Comparing Test Equating by Item Response Theory and Raw Score Methods with Small Sample Sizes on a Study of the ARTé: Mecenas Learning Game
Abstract:The purpose of the present research is to equate two
test forms as part of a study to evaluate the educational effectiveness
of the ARTé: Mecenas art history learning game. The researcher
applied Item Response Theory (IRT) procedures to calculate item,
test, and mean-sigma equating parameters. With the sample size
n=134, test parameters indicated “good” model fit but low Test
Information Functions and more acute than expected equating
parameters. Therefore, the researcher applied equipercentile equating
and linear equating to raw scores and compared the equated form
parameters and effect sizes from each method. Item scaling in IRT
enables the researcher to select a subset of well-discriminating items.
The mean-sigma step produces a mean-slope adjustment from the
anchor items, which was used to scale the score on the new form
(Form R) to the reference form (Form Q) scale. In equipercentile
equating, scores are adjusted to align the proportion of scores in each
quintile segment. Linear equating produces a mean-slope adjustment,
which was applied to all core items on the new form. The study
followed a quasi-experimental design with purposeful sampling of
students enrolled in a college level art history course (n=134) and
counterbalancing design to distribute both forms on the pre- and posttests.
The Experimental Group (n=82) was asked to play ARTé:
Mecenas online and complete Level 4 of the game within a two-week
period; 37 participants completed Level 4. Over the same period, the
Control Group (n=52) did not play the game. The researcher
examined between group differences from post-test scores on test
Form Q and Form R by full-factorial Two-Way ANOVA. The raw
score analysis indicated a 1.29% direct effect of form, which was
statistically non-significant but may be practically significant. The
researcher repeated the between group differences analysis with all
three equating methods. For the IRT mean-sigma adjusted scores,
form had a direct effect of 8.39%. Mean-sigma equating with a small
sample may have resulted in inaccurate equating parameters.
Equipercentile equating aligned test means and standard deviations,
but resultant skewness and kurtosis worsened compared to raw score
parameters. Form had a 3.18% direct effect. Linear equating
produced the lowest Form effect, approaching 0%. Using linearly
equated scores, the researcher conducted an ANCOVA to examine
the effect size in terms of prior knowledge. The between group effect
size for the Control Group versus Experimental Group participants
who completed the game was 14.39% with a 4.77% effect size
attributed to pre-test score. Playing and completing the game
increased art history knowledge, and individuals with low prior
knowledge tended to gain more from pre- to post test. Ultimately,
researchers should approach test equating based on their theoretical
stance on Classical Test Theory and IRT and the respective assumptions. Regardless of the approach or method, test equating
requires a representative sample of sufficient size. With small sample
sizes, the application of a range of equating approaches can expose
item and test features for review, inform interpretation, and identify
paths for improving instruments for future study.
 M. J. Kolen and R. L. Brennan, Test Equating, Scaling, and Linking:
Methods and Practices. New York: Springer, 2014.
 F. M. Lord, Applications of Item Response Theory to Practical Testing
Problems. Hillsdale, NJ: Erlbaum, 1980.
 R. K. Hambleton, H. Swaminathan, and H. J. Rogers, Fundamentals of
Item Response Theory, vol. 2, New York: Sage, 1991.
 L. L. Cook and N. S. Paterson, “Problems related to the use of
conventional and item response theory equating methods in less than
optimal circumstances,” Applied Psychological Measurement, vol. 11,
no. 3, pp. 225-244, 1987.
 J. González, “SNSequate: Standard and nonstandard statistical models
and methods for test equating,” Journal of Statistical Software, vol. 59,
no. 7, pp. 1-30, 2014.
 S. A. Livingston, “Equating Test Scores (Without IRT).” Princeton, NJ:
Educational Testing Service, 2004.
 M. J. Kolen, “Effectiveness of analytic smoothing in equipercentile
equating,” Journal of Educational Statistics, vol. 9, no. 1, pp. 25-44,
 J. P. Meyer and S. Zhu, “Fair and equitable measurement of student
learning in MOOCs: An introduction to item response theory, scale
linking, and score equating,” Research & Practice in Assessment, vol. 8,