Calculates least significant intervals (M.C. Hannah).

### Options

`PRINT` = string tokens |
What to print (`delta` , `lsi` , `fittedsed` , `discrepancy` , `maxdiscrepancy` , `%discrepancy` ); default `delta` , `lsi` , `maxd` |
---|---|

`METHOD` = string token |
Selects the method for computing the deltas (`leastsquares` , `max` , `maxpse` ); default `leas` |

`PLOT` = string tokens |
What to plot (`sed` , `lsi` ); default `sed` , `lsi` |

`CHECKFIT` = string token |
Which pairwise contrasts to use in printed output or plots involving the fitted SEDs (`specified` , `all` ); default `spec` |

`PROBABILITY` = scalar |
Significance level for the least significent intervals; default 0.05. |

`DF` = scalar |
Degrees of freedom for the t-distribution use in calculation of the least significent intervals; default `*` assumes an infinite number of degrees of freedom (i.e. a Normal rather than a t-distribution) |

`WINDOW` = scalar |
Window in which to plot the graphs |

`TITLE` = text |
Title for the graphs; default `'Estimates with LSIs by Treatment'` |

`YTITLE` = text |
Title for the y-axis; default `'Estimates'` |

### Parameters

`ESTIMATES` = tables or variates |
Parameter estimates; if these are not supplied `SEDLSI` can calculate the parameters {δ_{i}} but not the LSIs |
---|---|

`SED` = symmetric matrices |
Matrix containing standard errors of (pairwise) differences between estimates |

`VCOVARIANCE` = symmetric matrices |
Matrix containing variances and covariances of estimates |

`WEIGHTS` = symmetric matrices |
Weight (or importance) to be used for each pairwise difference; default is a matrix of ones (i.e. all pairwise differences of equal interest) |

`LABELS` = texts |
Text vector (e.g. treatment labels) for labelling output; default takes the labels of levels of the factor classifying an `ESTIMATES` table or (if `ESTIMATES` is a variate or unset) row labels from `SED` or `VCOVARIANCE` |

`DELTA` = variates |
Saves the estimated parameters {δ}_{i} |

`LSI` = pointers |
Saves details of the least significant intervals |

`FITTEDSED` = symmetric matrices |
Saves the fitted SED matrices |

### Description

Least significant intervals (LSIs) are used for comparing a set of estimates (e.g. predicted means from `ANOVA`

or regression) graphically, especially when their SEDs differ. LSIs are intervals (or error bars) that are designed to overlap where there is no significant difference between estimates, and to be disjoint (i.e. not to overlap) where there are significant differences.

Presentation of results can be problematic when standard errors of differences vary appreciably due to unequal replication or an unbalanced design. LSIs attempt to address this difficulty, and are suitable for graphical presentation (Snee 1981). They can also be useful for presentation of results following a transformation of scale. Intervals can be formed on the scale on which the analysis of data took place (or the scale of a linear predictor for a generalized linear model) and be back-transformed, along with point estimates, to the original measurement scale for graphical presentation (see e.g. Hannah & Quigley 1996).

The SEDs can be supplied, in a symmetric matrix, using the `SED`

parameter. Alternatively, you can provide a (symmetric) variance-covariance matrix, using the `VCOVARIANCE`

parameter. `SEDLSI`

uses these to compute parameters {δ_{i}} such that δ_{i} + δ_{j} is approximately equal to SED_{ij}. The delta values can be saved, in a variate, using the `DELTA`

parameter.

You can also supply parameter estimates (e.g. treatment means), in either a variate or a table, using the `ESTIMATES`

parameter. If `ESTIMATES`

is a variate, you may want to use the `LABELS`

option to supply a text of labels. If estimates are available, `SEDLSI`

can also construct least significant intervals (LSIs)

`lower_LSI = ESTIMATES - EDT(1 - PROBABILITY/2; DF) * DELTA`

`upper_LSI = ESTIMATES + EDT(1 - PROBABILITY/2; DF) * DELTA`

where the significance probability is specified by the `PROBABILITY`

option (default 0.05), and the degrees of freedon are specified by the `DF`

option. If `DF`

is not set, the number of degrees of freedom is assumed to be infinite (and so `SEDLSI`

uses a Normal rather than a t-distribution).

When the SEDs are all equal the calculation is trivial; δ = SED/2 (Snee 1981). When SEDs depend on the treatment pair, estimation of δ is more difficult and there may not be an exact solution. However, there is usually an adequately approximate or a conservative solution. `SEDLSI`

offers three methods for estimating delta, requested using the `METHOD`

option. The first method (`leastsquares`

, the default) provides least-squares estimates such that δ_{i} + δ_{j} is approximately equal to SED_{ij}. The second method (`max`

) provides estimates such that δ_{i} + δ_{j} is greater than or equal to SED_{ij}. For `METHOD`

settings `leastsquares`

and `max`

at least one of the `SED`

or the `VCOVARIANCE`

parameters must be set. The third method (`maxpse`

) is similar to the `max`

method but the δ’s are constrained to be proportional to the standard errors of the estimates, `SQRT(DIAG(VCOVARIANCE))`

. For this method, the `VCOVARIANCE`

parameter must be set. This method may be considered desirable as it apparently constrains the width of resulting LSIs to reflect the relative precisions of the estimates more faithfully. However, it is often highly conservative, with some δ_{i} + δ_{j} values much greater than SED_{ij}, and it often neglects an exact solution.

Usually only comparisons between certain pairs of means are of genuine interest. To restrict attention just to these pairwise differences, a symmetric matrix corresponding to the `SED`

or `VCOVARIANCE`

matrix can be supplied using the `WEIGHTS`

parameter. This should contain zero in the positions of the contrasts that are not of interest, and one elsewhere. This then weights-out irrelevant SEDs from the calculation and thus avoids the δ’s being unnecessarily large (conservative) for the purpose at hand. For example, it could be that the only contrasts of interest are those between each treatment and a control treatment. This is specifed by a weights matrix with the row and column corresponding to the control containing ones, and with zeros elsewhere. By default all the weights are one (signifying all pairwise comparisons of interest). For the `leastsquares`

or `max`

methods, the weights can be any non-negative numeric values to reflect the (subjective) importance of particular pairwise contrasts.

Printed output is controlled by the `PRINT`

option, with settings:

`delta` |
prints the parameters {δ_{i}}, |
---|---|

`lsi` |
prints the least significant intervals, |

`fittedsed` |
prints the matrix [δ_{i} + δ_{j}] of fitted SEDs, |

`discrepancy` |
prints the difference between [δ_{i} + δ_{j}] and [SED_{ij}], |

`maxdiscrepancy` |
prints the maximum difference between [δ_{i} + δ_{j}] and [SED_{ij}], |

`%discrepancy` |
prints the difference as a percentage. |

The default is `PRINT=delta,lsi,maxd`

.

The `PLOT`

option produces graphs:

`lsi` |
plots the least significant interval for each estimate, and |
---|---|

`sed` |
plots the difference between [δ_{i} + δ_{j}] and [SED_{ij}]. |

By default `PLOT=lsi,sed`

. The `WINDOW`

option allows you to specify the window in which to plot the LSIs. By default a window is defined internally, within `SEDLSI`

, to fill the whole screen. The `TITLE`

option supplies the title for the plot (default `'Estimates with LSIs by Treatment'`

), and the `YTITLE`

option supplies a title for the y-axis (default `'Estimates'`

).

If the δ’s do not reproduce the SEDs exactly, it is recommended that the success of the approximation be checked, by examining the fitted SEDs, the differences, or the percent differences. By default, these outputs are produced only for differences of interest (indicated by non-zero weights in the `WEIGHTS`

matrix). If you also wish to check how well the solution applies to contrasts that had weight zero, you can set option `CHECKFIT=all`

to retain all the fitted SED values, provided their corresponding SED_{i} values were non-missing. (Note, though, that `CHECKFIT`

controls only what contrasts are printed or plotted, not the ones that are used to estimate the deltas.)

The information defining the LSIs can be saved, in a pointer, using the `LSI`

parameter. The components of the pointer are `'Label'`

, `'lowLSI'`

, `'estimate'`

and `'upLSI'`

; each is a variate except for `'Label'`

which is a text. The `LSI`

pointer can be used as input to the `LSIPLOT`

procedure, to plot the LSIs on a later occasion.

Options: `PRINT`

, `METHOD`

, `PLOT`

, `CHECKFIT`

, `PROBABILITY`

, `DF`

, `WINDOW`

, `TITLE`

, `YTITLE`

.

Parameters: `ESTIMATES`

, `SED`

, `VCOVARIANCE`

, `WEIGHTS`

, `LABELS`

, `DELTA`

, `LSI`

, `FITTEDSED`

.

### Method

Approximate least significant intervals are calculated as

`lower_LSI = ESTIMATES + EDT(1 - PROBABILITY/2; DF) * DELTA`

`upper_LSI = ESTIMATES + EDT(1 - PROBABILITY/2; DF) * DELTA`

where

`EDT(1 - PROBABILITY/2; DF)`

is the

`1 - PROBABILITY/2`

quantile of the t-distribution with `DF`

degrees of freedom.

`SEDLSI`

offers three methods of estimating δ. The first method (`leastsquares`

, the default) provides weighted least squares estimates based on the model

SED_{ij} = δ_{i} + δ_{j}

with weights optionally provided in the `WEIGHTS`

parameter.

The second method (`max`

) described in Hannah & Quigley (1996) uses

δ_{i} = max {SED_{ij} / δ_{oi} + δ_{oj}: *j*} δ_{oi}

where the parameters {δ_{oi}} are those obtained from the ordinary least-squares method.

The third method (`maxpse`

) is the same as the second method but with the parameters {δ_{oi}} being standard errors of estimates, obtained as square roots of the diagonal of the variance-covariance matrix for the estimates.

The `leastsquares`

method generally gives closer approximations to SEDs, but may be anti-conservative for some comparisons and conservative for others. Maximum SED methods are never anti-conservative but can be excessively conservative. If an exact solution to δ_{i} + δ_{j} = SED_{ij} exists, the `leastsquares`

and `max`

methods should find it.

If there is no contrast of interest for a particular estimate, due either to missing values in the `SED`

or `VCOVARIANCE`

matrix (zeros are interpreted as missing values here), or zeros specified in the `WEIGHTS`

matrix, the corresponding δ is not estimated. `SEDLSI`

also checks for missing values in the `ESTIMATES`

parameter and sets SED elements corresponding to these as missing. If the only contrasts of interest are those between each treatment and a control treatment, the number of relevant SEDs is one fewer than the number of delta values requiring estimation. `SEDLSI`

detects this treatments-verses-control scenario and, if `METHOD=leastsquares`

, it imposes the arbitrary constraint δ_{control} = SE_{control} if `VCOVARIANCE`

is set, or δ_{control} = min(SE_{control,i}) otherwise.

### References

Snee, R.D. (1981). Graphical display and assessment of means. *Biometrics*, 37, 835-836.

Hannah, M.C. & Quigley, P. (1996). Presentation of ordinal regression analysis on the original scale. *Biometrics*, 52, 771-775.

Hannah, M.C. (1999). Usefully combining a series of unreplicated cheesemaking experiments. *Journal of Dairy Research*, 66, 365-374.

### See also

Commands for: Calculations and manipulation.

### Example

CAPTION 'SEDLSI example',\ !t('Experiment on foster feeding of rats from Scheffe (1959)',\ 'The Analysis of Variance; also see McConway, Jones & Taylor (1999)',\ 'Statistical Modelling using GENSTAT, Example 7.6.'); STYLE=meta,plain FACTOR [NVALUES=61; LABELS=!t('A','B','I','J')] litter READ litter; FREPRESENTATION=labels A A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B I I I I I I I I I I I I I I J J J J J J J J J J J J J J J : FACTOR [NVALUES=61; LABELS=!t('A','B','I','J')] mother READ mother; FREPRESENTATION=labels A A A A A B B B I I I I J J J J J A A A A B B B B B I I I I J J A A A B B B I I I I I J J J A A A A B B B I I I J J J J J : VARIATE [NVALUES=61] littwt READ littwt 61.5 68.2 64 65 59.7 55 42 60.2 52.5 61.8 49.5 52.7 42 54 61 48.2 39.6 60.3 51.7 49.3 48 50.8 64.7 61.7 64 62 56.5 59 47.2 53 51.3 40.5 37 36.3 68 56.3 69.8 67 39.7 46 61.3 55.3 55.7 50 43.8 54.5 59 57.4 54 47 59.5 52.8 56 45.2 57 61.4 44.8 51.5 53 42 54 : MODEL littwt FIT [PRINT=accumulated; FPROBABILITY=yes] litter*mother RKEEP DF=rdf PREDICT [PREDICTIONS=mean; VCOVARIANCE=var] mother SEDLSI [DF=rdf; TITLE='Means and least significant intervals';\ YTITLE='Weight'] mean; VCOVARIANCE=var