Forms predictions from an unbalanced analysis of variance, performed by
AUNBALANCED (R.W. Payne).
|What to print (
||Model to use to calculate the predictions; default * i.e. full model fitted by
||Limit on number of factors or variates in each term specified by
||Factor combinations for which to form predicted means (
||Type of adjustment to be made when predicting means (
||Weights classified by some or all of the factors in the model|
||Saves predictions; default
||Saves standard errors of predictions; default
||Saves matrices of standard errors of differences between predictions; default
||Saves effective standard errors; default
||Saves least significant differences between predictions; default
||Significance level (%) for least significant differences; default 5|
||Saves variance-covariance matrices of predictions; default
||Save structure (from
||Variates and/or factors to classify table of predictions|
||To specify values of variates, levels of factors|
AUPREDICT can produce predicted means following an analysis of variance of an unbalanced design by
AUNBALANCED. The predictions are calculated using the
PREDICT directive. The first step (A) of the calculation forms a 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. The
WEIGHTS options then define how the averaging is done in Step B. The
WEIGHTS option allows you to specify your own table of weights to use in the averaging. Alternatively, if
WEIGHTS is not set, the weights are formed automatically according to the setting of the
ADJUSTMENT option. The default setting,
ADJUSTMENT 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.
Printed output, which extends the output available from
PREDICT, is controlled by settings of the
||standardization policies used when forming the predictions,|
||predictions and standard errors,|
||standard errors for differences between the predictions,|
||summary of the standard errors for differences between the predictions,|
||least significant differences between the predictions,|
||summary of the least significant differences between the predictions,|
||approximate effective standard errors – these are formed by procedure
||variance and covariances of the predictions.|
The default is to print predictions and a summary of the standard errors of differences. The standard errors (and sed’s) are relevant for the predictions when considered as means of those data that have been analysed, with the means formed according to the averaging policy defined by the options of
PREDICT. The word prediction is used because these are predictions of what the means would have been if the factor levels been replicated differently in the data; see Lane & Nelder (1982) for more details. The
LSDLEVEL option specifies the significance level (%) to use in the calculation of least significant differences (default 5%).
Another extension in
AUPREDICT is that you can produce predictions using a smaller model than the full model that has been fitted by
AUNBALANCED. This can be useful if the full model contains many parameters. A substantial amount of time and computer workspace may then be needed to calculate the predictions and standard errors. Very large models may even exceed the capacity of some PCs.
You might choose to omit a term from the full model when forming a particular table of predictions if the term is orthogonal to all the terms involved in the table. For example, you might omit the term
blocks when forming an
B table of predictions if each combination of levels of the factors
B is replicated the same number of times in every block. The justification is that an orthogonal term cannot affect the size of any of the differences between predictions. Different weighting of the levels of the orthogonal term may affect the overall mean of the predictions, but this is usually unimportant. If you omit the term, it is though you had included it with weightings based on the observed replication of its levels in the data set – and in any well-designed data set these should provide a satisfactory outcome. You might also omit a term if it is nearly orthogonal to the terms involved in the table, and you are happy to ignore its effect on the predictions.
The model is specified by the
MODEL option. The
FACTORIAL option sets a limit on number of factors or variates in each term specified by
MODEL; default 3.
VCOVARIANCE options allow the results of the prediction to be save in appropriate Genstat data structures.
SAVE option allows you to specify save structure from the analysis for which further output is required. If
SAVE is not set, output will be produced for the most recent analysis from
AUNBALANCED; however, none of the Genstat regression directives (
DROP and so on) must then have been used in the interim.
The predictions are produced using the
Lane, P.W. & Nelder, J.A. (1982). Analysis of covariance and standardization as instances of prediction. Biometrics, 38, 613-621.
Commands for: Analysis of variance.
CAPTION 'AUPREDICT example',\ 'Data from Genstat 5 Release 1 Reference Manual, page 340.';\ STYLE=meta,plain FACTOR [NVALUES=36; LEVELS=3; VALUES=12(1...3)] Block FACTOR [NVALUES=36; LABELS=!t(baresoil,emerald,emergo)] Leachate & [LABELS=!t('1','1/4','1/16','1/64')] Dilution VARIATE [NVALUES=36] Nhatch,Nnohatch READ Leachate,Dilution,Nhatch,Nnohatch 1 2 109 318 3 4 54 350 3 1 * 415 2 2 783 212 3 3 652 1375 2 4 490 816 1 3 95 1219 2 1 1012 66 1 4 166 943 3 2 1059 313 1 1 257 1006 2 3 1058 234 2 4 507 1119 1 2 194 840 1 3 175 1707 1 1 326 609 3 4 142 980 2 3 286 230 3 2 546 313 2 2 * 301 2 1 2471 112 3 3 76 489 1 4 208 503 3 1 * 325 1 1 322 913 1 2 255 2246 3 2 1774 1446 2 2 999 193 2 4 388 1836 3 4 221 1800 1 3 220 1902 2 1 2821 187 3 1 1486 463 3 3 717 1473 1 4 143 941 2 3 968 550 : CALCULATE Logit%h = LOG(Nhatch/Nnohatch) BLOCKSTRUCTURE Block TREATMENTSTRUCTURE Leachate*Dilution AUNBALANCED [PRINT=aovtable] Logit%h AUPREDICT Leachate & Dilution & Leachate,Dilution