What to print (delta, lsi, fittedsed, discrepancy, maxdiscrepancy, %discrepancy); default delta, lsi, maxd

Selects the method for computing the deltas (leastsquares, max, maxpse); default leas

What to plot (sed, lsi); default sed, lsi

Which pairwise contrasts to use in printed output or plots involving the fitted SEDs (specified, all); default spec

Significance level for the least significent intervals; default 0.05.

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 in which to plot the graphs

Title for the graphs; default 'Estimates with LSIs by Treatment'

Title for the y-axis; default 'Estimates'

Parameters

Parameter estimates; if these are not supplied SEDLSI can calculate the parameters {δ_i} but not the LSIs

Matrix containing standard errors of (pairwise) differences between estimates

Matrix containing variances and covariances of estimates

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

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

Saves the estimated parameters {δ_i}

Saves details of the least significant intervals

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)

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.

prints the parameters {δ_i},

prints the least significant intervals,

prints the matrix [δ_i + δ_j] of fitted SEDs,

prints the difference between [δ_i + δ_j] and [SED_ij],

prints the maximum difference between [δ_i + δ_j] and [SED_ij],

prints the difference as a percentage.

plots the least significant interval for each estimate, and

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.

Parameters: ESTIMATES, SED, VCOVARIANCE, WEIGHTS, LABELS, DELTA, LSI, FITTEDSED.

Method

SEDLSI offers three methods of estimating δ. The first method (leastsquares, the default) provides weighted least squares estimates based on the model

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.

SEDLSI procedure

Options

Parameters

Description

Method

References