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# HGGRAPH procedure

Draws a graph to display the fit of an HGLM or DHGLM analysis (R.W. Payne, Y. Lee, J.A. Nelder & M. Noh).

### Options

`GRAPHICS` = string token Type of graphics to use (`lineprinter`, `highresolution`); default `high` Title for the graph; default `*` sets an appropriate title automatically Which high-resolution graphics window to use; default 4 (redefined if necessary to fill the frame) Whether to clear the graphics screen before plotting (`clear`, `keep`); default `clea` What back-transformation to make (`link`, `none`, `axis`); default `none` Whether to omit the adjusted response values (`no`, `yes`); default `no` Specifies the save structure (from `HGANALYSE`) of the analysis from which to predict; default uses the most recent analysis

### Parameters

`INDEX` = variates or factors Which variate or factor to display along the x-axis; default `*` if `GROUPS` is set, otherwise `INDEX` is set to the first variate in the fixed model Factor to define groups of points to display; default `*` if `INDEX` is set, otherwise `GROUPS` is set to the first factor in the fixed model

### Description

`HGGRAPH` is one of several procedures with the prefix HG, which provide tools for fitting the hierarchical and double hierarchical generalized linear models (HGLMs and DHGLMs) defined by Lee & Nelder (1996, 2001, 2006) and described by Lee, Nelder & Pawitan (2006). The models are defined by the `HGFIXEDMODEL`, `HGRANDOMMODEL` and `HGDRANDOMMODEL` procedures, and fitted by the `HGANALYSE` procedure. `HGGRAPH` has a similar role to the `RGRAPH` procedure in ordinary regression and generalized linear models. It displays the fitted model in one or two dimensions. It usually also displays the observed response values, adjusted for any other explanatory terms in the model, but these can be omitted by setting option `OMITRESPONSE=yes`.

The dimensions to display are specified by the `INDEX` and `GROUPS` parameters. The `INDEX` vector, which can be either a variate or a factor from the fixed model of the HGLM, defines the x-axis of the plot. (The y-axis corresponds to the response scale.) The `GROUPS` parameter can be set to another factor from the fixed model. A set of points is then plotted for each level of `GROUPS`, so that you can study the interaction between `GROUPS` and `INDEX`. If `INDEX` and `GROUPS` are not set, `HGGRAPH` takes the first variate (if any) and the first factor in the fixed model.

The relationship is usually plotted on the scale of the linear predictor. However, with a conjugate HGLM, you can set option `BACKTRANSFORM=link` to use the original scale of the response. Alternatively, you can set `BACKTRANSFORM=axis` to include axis markings, back-transformed onto the natural scale, on the right-hand side of the y-axis. However, this is not available for the reciprocal link.

The `TITLE` option can be used to supply a title for the graph. By default the graph is plotted on the current high-resolution device, but the `GRAPHICS` option can be set to `line` for a line printer plot. The `WINDOW` option can be used to select a pre-defined window for high-resolution plots; otherwise window 4 is used, and is redefined if necessary to fill the frame. The SCREEN option allows the graph to be added to an existing high-resolution plot. The colours and symbols used in the displays can be controlled by setting the attributes of the following pens with the `PEN` directive before calling the procedure:

    pen 1 labels for lines when drawn for each level of a factor, fitted lines and means, points, and back-transformed axis marks and labels.

Options: `GRAPHICS`, `TITLE`, `WINDOW`, `SCREEN`, `BACKTRANSFORM`, `OMITRESPONSE`, `SAVE`.

Parameters: `INDEX`, `GROUPS`.

### Method

`HGGRAPH` calculates the points using the `HGPREDICT` procedure.

### References

Lee, Y. & Nelder, J.A. (1996). Hierarchical generalized linear models (with discussion). J. R. Statist. Soc. B, 58, 619-678.

Lee, Y. & Nelder, J.A. (2001). 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. (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.

Procedures: `HGANALYSE`, `HGDISPLAY`, `HGDRANDOMMODEL`, `HGFIXEDMODEL`, `HGFTEST`, `HGKEEP`, `HGNONLINEAR`, `HGPLOT`, `HGPREDICT`, `HGRANDOMMODEL`, `HGRTEST`, `HGSTATUS`, `HGTOBITPOISSON`, `HGWALD`.

Commands for: Regression analysis.

### Example

```CAPTION  'HGGRAPH example',!t(\
'Breaking angles of cake baked from 3 recipes at 10 temperatures',\
'(Cochran & Cox, 1957, Experimental Designs, page 300).',\
'Data values are assumed to follow a GLM with a gamma distribution',\
'random variables, with inverse gamma distributions and reciprocal',\
'link, for replicates and batches of cake mixture.');\
STYLE=meta,plain
FACTOR   [NVALUES=270; LEVELS=3] Recipe
&        [LEVELS=15] Replicate
&        [LEVELS=!(175,185...225)] Temperature
GENERATE Recipe,Replicate,Temperature
VARIATE  [NVALUES=270] Angle
42 46 47 39 53 42 47 29 35 47 57 45 32 32 37 43 45 45
26 32 35 24 39 26 28 30 31 37 41 47 24 22 22 29 35 26
26 23 25 27 33 35 24 33 23 32 31 34 24 27 28 33 34 23
24 33 27 31 30 33 33 39 33 28 33 30 28 31 27 39 35 43
29 28 31 29 37 33 24 40 29 40 40 31 26 28 32 25 37 33
39 46 51 49 55 42 35 46 47 39 52 61 34 30 42 35 42 35
25 26 28 46 37 37 31 30 29 35 40 36 24 29 29 29 24 35
22 25 26 26 29 36 26 23 24 31 27 37 27 26 32 28 32 33
21 24 24 27 37 30 20 27 33 31 28 33 23 28 31 34 31 29
32 35 30 27 35 30 23 25 22 19 21 35 21 21 28 26 27 20
46 44 45 46 48 63 43 43 43 46 47 58 33 24 40 37 41 38
38 41 38 30 36 35 21 25 31 35 33 23 24 33 30 30 37 35
20 21 31 24 30 33 24 23 21 24 21 35 24 18 21 26 28 28
26 28 27 27 35 35 28 25 26 25 38 28 24 30 28 35 33 28
28 29 43 28 33 37 19 22 27 25 25 35 21 28 25 25 31 25 :
FACPRODUCT    !p(Replicate,Recipe); Batch
CALCULATE     xTemperature = Temperature