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RQUADRATIC procedure

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, 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 which operate similarly to those of FIT, except that PRINT has an additional setting stationary to print the stationary point.

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.

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
Updated on March 5, 2019

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