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10 Tabulation of the data and prediction from the model 159For covariate terms (fixed or random) the associated effect represents the coefficientof a linear trend in the data with respect to the covariate values. Theseterms should be evaluated at a given value of the covariate, or averaged over severalgiven values. Omission of a covariate from the predictive model is equivalentto predicting at a zero covariate value, which is often inappropriate.Interaction terms constructed from factors generate an effect for each combinationof the factor levels, and behave like single factor terms in prediction. Interactionsconstructed from covariates fit a linear trend for the product of the covariatevalues and behave like a single covariate term. An interaction of a factor and acovariate fits a linear trend for the covariate for each level of the factor. For bothfixed and random terms, a value for the covariate must be given, but the factormay be evaluated at a given level, averaged over or (for random terms) omitted.Before considering some examples in detail, it is useful to consider the conceptualsteps involved in the prediction process. Given the explanatory variables used todefine the linear (mixed) model, the four main steps are(a) Choose the explanatory variable(s) and their respective value(s) for whichpredictions are required; the variables involved will be referred to as the classifyset and together define the multiway table to be predicted.(b) Determine which variables should be averaged over to form predictions. Thevalues to be averaged over must also be defined for each variable; the variablesinvolved will be referred to as the averaging set. The combination of the classifyset with these averaging variables defines a multiway hyper-table. Note thatvariables evaluated at only one value, for example, a covariate at its mean value,can be formally introduced as part of the classifying or averaging set.(c) Determine which terms from the linear mixed model are to be used in formingpredictions for each cell in the multiway hyper-table in order to give appropriateconditional or marginal prediction.(d) Choose the weights to be used when averaging cells in the hyper-table toproduce the multiway table to be reported.Note that after steps (a) and (b) there may be some explanatory variables in thefitted model that do not classify the hyper-table. These variables occur in termsthat are ignored when forming the predicted values. It was concluded above thatfixed terms could not sensibly be ignored in forming predictions, so that variables