Learning Java

In my last few blogs I have used Java to generate a dataset simulating 64 students sitting a 64 item test. I have talked about the "prox" method of estimation discussed in the Winsteps documentation. I shall now show the results of seven iterations through the dataset using this formula: d i ' = m i - root(1 + s i 2 /2.9)d i where d i ' is the revised item difficulty and d i is the previous estimate of item difficulty, m i is the mean ability of the children from the most recent estimate and s i is the standard deviation of those abilities. The chart immediately below shows the raw mean item difficulty as di0 and the mean item difficulty from 6 subsequent iterations. The chart below that shows the variance of raw item difficulty as di0 and the variance from 6 subsequent iterations. It is clear from these charts that the best estimate of difficulty and the tightest distribution is the original dataset. Four addition iterations were carried out but not char...

The essence of Rasch-based estimation is that observed scores are adjusted to take into account other observations. So for example item scores may be adjusted to take into account the observed abilities of the candidates (who in my research are usually children). And the candidate or child scores may be adjusted to take into account the observed difficulties of the items. To explain this process I shall refer to the Winsteps documentation because the Winsteps software is the only computer based estimation tool I have used, and because its creators are recognised academically as being at the core of the objective measurement movement (to the extent that such a thing exists). The formula given by them for adjusting item scores is: d i = m i - root(1 + s i 2 /2.9)ln(r i /(n i - r i )) where d i is the revised item difficulty, m i is the mean ability of the children attempting item i and s i is the standard deviation of those abilities, r i is the observed raw score on...

In my last blog I used a few lines of Java to simulate the selection of beans from a set of 64 boxes each containing 64 beans in which respectively 1 to 64 were red. The idea was to simulate the results from a test comprising 64 items with smoothly graduated difficulty. For the time being the test candidates have been assumed to have a neutral effect on the results - I pictured children randomly selecting beans from the boxes. In the chart below I have charted items scores against item number as a simple scatter, and before carrying out any Rasch analysis, I ran a simple regression on the data. The coefficient was unity (to 2 decimal places) and the intercept was 0.3. R squared was 0.97. This is a pretty good fit to the theoretical line predicted in the previous blog. When data from a Probabilistic Model fits well with a theoretical prediction it indicates that the generated dataset is large enough to be useful. Next I ran the "transformation" described in ...

Imagine a box containing 64 beads, of which 32 are red and 32 are blue. Imagine selecting beans one at a time, noting their colour and returning them to the box. We can synthesise this with a few lines of Java as follows: public class prob1 { public static void main(String[] args) { int num1 = 0; String beancolor = new String("blue"); System.out.println("Generated from box 32"); int count = 1; while (count < 11) { num1 = (int)(Math.random() * 64); System.out.println("random no: " + num1); if (num1>32) { beancolor = "red"; } else { beancolor = "blue"; } System.out.println("bean colour: " + beancolor); count++; } } } And the output might look something like: Generated from box 32 random no: 42 bean colour: red random no: 12 bean colour: blue random no: 5 bean colour: blue random no: 48 bean colour: red random no: 14 bean colour: blue random no: 60 bean colour: red random no: 35 bean colour: ...

If we imagine a box containing 64 beans, of some are red and some are blue, the probability of pulling out a red bean is directly proportional to the number of red beans in the box. You could plot this as a graph of probability, stated either as a fraction or as a percentage, against the number of beans, ranging from zero to 64, and the graph would be a straight line, from the origin. The essence of Rasch is that the probability of a child answering an item correctly is a function of the ability of the child. I am not sure that there is any need to complicate this. It seems fine just as it is. Yet if you read the iterations of Winsteps , it is made complicated. Before looking more closely the iterations of Winsteps , I should like to spend a little more time with my simple model. Imagine a test comprising 64 items, each represented by a box of 64 beads, being sat by 64 children, who are also each represented by a box of 64 beads. Imagine the difficulty of the items being perfec...

A problem sometimes with high theory is that even if the original theorist never lost track of reality in his own mind, the readers and followers sometimes do. I shall look at what Rasch was doing in terms of beads in a box, because I find it helpful. I don't have his book in front of me, but from memory, he began by suggesting that the probability of a child j answering item i in a test correctly might be a function of the ability of the child j and the difficulty item i. He then launched straight into his mathematical method for estimating underlying probabilities from observations. That's all very well if you really understand what's going on, and I'm sure he did. But for someone (such as myself) encountering the argument for the first time, it's easy to get bogged in the mathematics and lose track of what it all means. I like to think of the probability of a child giving a correct answer to an item with "neutral" difficulty in terms of the propo...

Georg Rasch was a Danish Mathematician, born 1901, and sadly deceased 1980. He has a Wikipedia entry and a whole web site dedicated to his memory. His best known publication is a book entitled " Probabilistic Models for Some Intelligence and Attainment Tests ". It's a great read, but sadly out of print and not widely available second hand. When people ask what Rasch methodology is all about, I describe it as chaos theory for social scientists or more specifically educationalists and psychometricians. Traditional models in both natural and social sciences were deterministic. An example of a deterministic expression is: y = f(x) where y is a function of x. The function might be x + 2, 2x, x 2 , or x 2 + 2x + 2, it doesn't matter. In a deterministic model, if the independent variable (in this case x) is known, the dependent variable (in this case y) can be precisely calculated. Stochastic, or probabilistic, models take a less cer...