Predict Directive

Underlying Principles

Our approach to prediction is a generalisation of that of Lane and Nelder (1982) who consider fixed effects models. They form fitted values for all combinations of the explanatory variables in the model, then take marginal means across the explanatory variables not relevent to the current prediction. Our case is more general in that we also consider the case of associated factors and options for random effects that appear in our (mixed) models. A full description can be found in Gilmour et al. (2004) and Welham et al. (2004).

Terms in the model may be fitted as fixed or random, and are formed from explanatory variables which are either factors or covariates. For this exposition, we define a fixed factor as an explanatory variable which is a factor and appears in the model in terms that are fixed (it may also appear in random terms), a random factor as an explanatory variable which is a factor and appears in the model only in terms that are fitted as random effects. Covariates generally appear in fixed terms but may appear in random terms as well (random regression). In special cases they may appear only in random terms.

Random factor terms may contribute to predictions in several ways. They may be evaluated at values specified by the user, they may be averaged over, or they may be ignored (omitting all model terms that involve the factor from the prediction). Averaging over the set of random effects gives a prediction specific to the random effects observed. We call this a `conditional' prediction. Omitting the term from the model produces a prediction at the population average (zero), that is, substituting the assumed population mean for an unknown random effect. We call this a `marginal' prediction. Note that in any prediction, some terms may be evaluated as conditional and others at marginal values, depending on the aim of prediction.

For fixed factors there is no pre-defined population average, so there is no natural interpretation for a prediction derived by omitting a fixed term from the fitted values. Therefore any prediction will be either for specific levels of the fixed factor, or averaging (in some way) over the levels of the fixed factor. The prediction will therefore involve all fixed model terms.

Covariates must be evaluated at specified values. If interest lies in the relationship of the response variable to the covariate, predict a suitable grid of covariate values to reveal the relationship. will be across a suitable grid of covariate values to reveal the relationship. Otherwise, predict at an average or typical value of the covariate. Omission of a covariate from the predictive model is equivalent to predicting at a zero covariate value, which is often not appropriate (unless the covariate is centred).

Before considering some examples in detail, it is useful to consider the conceptual steps involved in the prediction process. Given the explanatory variables used to define the linear (mixed) model, the four main steps are
  • a: Choose the explanatory variable(s) and their respective value(s)/level(s) for which predictions are required; the variables involved will be referred to as the classify set and together define the multiway table to be predicted. Include only one from any set of associated factors in the classify set.
  • b: Note which of the remaining variables will be averaged over, the averaging set, and which will be ignored, the ignored set. The averaging set will include all remaining variables involved in the fixed model but not in the classify set. Ignored variables may be explicitly added to the averaging set. The combination of the classify set with these averaging variables defines a multiway hyper-table. All associated factors appear in this hypertable regardless of whether they are fitted as fixed or random. Note that variables evaluated at only one value, for example, a covariate at its mean value, can be formally introduced as part of the classify or averaging set.
  • c: Determine which terms from the linear mixed model are to be used in forming predictions for each cell in the multiway hyper-table in order to give appropriate conditional or marginal prediction. That is, you may choose to ignore some random terms in addition to those ignored because they involve variables in the ignored set.
  • d: Choose the weights to be used when averaging cells in the hyper-table to produce the multiway table to be reported. Operationally, ASReml does the averaging in the prediction design matrix rather than actually predicting the cells of the hypertable and then averaging them.

    The main difference in this prediction process compared to that described by Lane and Nelder (1982) is the choice of whether to include or exclude model terms when forming predictions. In linear models, since all terms are fixed, terms not in the classify set must be in the averaging set, and all terms must contribute to the predictions.
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