Linear terms in other models
Consider an experiment in which a numerical factor has three or more levels that is modelled by an equation of the form
yij = µ + ... + βi + ... + εij
where the numerical factor has level i. The experiment may have blocks and use other factors that are explained by other terms in the model. In any such model, it may be possible to replace the term for the factor with a linear term that imposes smoothness on effect of the factor on the mean response.
yij = µ + ... + β xi + ... + εij
Analysis of variance to assess linearity
The sum of squares explained by any such factor can be split into two components — a sum of squares for the linear effect of the factor and a sum of squares for nonlinearity. The sum of squares for nonlinearity can be used to test whether there is any evidence of nonlinearity in the relationship.
Soybean yield and trace elements
An experiment was conducted to assess how different applications of manganese (Mn) and copper (Cu) affect the yield of soybeans.
In the experiment, a large field was subdivided into 32 plots and two were randomly allocated to each combination of Mn rate and Cu rate — i.e. there were 2 replicates for each of the 16 treatments. Soybeans were planted in rows 1 metre apart and the yield of soybeans (in kg per hectare) was recorded from each plot.
Initially the analysis of variance table ignores the fact that both factors are numerical and the model therefore imposes no smoothness on the effect of Cu and Mn on the yield. Click the checkbox Split Cu rates to split the sum of squares for Cu into a linear component and a nonlinear component. The p-value associated with nonlinearity in the effect of Cu is 0.6555, so there is no evidence of nonlinearity. (The p-value for the linear term is also large here, so can also conclude that there is no evidence that Cu affects the yield at all.)
Now click Split Mn rates to split the sum of squares for Mn into similar components. The sum of squares for nonlinearity in the effect of Mn has a p-value 0.0001, so we would conclude that there is nonlinearity in the effect of Mn on soybean yield.
The diagram below shows these models graphically.
Use the pop-up menus under the diagram to display the least squares fit of models with linear and general factor terms for the two factors.