Comparing models
As in other situations with a hierarchy of possible models, an analysis of variance table is used to find the simplest model that is consistent with the data. This is based on the analysis of variance table for the model in which the factor is treated as categorical, but the sum of squares explained by the factor is split into sums of squares explained by linear and quadratic models, and by 'lack of fit' of the quadratic model.
As in other analysis of variance tables, mean sums of squares are calculated for each explained sum of squares and these are divided by the mean residual sum of squares to give F ratios. A p-value tests whether each F ratio is larger than would be expected by chance.
The method will be clearer with an example.
Antibiotic effectiveness
The anova table below initially treats the concentration of antibiotic as categorical (with 6 levels) so there is a single row for the explained variation (with 5 degrees of freedom).
Click Split ssq for factor to split the explained sum of squares into sums of squares for linear, quadratic and 'lack of fit' rows.
The p-values are interpreted as follows:
Interpretation
We therefore conclude that a quadratic model fits data adequately.