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# AOVANYHOW procedure

Performs analysis of variance using `ANOVA`, regression or `REML` as appropriate (R.W. Payne).

### Options

`PRINT` = string tokens Controls printed output from the analysis (`aovtable`, `information`, `means`, `residuals`); default `aovt`, `info`, `mean` Whether to complete the analysis or just form a recommendation (`analyse`, `recommend`); default `anal` Limit on number of factors in a treatment term; default 3 Printing of probabilities for variance ratios in the analysis-of-variance table (`yes`, `no`); default `no` Which residual plots to provide (`fittedvalues`, `normal`, `halfnormal`, `histogram`); default `*` i.e. none Factor combinations for which to form predicted means (`present`, `estimable`); default `esti` Type of adjustment to be made when predicting means (`marginal`, `equal`, `observed`); default `marg` Types of standard errors to be printed with the predicted means (`differences`, `alldifferences`, `lsd`, `alllsd`, `means`; default `diff` Weights for each unit; default `*` i.e. all units with weight one Significance level (%) for least significant differences; default 5 Maximum loss of efficiency occurring on any treatment contrast if the analysis is done by regression Limit on the loss of efficiency for the analysis to be done by regression; default 0.1 Exit code indicating the recommended method of analysis

### Parameters

`Y` = variates Data values to be analysed Variate to save the residuals from each analysis Variate to save the fitted values from each analysis To save details of each analysis to use subsequently with the `AOVDISPLAY` procedure

### Description

`AOVANYHOW` assesses a data set to select the most appropriate method for analysis of variance. If the design is orthogonal or balanced it uses the `ANOVA` directive. Otherwise, if there is no blocking in the design (i.e. there is only one random term) it uses the Genstat regression facilities through procedure `A2WAY` or `AUNBALANCED`. Finally, if there are additional random terms, it looks to see if these contain any useful information about the treatments in order to choose between regression and `REML`. The `EFLIMIT` option sets a limit on the amount of information that may be lost on any of the treatment contrasts if the analysis to be done by regression instead of `REML`; the default of 0.1 implies that no more than 10% of the information on any contrast may be estimated between the random terms.

The method of use is similar to that for `ANOVA`. The treatment terms to be fitted must be specified, before calling the procedure, by the `TREATMENTSTRUCTURE` directive. Similarly, any covariates must be indicated by the `COVARIATE` directive. Any blocking structure must be specified by the `BLOCKSTRUCTURE` directive.

The parameters of the procedure are identical to those of `ANOVA`. The variates to be analysed are specified by the `Y` parameter. Residuals and fitted values can be saved using the `RESIDUALS` and `FITTEDVALUES` parameters respectively. Finally, the `SAVE` parameter allows details of the analysis to be saved so that further output can be obtained using the `AOVDISPLAY` procedure.

Printed output is controlled by the `PRINT` option. The settings are limited to those that can produce analogous output from any of the analysis methods:

    `aovtable` analysis-of-variance table from `ANOVA` or regression, or Wald and F tests for fixed effects from `REML`, design type, efficiency factors and name of the command used for the analysis, tables of (predicted) means, and residuals (fitted values are printed too for analyses by regression or `REML`).

Probabilities can be printed for variance ratios by setting option `FPROBABILITY=yes`.

The `SAVE` parameter allows you to save a pointer containing information about the analysis. You can use this as the input for the `SAVE` parameter of the `AOVDISPLAY` procedure to print (or reprint) any of the information provided by the `PRINT` option above. Alternatively, the first element of the pointer is the save structure from the command that was used for the analysis. So, if you use this with the display commands associated with that analysis command, you can display the more specialized output from the command (for example, variance components from `REML`).

Tables of means from regression and `REML` are calculated using the `PREDICT` and `VPREDICT` directives, respectively. The first step (A) of their calculations forms the full table of predictions, classified by every factor in the model. The second step (B) averages the full table over the factors that do not occur in the `table of means. `The `COMBINATIONS` option specifies which cells of the full table are to be formed in Step A. The default setting, `estimable`, fills in all the cells other than those that involve parameters that cannot be estimated, for example because of aliasing. Alternatively, setting `COMBINATIONS=present` excludes the cells for factor combinations that do not occur in the data. The `ADJUSTMENT` option then defines how the averaging is done in Step B. The default setting, `marginal`, forms a table of marginal weights for each factor, containing the proportion of observations with each of its levels; the full table of weights is then formed from the product of the marginal tables. The setting `equal` weights all the combinations equally. Finally, for regression analyses, the setting `observed` uses the `WEIGHTS` option of `PREDICT` to weight each factor combination according to its own individual replication in the data.

The `PSE` option controls the types of standard errors that are produced to accompany the tables of means, with settings:

    `differences` summary of standard errors for differences between pairs of means, standard errors for differences between all pairs of means, summary of least significant differences between pairs of means, least significant differences between all pairs of means, effective standard errors for analyses by `ANOVA`, or approximate effective standard errors for analyses by regression or `REML` – these are formed by procedure `SED2ESE` with the aim of allowing good approximations to the standard errors for differences to be calculated by the usual formula of sedi,j = √( esei2 + esej2 ).

The default is `differences`. The `LSDLEVEL` option sets the significance level (as a percentage) for the least significant differences.

The `PLOT` option allows various residual plots to be requested: `fittedvalues` for a plot of residuals against fitted values, `normal` for a Normal plot, `halfnormal` for a half Normal plot, and `histogram` for a histogram of residuals.

The `FACTORIAL` option sets a limit on the number of factors that a higher-order term, such as an interaction, can contain; any terms with more factors are deleted from the analysis. The `WEIGHTS` option allows a variate of weights to be specified for a weighted analysis of variance.

You can save a scalar indicating the recommended method of analysis by using the `EXIT` option. The scalar can take values with the following meanings.

0.   The design is orthogonal. Analyse by `ANOVA`.

1.   The design is balanced. Analyse by `ANOVA`.

2.   The design unbalanced. It has 1 or 2 treatment factors and no blocking. Analyse by `A2WAY`.

3.   The design unbalanced and has 1 or 2 treatment factors. No more than a proportion defined by the `EFLIMIT` option of the information on any treatment contrast is estimated between block terms. Analyse by `A2WAY`.

4.   The design unbalanced, and there are either weights or more than 2 treatment factors. There is no blocking. Analyse by `AUNBALANCED`.

5.   The design is unbalanced, and there either are weights or more than 2 treatment factors. No more than a proportion defined by the `EFLIMIT` option of the information on any treatment contrast is estimated between block terms. Analyse by `AUNBALANCED`.

6.   The design unbalanced with several block (i.e. random) terms. Analyse by `REML`.

The `EFLOSS` option can save the maximum loss of efficiency that would occur on any treatment contrast if the analysis is done by regression.

You can set option `METHOD=recommend` to request that `AOVANYHOW` will just form a recommendation for the command to be used if the analysis cannot be done by `ANOVA`. The only available `PRINT` option is then `information`, which tells you which command is recommended. You can still use the `EXIT` and `EFLOSS` options, but residuals and fitted values will be saved (by the `RESIDUALS` and `FITTEDVALUES` parameters) only if the analysis should be done by `ANOVA`.

Options: `PRINT`, `METHOD`, `FACTORIAL`, `FPROBABILITY`, `PLOT`, `COMBINATIONS`, `ADJUSTMENT`, `PSE`, `WEIGHTS`, `LSDLEVEL`, `EFLOSS`, `EFLIMIT`, `EXIT`.

Parameters: `Y`, `RESIDUALS`, `FITTEDVALUES`, `SAVE`.

### Method

The `EXIT` option of the `ANOVA` directive is used to ascertain whether or not the design is orthogonal or balanced; if so it can be analysed by `ANOVA`. (For details, see the Guide to the Genstat Command Language, Part 2 Statistics, Section 4.7.) If the design is not orthogonal or balanced and there are several random terms, the `AEFFICIENCY` procedure is used to calculate the efficiency factors for the treatment terms, in order to decide whether to use regression of `REML`.

### Action with `RESTRICT`

If the `Y` variate or any of the factors or covariates is restricted, only the units not excluded by the restriction will be analysed.

Directives: `ANOVA`, `REML`.

Procedures: `AOVDISPLAY`, `AN1ADVICE`, `AUNBALANCED`, `A2WAY`.

Commands for: Analysis of variance.

### Example

```CAPTION  'AOVANYHOW example 1',\
'Split plot design, see Guide to Genstat, Part 2, Section 4.2.1.';\
STYLE=meta,plain
FACTOR   [NVALUES=72; LEVELS=6] Blocks
&        [LEVELS=3] Wplots
&        [LEVELS=4] Subplots
GENERATE Blocks,Wplots,Subplots
FACTOR   [LABELS=!T('0 cwt','0.2 cwt','0.4 cwt','0.6 cwt')] Nitrogen
&        [LABELS=!T(Victory,'Golden rain',Marvellous)]      Variety
VARIATE  Yield; DECIMALS=2; EXTRA=' of oats in cwt. per acre'
4 3 2 1 1 2 4 3 1 2 3 4 3 1 2 4 4 1 2 3 2 1 3 4
2 3 4 1 4 2 3 1 1 4 2 3 3 4 1 2 1 3 4 2 2 3 4 1
4 1 3 2 3 4 1 2 3 4 2 1 3 1 4 2 4 3 1 2 1 2 3 4 :
3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2
2 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 2 2 2 2 1 1 1 1
2 2 2 2 1 1 1 1 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 :
156 118 140 105 111 130 174 157 117 114 161 141
104  70  89 117 122  74  89  81 103  64 132 133
108 126 149  70 144 124 121  96  61 100  91  97
109  99  63  70  80  94 126  82  90 100 116  62
96  60  89 102 112  86  68  64 132 124 129  89
118  53 113  74 104  86  89  82  97  99 119 121 :
" Define the treatment structure: factorial effects of V and N."
TREATMENTS Variety*Nitrogen
" Subplots nested within whole-plots nested within blocks."
BLOCK      Blocks/Wplots/Subplots
AOVANYHOW  [PRINT=aovtable,information] Yield

CAPTION    'AOVANYHOW example 2',\
'Unbalanced design with almost all information within blocks.';\
STYLE=meta,plain
BLOCKSTRUCTURE day
TREATMENTSTRUCTURE A*B*C
AOVANYHOW  [PRINT=aovtable,information] Y
```
Updated on June 20, 2019