Introduction

This chapter explains simple linear regression analysis. The different uses of correlation and regression were contrasted in Chapter 15. Correlation examines if two variables are related. Regression describes the functional relationship between a dependent and an independent variable.

Linear regression

Linear regression analysis is often used by earth scientists. For example, the equation for the regression of one variable on another may suggest hypotheses about why the two variables are related. More practically, regression can be used in situations where the dependent variable is difficult, expensive or impossible to measure, but its values can be predicted from another easily measured variable to which it is functionally related. Here is an example.

It can be quite difficult to measure the temperature of an erupting magma. In situ measurements can be made if you can safely get close enough to lower a sheathed thermocouple into the hot lava, but this is a dangerous undertaking. Optical pyrometers can be used to estimate magma temperatures from any distant position with a direct line of sight, but corrections for distance, elevation and air temperatures must be applied.

Fortunately, it has been shown that the SiO2 content of the magma varies inversely with eruption temperature: basaltic magmas tend to erupt at hotter temperatures (~1200–1300 °C) and more silicic ones at cooler temperatures (~700–800 °C). So eruption temperature can be predicted from the SiO2 content of the cooled magma, which can be accurately (and safely) measured in the laboratory after the eruption.