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# AONEWAY procedure

Performs one-way analysis of variance (R.W. Payne).

### Options

`PRINT` = string tokens Controls printed output from the analysis of variance (`aovtable`, `information`, `covariates`, `effects`, `residuals`, `contrasts`, `means`, `cbeffects`, `cbmeans`, `stratumvariances`, `%cv`, `missingvalues`, `homogeneity`, `permutationtest`); default `aovt`, `mean`, `miss` Defines the treatments for the analysis Covariates (if any) for analysis of covariance Which residual plots to provide (`fittedvalues`, `normal`, `halfnormal`, `histogram`, `absresidual`); default `fitt`, `norm`, `half`, `hist` Type of graphs (`lineprinter`, `highresolution`); default `high` Probabilities for variance ratio (`yes`, `no`); default `no` Types of standard errors to be printed with the means (`differences`, `lsd`, `means`); default `diff` Significance level (%) for least significant differences; default 5 Number of random allocations to make when `PRINT=perm`; default 999 Seed for the random number generator used to make the allocations; default 0 continues from the previous generation or (if none) initializes the seed automatically

### Parameters

`Y` = variates Each of these contains the data values for an analysis Saves the residuals from each analysis Saves the fitted values from each analysis

### Description

`AONEWAY` provides customized facilities and output for one-way analysis of variance. For example, if the treatments have unequal replication, a standard error is printed for each mean, rather than the summary for comparisons of means with minimum and maximum replication as given by `ANOVA`. Similarly, any missing values are excluded from the analysis by `AONEWAY`. Conversely, in `ANOVA` they need to be included, to ensure balance in the more general situations that it covers, and are estimated as part of the analysis. In addition, `AONEWAY` provides residual plots directly, instead of requiring you to use procedure `APLOT` after the analysis, and it can test the homogeneity of the variances within the groups.

The `Y` parameter supplies a variate containing the data values to be analysed. The factor defining the groups to be compared is supplied by the `GROUPS` option. You can either specify just the factor to produce a simple one-way anova, or you can put it within a `POL`, `REG` or `COMPARISON` function to fit some contrasts at the same time. There is also a `COVARIATES` option which can supply one or more variates to be used as covariates in an analysis of covariance.

Printed output is requested by listing the required components with the `PRINT` option. The most relevant settings are:

    `aovtable` to print the analysis-of-variance table; to print the table of means; to print the effects (means minus grand mean); to print the coefficient of variation; to print estimates for missing values (if any); to print tests for the homogeneity of the variances within the groups; and analysis-of-variance table with the probabilities calculated by a random permutation test.

However, for compatibility, all the settings of the `PRINT` option of `ANOVA` are included. By default, when `PRINT=perm`, `AONEWAY` makes 999 random allocations of the data to the two samples (using a default seed), and determines the probabilities of the variance ratios from their distribution over these randomly generated datasets. (It therefore makes no assumptions about the distribution of the data values.) The `NTIMES` option allows you to request another number of allocations, and the `SEED` option allows you to specify another seed. `AONEWAY` checks whether `NTIMES` is greater than the number of possible ways in which the data values can be allocated. If so, it does an exact test instead, which takes each possible allocation once.

The `FPROBABILITY` option can be set to `yes` to print of probabilities for variance ratios in the analysis-of-variance table. The `PSE` option controls the standard errors printed with the tables of means. The default setting is `differences`, which gives standard errors of differences of means. The setting `means` produces standard errors of means, `LSD` produces least significant differences, and by setting `PSE=*` the standard errors can be suppressed altogether. The significance level to use in the calculation of the least significant differences can be changed from the default of 5% using the `LSDLEVEL` option.

The `PLOT` option allows up to four of the following residual plots to be requested:

    `fittedvalues` for a plot of residuals against fitted values; for a Normal plot; for a half-Normal plot; for a histogram of residuals; and for a plot of the absolute values of the residuals against the fitted values.

By default the first four are produced. The `GRAPHICS` option determines the type of graphics that is used, with settings `highresolution` (the default) and `lineprinter`.

Variates of residuals and fitted values can be saved using the `RESIDUALS` and `FITTEDVALUES` parameters, respectively. Directive `AKEEP` can be used to save other information from the analysis of the last data variate to be analysed by the procedure (see the Guide to the Genstat Command Language, Part 2, Section 4.6.1 for details).

Options: `PRINT`, `GROUPS`, `COVARIATES`, `PLOT`, `GRAPHICS`, `FPROBABILITY`, `PSE`, `LSDLEVEL`.

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

### Method

`AONEWAY` uses the standard Genstat facilities for analysis of variance, except that the standard errors and lsd’s for the means are saved by `AKEEP` and then printed, rather than being printed directly (as just a summary) by `ADISPLAY`. Permutation tests are performed by `APERMTEST`, residual plots are produced by `APLOT`, and the homogeneity of variances is tested by `VHOMOGENEITY`.

### Action with `RESTRICT`

If the `Y` variate is restricted, only the units not excluded by the restriction will be analysed.

Directive: `ANOVA`.

Procedure: `A2WAY`.

Commands for: Analysis of variance.

### Example

```CAPTION 'AONEWAY example',\
!t('Data from Snedecor & Cochran (1980), Statistical Methods',\
'(7th edition), page 216 and also see page 252.');\
STYLE=meta,plain
FACTOR  [LEVELS=4; VALUES=(1...4)6] Fat
VARIATE [VALUES=64,78,75,55, 72,91,93,66, 68,97,78,49,\
77,82,71,64, 56,85,63,70, 95,77,76,68] Absorbed
AONEWAY [PRINT=aov,means,homogeneity; GROUPS=Fat; FPROBABILITY=yes] Absorbed
```
Updated on June 20, 2019