W Difference Scores

The W scale was developed by Richard Woodcock and Marshall Dahl in consultation with Benjamin Wright. The W scale is simply a transformation of the ability/item score from a Rasch analysis that uses a logarithm with base 9 (log9) instead of the more common base e (ln). Base 9 was used because Woodcock believed it aided in interpreting the difference between personal ability and item difficulty values.

In the simplest Rasch model, the probability that person n correctly answers item i, Pni, is

$$P_{ni}=\frac{\exp (B_n-D_i)}{1+\exp (B_n-D_i)},$$

where Bn is person n’s ability (on a logit scale) and Di is the item’s difficulty (on the same logit scale as Bn). Equation 1 can be rearranged to isolate the relation between B and D:

$$\text{In}\left(\frac{P_{ni}}{1-P_{ni}}\right)=B_n-D_i.$$

Converting B and D to the W scale simply involves the following linear transformation:

$$W=9.1024\left(A\right)+C,$$

where A is either the person’s ability (B) or the item difficulty (D), and C is some arbitrary constant used to reduce the likelihood of having a negative value. Originally, C was 100, but a value of 500 is used in most current applications. A value of 500 is customarily set to be the ability on the measured trait associated with a student beginning fifth grade (grade norms) or a child of age 10 years, 0 months (age norms). This adjustment is made to set a reference point for proficiency, as W ability scores are intended to be used for measuring change in proficiency over time.

The 9.1024 multiplier changes the scale of the ln logit in a way that is equivalent to using a value of 20 for the base 9 logit. Using the logarithm change-of-base formula, this value can be derived as:

$$\frac{20{\log }_{9}M}{\text{In\hspace{0.17em}}M}=\frac{20}{\text{In}9}\approx 9.104,$$

for any value of M > 0.

The version of Equation 1 that uses W scores is

$$P_{ni}=\frac{\exp \left(\frac{W_{B_n}-W_{D_i}}{9.104}\right)}{1+\exp \left(\frac{W_{B_n}-W_{D_i}}{9.104}\right)},$$

where W* is the result of applying Equation 3 to either Bn or Di.

If, say, $W_{B_n}-W_{D_i}=0$ (i.e., the person’s ability is the same as the item difficulty), the person has a 50% chance of answering the item correctly. When a $W_{B_n}-W_{D_i}>0$, the probability of a correct response is greater than .50. Likewise, when $W_{B_n}-W_{D_i}<0$, the probability of a correct response is less than .50. Differences of +10, +25, and +50 correspond to probabilities of .75, .94, and .995, respectively, while differences of −10, −25, and [Page 1802]−50 correspond to probabilities of .25, .06, and .004, respectively.

For a given group (e.g., age, grade), one can calculate the median W score, which is typically called the Reference W. The difference between a person n’s W score and the Reference W score is called the W Difference score:

$$W\text{\hspace{0.17em}Difference}=W_{B_n}-\text{Reference\hspace{0.17em}}W.$$

W difference scores are used in a number of commercially available tests, perhaps most notably the Woodcock-Johnson IV Tests of Achievement (Schrank, Mather, & McGrew, 2014), Woodcock-Johnson IV Tests of Cognitive Abilities (Schrank, McGrew, & Mather, 2014), and Stanford–Binet Intelligence Scales-Fifth Edition (Elliott, 2007). The W difference score is the value from which other scores (e.g., standard scores, percentile ranks) are derived. Adjustments to the formula for calculating W difference scores can be used to obtain other useful measures of growth. For example, the relative proficiency index is a modification of the W difference score, where the reference W is set at a value of 20 W units below the median. This adjustment facilitates prediction of success with items that same age or same grade peers answer correctly 90% of the time. The relative proficiency index is expressed as a fraction where the denominator is set to 90 to represent the probability of success for same-age or same-grade peers, depending on the type of norms used, and the numerator represents the probability of success for a given examinee. Thus, a ratio of 40:90 would indicate that a person has a 40% chance of responding correctly when same-age or same-grade peers have a 90% chance of responding correctly.

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