Fits regressions 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 |
|---|---|
CALCULATION = expression structures |
Calculation of explanatory variates involving nonlinear parameters |
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 |
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 |
CMETHOD = string token |
Estimation method (maximumlikelihood, reml); default maxi |
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 |
|---|
Description
RAR1 allows you to fit regression 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. The parameter phi of the AR1 or power-distance model is estimated within RAR1, 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, RAR1 is used much like FIT. It must be preceded by a MODEL statement. You can also give an RCYCLE statement first if you want to estimate nonlinear parameters. The MODEL statement must have the WEIGHT option set to a symmetrix matrix, which need not have any values defined. RAR1 will set the values according to the distances (CPOSITIONS), groups (CGROUPS) and estimated parameter phi. These values remain set after RAR1. So you can display or save further output using RCHECK, RDISPLAY, RGRAPH or RKEEP, in the usual way. You could also, for example, use RAR1 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. RAR1 has a TERMS parameter to specify the terms to be fitted, like the parameter of FIT. It also has options CALCULATION, CONSTANT, FACTORIAL, POOL, DENOMINATOR, NOMESSAGE, FPROBABILITY, TPROBABILITY, SELECTION and SELINEAR which operate like those of 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, |
|---|---|
cmonitoring |
provides monitoring information for the estimation of phi, |
cplot |
plots the likelihood (or REML 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 CMETHOD option controls whether phi is estimated for regression models by REML or by maximum likelihood (default maxi); with nonlinear models only maximum likelihood is available. 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, CALCULATION, CONSTANT, FACTORIAL, POOL, DENOMINATOR, NOMESSAGE, FPROBABILITY, TPROBABILITY, SELECTION, SELINEAR, WEIGHTS, CMETHOD, CPARAMETER, CPOSITIONS, CGROUPS, MAXCYCLE, TOLERANCE.
Parameter: TERMS.
Method
To estimate phi RAR1 uses procedure MIN1DIMENSION, which calls a procedure _MIN1DFUNCTION, which is loaded automatically with RAR1. _MIN1DFUNCTION uses the FIT directive to fit the regression model for a particular value of phi, and then evaluates the likelihood or REML likelihood (according to the setting of the CMETHOD option).
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
Directive: VSTRUCTURE.
Procedure: NLAR1.
Commands for: Repeated measurements, Regression analysis.
Example
CAPTION 'RAR1 example',!t('Regress daily gas demand on coldness,',\
'using an AR(1) model for errors, with phi estimated by REML;',\
'see Guide to Genstat, Part 2, Example 7.4a'); STYLE=meta,plain
READ [SETNVALUES=yes] Demand
324.04 333.30 345.96 403.74 391.84 361.36
358.26 356.44 360.60 326.56 341.74 349.14
363.76 352.46 368.34 382.50 381.26 351.14 348.04 359.46
364.76 338.44 339.20 342.06 396.84 418.24
457.46 463.36 440.34 443.80 419.06 390.34 383.14 415.26
449.86 471.84 458.70 408.46 345.34 309.04
283.56 287.36 314.44 303.40 286.16 298.54 308.44 309.16
336.86 337.64 334.10 302.66 303.84 289.94
251.86 239.26 288.84 319.40 311.46 308.74 361.74 390.16
367.76 365.14 364.40 367.56 363.84 340.34
319.76 319.06 322.24 281.10 312.96 295.64 273.74 321.16
313.16 286.64 348.60 359.46 338.94 322.44
338.96 359.26 357.24 404.00 386.56 391.14 411.14 405.46
377.86 371.74 333.20 354.06 319.64 310.04
336.86 301.36 295.24 264.90 312.46 363.34 362.04 340.26 :
READ [SETNVALUES=yes] Coldness
-33.5 -21.8 -13.8 -12.4 -0.9 -38.7 -32.7 -49.0
-34.2 -59.2 -40.9 -34.8 -29.8 -34.9 -10.8 -26.2
1.6 -50.3 -53.0 -40.5 -32.2 -60.2 -50.5 -47.5
9.1 32.9 42.6 29.2 16.8 6.5 -0.8 -14.6
-14.2 -9.1 8.9 13.8 11.3 -34.1 -67.9 -93.3
-100.9 -91.7 -69.3 -82.8 -79.6 -79.6 -57.4 -74.8
-58.8 -53.0 -63.8 -68.1 -91.1 -76.0 -115.9 -114.2
-93.5 -68.3 -74.9 -78.4 -21.4 -27.1 -38.7 -55.8
-64.1 -46.2 -59.0 -49.7 -59.8 -87.9 -51.5 -90.9
-85.5 -87.5 -94.1 -55.8 -75.7 -104.1 -60.9 -48.9
-59.1 -56.8 -50.3 -59.1 -56.0 -38.7 -13.7 -47.0
-24.1 -31.8 -44.8 -65.4 -84.2 -47.9 -70.2 -98.4
-76.1 -104.2 -97.0 -117.3 -62.0 -44.8 -47.3 -64.3 :
CALCULATE nunits = NVALUES(Demand)
VARIATE [VALUES=1...nunits] Time
SYMMETRIC [ROWS=nunits] wmat
MODEL [WEIGHTS=wmat] Demand
RAR1 [CMETHOD=reml; CPOSITIONS=Time] Coldness
" compare with REML "
GROUP Time; Timefactor
VCOMPONENTS [FIXED=Coldness; CADJUST=none] Timefactor
VSTRUCTURE [TERM=Timefactor] AR; ORDER=1
REML [PRINT=effects,components,deviance] Demand