HGDRANDOMMODEL procedure

Defines the random model in a hierarchical generalized linear model for the dispersion in a double hierarchical generalized linear model (R.W. Payne, Y. Lee, J.A. Nelder & M. Noh).

Options

`DISTRIBUTION` = string token	Distribution for the random model (`beta`, `normal`, `gamma`, `inversegamma`); default `norm`
`LINK` = string token	Link for the random model (`identity`, `logarithm`, `logit`, `reciprocal`); default `iden`
`RANDOMTERM` = formula	Random term whose dispersion is being modelled; if unset, the model is assumed to be for the residual dispersion parameter (phi)
`PHIMETHOD` = string token	Whether to fix or estimate the residual dispersion parameter in the dispersion HGLM (`fix`, `estimate`); default `fix`

Parameters

`TERMS` = formula	Random model
`DLINK` = string tokens	Link for the dispersion model for each random term (`logarithm`, `reciprocal`); default `loga`
`DFORMULA` = formula structures	Dispersion model for each random term; default `*` i.e. none
`DOFFSET` = variates	Offset variate for dispersion model for each random term; default `*` i.e. none
`LMATRIX` = matrices	Linear transformation to apply to design matrix Z of each random term, in order to define correlations between its effects; default `*` i.e. none
`DDISPERSION` = scalar	Dispersion parameter to use in the dispersion model for each random term; default 1
`FDISPERSION` = scalar	Fixed value for the dispersion parameter of each random term; default `!s(*)` i.e. dispersion is estimated

Description

HGDRANDOMMODEL lets you extend a hierarchical generalized linear model (HGLM) to become a double hierarchical generalized linear model (DHGLM); see Lee & Nelder (1996, 2001a, 2006) or Lee, Nelder & Pawitan (2006). This is done by adding some random terms to one of the generalized linear models that is to model the dispersion, so that this becomes an HGLM. By default the residual dispersion of this HGLM is fixed, but you can set option PHIMETHOD=estimate to estimate it. The random term whose dispersion is to be modelled by the HGLM is indicated by the RANDOMTERM option. If RANDOMTERM is omitted, the dispersion model is assumed to be for the residual dispersion parameter (phi) of the original HGLM.

The TERMS parameter defines the additional random terms, and the LINK and DISTRIBUTION options specify their distribution and link function respectively. You can specify a generalized linear model (GLM) to model the dispersion parameter for any of these additional random terms by specifying a Genstat formula structure, containing the (fixed) terms to be fitted in the GLM, using the DFORMULA parameter (which runs in parallel with the list of random terms supplied by the TERMS parameter). The DLINK parameter specifies the link to use with each dispersion model, the DOFFSET parameter allows you to specify an offset variate, and the DDISPERSION parameter defines the dispersion parameter for the dispersion GLM (default 1). Alternatively, if you do not define a dispersion model for a random term, you can use the FDISPERSION parameter to fix its dispersion at a specific value.

The LMATRIX parameter allows correlation structures to be defined for random terms, using the method described by Lee & Nelder (2001b). This is done by setting LMATRIX to a matrix L that is used as a post-multiplier for the Z matrix of the random term concerned. Lee & Nelder (2001b) give examples illustrating the types of model that can be defined.

Options: DISTRIBUTION, LINK, RANDOMTERM, PHIMETHOD.

Parameters: TERMS, DLINK, DFORMULA, DOFFSET, LMATRIX, DDISPERSION, FDISPERSION.

Method

The information is stored in a workspace G5PL_HG (accessed using the WORKSPACE directive) for later use by HGANALYSE.

References

Lee, Y., & Nelder, J.A. (1996). Hierarchical generalized linear models (with discussion). Journal of the Royal Statistical Society, Series B, 58, 619-678.

Lee, Y., & Nelder, J.A. (2001a). Hierarchical generalized linear models: a synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika, 88, 987-1006.

Lee, Y. & Nelder, J.A. (2001b). Modelling and analysing correlated non-normal data. Statistical Modelling, 1, 3-16.

Lee, Y. & Nelder, J.A. (2006). Double hierarchical generalized linear models (with discussion). Appl. Statist., 55, 139-185.

Lee, Y., Nelder, J.A. & Pawitan, Y. (2006). Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood. Chapman and Hall, Boca Raton.

Example

CAPTION 'HGDRANDOMMODEL example','Crack-growth data (Hudak et al. 1978)';\
        STYLE=meta, plain
FACTOR  [NVALUES=241; LEVELS=21] g
READ    g
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8 8
8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10
10 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12
13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 14 14 14 15
15 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 17 17
17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19
19 19 19 19 19 19 19 19 19 20 20 20 20 20 20 20 20 20 20 20 20 21 21 21 21
21 21 21 21 21 21 21 21 :
VARIATE [NVALUES=241] t
READ    t
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.01 0.02 0.03 0.04 0.05 0.06
0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.01 0.02 0.03 0.04
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.09 0.1 0.11 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.01
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.01 0.02 0.03 0.04 0.05
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.09 0.1 0.11 0.12 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
0.12 :
VARIATE [NVALUES=241] y
READ    y
0.05 0.05 0.05 0.07 0.07 0.08 0.08 0.13 0.16 0.04 0.04 0.05 0.05 0.06 0.07
0.07 0.09 0.1 0.13 0.04 0.04 0.05 0.05 0.05 0.06 0.07 0.09 0.11 0.12 0.19
0.04 0.04 0.05 0.04 0.05 0.07 0.06 0.09 0.09 0.12 0.18 0.04 0.04 0.05 0.04
0.05 0.07 0.05 0.1 0.09 0.12 0.16 0.04 0.04 0.05 0.04 0.05 0.06 0.05 0.1
0.08 0.1 0.17 0.04 0.04 0.04 0.05 0.04 0.06 0.06 0.09 0.09 0.11 0.14 0.03
0.04 0.03 0.06 0.05 0.06 0.06 0.07 0.09 0.1 0.13 0.02 0.05 0.04 0.04 0.04
0.06 0.06 0.07 0.08 0.08 0.11 0.17 0.02 0.04 0.04 0.04 0.04 0.05 0.06 0.07
0.08 0.08 0.1 0.15 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.06 0.07 0.08 0.1
0.16 0.03 0.04 0.03 0.03 0.04 0.03 0.06 0.06 0.07 0.08 0.11 0.16 0.02 0.05
0.02 0.04 0.03 0.04 0.04 0.06 0.06 0.05 0.09 0.12 0.03 0.03 0.04 0.03 0.04
0.05 0.04 0.04 0.06 0.04 0.07 0.08 0.02 0.04 0.03 0.04 0.03 0.04 0.06 0.05
0.06 0.06 0.07 0.09 0.02 0.03 0.02 0.03 0.03 0.04 0.04 0.05 0.06 0.04 0.07
0.07 0.03 0.03 0.01 0.03 0.05 0.03 0.03 0.05 0.04 0.04 0.08 0.06 0.02 0.02
0.03 0.04 0.03 0.03 0.02 0.05 0.05 0.04 0.05 0.07 0.02 0.02 0.03 0.02 0.03
0.03 0.03 0.04 0.04 0.04 0.05 0.06 0.02 0.02 0.03 0.02 0.03 0.03 0.03 0.04
0.04 0.03 0.05 0.05 0.02 0.02 0.03 0.02 0.03 0.02 0.03 0.04 0.03 0.04 0.04
0.05 :
VARIATE [NVALUES=241] y0
READ    y0
0.9 0.95 1 1.05 1.12 1.19 1.27 1.35 1.48 0.9 0.94 0.98 1.03 1.08 1.14 1.21
1.28 1.37 1.47 0.9 0.94 0.98 1.03 1.08 1.13 1.19 1.26 1.35 1.46 1.58 0.9
0.94 0.98 1.03 1.07 1.12 1.19 1.25 1.34 1.43 1.55 0.9 0.94 0.98 1.03 1.07
1.12 1.19 1.24 1.34 1.43 1.55 0.9 0.94 0.98 1.03 1.07 1.12 1.18 1.23 1.33
1.41 1.51 0.9 0.94 0.98 1.02 1.07 1.11 1.17 1.23 1.32 1.41 1.52 0.9 0.93
0.97 1 1.06 1.11 1.17 1.23 1.3 1.39 1.49 0.9 0.92 0.97 1.01 1.05 1.09 1.15
1.21 1.28 1.36 1.44 1.55 0.9 0.92 0.96 1 1.04 1.08 1.13 1.19 1.26 1.34 1.42
1.52 0.9 0.93 0.96 1 1.04 1.08 1.13 1.18 1.24 1.31 1.39 1.49 0.9 0.93 0.97 1
1.03 1.07 1.1 1.16 1.22 1.29 1.37 1.48 0.9 0.92 0.97 0.99 1.03 1.06 1.1 1.14
1.2 1.26 1.31 1.4 0.9 0.93 0.96 1 1.03 1.07 1.12 1.16 1.2 1.26 1.3 1.37 0.9
0.92 0.96 0.99 1.03 1.06 1.1 1.16 1.21 1.27 1.33 1.4 0.9 0.92 0.95 0.97 1
1.03 1.07 1.11 1.16 1.22 1.26 1.33 0.9 0.93 0.96 0.97 1 1.05 1.08 1.11 1.16
1.2 1.24 1.32 0.9 0.92 0.94 0.97 1.01 1.04 1.07 1.09 1.14 1.19 1.23 1.28 0.9
0.92 0.94 0.97 0.99 1.02 1.05 1.08 1.12 1.16 1.2 1.25 0.9 0.92 0.94 0.97
0.99 1.02 1.05 1.08 1.12 1.16 1.19 1.24 0.9 0.92 0.94 0.97 0.99 1.02 1.04
1.07 1.11 1.14 1.18 1.22 :
HGRANDOMMODEL  [DISTRIBUTION=inversegamma; LINK=log] g
HGFIXEDMODEL   [DISTRIBUTION=gamma; LINK=log; DTERMS=t] y0
HGDRANDOMMODEL [DISTRIBUTION=normal; LINK=identity] g
HGANALYSE      y

Updated on March 7, 2019

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