Performs pairwise comparisons between REML means (D.M. Smith).

### Options

`PRINT` = string tokens |
Controls printed output (`comparisons` , `critical` , `description` , `lines` , `letters` , `plot` , `mplot` , `pplot` ); default `lett` |
---|---|

`METHOD` = string token |
Test to be performed (`fplsd` , `fulsd` , `bonferroni` , `sidak` ); default `fuls` |

`FACTORIAL` = scalar |
Limit on the number of factors in each term; default 3 |

`DIRECTION` = string token |
How to sort means (`ascending` , `descending` ); default `asce` |

`PROBABILITY` = scalar |
The required significance level; default 0.05 |

`STUDENTIZE` = string token |
Whether to use the alternative LSD test where the Studentized Range statistic is used instead of Student’s t (`yes` , `no` ); default `no` |

`DFMETHOD` = string token |
Specifies which degrees of freedom to use for the tests (`fddf` , `given` , `tryfddf` ); default `fddf` |

`DFGIVEN` = scalar |
Specifies the number of degrees of freedom to use for the tests when `DFMETHOD=given` , or if d.d.f. are unavailable when `DFMETHOD=tryfddf` |

`FMETHOD` = string token |
Controls how to calculate denominator degrees of freedom for the F-statistics, if these are not already available in the `REML` save structure (`automatic` , `algebraic` , `numerical` ); default `auto` |

`SAVE` = `REML` save structure |
Save structure to provide the table of means and associated information; default uses the save structure from the most recent `REML` |

### Parameters

`TERMS` = formula |
Treatment terms whose means are to be compared |
---|---|

`MEANS` = pointer or variate |
Saves the (sorted) means |

`DIFFERENCES` = pointer or symmetric matrix |
Saves differences between the (sorted) means |

`LABELS` = pointer or text |
Saves labels for the (sorted) means |

`LETTERS` = pointer or text |
Saves letters indicating groups of means that do not differ significantly |

`SIGNIFICANCE` = pointer or symmetric matrix |
Indicators to show significant comparisons between (sorted) means |

`CIWIDTH` = pointer or symmetric matrix |
Saves the width of the confidence interval for the absolute differences between the (sorted) means |

### Description

`VMCOMPARISON`

calculates comparisons between means estimated in a `REML`

analysis, and tests them with t-statistics using the approximate numbers of residual degrees of freedom that can be printed by `REML`

with the Wald statistics. This corresponds to Fisher’s unprotected LSD test, or you can set option `METHOD=fplsd`

to request Fisher’s protected LSD test (so that the comparisons are not tested if the fixed term generating the means is not significant). Alternatively, the `METHOD`

settings `bonferroni`

or `sidak`

allow you to use adjusted critical probability values for the t-statistic that take account of the numbers of comparisons that are being made; see Hsu (1996) page 65.

The `TERMS`

parameter specifies a model formula to define the fixed terms whose predicted means are to be compared. The means (and the necessary associated information) are usually taken from the most recent analysis performed by `REML`

, but you can set the `SAVE`

option to a save structure from another `REML`

if you want to examine means from an earlier analysis. As in `VCOMPONENTS`

, the `FACTORIAL`

option sets a limit on the number of factors in each term (default 3).

The `DFMETHOD`

option specifies how to obtain the degrees of freedom for the tests. The default is to use the numbers of denominator degrees of freedom printed by `REML`

in the `d.d.f.`

column in the table of tests for fixed tests (produced by setting option `PRINT=wald`

). The degrees of freedom are relevant for assessing the fixed term as a whole, and may vary over the contrasts amongst the means of the term. So the results should be used with caution. (If you are interested in a specific comparison, you should set up a 2-level factor to fit this explicitly in the analysis.) The `FMETHOD`

option controls how the denominator degrees of freedom should be calculated, if they are not already available in the `REML`

save structure (e.g. because they were printed in the original analysis). The settings are the same as in the `REML`

and `VKEEP`

directives, except that there is no `none`

setting. (You would set this option only if you really do want to calculate them.)

In some of the more complicated analyses, `REML`

may be unable to calculate the denominator degrees of freedom. You might then want to supply the number of degrees of freedom yourself, using the `DFGIVEN`

option, rather than having no tests at all. For example, you could use the number of denominator degrees of freedom from the analysis of an earlier similar design. However, the results will only be as good as the degrees of freedom that you have supplied, and thus should be used with caution! You can set option `DFMETHOD=tryfddf`

to use the denominator degrees of freedom, if these can be calculated, or those specified by `DFGIVEN`

otherwise. The setting `DFMETHOD=given`

always uses the degrees of freedom specified by `DFGIVEN`

.

Printed output is controlled by the `PRINT`

option, with settings:

`comparisons` |
prints the differences between the pair of means, upper and lower confidence limits for the differences, t-statistics and an indication of whether or not they are significant; |
---|---|

`critical` |
gives critical values for the t-statistic; |

`description` |
provides a description including information such as the experiment-wise and compartment-wise error rates; |

`lines` |
gives the means, with lines joining those that do not differ significantly; |

`letters` |
gives the means, with identical letters (a, b etc.) alongside those that do not differ significantly; |

`mplot` |
does a mean-mean scatter plot (synonym `plot` ); |

`pplot` |
displays the probabilities in a shade plot. |

By default, `PRINT=letters`

.

The means are usually sorted into ascending order, but you can set option `DIRECTION=descending`

for descending order, or `DIRECTION=*`

to leave them in their original order. Note, though, that the lines joining means with non-significant differences may then be broken.

In most `REML`

analyses the standard errors for the differences between the means will be unequal, and the memberships of the groups defined by the lines or letters may then be inconsistent. Suppose, for example, you have ordered means A, B and C. If the s.e.d. for A vs. C is large compared to those for A vs. B and B vs C, you might find that there is no significant difference between A and C, but there are significant differences between A and B, and between B and C. So treatments A and B and treatments B and C would be in different groups. However, treatments A and C (which are further apart) would be in the same group. This contradicts the idea behind multiple comparisons, where you expect that if two means are in the same group, than any mean between them should be in that group too. If `VMCOMPARISON`

finds inconsistencies like this, it gives a diagnostic and suppresses the printing of lines and letters (but not the other types of output).

The mean-mean scatter plot allows you to assess the confidence region for the difference between each pair of means visually. It has grid lines from both the x- and y-axis at the position of each mean, and a diagonal line at 45 degrees marking y=x. The confidence interval for each pair of means is plotted as a line at an angle of -45 degrees and centred on the intersection above the line y=x of the grid lines for the two means (so the y grid line is for the larger of the two means, and the x grid line is for the smaller mean). The difference between the means is significant if their confidence line does not intersect the line y=x. For more details, see Hsu (1996) pages 151-153.

The shade plot displays the probabilities in a symmetric matrix. The colour of each cell

represents the probability for the difference between the means for the treatments in the

corresponding row and column.

The `PROBABILITY`

option allows the experiment-wise significance level for the intervals from the Bonferroni and Sidak tests to be changed from the default 0.05 (e.g. to 0.01). For the Fisher’s tests, it changes the pair-wise significance level. The `STUDENTIZE`

option can specify that the tests should use the Studentized Range statistic rather than Student’s t (for further information see Hsu 1996, page 139).

The `MEANS`

parameter can save the means, sorted according to the `DIRECTION`

option and omitting any that were non-estimable. If the `TERMS`

parameter specifies a single term, `MEANS`

should be set to a variate. If `TERMS`

specifies several terms, you must supply a pointer which will then be set up to contain as many variates as there are terms. Similarly the `LABELS`

parameter can save labels to identify the means, in either a text (for a single term) or in a pointer of texts (for several). Likewise the `LETTERS`

parameter can save texts with the letters identifying means that do not differ significantly, and the `SIGNIFICANCE`

parameter can save symmetric matrices containing ones or zeros according to whether the various comparisons were significant or non-significant. The `DIFFERENCES`

parameter can save symmetric matrices containing the differences between the (sorted) means, and the `CIWIDTH`

parameter can save symmetric matrices containing the widths of the confidence intervals for the differences.

Options: `PRINT`

, `METHOD`

, `FACTORIAL`

, `DIRECTION`

, `PROBABILITY`

, `STUDENTIZE`

, `DFMETHOD`

, `DFGIVEN`

, `FMETHOD`

, `SAVE`

.

Parameters: `TERMS`

, `MEANS`

, `DIFFERENCES`

, `LABELS`

, `LETTERS`

, `SIGNIFICANCE`

, `CIWIDTH`

.

### Method

The methodology implemented is based on that described and reviewed in Hsu (1996).

### Reference

Hsu, J.C. (1996). *Multiple Comparisons Theory and Methods*. Chapman & Hall, London.

### See also

Directive: `VDISPLAY`

.

Procedures: `VLSD`

, `AMCOMPARISON`

, `AUMCOMPARISON`

, `MCOMPARISON`

.

Commands for: REML analysis of linear mixed models.

### Example

CAPTION 'VMCOMPARISON example',\ !t('Experiment to study 5 diets for rats, using',\ 'a randomized-block design where the blocks are',\ '8 litters each of 5 rats (John & Quenouille, 1977,',\ 'Experiments Design and Analysis, page 32)');\ STYLE=meta,plain FACTOR [NVALUES=40; LEVELS=8] Litter & [LEVELS=5] Rat & [LABELS=!t(A,B,C,D,E)] Diet GENERATE Litter,Rat READ Diet,Gain; FREPRESENTATION=labels E 76.0 C 70.7 D 68.3 A 57.0 B 64.8 A 55.0 D 67.1 B 66.6 C 59.4 E 74.5 C 64.5 A 62.1 D 69.1 E 76.5 B 69.5 D 72.7 B 61.1 A 74.5 C 74.0 E 86.6 A 86.7 E 94.7 B 91.8 D 90.6 C 78.5 B 51.8 C 55.8 E 43.2 A 42.0 D 44.3 D 53.8 A 71.9 C 63.0 B 69.2 E 61.1 E 54.4 D 40.9 B 48.6 C 48.1 A 51.5 : VCOMPONENTS [FIXED=Diet] Litter REML Gain VMCOMPARISON Diet