Step Function in Winsteps

When a test includes partial credit items, Winsteps can output a file called step function to indicate the difficulty level of each step. For example, if an item is worth 4 points, there will be five steps as shown in the following:

Between each level, there is a difficulty estimate (How hard is it to reach 1 point from 0? How hard is it to reach 2 points from 1? …etc). The technical term of this estimate is threshold. To make it easy to comprehend, we can call it difficulty.

The difficulty estimate uses a scaling method called logit, which is the natural log of the odd ratio. Distances in logit are comparable. For instance, if step3 – step2 = 0.1 and step2 – step1 = 0.1, then the two numbers are indeed equal. The decision rule of this estimate is as follows (I made up these nice and clean numbers for the ease of illustration):

Score	Frequency	Step Difficulty	Step	Decision
0	50	NA	NA	NA
1	30	.60	1. Difficulty of reaching 1 point from 0	Relatively Difficult
2	40	-.40	2. Difficulty of reaching 2 points from 1	Relatively Easy
3	40	0	3. Difficulty of reaching 3 points from 2	Average
4	5	.90	4. Difficulty of reaching 4 points from 3	Relatively Difficult

If the number is around zero, it is considered average. If the number is above 0.1, it is hard. If the number is below zero, it is easy. Typically going from 0 to 1 it is challenging, reaching middle steps is easier, and then reaching the top (step four in our example) becomes difficult again. For example, for a Chinese speaker who doesn’t know anything about English, it will be challenging for him/her to start from no prior knowledge to handling the basic English grammar. However, after he/she has grabbed the foundational knowledge, it will be easier to gradually improve his/her proficiency from grade 3 to 4, from grade 4 to 5, and so on. But for him/her to master the English language at the level of Shakespeare, it becomes difficult again.

In the previous example, fifty examinees scored 0 and thirty got 1 point. There are less examinees in the "score=0" group than those in the "score=1" group, and thus "climbing up" from 0 to 1 is relatively difficult. Forty examinees earned 2 points whereas thirty earned 1 point. Needless to say, those forty people who reached Step 2 and scored 2 points must also pass Step 1, just like that if my age is 20 then I must have been 19 a year ago. Since forty out of seventy people who reached Step1 were able to reach Step 2 while only thirty out of seventy stayed at Step 1, reaching Step 2 seems to be relatively easy. The same number of examinees scored 2 or 3 points, respectively. It means half of the examinees who were in Step 2 were able to climb up one level, and thus the difficulty level is average, or just about right. Following the same logic, reaching the last step seems to be the most difficult.