Nonparametric Description of Relationship

A straight line does not describe the relationship between the weights and lengths of the slugs. Before considering other types of equation to model this relationship, we present a graphical description that is based on smoothing the data. This approach does not rely on any assumptions about the shape of the relationship.

Lowess Line

A lowess line is a commonly used way to draw a smooth line on a scatterplot. In the context of the slug data, a 'smoothed' slug weight is found for each slug based on several slugs with similar lengths; the lowess line joins these smoothed values.

(Page 4.2.6 of the introductory CAST e-book gives more information about how lowess lines are drawn.)

The diagram below shows a scatterplot of slug weight vs length and a scatterplot of the least squares residuals.

Click Show Lowess Smooth above to add lowess lines to the two scatterplots. The lowess lines clearly show the nonlinearity of the relationship between slug weight and length and indicate the shape of curve that we should be trying to use.

Problems

Although a lowess line can show the shape of a relationship well, a parametric equation is much easier to use to predict the response (slug weight here) from the value of the explanatory variable (slug length). Also, the form of the equation can help us to understand the relationship better.

We therefore look further for an equation that describes the relationship better.