Computerizing Woodcock's Test Construction Nomograph

Woodcock's Test Design Nomograph (RM 6:3 p.244) is a handy graph for obtaining the expected measurement precision of a uniform test. For those designing tests by computer, I suggest a simple formula that performs the same function.

For a uniform test of L items, with logit range W = (Dh -Dl ) between the hardest and easiest items, the average item spacing is given by D = W/(L-1). An estimated person logit standard error at test center is

SE = sqrt( D / Tanh (LD/4))
where tanh() is the hyperbolic tangent. If tanh() is not available to you, compute

tanh x = (exp(x) - exp(-x)) / (exp(x) + exp(-x))

For a 75 item test of range 3 logits, this estimates a central precision of 0.25 logits as does Woodcock's Nomograph.

This formula can be applied iteratively to find the number of items required for a particular precision. How many items are required for a test of 4 logits range to measure with a precision of 0.3 logits? Start with test length at 100 items, then increase or decrease the number of items till a match to the desired precision is found. Following this procedure, the formula estimates a test length of 60 items. This result is also in agreement with Woodcock's Nomograph.

(For further discussion of the design and use of uniform tests, see Wright & Stone's Best Test Design, 1978, p. 137-140, 144-146, 212, 214.)

Computerizing Woodcock's Nomograph, P Pedler … Rasch Measurement Transactions, 1993, 6:4 p. 255

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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