STAT 778 – Homework 7
Note that many of the answers for the problems below are not necessarily dealt with directly in the reading.
This equating assignment involves three data sets:
 http://www.stat.sc.edu/~habing/courses/data/mg1.txt consists of the responses of 1,388 examinees to 21 items
selected from form B of the math placement test. The first 10 items of this form are the same as the first 10
items of form mg2 that was given to a sample of examinees that are not believed to be equivalent to this group.
 http://www.stat.sc.edu/~habing/courses/data/mg2.txt consists of the responses of 1,254 examinees to 21 items
selected from form B of the math placement test. The first 10 items of this form are the same as the first 10
items of form mg1 that was given to a sample of examinees that are not believed to be equivalent to this group.
 http://www.stat.sc.edu/~habing/courses/data/mgx.txt consists of the responses of 1,060 examinees to 25 items
selected from form A of the math placement test. Assume that the examinees taking mgx are randomly
equivalent to those taking mg1 described above.
Equating mgx to mg1
One of the easiest forms of equating when for the random groups design is “linear equating” where the z-scores
from the two forms are set equal to each other: (x-(X) )/(X) = (y-(Y) )/(Y) . Solving this equation produces
the formula: eqY(x)=x(Y)/(X)+[(Y)-(X)(Y)/(X)]. So, to convert a score x to the Y scale we simply apply
the above formula to the x score.
1) Construct a table showing what each score from 0 to 25 on the mgx form equates to on the mg1 form (i.e. make
a table showing the function eqmg1(mgx) ).
2) What practical problem in explaining the test scores to the students and parents might occur in reporting the
scores to the students as the mg1 total if that is the form they took and eqmg1(mgx) if they took mgx?
3) What is an obvious solution to the difficulty noted in 2? Which equating property would this solution likely
violate?
4) Table 8.5 in Kolen and Brennan suggests that “linear equating” this method works well near the mean. Briefly
say why it seems reasonable that it would work well near the mean and not for more extreme values.
A more complicated method of equating between random groups is “equipercentile equating” where the scores on
the two-forms are set equal to the score that would have the same percentile on the other form. Mathematically, if
X and Y were continuous variables with cumulative distribution functions F(x) and G(y) respectively, then
eqY(x)=G-1(F(x)). The difficulty is that we generally have discrete samples instead of continuous populations. The
algorithm for performing the equipercentile equating is rather involved to program from scratch, but is given on
the R-templates page.
5) Construct a table showing what each score from 0 to 25 on the mgx form equates to on the mg1 form.
6) If you had scored 19 on the mgx form, would you prefer that equipercentile or linear equating were used?
7) What is it about the scores on form mgx that might cause the “flat region” on the graphical display of the
equipercentile equating?
Equating mg2 to mg1
One method of equating to exams with common items is “concurrent calibration” using item response theory. This
involves combining the two sets of items into one large data set where all of the examinees took the common
items, and each set is coded as “not presented” for the items those examinees didn’t see. Assume Bilog has
produced 3PL estimates for all of the examinee abilities and all of the item parameters, and that the model appears
to fit. (Kolen and Brennan note that this method may be particularly vulnerable if the assumptions are not met.)
8) How could you use this information to give each of the mg2 examinees a score on the mg1 scale?
9) What would happen to the examinees who performed very poorly if you used your answer in 8 to report the
scores for the mg2 students, but used the observed scores for the examinees on mg1?