Asymmetric Triangulation Scaling

For detailed information on asymmetric triangulation scaling (ATRISCAL), please click here. As a type of multidimensional scaling, ATRISCAL provides the global dependency relationship between test items. In this method, each item is given coordinates within a hemispherical dome as a result of analysis, and the positional relationship of those coordinates are used to explore inter-item dependency.

Model Overview

Stress

Stress is the value of the target function of the estimated model. There is no need, in particular, to investigate the magnitude of this value.

Correct triangle ratio

Given a perpendicular dropped from origin O to the segment joining the coordinates of two items, we consider that a correct triangle is formed if the foot of that perpendicular lies on that segment and that an incorrect triangle is formed if it does not. For n number of items, ATRISCAL adds an imaginary item so that there are n+1 items in total. As a result, the total number of item pairs is m=n(n+1)/2, which means that the total number of triangles formed by connecting those pairs to the origin is m. Now, the ratio of correctly formed triangles among these m triangles is the correct triangle ratio. A ratio value near 1.0 indicates a good fit between the model and data. As the number of items increases however, a value of 1.0 is difficult to approach.

Correct Triangle                   Incorrect Triangle

Mean Residual (MR)

This index is also called Mean Difference (MD). Among the item pairs used to form correct triangles, this is the average error between actual conditional correct response rate and estimated conditional correct response rate. A value of MD close to 0.0 indicates a good fit between model and data.

Standardized Root Mean Squared Residual (SRMR)

This index is also called Root Mean Squared Difference (RMSD). This is the square root of the mean of the square of the above error. A value of SRMR close to 0.0 indicates a good fit between model and data.

Analysis Setting

Estimation model

At the present, only binary data (correct/incorrect) can be analyzed.

Output Options

CCRR coloring

With this option, each element making up the conditional correct response rate matrix is given a color according to its magnitude.

Sorting by correct response rate

This option outputs items according to correct response rate in order from highest to lowest.

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