Fits curves with an AR1 or a power-distance correlation model (R.W. Payne).
Options
PRINT = string tokens |
What to print (model, deviance, summary, estimates, correlations, fittedvalues, accumulated, monitoring, cparameter, cmonitoring, cplot); default mode, summ, esti, cpar |
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CURVE = string token |
Which standard curve to fit (exponential, dexponential, cexponential, lexponential, logistic, glogistic, gompertz, ldl, qdl, qdq, fourier, dfourier, gaussian, dgaussian); default expo |
SENSE = string token |
Sense of a standard curve (right, left); default righ |
ORIGIN = scalars |
Constrained origin for a standard curve; default * i.e. not constrained |
NONLINEAR = string token |
How to treat nonlinear parameters between groups in standard curves (common, separate); default comm |
CALCULATION = expression structures |
Define a nonlinear model involving explanatory variates and nonlinear parameters; default * implies that a standard curve is fitted |
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 |
SELECTION = string tokens |
Statistics to be displayed in the summary of analysis produced by PRINT=summary (%variance, %ss, adjustedr2, r2, seobservations, dispersion, %cv, %meandeviance, %deviance, aic, bic, sic); default %var, seob |
SELINEAR = string token |
Whether to calculate s.e.s for linear parameters when nonlinear parameters are also estimated (yes, no); default no |
WEIGHTS = variate |
Prior weights for the units |
CPARAMETER = scalars |
Correlation parameter |
CPOSITIONS = variate |
Correlation positions |
CGROUPS = factor |
Groupings of correlation positions |
MAXCYCLE = scalars |
Maximum number of iterations; default 100 |
TOLERANCE = scalars |
Convergence criterion; default 10-5 |
Parameter
TERMS = formula |
Terms to be fitted |
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Description
NLAR1 allows you to fit curves and nonlinear models to data, such as repeated measurements, where the residuals may follow an AR1 or a power-distance correlation model. The CPOSITIONS option specifies the coordinates of the observations in the direction (e.g. time) along which the correlation model operates. You can also use the CGROUPS option to specify a factor to define groups of observations for the model – the correlation model is then defined only over the observations that belong to the same groups. If you are fitting a standard curve, CPOSITIONS will take the x-variate for the curve as its default, and the group factor (if specified e.g. to define parallel curves) as the default for CGROUPS. NLAR1 also allows the data units to have unequal weights, which can be supplied in a variate using the WEIGHTS option.
The parameter phi of the AR1 or power-distance model is estimated within NLAR1, and is assumed to be the same for every group. (Note that the model will be AR1 if the observations are each one unit apart within each group – the power-distance model is the natural extension of the AR1 model to unequally-spaced data; see Method.) You can save the estimated value of phi, in a scalar, using the CPARAMETER option.
Otherwise, NLAR1 is used much like FITCURVE or FIT (which are used inside NLAR1 to fit the model). NLAR1 must be preceded by a MODEL statement. You must also give an RCYCLE statement first if you want to fit a user-defined nonlinear model (using FIT), rather than a standard curve (using FITCURVE). The MODEL statement must have the WEIGHT option set to a symmetrix matrix, which need not have any values defined. NLAR1 will set the values according to the distances (CPOSITIONS), groups (CGROUPS) and estimated parameter phi. These values remain set after NLAR1. So you can display or save further output using RCHECK, RDISPLAY, RGRAPH or RKEEP, in the usual way. You could also, for example, use NLAR1 to fit a full set of regression terms, and then use DROP to investigate smaller models while still using the phi estimate from the full model. NLAR1 has a TERMS parameter to specify the terms to be fitted, like the parameter of FIT and FITCURVE. It also has options CURVE, SENSE, ORIGIN, NONLINEAR, CALCULATION, CONSTANT, FACTORIAL, POOL, DENOMINATOR, NOMESSAGE, FPROBABILITY, SELECTION and SELINEAR which operate like those of FITCURVE and FIT. If the CALCULATION option is unset, then options CURVE, SENSE, ORIGIN, NONLINEAR define which standard curve to fit (using FITCURVE). Alternatively, if CALCULATION is set, those options are ignored, and the expressions specified by CALCULATION define a nonlinear model to be fitted (by FIT).
The PRINT option is also similar, except that it has three additional settings:
cparameter |
prints the estimated value of the correlation phi, together with a test for phi=0, |
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cmonitoring |
provides monitoring information for the estimation of phi, |
cplot |
plots the likelihood for phi. |
Note, the likelihood values omit some constant terms that depend only on the regression terms. The default is PRINT=model,summary,estimates,cparameter.
The other options control the estimation. The MAXCYCLE option defines the maximum number of iterations (default 100) used to estimate phi, and the TOLERANCE option specifies the convergence criterion i.e. the accurary to which phi is to be estimated (default 10-5).
Options: PRINT, CURVE, SENSE, ORIGIN, NONLINEAR, CALCULATION, CONSTANT, FACTORIAL, POOL, DENOMINATOR, NOMESSAGE, FPROBABILITY, SELECTION, SELINEAR, WEIGHTS, CMETHOD, CPARAMETER, CPOSITIONS, CGROUPS, MAXCYCLE, TOLERANCE.
Parameter: TERMS.
Method
To estimate phi NLAR1 uses procedure MIN1DIMENSION, which calls a procedure _MIN1DFUNCTION, which is loaded automatically with NLAR1. _MIN1DFUNCTION uses the FITCURVE or FIT directives to fit the regression model for a particular value of phi, and then evaluates the likelihood. If standard curves are fitted using FITCURVE for groups of observations, these groups must be independent. Otherwise FITCURVE will give a fault diagnostic. (Thus the default setting for the CGROUPS option with standard curves is the group factor, if one has been specified in the TERMS formula.)
The total degrees of freedom for the regression are decreased by one, to take account of the estimation of the correlation parameter phi, by setting a variable in the regression save structure (rsave[1][3]$[47]) to one.
Action with RESTRICT
Restrictions are not allowed.
See also
Directives: FITCURVE, FITNONLINEAR, VSTRUCTURE.
Procedure: RAR1.
Commands for: Repeated measurements, Regression analysis.
Example
CAPTION 'NLAR1 example'; STYLE=meta
VARIATE [VALUES=5...30] x
& [VALUES=1.30,3.55,5.13,6.48,7.85,8.96,9.84,10.91,11.29,11.76,\
12.12,12.55,12.70,13.14,13.47,13.78,14.01,14.11,14.55,14.71,\
14.57,14.30,14.67,14.68,15.03,15.00] y
SYMMETRIC [ROWS=26] wt
MODEL [WEIGHTS=wt] y
NLAR1 [CURVE=exponential] x