Parameter Estimation

Click the OK button in the Multidimensional Scaling Startup Panel to display the Parameter Estimation dialog box. Multidimensional Scaling is an implementation of nonmetric multidimensional scaling. After determining the starting configuration (e.g., via principal components analysis) STATISTICA will begin iterations under steepest descent (see, for example, Schiffman, Reynolds, and Young, 1981). The goal of these iterations is to minimize the raw stress (or raw Phi) and the coefficient of alienation (see Guttman, 1968). The raw stress is defined as:

where dij are the reproduced distances, given the current number of dimensions, and f(dij) represents the monotone transformation of the observed input data dij (deltaij).

The coefficient of alienation K is defined as:

In general, STATISTICA will attempt to minimize the differences between the reproduced distances and a monotone transformation of the input data, that is, the program will attempt to reproduce the rank-ordering of the input distances or similarities (hence, also the name nonmetric multidimensional scaling).

Note that under steepest descent the fitted values are calculated via the rank-image permutation procedure (see Guttman, 1968; or Schiffman, Reynolds, & Young, 1981, pp. 368-369). After each iteration under steepest descent, the program will perform up to five iterations using the monotone regression transformation procedure (see Kruskal, 1964; or Schiffman, Reynolds, & Young, 1981, pp. 367-368). This procedure is aimed at minimizing the standardized stress (S):

Before final convergence, STATISTICA will perform several monotone regression transformation iterations.

Note: D-stars and D-hats. D-stars are calculated via a procedure known as the rank-image permutation procedure (see Guttman, 1968; or Schiffman, Reynolds, & Young, 1981, pp. 368-369). In general, this procedure attempts to reproduce the rank order of differences in the similarity or dissimilarity matrix. D-hats are calculated via a procedure referred to as the monotone regression transformation procedure (see Kruskal, 1964; or Schiffman, Reynolds, & Young, 1981, pp. 367-368). In this procedure, STATISTICA attempts to determine the best monotone (regression) transformation to reproduce the similarities (or dissimilarities) in the input matrix. To view these values, use the D-hat values and D-star values options on the MDS Results - Advanced tab.

Summary box.  The Summary box contains summary information about the parameter estimation (described above).

Copy button. Click the Copy button to copy either the selected text (if text has been selected) in the Summary box or all of the text (if no text has been selected) to the Clipboard. Note that the copied text retains formatting information (such as font, color, etc.).

Contract/Expand button. Click the Contract/Expand button to contract or expand the Summary box. When contracted, you can see only one line of the Summary box text and can scroll through the text using a scroll bar. Note that when contracted the text is scrolled so that the first non-blank line is at the top. When expanded (the default setting), the entire Summary box will be displayed in the Parameter Estimation dialog box.

Cancel. Click the Cancel button to return to the Multidimensional Scaling Startup Panel.

OK. Click the OK button to display the MDS Results dialog box.