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# RCOMPARISONS procedure

Calculates comparison contrasts amongst regression means (R.W. Payne).

### Options

`PRINT` = string tokens Controls printed output (`aov`, `contrasts`); default `aov`, `cont` Factor combinations for which to form the predicted means (`full`, `present`, `estimable`); default `esti` Type of adjustment to be made when forming the predicted means (`marginal`, `equal`, `observed`); default `marg` Types of standard errors to be printed with the contrasts (`contrasts`, `differences`, `lsd`); default `cont` Weights classified by some or all of the factors in the model; default `*` Value of offset on which to base predictions; default mean of offset variate Method of forming margin (`mean, total`); default `mean` How to deal with aliased parameters (`fault`, `ignore`); default `faul` What back-transformation to apply to the values on the linear scale, before calculating the predicted means (`link, none`); default `link` Controls whether the variance of predictions is calculated on the basis of forecasting new observations rather than summarizing the data to which the model has been fitted (`data`, `new`); default `data` Which warning messages to suppress (`dispersion`, `nonlinear`); default `*` Value of dispersion parameter in calculation of s.e.s; default is as set in the `MODEL` statement Basis of estimate of dispersion, if not fixed by `DISPERSION` option (`deviance, Pearson`); default is as set in the `MODEL` statement Supplies the total number of trials to be used for prediction with a binomial distribution (providing a value n greater than one allows predictions to be made of the number of “successes” out of n, whereas the value one predicts the proportion of successes); default 1 Significance level (%) for least significant differences; default 5 Regression save structure for the analysis from which the comparison contrasts are to be calculated

### Parameters

`FACTOR` = factors Factor whose levels are compared Defines the comparisons to be estimated Number of comparisons to estimate; default is the number of rows of the `CONTRASTS` matrix Set if comparisons are to be made at different combinations of another factor or factors Saves the estimated contrasts in a variate if `GROUPS` is unset, or in a pointer to a set of tables Saves standard errors of the contrasts in a variate if `GROUPS` is unset, or in a pointer to a set of tables Pointer to a set of symmetric matrices to save standard errors for differences between the contrasts estimated for different levels of the `GROUPS` factor(s) Pointer to a set of symmetric matrices to save least significant differences for the contrasts estimated for different levels of the `GROUPS` factor(s) Saves sums of squares or deviances of the contrasts Saves degrees of freedom for the contrasts

### Description

`RCOMPARISONS` allows you to make comparisons between predicted means from a linear or generalized linear regression. The model should previously have been fitted by the `FIT` directive in the usual way. The `SAVE` option can be used to specify the regression save structure from the analysis for which the comparisons are to be calculated (see the `SAVE` option of the `MODEL` directive). If `SAVE` is not specified, the comparisons are calculated from the most recent regression analysis.

The factor amongst whose levels the comparisons are to be calculated is specified by the `FACTOR` parameter. The `CONTRASTS` parameter supplies a matrix to specify the comparisons to be calculated. This works in the same way as the matrix supplied as the third parameter of the `COMPARISON` function, with a column for each level of the `FACTOR`, and a row for each comparison. You can set the `ORDER` parameter to a scalar, n say, to indicate that only the comparisons in the first n rows of the `CONTRASTS` matrix are to be calculated (otherwise they are all calculated).

By default the comparisons are calculated between the means in the one-way table classified by `FACTOR`. However, you can set the `GROUPS` parameter to some other factor to indicate that the comparisons are to be made for each level of that factor, or you can set it to a pointer of factors to make the comparisons for every combination of the levels of those factors.

`RCOMPARISONS` calculates the means using the `PREDICT` directive. The first step (A) of the calculation 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, or `COMBINATIONS=full` uses all the cells. 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, the setting `observed` uses the `WEIGHTS` option of `PREDICT` to weight each factor combination according to its own individual replication in the data. Alternatively, you can supply your own table of weights, using the `WEIGHTS` option. There are also options `OFFSET`, `METHOD`, `ALIASING`, `BACKTRANSFORM`, `SCOPE`, `NOMESSAGE`, `DISPERSION`, `DMETHOD` and `NBINOMIAL` to control further aspects of the calculations; these operate exactly as in the `PREDICT` directive.

The `PRINT` option controls printed output, with settings:

    `aov` to print an analysis of variance (for an ordinary linear regression) or an analysis of deviance (for a generalized linear model), giving the sums of squares (or deviances) and so on for the comparisons; to print the contrasts.

By default these are both printed. The `PSE` option controls the types of standard errors that are produced to accompany the contrasts, with settings:

    `contrasts` for standard errors of the contrasts; for standard errors for differences between pairs of contrasts calculated for the different GROUPS; for least significant differences for contrasts calculated for the `GROUPS`.

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

The `ESTIMATES` parameter allows you to save the estimated contrasts. These are in a variate if `GROUPS` is unset, or in a pointer containing a table classified by `GROUPS` for each comparison otherwise. The `SE` parameter saves the standard errors of the contrasts, in a variate or pointer similarly to `ESTIMATES`. If `GROUPS` is set, you can also save standard errors for differences between the contrasts estimated for different levels of the `GROUPS` factor(s). This is again a pointer, with a symmetric matrix for each comparison. Finally, the `DF` parameter can save a variate containing the degrees of freedom of the contrasts, and the `DEVIANCES` parameter can save a variate with their deviances (for a generalized linear model) or sums of squares (for an ordinary linear regression).

Options: `PRINT`, `COMBINATIONS`, `ADJUSTMENT`, `PSE`, `WEIGHTS`, `OFFSET`, `METHOD`, `ALIASING`, `BACKTRANSFORM`, `SCOPE`, `NOMESSAGE`, `DISPERSION`, `DMETHOD`, `NBINOMIAL`, `LSDLEVEL`, `SAVE`.

Parameters: `FACTOR`, `CONTRASTS`, `ORDER`, `GROUPS`, `ESTIMATES`, `SE`, `SED`, `LSD`, `DEVIANCES`, `DF`.

### Method

The predicted means and their variances and covariances are calculated using the `PREDICT` directive. The comparisons, their standard errors and sums of squares are then calculated using Genstat’s table and matrix calculation facilities.

Directive: `PREDICT`.

Procedures: `FCONTRASTS`, `RTCOMPARISONS`, `VTCOMPARISONS`.

Commands for: Regression analysis.

### Example

```CAPTION      'RCOMPARISONS example',\
!t('3x2 factorial experiment (Snedecor & Cochran, 1980,',\
'Statistical Methods, seventh edition, p. 305).');\
STYLE=meta,plain
FACTOR       [NVALUES=60; LABELS=!T(high,low); VALUES=3(1,2)10] Amount
&            [LABELS=!T(beef,cereal,pork); VALUES=(1...3)20] Source
VARIATE      [NVALUE=60] Gain
73  98  94  90 107  49
102  74  79  76  95  82
118  56  96  90  97  73
104 111  98  64  80  86
81  95 102  86  98  81
107  88 102  51  74  97
100  82 108  72  74 106
87  77  91  90  67  70
117  86 120  95  89  61
111  92 105  78  58  82 :
MODEL        Gain
FIT          Source*Amount
MATRIX       [ROWS=!T('animal vs cereal','beef vs pork'); COLUMNS=3;\
VALUES=0.5,-1,0.5,1,0,-1] Compare
RCOMPARISONS Source; CONTRASTS=Compare
&            [PSE=contrasts,differences,lsd]\
Source; CONTRASTS=Compare; GROUPS=Amount
```
Updated on June 18, 2019