ABSTRACT
Although the linear models discussed in Chapters 3 through 6 will be adequate for the majority of data analyses that are undertaken, there are situations when it is clearly necessary to use a more complex non-linear model. A common case is when the relationship between a covariate and the response is “sigmoidal.” That is, initially the response changes very slowly as the covariate increases, then a stronger covariate-response relationship occurs, and ˜nally this slows again as some sort of plateau is reached. In the biological sciences, when dose or concentration-response relationships are being assessed, a commonly used model to describe this is the Emax model. This model has four parameters: the response when the dose is zero, E0, the maximum possible effect, Emax, the dose that produces 50% of this maximum, ED50, and a shape parameter, Hill, which determines the steepness of the dose response. The Hill parameter is named after the scientist who ˜rst showed the relevance of this equation to pharmacology, Hill (1910). The Emax model formula is
E Emax X / ED X Hill
5 Hill Hill0 0+ +( ) ( )
where X is the dose, or more generally the value of the covariate. This is a non-linear model since the formula given above implies that the mean response is not a weighted linear combination of the unknown parameters. Figure 7.1 shows the shape of the Emax model, and how it is affected by the Hill parameter. The E0 and Emax parameters simply shift and stretch the curves vertically, respectively, while the ED50 parameter stretches the curves horizontally, in an obvious manner. Although the curves plotted all show an increasing response with increasing dose, a decreasing response is implied by a negative Emax parameter; the parameterization adopted by BugsXLA forces Hill to be positive in both cases. Note that between the doses that give 20% and 80% of the effect, the relationship is very close to being linear on the log-dose scale. For this reason, it is usually much easier to interpret plots of the data using the log-scale for the dose. Also, if it is known that the experiment is being run in this central region, a much simpler linear model, with log-dose as the covariate, should be adequate. In this case, the models discussed in the earlier chapters could be used.
