Wednesday, June 20, 2012
Non-Iterative PROX Algorithm
(Single-cycle Rasch model measures estimation)
How student scores and item difficulty can be re-plotted onto one scale was considered graphically in Rasch estimated measures. The item difficulty mean was placed in register with the student ability zero location on the logit scale. The item difficulty zero location was then in register with the student ability mean location. Equivalent portions of the scale for the two distributions (item to student: mean to zero and zero to mean) were in register with one another.
This same thing can be done by capturing the required properties in numbers. These estimates can be made with PROX for a data set with no missing marks. (This is no problem using traditional right count scoring when omits are scored as wrong.) Catherine E. Cantrell published in 1997 the step-by-step calculations for PROX.
These estimates are summarized in PUP Table 10. The table lists values for right and wrong counts, and ability and difficulty measures. The table provides an insight into how PROX performs. It is the source for several charts.
[Plotting the tally column by the expanded measures columns yields the student ability–item difficulty (Winsteps bar) tally. Plotting the black box output column (Winsteps expected student scores) by the expanded measures columns yields the test characteristic curve (TCC). And plotting the output columns by the input columns yields the black box audit tool that is responsive to all changes made.]
PROX makes a tally of the student right mark counts and the item wrong mark counts (columns one and seven). The observed scores are converted into natural log ratios (right/wrong for score ratios and wrong/right for item ratios) to obtain a nearly linear logit scale.
Now to shift the item difficulty mean to the student ability zero location on the logit scale. The logit scale starts at zero and radiates in either direction (-5 to +5 in this example). The initial item measure mean was 0.22 logits. This is subtracted from each item difficulty measure (column 9) to shift the item difficulty measure distribution into register with the student ability measure distribution (as was done graphically in Rasch estimated measures).
The final step is to apply an expansion factor. It is based on the variance within student score and item difficulty measures. The expansion factor chart shows that as the standard deviation for item difficulty grows, the greater is the resulting expansion factor. In general, the expansion factor for ability is about twice that for difficulty. It is normal for item difficulty to spread out in a wider distribution than student ability scores.
The expansion factor for student ability is obtained for PROX by taking the square root of the ratio of the variance within item difficulty measures (U) to the product of the variances for student ability and item difficulty measures. The expansion factor for item difficulty is based on the ratio of the variance within student ability measures (V) to the product of the same two variances.
After adding in constants to match the logistic and normal distributions (1.7, 2.89 = 1.7 squared, and 8.35 = 2.89 squared) the expansion factors become SQRT((1+(U/2.89))/(1-((U*V)/8.35))) for student ability and SQRT((1+(V/2.89))/1-((U*V)/8.35))) for item difficulty measures. The expand (expanded) table columns are the products of the student ability logit values or item difficulty shift values and their respective expansion factors.
[Multiplying pools the source variances U and V. Dividing their variances by the pool assigns a portion to each source. The larger portion is applied to the smaller source (which is normally the student score distribution). The expansion factors increase the spread of the ability and difficulty measure distributions each way (+ & -) from the zero measure location. The entire PROX process for estimating measures includes simple math and no pixy dust.]
The student ability and item difficulty expanded values are plotted on the ability-difficulty tally chart. The goal is, to have a student ability to mark a correct answer 50% of the time, match an item with a difficulty of equal magnitude. By definition this happens at the zero logit (measure) location. Does this continue to occur as one moves further away from the zero logit location?
Cantrell, Catherine E. (1997). Item Response Theory: Understanding the One-Parameter Rasch Model. 42 p. Paper presented at the Annual Meeting of the Southwest Educational Research Association (Austin, TX, January 23, 1997). EDRS: ED 415 281.