability is estimated at 5%. I've initially chosen a low value, because Rasch uses Poisson estimation, which relies on low values. That returns a value of 20 for ζ . Rasch then looks for a person with unit ability. Rasch uses this conceptual person to estimate test/item difficulty as follows: θ 0i = δ i (17) He then adds the verbal definition: "the difficulty of a test is the probability that the standard person makes a mistake in the test...

Rasch was very clear on one thing. All these measures are relative. "Neither δ nor ζ can be determined absolutely" ( op. cit. page 16). Rasch emphasises the arbitrary nature of units, even in physical science: "1 Ft = the length of the king's foot" ( op. cit. page 16). This answers my own question posed in the fourth paragraph of this blog , which was that if the difficulty parameters for three items were found to be 1, 2 and 5 using one population, would they be exactly the same for another population or in the same proportion. The answer is the latter. It also confirms my own gut feeling that those writers who have developed the habit of talking about "logits", in the context of Rasch parameters, should desist from doing so, because stipulating units flies in the face of the original Rasch argument. Sometimes I think Rasch is oversold by the enthusiasts. If all you want to do, with say three children, Flossy, Gertrude, and Samantha, is esti...

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. θ vi = δ i /ζ v (12) where θ vi 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 ...

In Chapter two of his book , Rasch jumps from his Equation 6.1 to an approximation, which he attributes to Poisson, but he not only provides no derivation of the approximation, but also omits to write out the approximation in general terms. I have found a page on the web , which sets out a cursory but satisfactory explanation of the approximation. Rasch calls the approximation Poisson's Law. The page on the web calls it the Poisson distribution or the Law of Rare Events. Let's begin with my Equation 3 from my previous blog. p{a|n} = (n!/((n-a)!a!))θ a (1-θ) n-a (3) Then according to the argument, if you focus on the right hand term, (1-θ) n-a , you can approximate that as: (1-θ) n-a ≈ e -n θ (4) where ≈ means is approximately equal to, θ is assumed to be low. The derivation involves isolating the left hand of the equation and taking the natural log: ln(1-θ) n-a = (n-a)ln(1-θ)...

Towards the end of his introduction, Rasch describes a simple probabilistic model involving mice. I shall not use his exact notation because it is not convenient for me to put bars over characters. So where Rasch describes and outcome A or à I shall describe the outcomes as 1 or 0. The probability of outcome 1 is q , and the probability of outcome 0 is 1- q . The thing about probability is not that it is hard, but that it is tedious. What makes it hard is that people who work with probability all the time, to reduce the tedium, have developed annotations. They also jump steps. I guess for them the fewer the steps, the lower the tedium, for for people (like me) who don't work with it a lot, it has the effect of combining tedium with headache. I am sure this is one of the reasons why Rasch is not well known by people who walk up and down high streets carrying shopping bags. Rasch jumps with his mouse model straight into an expression which looks to the layman like gibberis...