Modelling nonlinear relationships with time

The models considered above assume a linear relationship with time. In many applications the linear model described is sufficient for assessing whether there is a time trend, or whether the trend is varying across treatment groups. However, it is also possible to model non-linear relationships with time; for example, by using polynomial or exponential functions. Here, we will only consider models that can be fitted in PROC MIXED using polynomial functions of time. PROC NLMIXED can also be used for non-linear functions but that is outwith the coverage in this book.

We suggest that a model based on polynomials of time is built up by adding polynomials of increasingly higher order one at a time, both as fixed effects and random coefficients. If a variance component for a random coefficient is negative, the last random coefficient added to the model should be removed and no further random coefficients should be added. However, higher order polynomials can still be considered as fixed effects if appropriate. This model building process is illustrated in the worked example in Section 6.6.2 and readers may find it helpful to refer to this example and the example in Section 6.6.1 before considering the material in the remainder of this section.

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