link to polynomials inserted

1.2-analyses_steps.Rmd

@@ -48,10 +48,13 @@ d) Is it expected that a change of 1 at lower values for x has the same biologic
 
 4. Collinearity: Look at the correlation among the explanatory variables (pairs plot or correlation matrix). If the explanatory variables are correlated, go back to step 2 and add this relationship. Further, read our thoughts about [collinearity](#collinearity).
 
-5. Are interactions and polynomial terms needed in the model? If not already
-done in step 2, think about the relationship between each explanatory variable and the dependent variable. Is it linear or do polynomial terms have to be included in the model? If the relationship cannot be described appropriately by polynomial terms, think of a nonlinear model or a generalized additive model (GAM). May the effect of one explanatory variable depend on the value of
+5. Are [polynomial terms](#polynomials) needed in the model? If not already
+done in step 2, think about the relationship between each explanatory variable and the dependent variable. Is it linear or do polynomial terms have to be included in the model? If the relationship cannot be described appropriately by polynomial terms, think of a nonlinear model or a generalized additive model (GAM).  
+
+6. Interactions: May the effect of one explanatory variable depend on the value of
 another explanatory variable (interaction)?
 
+
 ## Data Structure  {#step5}
 After having taken into account all of the (fixed effect) terms from step 4: are the observations independent or grouped/structured? What random factors are needed in the model? Are the data obviously temporally or spatially correlated? Or, are other correlation structures present, such as phylogenetic relationships? 
 Our strategy is to start with a rather simple model that may not account for all correlation structures that in fact are present in the data. We first, only include those that are known to be important a priory. Only when residual analyses reveals important additional correlation structures, we include them in the model.

2.03-lm.Rmd

@@ -942,7 +942,7 @@ The group means are the same, independent of whether a factor is defined as orde
 Note that it is also possible to define own contrasts if we are interested in specific differences or trends. However, it is not trivial to find meaningful and orthogonal (= uncorrelated) contrasts.
 
 
-## Quadratic and Higher Polynomial Terms
+## Quadratic and Higher Polynomial Terms {#polynomials}
 
 The straight regression line for the biomass of grass species Ap *Alopecurus pratensis* dependent on the distance to the ground water does not fit well (Figure \@ref(fig:fgbiom)). The residuals at low and high values of water tend to be positive and intermediate water levels are associated with negative residuals. This points out a possible violation of the model assumptions.
 The problem is that the relationship between distance to water and biomass of species Ap is not linear. In real life, we often find non-linear relationships, but if the shape of the relationship is quadratic (plus, potentially, a few more polynomials) we can still use ‘linear modeling’ (the term 'linear' refers to the linear function used to describe the relationship between the outcome and the predictor variables: $f(x) = \beta_0 + \beta_1x + \beta_2x^2$ is a linear function compared to, e.g., $f(x) = \beta^x$, which is not a linear function). We simply add the quadratic term of the predictor variable, that is, water in our example, as a further predictor in the linear predictor: