On Thu, 15 Feb 1996, * AHIRU NO PEKKLE * wrote:

> > I'm looking for the underlying theorem of the option UNTIE in the > SAS procedure PRINQUAL. Is there any book or paper discussing it > and also its applications?? > > How does it deal with those missing data? >

There is a SAS Technical Report R-108 that can be ordered (probably <$5) from SAS publications. This report is entitled: "Algorithms for the PRINQUAL and TRANSREG Procedures."

On page 2, it reads:

" The PRINQUAL procedure (principal components of qualitative data) is a data transformation procedure that is based on the work of Kruskal and Shepard (1974); Young, Takane, and de Leeuw (1978); and Winsberg and Ramsay (1983)."

On page 3, it reads:

" * For all transformations, missing data can be estimated without constraint, with category constraints (that is, missing values within the same group get the same value), and with order constraints (that is, missing value estimates in adjacent groups can be tied to weakly preserve a specified ordering) (Gifi, 1981; Young, 1981; Kuhfeld and de Leeuw, in preparation)."

On page 4, it reads:

" The UNTIE transformation (Kruskal, 1964, primary approach to ties) uses the same algorithm on the means of the nonmissing values (1 2 3 4 6 4 5 6 7)' but with different results for this example: 1<2:OK, 2<3:OK, 3<4:OK, 4<6:OK, 6>4: average 6 and 4 and replace 6 and 4 by the average. The new means of the nonmising values are (1 2 3 4 5 5 5 6 7)'. The check resumes: 4<5:OK, 5=5:OD, 5=5:OK, 5<6:OK, 6<7:OK. If some of the special missing values are ordered, the upward checking, downward averaging method is applied to them too, independently of the other missing and nonmissing partitions. Once the means conform to any required category or order constraints, an optimally scaled vector is produced from the means. The following example results from a MONOTONE transformation:

X: (. . .A .A .B 1 1 1 2 2 3 3 3 4)' Y: (5 6 2 4 2 1 2 3 4 6 4 5 6 7)' result: (5 6 3 3 2 2 2 2 5 5 5 5 5 7)' "

References:

Gifi, A. (1981) _Nonlinear Multivariate Analysis_. Department of Data Theory, The Netherlands: The University of Leiden. (this has since been published as a book)

Kruskal, J.B. (1964) "Multidimensional Scaling By Optimizing Goodness of Fit to a Nonmetric Hypothesis," _Psychometrika_, 29: 1-27.

Kruskal, J.B., and Shepard, R.N. (1974) "A Nonmetric Variety of Linear Factor Analysis," _Psychometrika_, 38: 123-157.

Kuhfeld, W.F., and de Leeuw, J., "Optimal Scaling of Partitioned Variables," (in preparation).

Winsberg, S., and Ramsay, J.O. (1983) "Monotone Spline Transformations for Dimension Reduction," _Psychometrika_, 48: 575-595.

Young, F.W. (1981) "Quantitative Analysis of Qualitative Data," _Psychometrika_, 46: 357-388.

Young, F.W., Takane, Y., and de Leeuw, J. (1978) "The Principal Components of Mixed Measurement Level Multivariate Data: An Alternating Least Squares Method with Optimal Scaling Features," _Psychometrika_, 43: 279-281.

Raymond V. Liedka Department of Sociology University of New Mexico