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The TRANSREG Procedure |

The expanded set of independent variables generated from the POINT,
EPOINT, and QPOINT expansions can be used to perform ideal point
regressions (Carroll 1972) and compute ideal point coordinates for
plotting in a biplot (Gabriel 1981). The three types of ideal point
coordinates can all be described as transformed coefficients. Assume
that *m* independent variables are specified in one of the three point
expansions. Let **b**' be a 1 ×*m* row vector of
coefficients for these variables and one of the dependent variables.
Let **R** be a matrix created from the coefficients of the extra
variables. When coordinates are requested with the MPC, MEC, or MQC *
o-options*, **b**' and **R** are created from multiple regression
coefficients. When coordinates are requested with the CPC, CEC, or CQC
*o-options*, **b**' and **R** are created from canonical
coefficients.

If you specify the POINT expansion in the MODEL statement,
**R** is an *m* ×*m* identity matrix times the
coefficient for the sums of squares (_ISSQ_) variable. If you specify
the EPOINT expansion, **R** is an *m* ×*m* diagonal matrix of
coefficients from the squared variables. If you specify the QPOINT
expansion, **R** is an *m* ×*m* symmetric matrix of coefficients from
the squared variables on the diagonal and crossproduct variables off the
diagonal. The MPC, MEC, MQC, CPC, CEC, and CQC ideal point coordinates
are defined as -0.5 **b**'**R**^{-1}. When **R** is singular, the ideal
point coordinates are infinitely far away and are set to missing, so
you should try a
simpler version of the model. The version that is
simpler than the POINT model is the vector model where no extra
variables are created. In the vector model, designate all independent
variables as IDENTITY. Then draw vectors from the origin to the
COEFFICIENTS points.

Typically, when you request ideal point coordinates, the MODEL statement should consist of a single transformation for the dependent variables (usually IDENTITY, MONOTONE, or MSPLINE) and a single expansion for the independent variables (one of POINT, EPOINT, or QPOINT).

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