http://en.wikipedia.org/wiki/Granger_causality

http://en.wikipedia.org/wiki/Vector_autoregression

# General Analytic Strategies

Although there are many important issues and questions that need to be addressed in research on the effects of accommodations on test scores, some of the most central issues need to focus on the question: “Do accommodations change the nature of what is being measured by the test?” If the answer is “no,” then the scores obtained under nonstandard conditions can be placed on the same measurement scale used for all students, and the scores can be aggregated and compared. To address these score comparability or validity questions, differential item functioning, factor analytic, and criterion-related data analytic strategies need to be used. A brief description of each of these strategies follows.

It is important to note, however, that this list of analytic strategies is illustrative—not exhaustive. For example, there are many other accommodation, research design, and sampling issues (e.g., accommodation or treatment integrity, randomly selected samples) that are not addressed in this conceptual paper. Furthermore, differences between and within district or state testing programs most likely will require unique variations and modifications of these strategies.

## Item Response Theory (IRT)

Item Response Theory (IRT) strategies are recommended for evaluating the extent to which the abilities measured by the individual items of a test are changed substantially as a result of an accommodation (i.e., investigating differential item functioning). These strategies are explained using Figures 1 and 2 as examples that illustrate the application of IRT methods to four items from the 1995 NAEP Field Test.

In each figure, the x-axis represents the ability or trait being measured by the items in the test (in this case, mathematical ability). The scale is a continuum from less (left end) to more (right end) mathematical ability. The y-axis represents the probability of success on the item in question. By applying IRT procedures, an item characteristic curve (ICC) is developed for each item that visually represents the probability of success on that item (y-axis) as a function of ability (x-axis).

The two graphs in Figure 1 portray the ICCs for two mathematics items. The solid ICC lines represent the ICC obtained for the two items in the general population (i.e., how the items “behave” for most subjects who responded to the items under standard test conditions). Also plotted on the two graphs are the ICCs for the same items when given to students with disabilities under accommodated conditions. These curves are represented by the small dots. What one hopes to find, and what is represented in the two graphs in Figure 1, is a situation where the ICCs for standard and accommodated administrations are almost identical. A visual review of the two sets of ICCs for the two graphs shows that the ICCs are indeed similar. This means that the items appear to be “behaving” similarly regardless of whether they were administered with or without accommodations. Thus, they appear to be measuring the same trait or ability, and therefore can be placed on the same measurement scale.

Figure 1. Graphs Showing Similar Item Characteristic Curves for Standard and Accommodated Administrations

Figure 2 presents a contrasting finding. In both graphs, the ICCs for the standard and accommodated administrations are dramatically different. ICC plots such as these, and their associated empirical fit indicators, suggest that when the items are administered under nonstandard conditions, they “behave differently.” That is, the same items are not measuring the same trait or ability when an accommodation is introduced. The empirical relationship between the probability of success on these items and the traits or abilities being measured has been altered by the use of an accommodation. As a result, test scores generated from the combination of a large number of these “misbehaving” items cannot be placed on the same measurement scale as scores based on a combination of items administered under standard conditions.

Figure 2. Graphs Showing Different Item Characteristic Curves for Standard and Accommodated Administrations

## Factor Analysis

Factor analytic strategies are important for evaluating the construct validity of tests. These procedures help determine whether the underlying dimensions or constructs measured by a test are the same when administered under standard and nonstandard conditions. The two diagrams shown in Figure 3 illustrate the essence of these types of analyses.

Figure 3. Factor Analysis Models Indicating Different Factor Structures for Standard and Accommodated Administrations

The rectangles in the figure represent sub-scales A-F from a math test. Each subscale is constructed from a combination of math items that together measure a mathematics subskill. When given a set of variables or subscales (in this example, six math subscales), factor analytic procedures help determine the number of broader dimensions, factors, or constructs that account for the shared abilities of the subscales. In the first factor model, the six subscales (A-F) were found to be indicators of one general construct of math (viz., General Math). The circle represents this factor or construct. Assume that this single or general factor model was found when the math tests were administered to the general population (without accommodations) and the data were factor analyzed.

Next, assume that the same six subscales were administered under accommodated conditions. If the accommodations do not change the nature of the construct being measured, then the application of factor analytic procedures to the data from this sample should result in generally the same factor structure. That is, the construct being measured by the test under nonstandard conditions would be the same if a similar single or General Math factor was found to best represent the relationship between subscales A-F. Alternatively, if the accommodations changed the nature of the construct being measured, a different factor structure might emerge.

The second model displays factor analytic results that suggest that the structure or dimensions being measured by subscales A-F under accommodated conditions are best explained by two different broad math factors (Math X and Math Y). This finding, together with the finding of the General Math factor model in the general population, would indicate that this specific collection of math subscales is not measuring the same constructs under standard and accommodated conditions. Besides exploratory factor analysis, confirmatory factor analysis (LISREL) procedures can be used in these types of analyses. Confirmatory procedures are particularly well suited to evaluating the extent to which the constructs measured by a collection of variables or tests are similar (i.e., invariant) across different samples and conditions.

## Criterion-Related Analyses

Criterion-related analytic strategies are needed to investigate the extent to which accommodated test administrations change the relationship between test scores and other criteria. These procedures can help evaluate whether the criterion-related validity (often referred to as predictive validity) of a test is similar for different samples or for different versions of the same test (i.e., standard and accommodated test administrations). If a test is used to make predictions about a person’s performance on an important outcome criterion (e.g., potential success in college, mastery of a domain of skills), it is important to know whether the relationship that exists between the test score(s) (i.e., the predictor) and the important outcome criteria changes when the test is administered under accommodated conditions. That is, can prediction and classification decisions about a person be made with a similar degree of confidence for test scores administered under standard and accommodated conditions?

Although the specific data analytic method may vary depending on the nature of the predictor and outcome variables (e.g., correlation, multiple regression, classification agreement), most criterion-related analytic strategies are concerned with addressing the question represented in Figure 4, “Is this relationship the same?”

Figure 4. Representation of Criterion-Related Analystic Strategies

# Group Research Design Considerations

To use the general analytic strategies described above, research designs must be employed that meet certain characteristics. This section presents general design considerations for sampling methods and sampling size in group-based accommodations research. These are presented to provide an idea of design considerations that may be required to conduct research on the effects of accommodations on test scores.

## Sampling

Sampling issues are very complex and cannot be treated in detail in this paper. Ideally, the samples in each design matrix cell would be randomly selected from the appropriate population (e.g., Group 1 and Group 2, both randomly selected from all students with the particular characteristic or accommodation need being targeted in the larger population of interest).

## Size

Another important consideration is the size of the sample in each design cell. The general analytic strategies described above (factor analysis and IRT, in particular) require relatively large samples to obtain stable statistical estimates. Many measurement specialists would recommend sample sizes as large as 500 for each cell in each design matrix for IRT analyses. However, given the practical constraints of applied research, and the small number of students with disabilities who take tests with accommodations, smaller sample sizes are more realistic. We suggest that, at a minimum, 200 subjects per subsample (i.e., each cell in each design matrix) should be used for applied research employing the general data analytic strategies outlined in this paper.

# Group Research Designs

The four general group research designs presented in this section are ordered from the most optimal (Design # 1) to the least optimal (Design # 4). For illustrative purposes, only one type of accommodation group (e.g., students with reading difficulties or who need a specific type of accommodation) is presented in each design. Additional groups, or students with other characteristics (e.g., limited English proficiency), with parallel information in each cell, could be added to the design matrices. In addition, we have presented the simple version of each design. Any of the designs could be made more sophisticated by counterbalancing not only form of the test, but also order in which forms are presented, and so on. The designs that we present can be modified in many ways. It is also important to note that we do not define accommodation groups by disability category since category of disability does not define the need for accommodations. Nevertheless, it generally is helpful to select subjects meeting a specific criterion (e.g., reading problem identified by test score) from within a single disability category (e.g., learning disability) so that other complicating characteristics (e.g., visual disability) are less likely to complicate findings.

## Design 1

This design allows for the examination of the comparability of scores as a function of the presence/absence of a characteristic, the use of an accommodation, and the interaction of these two factors. The design requires equivalent forms (A & B) of the test. The effect of test order is controlled by counterbalancing the administration of forms A and B. Thus, Design 1 requires subjects who are willing to take two versions (with and without accommodations) of the same test. Subjects with and without disabilities who take the test without the accommodations could be drawn from the general testing population. Their scores could be randomly selected from the total test sample of all students who regularly take the version of Forms A and B. This design does not require that the samples from the two respective groups (Disability groups 1 and 2 and Non-disability groups 1 and 2) be exactly similar (i.e., matched) in important characteristics. Design 1 is illustrated in Table 2.

Table 2. Design 1: Comparability of Scores as a Function of the Presence/Absence of a Disability

 Disability Group 1* Disability Group 2* Non-Disability Group 1 Non-Disability Group 2 With Accommodation Test Form A Test Form B Test Form A Test Form B Without Accommodation Test Form B Test Form A Test Form B Test Form A

* Disability Groups 1 and 2 are students with a common characteristic (e.g., students with reading problems) or who have the same accommodation need (e.g., Braille edition).

An example of a study that used Design 1 is a recent multi-state study supported by the Technical Guidelines for Performance Assessment project, which received funding from the U.S. Department of Education, Office of Educational Research and Improvement (OERI). In this study, groups of students with reading disabilities and students without any special education designation were administered the equivalent of a state test using a videotape presentation of a math test and the test administered under typical conditions. Two forms of the test were given to both groups, in counterbalanced order, to begin to sort out the effects of the change in test administration procedures.

## Design 2

Design 2 also allows for the examination of the comparability of scores as a function of the presence/absence of a disability-related need, the use of an accommodation, and the interaction of these two factors. It is different from Design 1 in that this design requires that the respective samples of the students with disabilities groups (Groups 1 and 2) and students without disabilities (Groups 1 and 2) be equivalent in important characteristics (e.g., matched samples). If they are not, it is impossible to determine whether any differences between the score characteristics of the respective groups (Disability Group 1 vs. Disability Group 2; Non-Disability Group l vs. Non-Disability Group 2) are due to the effects of the accommodations, or are attributable to differences in sample characteristics. Design 2 does not require equivalent forms (A & B) of the test. Subjects with and without disabilities who take the test without accommodations can be drawn from the general testing population. Their scores can be randomly selected from the total test sample of all students who regularly take versions of Form A (see Table 3).

Table 3. Design 2: Comparability of Scores as a Function of the Presence/Absence of a Disability

 Disability Group 1* Disability Group 2* Non-Disability Group 1 Non-Disability Group 2 With Accommodation Test Form A Test Form A Without Accommodation Test Form A Test Form A

* Disability Groups 1 and 2 are students with a common characteristic (e.g., students with reading problems) or who have the same accommodation need (e.g., Braille edition).

A version of this design was used by Tindal, Hollenbeck, Heath, & Almond (1998), who had students take a statewide writing test that required them to write a composition. The students were allowed use of either paper and pencil or a computer over the three days devoted to writing the composition. In this study there was the additional condition that students could use the computer to (1) compose on all three days, (2) compose on the computer only the last day, or (3) compose with a spell-checker available. The compositions were compared on six traits (ideas-content, organization, voice, word choice, sentence fluency, conventions).

Phillips and Millman (1996) noted that beyond the selection of comparable students, there are additional concerns, such as the standardization of equipment and ensuring that students had adequate training in word processing:

Standardization of equipment is an issue because the study would probably rely on the use of computer equipment already present in the schools. Because different software programs offer a variety of options, it would be necessary to develop a list of permissible equipment and software, which is judged to provide the same basic features and ease of use. Spell-check, thesaurus and editing functions should be comparable. Finally, each student should be thoroughly familiar with the hardware and software to be used during testing and should have had sufficient practice time to develop facility with the software (p. 4).

## Design 3

Design 3 allows for the examination of score comparability as a function of accommodation use for only one disability group. This design requires the assumption (based on prior research) that the scores of subjects with disabilities who take the test without the accommodation are comparable to the scores of subjects without disabilities who take the test without the accommodation. It requires equivalent forms (A & B) of the test, and controls for the effect of test order by counterbalancing the administration of forms A and B. Design 3 also requires subjects who are willing to take two versions (with and without accommodations) of the same test. Subjects with disabilities who take the test without the accommodation can be drawn from the general testing population. Their scores can be randomly selected from the total test sample of all students with disabilities who regularly take the versions of Form A and B. Finally, Design 3 does not require that the two respective samples (Groups 1 and 2) be exactly similar (i.e., matched) in important characteristics. This design is illustrated in Table 4.

Table 4. Design 3: Examination of the Comparability of Scores as a Function of the Use of an Accommodation for a Single Disability

 Disability Group 1* Disability Group 2* With Accommodations Test Form A Test Form B Without Accommodations Test Form B Test Form A

* Disability Groups 1 and 2 are students with a common characteristic (e.g., students with reading problems) or who have the same accommodation need (e.g., Braille edition).

An example of a study that used something like Design 3 is one conducted by Tindal, Heath, Hollenbeck, Almond, and Harniss (1998). They had students complete reading and math multiple choice tests by either filling in the standard bubble sheets or by marking on the test booklet.

## Design 4

Design 4 allows for the examination of the comparability of scores as a function of the use of an accommodation for subjects with disabilities only. This design requires the assumption (based on prior research) that the scores for students with disabilities who take the test without the accommodation are comparable to those for regular education students who take the test without the accommodation. It also requires that the respective subjects with disabilities be equivalent in important characteristics (e.g., matched samples). If not, it is impossible to determine whether any differences in score characteristics between the respective groups are due to the effect of the accommodation or are attributable to differences in sample characteristics. Design 4 does not require equivalent forms (A & B) of the test (see Table 5). Subjects who take the test without the accommodation could be drawn from the general testing population. Their scores can be randomly selected from the total sample of all students with disabilities who regularly take versions of Form A.

Table 5. Design 4: Examination of the Comparability of Scores as a Function of the Use of an Accommodation for Subjects with Disabilities

 Disability Group 1* Disability Group 2* With Accommodation Test Form A Without Accommodation Test Form A

* Disability Groups 1 and 2 are students with a common characteristic (e.g., students with reading problems) or who have the same accommodation need (e.g., Braille edition).

A study using this design could take place during an actual large-scale testing session. As an example, Design 4 could be used in pre-selecting students with disabilities who would be participating in a large-scale assessment. Students could be matched by nature of disability (e.g., reading disability) and other important factors. Students in Group 1 would have the test read aloud while students in Group 2 would read the test to themselves. Scores could be compared for evidence of differences between groups. Of course, when using this design, researchers must ensure that students would not be denied accommodations they need, especially if the results would be used for high-stakes decision making.

http://www.cehd.umn.edu/nceo/OnlinePubs/Technical26.htm