Singular-value decomposition (SVD) is a technique for factoring a data matrix. Here it is used to look for patterns in the Rasch residuals which may indicate multidimensionality. SVD values are estimated by iterating through the residual data matrix. Each residual is explained, as much as possible, by a value attributed to the person (row) and a value attributed to the item (column). These values are multiplied.
The SVD estimation method implemented in Winsteps uses randomization for speed and flexibility, so the results reported for each estimation run are slightly different. For a description of the algorithm, see "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. Nathan Halko, Per-Gunnar Martinsson, Joel A. Tropp. SIAM Rev., Survey and Review section, Vol. 53, num. 2, pp. 217-288, June 2011. https://arxiv.org/pdf/0909.4061.pdf
SVDFILE=? opens a Browse window
The general form of the Rasch-SVD model is:
Xni = Eni + Vn*Ui ± ε
where Xni is the observation when person n meets item i. Eni is the Rasch-expected value. Vn is the SVD value for the person. Ui is the SVD value for the item. ε is the unexplained part of the residual.
After the first set of SVD values have been estimated, the process can be continued deeper down to further levels:
Xni = Eni + V1n*U1i + V2n*U2i ... ± ε
SVD control settings. These can be in the control file, or from the Output file menu, SVDFILE option.
Control |
Default |
Explanation |
SVDFACTORS= |
5 |
is the number of SVD terms (factors) to be estimated for each residual. SVD computation can be concluded with Ctrl+F. |
SVDEPOCHS= |
20 |
is the maximum number of epochs (iterations through the data) for estimating each SVD factor |
SVDMIN= |
.01 |
is the minimum improvement of the root-mean-square residuals based on the current SVD estimates at the end of an SVD iteration (epoch) through the data matrix of residuals in order for SVD estimation to finish for that SVD factor. |
SVDTYPE= |
R |
is the type of residual to be decomposed. R = the raw residual or S = the standardized residual. |
SVDFILE= |
|
name of the output file for the SVD values. |
SVD file format:
The first four lines are heading lines. To omit them, specify HLINES=NO or in the Output Files specifications dialog box. ROW1HEADING=N to omit the first line only.
RMSR = Root-mean-square-residual corresponding to the column of Rasch measures or SVD factors.
RMSS = Root-mean-square-standardized-residual corresponding to the column of Rasch measures or SVD factors.
MEASURE = Rasch measure in Table 14 etc.
1 2 3 4 ... are the columns of SVD values. For each SVD factor, every person and every item has an SVD value.
SVD factors corresponding to extreme scores have 0 values.
