Parameter Estimation

Click OK 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 using principal components analysis, Statistica begins iterations under steepest descent (see 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:

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

Note: 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 performs 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 performs 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 using 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.

Option Description
Copy Copies either the selected text in the Summary box or all of the text (if no text has been selected) to the Clipboard.

The copied text retains formatting information such as font, color, etc.

Contract/Expand Contracts or expands the Summary box.

In the default expanded view, the entire Summary box is displayed in the Parameter Estimation dialog box.

In contracted view, you can see only one line of the Summary box text and can scroll through the text using a scroll bar. When contracted, the text is scrolled so that the first nonblank line is at the top.

Cancel Returns to the Multidimensional Scaling Startup Panel.
OK Displays the MDS Results dialog box.