Calculates the Akaike and Schwarz (Bayesian) information coefficients for REML (R.W. Payne & V.M.Cave).
Options
PRINT = string tokens |
Controls printed output (deviance, aic, bic, sic, dffixed, dfrandom, changes); default aic |
|---|---|
INCLUDE = string |
When LMETHOD=residual which constants to include that depend only on the fixed model (determinant, pi); default pi |
DMETHOD = string token |
Method to use to calculate log(determinant(X′X)) (choleski, lrv); default chol |
†LMETHOD = string token |
Whether the residual or full log-likelihood is used to calculate the information coefficients (residual, full); default resi |
REPEAT = string token |
Whether to repeat output from the previous VAIC (yes, no); default no |
Parameters
DEVIANCE = scalars |
Saves the deviance |
|---|---|
AIC = scalars |
Saves the Akaike information coefficient |
SIC = scalars |
Saves the Schwarz (Bayesian) information coefficient |
DFFIXED = scalars |
Saves the number of parameters fitted in the fixed model |
DFRANDOM = scalars |
Saves the number of parameters fitted in the random model (and any covariance models) |
CHANGES = variates |
Saves changes since the previous VAIC; the units of the variates are labelled by the names of the coefficients (deviance, aic, sic, dffixed and dfrandom) |
SAVE = REML save structures |
Save structure for which to calculate the coefficients; default uses the save structure from the most recent REML |
Description
The Akaike and Schwarz (Bayesian) information coefficients are often used to compare the fit of models. Traditionally the residual log-likelihood is used to form the information coefficients, and these can be used to assess the appropriateness of random (and covariance) models in REML. However, for REML models with different fixed effects, the residual log-likelihoods are not comparable and hence information coefficients derived from the residual log-likelihood cannot be used. To compare models that differ in their fixed effects, information coefficients formed using the full log-likelihood evaluated at the REML estimates must be used. The LMETHOD option is thus provided to control whether the information coefficients are formed using the residual log-likelihood (the default) or the full log-likelihood.
When LMETHOD=residual, the information coefficients are calculated from the REML deviance:
aic = deviance + 2 × r
sic = deviance + log(n – p) × r
where n is the total number of usable units in the analysis, r is the number of parameters fitted in the random model (and any covariance models), and p is the number of parameters fitted in the fixed model.
An additional consideration is that the REML deviance omits some constants that depend on the fixed model. In fact the full deviance is given by
full deviance = deviance + (n–p)*log(2π) – log(det(X′X))
where X is the design matrix of the fixed model. Other software systems tend to include the first term, involving π, but omit the log-determinant term which is more time-consuming to calculate. The inclusion of these terms in the calculation is controlled by the INCLUDE option, with settings
determinant |
-log(det(X′X)) |
|---|---|
pi |
+(n-p)*log(2π) |
The DMETHOD option controls how -log(det(X′X)) is calculated when this is included. However, the default is INCLUDE=pi.
When LMETHOD=full, the information coefficients are calculated by:
aic = deviance + 2 × (r + p) + logdet
sic = deviance + log(n ) × (r + p) + logdet
where logdet is the log-determinant of the variance-covariance matrix for the full set of fixed and random effects. See Verbyla (2019) for more details. The options INCLUDE and DMETHOD are not relevant, and are ignored.
Printed output is controlled by the PRINT option, with settings:
deviance |
prints the deviance (adding the extra terms specified by INCLUDE when LMETHOD=residual); |
|---|---|
aic |
prints the Akaike information coefficient; |
bic or sic (synonyms) |
print the Schwarz (Bayesian) information coefficient; |
dffixed |
prints the number of parameters fitted in the fixed model; |
dfrandom |
prints the number of parameters fitted in the random model (and any covariance models); |
changes |
prints changes in the values of the coefficients since the previous use of VAIC, (provided the fixed model of the REML analysis has not also changed LMETHOD=residual) |
These can all be saved using the DEVIANCE, AIC, SIC, DFFIXED, DFRANDOM and CHANGES parameters. By default VAIC prints just the Akaike information coefficient.
By default, each time that you use VAIC, its record of the current and previous REML analyses is updated. However, you can set option REPEAT=yes to repeat output from the previous VAIC. The analysis record is then not updated, so the information required to calculate changes remains available.
The coefficients are usually calculated for the most recent REML analysis. However, you can use the SAVE parameter to specify the save structure from an earlier analysis.
Options: PRINT, INCLUDE, DMETHOD, LMETHOD, REPEAT.
Parameters: DEVIANCE, AIC, SIC, DFFIXED, DFRANDOM, CHANGES, SAVE.
References
Verbyla, A.P. (2019). A note on model selection using information criteria for general linear models estimated using REML. Australia & New Zealand Journal of Statistics, 61, 39-50.
See also
Directives: REML, VKEEP.
Procedure: VRACCUMULATE.
Commands for: REML analysis of linear mixed models.
Example
CAPTION 'VAIC example',\
'Example 5.3.6 from The Guide to Genstat, Part 2 Statistics';\
STYLE=meta,plain
FACTOR [NVALUES=322; LEVELS=27] Dam
& [NVALUES=322; LEVELS=18] Pup
FACTOR [NVALUES=322; LEVELS=2; LABELS=!T('M','F')] Sex
FACTOR [NVALUES=322; LEVELS=3; LABELS=!T('C','Low','High')] Dose
VARIATE [NVALUES=322] Littersize,Weight
READ Dose,Sex,Littersize,Dam,Pup,Weight; FREP=2(labels),4(levels)
C M 12 1 1 6.60
C M 12 1 2 7.40
C M 12 1 3 7.15
C M 12 1 4 7.24
C M 12 1 5 7.10
C M 12 1 6 6.04
C M 12 1 7 6.98
C M 12 1 8 7.05
C F 12 1 9 6.95
C F 12 1 10 6.29
C F 12 1 11 6.77
C F 12 1 12 6.57
C M 14 2 1 6.37
C M 14 2 2 6.37
C M 14 2 3 6.90
C M 14 2 4 6.34
C M 14 2 5 6.50
C M 14 2 6 6.10
C M 14 2 7 6.44
C M 14 2 8 6.94
C M 14 2 9 6.41
C F 14 2 10 5.92
C F 14 2 11 6.04
C F 14 2 12 5.82
C F 14 2 13 6.04
C F 14 2 14 5.96
C M 4 3 1 7.50
C M 4 3 2 7.08
C F 4 3 3 7.57
C F 4 3 4 7.27
C M 14 4 1 6.25
C M 14 4 2 6.93
C M 14 4 3 6.80
C M 14 4 4 6.69
C M 14 4 5 6.28
C M 14 4 6 6.27
C M 14 4 7 6.27
C M 14 4 8 6.47
C F 14 4 9 6.29
C F 14 4 10 5.98
C F 14 4 11 6.32
C F 14 4 12 6.28
C F 14 4 13 5.65
C F 14 4 14 5.57
C M 13 5 1 7.96
C M 13 5 2 6.84
C M 13 5 3 7.00
C M 13 5 4 8.10
C M 13 5 5 6.52
C M 13 5 6 7.23
C M 13 5 7 6.10
C M 13 5 8 7.31
C F 13 5 9 7.16
C F 13 5 10 7.09
C F 13 5 11 7.14
C F 13 5 12 5.02
C F 13 5 13 6.04
C M 9 6 1 8.26
C M 9 6 2 7.73
C M 9 6 3 8.33
C M 9 6 4 6.14
C M 9 6 5 7.75
C M 9 6 6 6.96
C F 9 6 7 7.26
C F 9 6 8 6.58
C F 9 6 9 3.68
C M 18 7 1 6.29
C M 18 7 2 6.32
C M 18 7 3 6.28
C M 18 7 4 6.24
C M 18 7 5 6.78
C M 18 7 6 6.63
C M 18 7 7 6.27
C M 18 7 8 6.29
C M 18 7 9 6.06
C F 18 7 10 6.16
C F 18 7 11 5.96
C F 18 7 12 6.26
C F 18 7 13 5.83
C F 18 7 14 6.11
C F 18 7 15 6.45
C F 18 7 16 6.25
C F 18 7 17 6.31
C F 18 7 18 5.74
C M 17 8 1 6.04
C M 17 8 2 5.84
C M 17 8 3 6.77
C M 17 8 4 5.59
C M 17 8 5 5.52
C M 17 8 6 6.42
C M 17 8 7 5.97
C M 17 8 8 6.34
C F 17 8 9 6.23
C F 17 8 10 5.95
C F 17 8 11 6.16
C F 17 8 12 6.19
C F 17 8 13 5.32
C F 17 8 14 5.00
C F 17 8 15 6.30
C F 17 8 16 5.00
C F 17 8 17 5.56
C M 17 9 1 5.37
C M 17 9 2 5.58
C M 17 9 3 5.51
C M 17 9 4 5.19
C M 17 9 5 5.34
C M 17 9 6 5.77
C M 17 9 7 5.17
C M 17 9 8 4.57
C M 17 9 9 5.39
C M 17 9 10 5.62
C M 17 9 11 5.40
C M 17 9 12 5.77
C M 17 9 13 5.24
C F 17 9 14 5.37
C F 17 9 15 5.33
C F 17 9 16 5.44
C F 17 9 17 5.14
C M 13 10 1 7.30
C M 13 10 2 6.60
C M 13 10 3 6.58
C M 13 10 4 6.68
C M 13 10 5 6.46
C M 13 10 6 6.38
C F 13 10 7 6.44
C F 13 10 8 6.67
C F 13 10 9 6.43
C F 13 10 10 6.53
C F 13 10 11 5.92
C F 13 10 12 6.52
C F 13 10 13 6.44
Low M 16 11 1 6.65
Low M 16 11 2 5.78
Low M 16 11 3 6.23
Low M 16 11 4 5.70
Low M 16 11 5 5.73
Low M 16 11 6 6.10
Low M 16 11 7 5.55
Low M 16 11 8 5.71
Low M 16 11 9 5.81
Low M 16 11 10 6.10
Low F 16 11 11 5.54
Low F 16 11 12 5.72
Low F 16 11 13 5.50
Low F 16 11 14 5.64
Low F 16 11 15 5.42
Low F 16 11 16 5.42
Low F 2 12 1 6.89
Low F 2 12 2 7.73
Low M 12 13 1 5.83
Low M 12 13 2 5.97
Low M 12 13 3 6.39
Low M 12 13 4 5.69
Low M 12 13 5 5.69
Low M 12 13 6 5.97
Low M 12 13 7 6.04
Low M 12 13 8 5.46
Low F 12 13 9 6.09
Low F 12 13 10 5.39
Low F 12 13 11 5.89
Low F 12 13 12 5.14
Low M 15 14 1 5.92
Low M 15 14 2 5.75
Low M 15 14 3 6.22
Low M 15 14 4 5.96
Low M 15 14 5 5.59
Low M 15 14 6 5.79
Low M 15 14 7 6.23
Low M 15 14 8 5.88
Low M 15 14 9 6.02
Low F 15 14 10 5.66
Low F 15 14 11 5.76
Low F 15 14 12 5.73
Low F 15 14 13 5.33
Low F 15 14 14 5.58
Low F 15 14 15 5.88
Low M 13 15 1 6.00
Low M 13 15 2 6.11
Low M 13 15 3 6.40
Low M 13 15 4 6.06
Low M 13 15 5 6.39
Low M 13 15 6 6.09
Low M 13 15 7 6.32
Low F 13 15 8 5.96
Low F 13 15 9 6.32
Low F 13 15 10 5.83
Low F 13 15 11 5.97
Low F 13 15 12 5.87
Low F 13 15 13 5.67
Low M 13 16 1 6.43
Low M 13 16 2 6.13
Low M 13 16 3 5.87
Low F 13 16 4 6.09
Low F 13 16 5 5.63
Low F 13 16 6 5.84
Low F 13 16 7 6.20
Low F 13 16 8 6.42
Low F 13 16 9 5.90
Low F 13 16 10 5.62
Low F 13 16 11 6.23
Low F 13 16 12 5.85
Low F 13 16 13 5.89
Low M 14 17 1 5.81
Low M 14 17 2 5.44
Low M 14 17 3 5.65
Low M 14 17 4 5.25
Low M 14 17 5 5.45
Low M 14 17 6 5.32
Low M 14 17 7 5.89
Low F 14 17 8 5.63
Low F 14 17 9 5.12
Low F 14 17 10 5.65
Low F 14 17 11 5.29
Low F 14 17 12 5.13
Low F 14 17 13 5.60
Low F 14 17 14 5.08
Low M 15 18 1 6.77
Low M 15 18 2 7.13
Low M 15 18 3 6.85
Low F 15 18 4 6.49
Low F 15 18 5 6.09
Low F 15 18 6 6.09
Low F 15 18 7 5.99
Low F 15 18 8 6.01
Low F 15 18 9 6.11
Low F 15 18 10 6.15
Low F 15 18 11 4.75
Low F 15 18 12 5.69
Low F 15 18 13 6.19
Low F 15 18 14 5.72
Low F 15 18 15 6.14
Low M 10 19 1 6.72
Low M 10 19 2 6.34
Low M 10 19 3 6.48
Low M 10 19 4 5.74
Low F 10 19 5 6.11
Low F 10 19 6 5.71
Low F 10 19 7 6.41
Low F 10 19 8 6.21
Low F 10 19 9 6.11
Low F 10 19 10 5.81
Low M 16 20 1 5.90
Low M 16 20 2 6.22
Low M 16 20 3 6.67
Low M 16 20 4 6.23
Low M 16 20 5 6.24
Low M 16 20 6 6.26
Low M 16 20 7 6.38
Low M 16 20 8 6.05
Low M 16 20 9 5.89
Low M 16 20 10 6.29
Low F 16 20 11 6.12
Low F 16 20 12 5.40
Low F 16 20 13 5.50
Low F 16 20 14 5.46
Low F 16 20 15 5.97
Low F 16 20 16 6.11
High M 14 21 1 5.09
High M 14 21 2 5.57
High M 14 21 3 5.69
High M 14 21 4 5.50
High M 14 21 5 5.45
High M 14 21 6 5.24
High M 14 21 7 5.36
High M 14 21 8 5.26
High M 14 21 9 5.36
High M 14 21 10 5.01
High M 14 21 11 5.03
High F 14 21 12 5.23
High F 14 21 13 5.13
High F 14 21 14 4.48
High M 10 22 1 5.30
High M 10 22 2 5.40
High M 10 22 3 5.55
High M 10 22 4 6.02
High M 10 22 5 5.27
High F 10 22 6 5.19
High F 10 22 7 5.42
High F 10 22 8 5.40
High F 10 22 9 5.12
High F 10 22 10 5.40
High M 3 23 1 7.70
High F 3 23 2 7.68
High F 3 23 3 6.33
High M 12 24 1 6.28
High M 12 24 2 5.74
High M 12 24 3 6.29
High F 12 24 4 5.68
High F 12 24 5 5.76
High F 12 24 6 6.03
High F 12 24 7 5.30
High F 12 24 8 5.55
High F 12 24 9 6.53
High F 12 24 10 5.76
High F 12 24 11 5.77
High F 12 24 12 5.49
High M 8 25 1 6.50
High M 8 25 2 7.10
High M 8 25 3 7.00
High M 8 25 4 7.00
High M 8 25 5 5.85
High F 8 25 6 6.10
High F 8 25 7 6.63
High F 8 25 8 6.33
High M 9 26 1 7.00
High M 9 26 2 6.15
High F 9 26 3 6.22
High F 9 26 4 6.20
High F 9 26 5 5.76
High F 9 26 6 6.21
High F 9 26 7 6.42
High F 9 26 8 6.42
High F 9 26 9 6.30
High M 9 27 1 5.64
High M 9 27 2 6.06
High M 9 27 3 6.56
High M 9 27 4 6.29
High M 9 27 5 5.69
High M 9 27 6 6.36
High F 9 27 7 5.93
High F 9 27 8 5.74
High F 9 27 9 5.74 :
VCOMPONENTS [FIXED=Littersize+Dose*Sex] RANDOM=Dam/Pup
REML [PRINT=model,components,wald] Weight
VAIC