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# FRTPRODUCTDESIGNMATRIX procedure

Forms summation, or relationship, matrices for model terms (C.J. Brien).

### Parameters

`TERM` = formula structures Model terms corresponding to design matrices whose summation matrices are required Saves the summation or relationship matrix for each term

### Description

This procedure forms the summation, or relationship, matrix for the design matrix of a model term. This is similar to the projection matrix, constructed by the `FPROJECTIONMATRIX` procedure. `FPROJECTIONMATRIX` calculates the mean of the units of a data variate with each effect of the term, and then forms a new data variate where each unit contains the mean calculated for the relevant effect. `FRTPRODUCTDESIGNMATRIX` calculates the sums of the data values instead of the means. The result is also a relationship matrix that indicates, using one and zero whether two units have the same levels of the factors in a term. One use for these matrices in synthesizing a variance matrix that involves a linear combination of variance components.

The term is specified (as a model formula) using the `TERM` parameter, and the matrix is saved (as a symmetric matrix) by the `MATRIX` parameter.

Options: none.

Parameters: `TERM`, `MATRIX`.

### Method

The summation or relationship matrix is

`D *+ TRANSPOSE(D)`

The design matrix `D` is formed using the `TERMS` directive.

### Action with `RESTRICT`

The factors must not be restricted, nor may they contain missing values.

Procedure: `FPROJECTIONMATRIX`.

Commands for: Analysis of variance, Calculations and manipulation.

### Example

```CAPTION                'FRTPRODUCTDESIGNMATRIX example'; STYLE=meta
FACTOR                 [NVALUES=15; LEVELS=5] Blocks
&                      [LEVELS=3] Plots
GENERATE               Blocks,Plots
FRTPRODUCTDESIGNMATRIX TERM=!f(Blocks),!f(Blocks.Plots); MATRIX=S_B,S_BP
PRINT                  [SERIAL=yes] S_B,S_BP; FIELDWIDTH=4; DECIMALS=1
CALCULATE              Variance = 20 * S_B + 5 * S_BP
PRINT                  Variance; FIELDWIDTH=4; DECIMALS=0
```
Updated on March 7, 2019