VFRESIDUALS procedure

Obtains residuals, fitted values and their standard errors from a REML analysis (S.J. Welham).

Options

RESIDUALS = variate	Saves the residuals
SERESIDUALS = variate	Saves standard errors of the residuals
FITTEDVALUES = variate	Saves the fitted values
SEFITTEDVALUES = variate	Saves prediction standard errors for the fitted values
RMETHOD = string token	Which random terms to use when calculating the residuals (final, all); default fina
MAXNUNITS = scalar	Maximum number of units for which the full variance-covariance matrix will be formed; default 1000
EXIT = scalar	Exit code set to zero if the saving was successful, one otherwise
SAVE = REML save structure	Save structure for the required analysis; default uses the save structure from the most recent REML

No parameters

Description

The VFRESIDUALS procedure saves residuals, fitted values and their standard errors from a REML analysis. The residuals are formed as differences between the data and the fitted model. The RMETHOD option controls which random terms are used to calculate the residuals, with settings:

all	uses all of the random effects, and
final	uses only the final random term (default).

The final setting thus provides conditional residuals, with the fitted model is calculated from all of the fixed and random terms in the model. The all setting provides marginal residuals, with the fitted model is calculated from the fixed terms alone.

The residuals and fitted values can be saved, in variates, using the RESIDUALS and FITTEDVALUES options, respectively. The SERESIDUALS option saves the standard errors of the residuals, and the SEFITTEDVALUES option saves the prediction standard errors of the fitted values (i.e. the square root of the prediction error variances).

The standard errors can be calculated in several different ways, and VFRESIDUALS will attempt to use the most efficient method. One method involves saving the full variance-covariance matrix for the data. This can be time-consuming for large data sets, so the MAXNUNITS option sets a limit (default 1000) on the size of data set for which this may be used.

By default, VFRESIDUALS forms the residuals etc. from the most recent REML analysis. However, you can form them from an earlier analysis, by using the SAVE option to specify its save structure (saved using the SAVE parameter of the REML command that performed the analysis).

VFRESIDUALS is currently unable to form standard errors for models containing spline terms.

Options: RESIDUALS, SERESIDUALS, FITTEDVALUES, SEFITTEDVALUES, RMETHOD, MAXNUNITS, EXIT, SAVE.

Parameters: none.

Method

The linear mixed model is

y = Xβ + Zu + ε

where

y is a vector of data,

β is a vector of fixed effects, with design matrix X,

u is a vector of random effects, with design matrix Z,

ε is a vector of random error

The conditional residuals take the form

ε^~_c = y – Xβ^{^} – Zu^~

with variance matrix

var(ε^~_c) = σ² (R – W C^-1 W′)

where

Wα = Xβ + Zu

σ² C^-1 = var(α^{^} – α)

and R is the matrix of variances and covariances fitted to the residual.

The standard errors of the residuals are given by the square root of the diagonal of the variance matrix. The diagonal of σ² (W C^-1 W′) can be obtained as the standard error of the predicted fitted values, and the matrix R can be derived from the fitted model.

The marginal residuals take the form

ε^~_m = y – Xβ^{^}

with variance matrix

var(ε^~_m) = σ² (H – X (X′ H^-1 X)^-1 X′)

where

σ² H = var(y)

Again, the standard errors are given by the square root of the diagonal of this matrix. The diagonal of σ² X (X′ H^-1 X)^-1 X′ can be obtained as the standard errors of the predicted fitted values. The matrix H can be derived from the fitted model, or obtained using the UVCOV option of VKEEP.

See also

Directives: REML, VCOMPONENTS, VKEEP.

Procedure: VCHECK, VPLOT.

Commands for: REML analysis of linear mixed models.

Example

CAPTION     'VFRESIDUALS example',\
            !t('Incomplete block design, data from Cochran & Cox',\
            '(1957), Experimental Designs, p.406.'); STYLE=meta,plain
FACTOR      [LEVELS=2; VALUES=25(1,2)] Reps
 &          [LEVELS=10; VALUES=5(1...10)] Blocks
 &          [LEVELS=25; VALUES=1...25,1,6...21,2,7...22,3,8...23,4,9...24,\
            5,10...25] Treats
VARIATE     [NVALUES=50] Yield
READ        Yield
   6   7   5   8   6  16  12  12  13   8
  17   7   7   9  14  18  16  13  13  14
  14  15  11  14  14  24  13  24  11   8
  21  11  14  11  23  16   4  12  12  12
  17  10  30   9  23  15  15  22  16  19  :
VCOMPONENTS [FIXED=Treats] RANDOM=Reps+Blocks
REML        [PRINT=*] Yield; SAVE=S
VFRESIDUALS [RESIDUALS=residual; SERESIDUALS=seresidual;\
            FITTEDVALUES=fittedvalue; SEFITTEDVALUES=sefittedvalue]
PRINT       Yield,fittedvalue,sefittedvalue,residual,seresidual
CAPTION     'The standard errors are equal, because the design is balanced.'