Estimating Testing Time Extension Ratios
for Students with Disabilities
from Item Cumulative Curves
Mamoru Fujiyoshi1 and Akio Fujiyoshi2
1Research Division, National Center for University Entrance Examinations,
2-19-23 Komaba, Meguro-ku, Tokyo 153-8501, Japan2
Department of Computer and Information Sciences, Ibaraki University, 4-12-1
Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan

Summary. This study was conducted to define item cumulative score ratio curves
and item cumulative completion ratio curves and to propose a new method toestimate fair extension ratios of testing time for students with disabilities fromthese curves. The proposed method estimates extension ratios of testing time of
regular tests under the time-limit method from experimentally collected test datautilizing the work-limit method. In this experiment we found that for students withvisual disabilities, the present extension ratio of testing time at the National Center
Test for University Admissions in Japan is fair in Mathematics and English andthat in order to improve the present testing method of the National Center Test,
further extension of testing time is necessary for some subjects with a large amountof reading questions, such as Japanese.

Key words: Student with disabilities, Testing time, Item cumulative curve,
Subject cumulative curve, Computer-based test

1 Introduction
The purpose of this study is to develop a new estimation method of fairextension ratios of testing time for students with disabilities in the National
Center Test for University Admissions. The National Center Test is the joint
first stage achievement test for admissions of all national and local public
universities as well as some private universities in Japan. Every year, about
600,000 students take it. As for disabled candidates, various kinds of special
treatment have been administered (Fujiyoshi 1995).
The extension of testing time has been administered for disabled students
according to the type and degree of their disabilities in order to eliminate thereflection of the individual’s impairment on assessment of achievement levelor aptitude (Fujiyoshi 1995; Laing and Farmer 1984; Willingham et al. 1988).
The present extension ratio of testing time at the National Center Test is 1.3
or 1.5, and the extension ratios of the ACT Assessment and the Scholastic

2Mamoru Fujiyoshi and Akio Fujiyoshi

Aptitude Test in the U.S. are from 1.8 to 4.8 (Bennett et al. 1985; Fujiyoshi
1995, 1997, 1999; Packer 1987; Ragosta and Wendler 1992; Willingham etal. 1988). However, the ground of the extension ratio of testing time haslong been based only on experience and lacked quantitative explanations.
It has been thought that multivariate or stochastic analysis cannot be usedbecause the number of candidates with disabilities is so few. It has also been
believed that defining fairness in extension ratios of testing time is delicate
issue because the distributions of achievement level or aptitude for disabledstudents and nondisabled students might be different.
Recently, an estimation method comparing the distributions of completiontime of nondisabled subjects and disabled subjects was developed (Fujiyoshi
1997; Packer 1987; Ragosta and Wendler 1992). In the estimation procedureof this method, subject cumulative completion ratio curves are employed.
Another estimation method comparing the distributions of completion timeweighted by score was also proposed (Fujiyoshi 1999). This method employs
subject cumulative score ratio curves. Because the type and degree of disabilities of disabled students are multifarious and the number of disabled
candidates is very small, it is still questionable whether the estimated extension ratios from these methods are fair.
In this study item cumulative score ratio curves and item cumulativecompletion ratio curves are re-defined (Fujiyoshi et al. 2001), and a new
method that enables the estimation of fair extension ratios even when applied
to a small number of subjects with disabilities is proposed. We also report
the estimated extension ratios of testing time calculated through the method
for partially sighted students who use large print test booklets and for blind
students who use braille format test booklets.

2 The Definitions of Subject Cumulative Curves andItem Cumulative Curves
In this section we will mathematically re-define the two types of subject
cumulative curves and the two types of item cumulative curves.
Suppose that there are n subjects (subject1 , subject2 , . . . , subjectn) andm items (item1 , item2 , . . . , itemm ). For each item1 , item2 , . . . , itemm , suppose also that the allotment of itemj is allotmentj . Let ti,j be the time that
subjecti completed the answer of itemj . If subjecti leaves itemj unanswered,
let ti,j = 0. Let ti be max{ti,j |j = 1, . . . , m}. Let ci,j be an integer such that
if subjecti answered itemj correctly, ci,j = 1, otherwise ci,j = 0. Let total benm
Σi=1
Σj=1ci,j · allotmentj .
2.1 Subject Cumulative Curves
Definition 2.1. The subject completion function for subjecti , SCi is a function such that if t < ti , then SCi (t) = 0, otherwise t ≥ ti , then SCi (t) = 1.

Estimating Testing Time Extension Ratios

3Definition 2.2. The distribution function of frequency of completed subjects F CS and the distribution function of score of completed subjects SCS
are defined as follows:nF CS(t) = Σi=1(1/n) · SCi (t)(1)

nm
SCS(t) = Σi=1
(Σj=1ci,j · allotmentj /total) · SCi (t)(2)

Definition 2.3. A subject cumulative completion ratio curve is a set ofpoints on a coordinate system with time t on the horizontal axis and cumulative relative frequency of completed subjects F CS(t) on the vertical axis
(Fujiyoshi 1997). A subject cumulative score ratio curve is a set of points on
coordinate system with time t on the horizontal axis and cumulative relativescore of completed subjects SCS(t) on the vertical axis (Fujiyoshi 1999).
2.2 Item Cumulative Curves
Definition 2.4. The item completion function for subjecti and itemj , ICi,jis a function such that if t < ti,j , then ICi,j (t) = 0, otherwise t ≥ ti,j , thenICi,j (t) = 1.
Definition 2.5. The distribution function of frequency of completed items
F CI and the distribution function of score of completed items SCI are defined as follows:nm
F CI(t) = Σi=1
Σj=1(1/mn) · ICi,j (t)(3)

nm
SCI(t) = Σi=1
Σj=1
(ci,j · allotmentj /total) · ICi,j (t)(4)

Definition 2.6. An item cumulative completion ratio curve is a set of points
on a coordinate system with time t on the horizontal axis and cumulative
relative frequency of completed items F CI(t) on the vertical axis. An item
cumulative score ratio curve is a set of points on coordinate system with time
t on the horizontal axis and cumulative relative score of completed items
SCI(t) on the vertical axis (Fujiyoshi et al. 2001).

3 Experimentation on the Estimation of Testing Time
Extension Ratios
3.1 Purpose
By analyzing test data acquired in this experiment, we will estimate testingtime extension ratios for students with disabilities from subject cumulativecompletion ratio curves, subject cumulative score ratio curves, item cumulative completion ratio curves and item cumulative score ratio curves.

4Mamoru Fujiyoshi and Akio Fujiyoshi

3.2 Method
We developed the computer-based test (CBT) to record the answering process of a conventional paper-and-pencil test (PPT) because the National Center Test is a conventional PPT which uses optical readable marking sheets(Fujiyoshi et al. 2001). Employing a pen computer (Amity SV, Mitsubishi
Electric Corporation), the CBT was designed to simulate the answering process of the PPT as faithfully as possible. The PPT sheet is displayed on thecomputer screen, and the electronic pen replaces the pencil.
Subjects are freshmen and sophomores of four-year universities. Thenondisabled group consisted of 99 students. The partially sighted group consists of 8 students who use large print test booklets. Their corrected visualacuity of common eyesight is from 0.01 to 0.1, and its median is 0.1. The
blind group included 17 students who used braille format test booklets.
Nondisabled students took the CBT. On the other hand, partially sighted
students were given large print test booklets and blind subjects were given
braille format test booklets. The behavior of disabled students was observedby test monitors, and the answering process of disabled students was input
by the monitors into the recording system used on the CBT.
The test sets were prepared from questions previously used in the National
Center Test. Calculating from the quantity of exam questions, testing timerequired for each of the three subject was 40 minutes.
The test procedure started with issuing instructions; after that the subjects answered test questions in accordance with the work-limit method.
3.3 Results
Score Distributions: Fig. 1 is Box-and-whisker plots representing the score
distributions of the three groups for Japanese, Mathematics and English. Thevertical lines in the middle of the box plots indicate the medians. The ‘+’symbols in the boxes are the means.
The score distributions are almost similar among the three groups for eachof the three subjects. Though the median of the nondisabled group is from
3 to 28 points (out of 80 points) higher than that of the partially sighted
group and the blind group, there are no significant differences among themedians of the three groups for each of the three subjects according to theMann-Whitney test.
Item Cumulative Score Ratio Curves: Firstly, we drew item cumulativescore ratio curves from the experimental test data. Fig. 2 shows the curves
for Japanese. The left curve made up of ‘×’ symbols shows the curve forthe nondisabled group. The thin line is the curve of the Compound Weibull
distribution function, which is used for smoothing and can be expressed by asimple formulation. The middle curve made up of ‘∗’ shows the curve for thepartially sighted group. The bold line is the curve of the Compound Weibull

Estimating Testing Time Extension Ratios

Japanese

Nondisabled

5(n=99)Partially sighted (n=8)

Mathematics

Blind(n=15)

Nondisabled(n=99)

Partially sighted (n=4)

EnglishBlind(n=7)

Nondisabled(n=99)

Partially sighted (n=8)
Blind(n=17)04080(points)

Fig. 1. Box-and-whisker plots of score distributions
1.000
0.936

Score ratio

Nondisabled
Partially sighted

Blind

0.000
040.00

67.78

91.32

120.00 (min)Completion time

Fig. 2. Item cumulative score ratio curves for Japanesedistribution function. The right curve made up of ‘+’ shows the curve forthe blind group. The broken line is the curve of the Compound Weibull
distribution function.
Author’s Policy and Estimation Method: The author’s policy of theestimation of extension ratio of testing time is to guarantee the partiallysighted group and the blind group sufficient testing time that enables themto arrive at the same score ratio or completion ratio of the nondisabled groupat the time limit of the regular tests.
Based on the Japanese curves in Fig. 2, we will show how to estimate
the extension ratios of testing time from item cumulative score ratio curves.
In order to formulate the estimation process of extension ratios, the curvesof the compound Weibull distribution function are used instead of the itemcumulative curves themselves.

6Mamoru Fujiyoshi and Akio Fujiyoshi

We first checked the score ratio of the nondisabled group at the time limitof 40 minutes. In the case of Japanese, the score ratio to guarantee is 0.936,
that is, the vertical coordinate of the intersection of the nondisabled group’s
curve and the vertical line of 40 minutes.
Next, we found the horizontal coordinates of the intersection of the horizontal line of 0.936 and the disabled group’s curves. The completion time is
40 minutes for the nondisabled group, 67.78 minutes for the partially sightedgroup and 91.32 minutes for the blind group.
The estimation value of extension ratio of testing time is defined as theratio of the completion time for the disabled group to completion time for thenondisabled group at the score ratio to guarantee. In the case of Japanese,the extension ratios of testing time are estimated to be 1.696 (67.78/40.00)
for the partially sighted group and 2.284 (91.32/40.00) for the blind group.
Estimated Extension Ratios of Testing Time: We estimated the extension ratios of testing time for the disabled groups by means of the following
four estimation methods:(Method 1) Estimation from item cumulative score ratio curves: Thismethod is explained in Section 3.3.3.
(Method 2) Estimation from item cumulative completion ratio curves: Thismethod is obtained from Method 1 exchanging score ratio for completion
ratio.
(Method 3) Estimation from subject cumulative score ratio curves: Thismethod is obtained from Method 1 exchanging item cumulative curves
for subject cumulative curves; this method was introduced in (Fujiyoshi
1999).
(Method 4) Estimation from subject cumulative completion ratio curves:
This method is obtained from Method 3 exchanging score ratio for completion ratio; this method was introduced in (Fujiyoshi 1997).
Table 1 shows the score ratios (Method 1 and 3) and the completion
ratios (Method 2 and 4) of the nondisabled group at 40 minutes. Table 2
shows the completion time and the extension ratios estimated from the item
cumulative score ratio curves (Method 1), the item cumulative completion
ratio curves (Method 2), the subject cumulative score ratio curves (Method 3)(Fujiyoshi 1999), and the subject cumulative completion ratio curves (Method4) (Fujiyoshi 1997).
The estimated extension ratios from the four methods are similar.
Table 1. Score ratios (Method 1 and 3) and completion ratios (Method 2 and 4)
JapaneseMathematics
EnglishMethod 1 Method 2 Method 3 Method 4
0.936
0.937
0.690
0.693
0.920
0.888
0.549
0.549
0.942
0.953
0.664
0.664

Estimating Testing Time Extension Ratios

7

Table 2. Completion times and estimated extension ratios from the four methods
Method 1 Method 2 Method 3 Method 4
time ratio time ratio time ratio time ratio
Nondisabled
40.00 1.000 40.00 1.000 40.00 1.000 40.00 1.000
Japanese
Partially sighted 67.78 1.696 65.24 1.629 61.53 1.538 57.14 1.428
Blind91.32 2.284 96.17 2.401 92.97 2.324 92.83 2.321
Nondisabled
40.00 1.000 40.00 1.000 40.00 1.000 40.00 1.000
Mathematics Partially sighted 68.30 1.706 82.58 2.062 51.05 1.276 59.80 1.495
Blind68.10 1.701 70.05 1.749 60.68 1.517 65.30 1.633
Nondisabled
40.00 1.000 40.00 1.000 40.00 1.000 40.00 1.000
EnglishPartially sighted 41.95 1.049 45.33 1.133 45.31 1.133 44.36 1.116
Blind64.43 1.611 73.99 1.850 66.99 1.675 69.46 1.737

3.4 Discussion
We estimated extension ratios of testing time for students with disabilities
from the four methods using the test data acquired in this experiment. In theNational Center Test for University Admissions, the present extension ratio oftesting time for all subjects has been 1.3 for partially sighted students whose
corrected visual acuity of common eyesight is less than 0.15 since 1988. For
blind students, the present extension ratio has been 1.5 since 1979.
The estimated extension ratios of testing time for disabled students arealmost the same as the present extension ratio for Mathematics and English.
However, the estimated extension ratios for Japanese are much greater thanthe present extension ratio.
For Mathematics, the estimated extension ratio for the partially sighted
group seems much greater than the present extension ratio, 1.3. However, it
is not a reliable figure because the number of subjects was too small.

4 Conclusion
We proposed methods to estimate fair extension ratios of testing time fordisabled students according to author’s policy of fairness from the item cumulative score ratio curves and the item cumulative completion ratio curves.
It was experimentally shown that the extension ratios for disabled students
in regular tests under the time-limit method can be estimated from the experimental test data under the work-limit method.
Though the estimated extension ratios from the four methods are similar,
we think that the estimation method by item cumulative score ratio curves
surpasses the others because of the following features in methodology:Firstly, as item cumulative curves have more measured points than subject
cumulative curves, item cumulative curves are smooth and stable even with a8

Mamoru Fujiyoshi and Akio Fujiyoshi

small number of subjects. Secondly, the estimation method by score ratio canbe applied even if the score distributions of two groups are different because
the estimation is made from the distributions of completion time weighted
by score. Thirdly, the estimation of extension ratios of testing time is fairbecause sufficient testing time is guaranteed for disabled students to arriveat the same score ratio of nondisabled students at the time limit of regulartests.
This study revealed that the estimated extension ratios of testing time forMathematics and English for students with visual disabilities are almost thesame as the present extension ratios of the National Center Test. This meansthat the present extension ratios of testing time are fair in Mathematics
and English. In order to improve the present testing method of the National
Center Test, further extension of testing time is necessary for some subjects
which include a large amount of reading questions, such as Japanese.

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