Fits a quadratic surface and estimates its stationary point (R.W. Payne).

### Options

`PRINT` = string tokens |
What to print (`model` , `deviance` , `summary` , `estimates` , `correlations` , `fittedvalues` , `accumulated` , `monitoring` , `confidence` , `predictions` , `stationary` ); default `mode` , `summ` , `esti` |
---|---|

`CONSTANT` = string token |
How to treat the constant (`estimate` , `omit` ); default `esti` |

`FACTORIAL` = scalars |
Limit for expansion of model terms; default 3 |

`POOL` = string token |
Whether to pool ss in accumulated summary between all terms fitted in a linear model (`yes` , `no` ); default `no` |

`DENOMINATOR` = string token |
Whether to base ratios in accumulated summary on rms from model with smallest residual ss or smallest residual ms (`ss` , `ms` ); default `ss` |

`NOMESSAGE` = string tokens |
Which warning messages to suppress (`dispersion` , `leverage` , `residual` , `aliasing` , `marginality` , `vertical` , `df` , `inflation` ); default `*` |

`FPROBABILITY` = string token |
Printing of probabilities for variance and deviance ratios (`yes` , `no` ); default `no` |

`TPROBABILITY` = string token |
Printing of probabilities for t-statistics (`yes` , `no` ); default `no` |

`SELECTION` = string tokens |
Statistics to be displayed in the summary of analysis produced by `PRINT=summary` , seobservations is relevant only for a Normally distributed response, and `%cv` only for a gamma-distributed response (`%variance` , `%ss` , `adjustedr2` , `r2` , `seobservations` , `dispersion` , `%cv` , `%meandeviance` , `%deviance` , `aic` , `bic` , `sic` ); default `%var` , `seob` if `DIST=normal` , `%cv` if `DIST=gamma` , and `disp` for other distributions |

`PROBABILITY` = scalar |
Probability level for confidence intervals for parameter estimates; default 0.95 |

`STATIONARY` = scalars |
Saves the estimated value of y at the stationary point |

`SESTATIONARY` = scalars |
Saves the standard error of the estimated value of y at the stationary point |

`TYPESTATIONARY` = scalars |
Identifies the type of stationary point (2 for maximum, 1 for maximum on a ridge, -2 for minimum, -1 for minimum on a ridge, or 0 for saddle point) |

`PREDICTIONS` = matrix |
Saves predictions |

`PLOT` = string tokens |
What to plot (`contour` , `surface` ); default `*` i.e. nothing |

`COLOURS` = text or variate |
Colours for the plots |

### Parameters

`X` = variates |
X-variates whose linear, quadratic and product terms define the quadratic surface |
---|---|

`ESTIMATE` = scalars |
Estimated value of each x-variate at the stationary point |

`SE` = scalars |
Standard error of the estimated value of each x-variate at the stationary point |

`LEVELS` = variates |
Values at which to evaluate each `X` for plots and predictions |

### Description

`RQUADRATIC`

fits a quadratic surface of several variates, and estimates the stationary point. It is used similarly to `FIT`

. It must be preceded by a `MODEL`

statement, and can be followed by `RCHECK`

, `RDISPLAY`

, `RGRAPH`

, `RKEEP`

, `ADD`

, `DROP`

, `SWITCH`

and so on. It also has options `PRINT`

, `CONSTANT`

, `FACTORIAL`

, `POOL`

, `DENOMINATOR`

, `NOMESSAGE`

, `FPROBABILITY`

, `TPROBABILITY`

, `SELECTION`

and `PROBABILITY`

. These operate similarly to those of `FIT`

, except that `PRINT`

has an additional setting `stationary`

to print the stationary point, and an additional setting predictions to print the predictions (see the `PREDICTIONS`

option below).

The x-variates whose linear, quadratic and product terms define the quadratic surface are specified by the `X`

parameter. There are also parameters `ESTIMATE`

and `SE`

to save the estimated value of each x-variate, and its standard error, at the stationary point. The y-value at the stationary point, and its standard error, can be saved by the `STATIONARY`

and `SESTATIONARY`

options. The `TYPESTATIONARY`

option saves a scalar, with one of the following values to identify the type of stationary point: 2 maximum, 1 maximum on a ridge, -2 minimum, -1 minimum on a ridge, or 0 saddlepoint.

The `PREDICTIONS`

option can save predictions from the fitted quadratic model. The `LEVELS`

parameter specifies a variate for each `X`

, to specify the values at which to form predictions. The predictions are stored in a matrix. The final column contains the predictions, and the earlier columns (one for each `X`

variate) store the set of x-values at which each prediction was made.

The `PLOT`

option specifies which plots to display, with settings:

`contour` |
for a contour plot, and |
---|---|

`surface` |
for surface plot. |

By default nothing is plotted. The `COLOURS`

option specifies a text or variate to define the colours to use. (This is used as the setting of the `PENFILL`

parameter of `DCONTOUR`

and `DSURFACE`

.) The default is a text containing the values `'darkgreen'`

and `'yellow'`

.

Options: `PRINT`

, `CONSTANT`

, `FACTORIAL`

, `POOL`

, `DENOMINATOR`

, `NOMESSAGE`

, `FPROBABILITY`

, `TPROBABILITY`

, `SELECTION`

, `PROBABILITY`

, `STATIONARY`

, `SESTATIONARY`

, `TYPESTATIONARY`

, `PREDICTIONS`

, `PLOT`

, `COLOURS`

.

Parameters: `X`

, `ESTIMATE`

, `SE`

, `LEVELS`

.

### Method

`RQUADRATIC`

forms variates with the quadratic and product terms of the x-variates, and fits these together with the x-variates themselves. The `RFUNCTION`

directive is then used to estimate the x- and y-values at the stationary point, with their standard errors. The type of stationary point is identified by an eigenvalue decomposition of the symmetric matrix of estimated regression coefficients of the product and quadratic terms, as described in Section 9.4 of Wu & Hamada (2000).

### Action with `RESTRICT`

As in `FIT`

, the y-variate (specified in an earlier `MODEL`

directive) can be restricted to analyse a subset of the data.

### Reference

Wu, C.F.J & Hamada, M. (2000). *Experiments: Planning, Analysis, and Parameter Design Optimization*. Wiley, New York.

### See also

Directive: `AFRESPONSESURFACE`

.

Procedures: `AGBOXBEHNKEN`

, `AGCENTRALCOMPOSITE`

. `VSURFACE`

Commands for: Regression analysis.

### Example

CAPTION 'RQUADRATIC example',\ 'Second-order Ranitidine experiment (Wu & Hamada 2000, Table 9.12)';\ STYLE=meta,plain VARIATE [VALUES=0,1,-1.41,0,0,0,1,1.41,0,0,-1,-1,0] A & [VALUES=-1.41,-1,0,0,0,0,1,0,1.41,0,-1,1,0] B & [VALUES=6.943,6.248,2.100,2.034,2.009,2.022,\ 3.252,9.445,1.781,1.925,2.390,2.066,2.113] lnCEF PRINT A,B,lnCEF; DECIMALS=2,2,3 MODEL lnCEF RQUADRATIC [PLOT=surface] A,B