One test at the same time point
Measurement problems are magnified when a specific cut point is used for identification purposes. This problem affects any psychometric approach to LD identification, particularly when the test score is not criterion referenced, and score distributions have been smoothed to create a normal univariate distribution. If there was a natural discontinuity in the score distribution, setting cut points would be straightforward. But as I indicated above, natural breaks are not usually apparent in achievement distributions. If the distribution of achievement in the population reflects the combination of achievement distributions from different populations (e.g., LD and non-LD individuals), it is possible to estimate the parameters of the multiple populations and the mixing proportions, and ultimately to assign individuals to the different populations. However, with only one (e.g., Achievement) or two measures (e.g., IQ and Achievement, or achievement at two time points), the model is not identified, and would require untenable assumptions to achieve a solution. For example, in the hypothetical mixture distribution of Figure 1, the observed distribution could be the result of combining two unobserved normal distributions with different means and unequal mixing proportions, or could reflect a natural skewing process such as a test with an inadequate ceiling (See Figure 1).
Figure 1: Hypothetical Probability Distribution (Mixture) Involving Two Latent Classes
Note: This distribution was created from mixing two normal distributions with different means, but equal standard deviations, and mixing proportions of 0.1 and 0.9. Thus, in the present case, a latent class underlies the observed distribution. However, without additional information, this situation is difficult to distinguish from one where the test has more room at the floor than at the top.
Regardless of normality, there is measurement error in any psychometric procedure (Shepard, 1980). Because of measurement error, any cut point set on the observed distribution will lead to instability in the identification of people as LD as scores fluctuate around the cut point with repeat testing, even for a decision as transparently straightforward as demarcating low achievement. This fluctuation is not a problem of repeat testing, nor is it a matter of selecting the most valid cut-score. Rather, the problem is that no single observed score can perfectly capture a student's ability on an imperfectly measured latent variable. The fluctuation in identifications will also vary across tests, depending in part on the measurement error and the cut-score because tests will vary in their precision at different points on the ability scale. As the cut point moves away from the center of the score distribution, where the standard error of measurement is generally smallest, this problem becomes more apparent. Tests are typically constructed to yield the greatest precision in the middle of a distribution.
Approaches in a variety of areas in which a normal distribution is subdivided to create groups and the groups are subsequently compared on external variables - the epitome of research on LD - have been widely criticized in the statistical literature because of the imposition of an arbitrary group structure on continuous distributions (Cohen, 1983; Maxwell & Delaney, 1993). Making categories of continuous distributions artificially constrains the within group variability, reducing the range of measurement and distorting the relative importance of the underlying dimensions to performance on other measures. This problem directly impacts attempts to identify children as LD based solely on low achievement cut points. In both real and simulated data sets, fluctuations up to 35% of cases are found even when a single test is used to identify a cut point. Similar problems are apparent if a discrepancy model is used (Francis et al., in press; Shaywitz et al., 1992).
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