Advanced Statistics 1 - Non-linear modelling
The living world is not always linear. More precisely, relationships between changes in an external driver and the reaction of a biological system are most often not linear. Some prominent examples are the Michaelis-Menten kinetics, the functional response by Holling, the hump-shaped optimal foraging curve or the S-shaped logistic growth of isolated populations. Also, other relationships that we treat as linear after a data transformation are non-linear — for instance, the power-law relationships of metabolic rates with body mass or the exponential relationships with temperature. Especially the last is used to describe biological dependencies from the cell to whole ecosystems. Statistics, however, were classically bound to linear relationships that can be solved “analytically" (that means with a brain, a pen and a paper). Non-linear relationships, however, can not be solved analytically any more, but need an extension to the human brain, the computer, to solve them iteratively.
Another feature in classic linear statistics is that data has to be normally distributed to confirm the underlying assumptions of the least squares fit. To correct for normality, the data is most often transformed to fit these assumptions (e.g. the log-transformation) but this goes hand in hand with a change of the underlying model (e.g. if you apply the log-transformation to both, the x-axis and the y-axis, you investigate a power law, not a linear relationship any more). This transformation may work to a subset of data we are investigating (a power law can describe, e.g. metabolism that depends on body mass, and the data is nearly lognormal distributed), but not for all. By using Maximum Likelihood methods, we can overcome these.
Didactic aims/competencies gained
Here, we will discover how to implement a Maximum Likelihood function into R and how to choose the most appropriate distribution that describes our data. Subsequently, we will work us through different biological problems that require more than a linear statistic starting with the power-law distributions (e.g. abundance-body mass relationships), hump-shaped curves (e.g. feeding-body-mass relations), the logistic growth function, infection essays, and different functional response models.
Prior knowledge needed
Basic knowledge in the statistical software R is required, knowledge of linear statistics and theory is of advantage but not required.