**ERIC Identifier:** ED447147

**Publication Date:** 2000-09-00

**Author: **Russell, Michael

**Source: **ERIC Clearinghouse on
Assessment and Evaluation College Park MD.

## Using Expected Growth Size Estimates To Summarize Test Score
Changes. ERIC/AE Digest.

An earlier digest described the shortcomings of three methods commonly used
to summarize changes in test scores (Russell, 2000). This article describes two
less commonly used approaches for examining change in test scores, namely
Standardized Growth Estimates and Effect Sizes. Aspects of these two approaches
are combined and applied to the Iowa Test of Basic Skills (ITBS) to demonstrate
the utility of using a third method, termed Expected Growth Size, to examine
change in test scores.

### STANDARDIZED GROWTH ESTIMATES

Stenner, Hunter, Bland, &
Cooper describe a standardized growth expectation (SGE) as "the amount of growth
(expressed in standard deviation form) that a student must demonstrate over a
given treatment interval to maintain his/her relative standing in the norm
group" (1978,p. 1). To determine an SGE, Stenner et. al. proposed the following
three-step method.

Step 1. The scale score associated with the 50th percentile for a given grade
level or the pre-test is identified.

Step 2. The percentile rank for the following grade level or the post-test
associated with this scale score is found.

Step 3. The difference between the 50th percentile and the post-test
percentile is calculated.

To determine this difference, a unit normal deviate table is used to convert
percentiles to z-scores and the z-score for the post-test is subtracted from the
z-score for the pre-test.

The difference between the pre-test and post-test z-scores is the SGE and
expresses "the amount of loss in relative standing that such a student would
suffer if he/she learned nothing during the time period" (Stenner, et. al.,
1977, p. 1).

As an example, to determine the SGE for grade 3, Table 1 indicates that the
scale score associated with the 50th percentile for grade 3 on the ITBS Language
sub-test is 174. The percentile rank for grade 4 that corresponds to a scale
score of 174 is 26. If a student received the same scale score in grades 3 and
4, their percentile rank would drop from 50 to 26. After both percentiles are
converted to z-scores and subtracted, the difference between the two z-scores
represents the SGE. In this case, the z-scores corresponding to percentile ranks
of 50 and 26 are 0 and -.64, respectively. Thus, the SGE is .64, which indicates
a relative loss of .64 standard deviations for a student who shows no change in
his/her test score.

### EFFECT SIZES

When applying Stenner et. al.'s method for
calculating SGEs, Haney, Madaus and Lyons (1993, p. 231-32) point out that the
idea of a SGE is analogous to an effect size in that each represents the
difference in mean performance of two groups expressed in standard scores. As
Glass, McGaw and Smith (1981) describe, an effect size represents the difference
between two groups in standard deviations. To calculate an effect size, the
difference between the mean of the control group and the experimental group is
divided by the standard deviation of the control group. Conceptually, the only
difference between an effect size and an SGE is that an effect size is used to
compare the means of a "control" group and an "experimental" group while a SGE
compares the performance of groups of students at various grade levels.

### -----

See TABLE 1 at end of digest.

###
-----

EXPECTED GROWTH SIZE

Although an SGE and an effect size are similar,
there is one important difference: an SGE focuses on the standing lost when
there is no change in test score, while the effect size focuses on the amount of
change in a test score necessary to maintain one's standing. When applied in
this manner, the effect size method provides an estimate of the expected growth
size between two time periods. In the example above, the expected growth size
(EGS) between grade three and grade four on the ITBS Composite Language test is
.89 standard deviations.

### DEFINING THE BASE YEAR OR CONTROL GROUP

In a well-designed
experiment, there is little question as to which group is defined as the control
group and which is the experimental group. However, when applying the concept of
an effect size to change in test scores between two grade levels, one could
reference growth to the pre-test or the post-test distribution.

In the case of SGEs, the post-test distribution is used to reference
"growth". Note, however, that although SGEs employ the term growth, the
methodology actual provides a measure of loss assuming that a student
experiences no growth whatsoever. In this way, using the post-test distribution
to reference "growth" is fundamentally flawed in that change is placed in the
context of where a student is expected to be rather than from where they
started. The situation is analogous to describing someone's progress on trip in
relation to how far they still must go in order to reach their destination
rather than from how far they have traveled since their departure.

In the case of using an effect size to express growth between two grade
levels, one might argue that the pooled standard deviation be employed in lieu
of the standard deviation of the control group. However, the difficulty of
obtaining an estimate of the pooled standard deviation for most standardized
tests forces a choice between designating the pre-test or the post-test as the
control group. Given the desire to measure change or growth from where a group
begins at one point in time to where they end at a second point in time, the EGS
methodology references change to the pre-test distribution. For this reason, the
pre-test distribution is assigned as the control group.

### ADVANTAGES OF AN EXPECTED GROWTH SIZE

Although an expected
growth size is more difficult to calculate, it offers three advantages. First,
by expressing change in relation to the standard deviation, growth rates for
different tests and different grade levels can be compared directly. Table 2
presents expected growth sizes for grades 1 through 8 for several portions of
the ITBS. Examining Table 2, one can see that the expected growth sizes differ
for each portion of the ITBS. Table 2 also shows an inverse relationship between
grade level and size of expected growth. As the grade level increases, the
amount of growth students experience decreases.

### -----

See TABLE 2 at end of digest.

### -----

Similarly, within each grade level, the amount of
growth students experience varies by percentile ranks. Students scoring at the
25th percentile experience less growth than students scoring at the mean. And
students scoring at the mean experience less growth than students scoring at the
75th percentile. This pattern explains why the standard deviation for most
standardized tests increases as the grade level progresses.

Second, once expected growth sizes are calculated for a given test, they can
be easily transformed to more common measurement scales. As an example,
multiplying the expected growth size by the standard deviation of an Normal
Curve Equivalent, NCE, (21.06) provides the number of NCE points a student's
score increases during a given time period relative to the student's initial
norm group when s/he maintains his/her current standing. For the ITBS Language
test, the score for a student who maintains a 50th percentile ranking increases
18.74 NCEs between the third and fourth grade.

Third, once expected growth sizes are transformed to an NCE scale, changes in
an individual's or a group's mean score can be reported in relation to expected
growth. Performance on most standardized tests is reported relative to the Norm
Group for a student's current grade. If the student grows at the same rate as
other students in the Norm Group, his/her percentile rank and NCE will remain
the same across two years. However, if the student's rate of growth differs from
that of the Norm Group, his/her NCE and percentile rank will change.

The expected growth size can be used to determine the extent to which the
student's growth exceeded or fell short of the expected growth size. To do so,
the student's current NCE is subtracted from his/her previous NCE and divided by
the expected NCE growth rate. As an example, consider a student whose NCE for
the ITBS Language test increased from 50 in grade 3 to 55 in grade 4. When
divided by the expected NCE growth size for third grade (18.74), this five point
increase represents 1.27 years of growth. Thus, the student's score increased
27% more than expected.

As Table 2 indicates, growth sizes vary across grade levels. Expressing
change in test scores in relation to expected growth size takes these
differences in growth rates into consideration. The extent to which performance
changes is placed in the context of how scores generally change for students in
a given grade. As a result, a more accurate measure of how a student changes
relative to other students in his/her grade is produced. As an example, Table 2
shows that students in grade 2 experience about twice as much growth in their
test scores compared to students in grade 5. For this reason, an increase of 5
NCEs on the ITBS Composite Math test represents larger growth relative to
expected growth for a student in grade 5 than for a student in grade 2.

### LIMITATIONS OF EXPECTED GROWTH SIZES

Although expected
growth sizes provided a sounder approach for summarizing change in test scores
than some of the more commonly used approaches, their use is limited to norm
referenced standardized tests. Moreover, the EGS methodology assumes that the
tests have been vertically equated. When comparing change across multiple years,
the methodology also assumes that the tests administered each year provide
measures of the same construct based on identical content. Although most
norm-referenced tests attempt to meet both assumptions vertical equating and
measures of the same construct the extent to which they fail to meet these
assumptions impacts the accuracy of estimates yielded by the EGS methodology.
Finally, as with all comparisons of change over time, the EGS method is also
limited by the reliability of the scores used to calculate change. Although
there is considerable debate over the extent to which low score reliability
impacts the meaningfulness of change scores, caution is advised when employing
the EGS method for tests with low reliability (see Willet, 1988 for fuller
discussion on reliability and change scores).

### USING EXPECTED GROWTH SIZES FOR YOUR STUDENTS

To apply
expected growth sizes to examine change in the performance of your students,
readers are encouraged to use the attached spreadsheet. The spreadsheet provides
an easy-to-use template that allows users to calculate expected growth sizes for
most standardized tests. In addition, the spreadsheet translates expected growth
sizes into expected changes in NCE scores for each grade level.

As the attached instructions indicate, two pieces of information are required
to use the spreadsheet: 1. Standard Score to Percentile Rank Conversion tables
for the standardized test; and 2. The standard deviation for the standard score
for each grade level. This information is available in the Technical Report(s)
for each standardized test.

Although expected growth sizes are more complicated to calculate, they
provide a more accurate and comparable method of examining change in test scores
within and across grade levels and on different tests.

### REFERENCES

Glass, G., McGaw, B. & Smith, M. L. (1981).
Meta-analysis in Social Research. Beverly Hills: Sage.

Haney, W., Madaus, G., & Lyons, R. (1993). The Fractured Marketplace for
Standardized Testing. Boston, MA: Kluwer Academic Publishers.

Russell, M. (2000). Summarizing change in test scores: shortcomings of three
common methods. ERIC Digest Series. Also in Practical Assessment, Research and
Evaluation, 7(5). [Available online:
http://ericae.net/pare/getvn.asp?v=7&n=5 ].

Stenner, A. J., Hunter, E. L., Bland, J. D., & Cooper, M. L. (1978). The
standardized growth expectation: Implications for educational evaluation. Paper
presented at the Annual Conference of the American Educational Research
Association, Toronto, Canada. (ERIC Document Reproduction Service Number ED 169
072.)

Willett, J. (1988). Questions and answers in the measurement of change. In E.
Z. Rothkopf (Ed.), Review of Research in Education 15 (pp. 345-422). Washington,
DC: American Educational Research Association.

This Digest is based on Russell, Michael (2000). Using Expected Growth Size
Estimates to Summarize Test Score Changes. Practical Assessment, Research &
Evaluation, 7(6). Available online:
http://ericae.net/pare/getvn.asp?v=7&n=6.

TABLE 1 |

==========================================================
Percentile Rank, Standard Score and Standard Deviations for the
Iowa Test of Basic Skills Language Sub-test Percentile Rank |

Standard Score |
Grade 3 |
Grade 4 |

174 |
50 |
26 |

175 |
52 |
27 |

176 |
54 |
29 |

189 |
78 |
47 |

190 |
79 |
48 |

191 |
81 |
50 |

St. Dev. |
19.05 |
24.25 |

* In the SGE example above, the third grade is designated as
the control group and the fourth grade is the experimental group. To
determine the effect size or amount of growth between grade three and
grade four, the standard score associated with the 50th percentile rank
for grade three is subtracted from the standard score associated with the
same percentile rank for |