Nonlinearities

Non-Linear Relationships

In this section, the speaker discusses non-linear relationships and how they can be dealt with in the context of general linear models. The speaker provides examples of common non-linear relationships encountered in nature and explains how they can be linearized through transformations or other tricks.

Linear vs. Non-Linear Relationships

• General linear models assume that predictors influence the response in a linear way. However, relationships in nature are often non-linear.

Linearizing Non-Linear Relationships

• Fundamentally non-linear relationships from nature can be linearized through some form of transformation or other trick to model them in our general linear model.

Common Non-Linear Relationships

• A table is presented showing some common non-linear relationships encountered in nature, including their descriptions, differential forms, integrated forms, and examples.

Exponential or Logarithmic Relationship

• An example of a fundamentally non-linear relationship like an exponential or logarithmic relationship is discussed where the rate of change of y with respect to x is not constant but instead proportional to y. This relationship can be linearized by taking the log transformation of our response variable so that we're modeling the percent rate of change of y by x in some linear way.

Power Law Relationship

• Another fundamentally non-linear relationship that can be linearized using a transformation is a situation where the percent rate of change of y is related to the percent rate of change of x which corresponds to a power law relationship shown here which we can linearize by taking both the log of our response variable as well as the log of our predictor variable.

Non-parametric Analyses

• Sometimes biological patterns take on non-linear relationships that don't follow any particular pattern. In this case, non-parametric analyses can be performed using some form of localized regression either a cubic spline or what's called a low s regression which is just a localized regression that allows our regression to not follow a linear relationship but to follow this smooth curve that's driven by the data points in a particular region of the overall scope of observations.

Understanding Non-Linear Relationships

In this section, the speaker discusses how to interpret non-linear relationships in data and how to use quadratic or polynomial terms to describe them.

Linear vs. Non-Linear Relationships

• Most of the data shows a linear relationship between variables.

• Wiggles in the relationship can occur when there are very few data points for certain predictor variable regions.

• A spline or low s approach is helpful for visualizing non-linear relationships but doesn't help with making predictions.

Quadratic and Polynomial Terms

• Quadratic or polynomial terms can be used to describe wiggly relationships using a hum-shaped equation.

• Quadratic relationships have their own biological meaning and can be used in selection analysis to quantify natural selection.

• The quadratic term represents a non-linear relationship between x and y where the effect of x on y is not constant; it depends on the value of x.

Interpreting Quadratic Effects

• We can calculate predicted values for a given value of x1 by plugging in parameters from our linear model.

• The effect of x on y is not constant, meaning that as we get farther along the x-axis, this positive slope becomes shallower and shallower.

• We can interpret the quadratic part of our overall linear model as an interaction between x1 and itself, which tells us that the effect of x1 on y is not constant and depends on the value of x1.

General Linear Models

In this section, the speaker explains how to use general linear models to predict woody species richness based on the percent of matrix habitat. They also discuss how to deal with non-linear biological effects in a general linear model.

Creating a New Variable for Predicting Woody Species Richness

• Use the cut command in R to create a new variable that categorizes observations as being either below or above a defined threshold value.

• Include this new factor in the statistical model along with the predictor variable of percent of matrix habitat and an interaction term between them.

Testing for Non-Linear Relationships

• Use quadratic or polynomial terms to represent humped or squiggly relationships between a predictor and response.

• Use break point regressions where the relationship between some predictor and our response changes dramatically below or above some hypothesized threshold.

Interpreting Linear Models

• Even when dealing with non-linear biological effects, interpretation of linear models and their parameters remains the same.