## Basic Analyses

##### Regression Analysis

Authors: G.W. Suter II, P. Shaw-Allen, L.L. Yuan, S.M. Cormier

### Regression Analysis

Linear regression is an approach for quantifying the relationship between a dependent (response) variable and an independent (explanatory) variable. The relationship is often assumed to be a straight line, but may also be curvilinear or nonlinear.

#### How do I run a regression analysis?

A regression tool is available in CADStat. Linear regression tools are also available in most spreadsheets and statistical programs.

#### What do regression results mean?

After running a linear regression, most programs will provide statistics that describe the characteristics of the estimated fit to the data. These statistics include estimated values for the coefficients, the standard errors and p-values for those coefficients (see discussion of interpreting p-values), and a measure of the degree the model accounted for observed variability in the response relative to a constant, null model (R2). Several existing resources provide complete explanations for these different statistics (e.g., the Handbook of Biological Statistics ).

Figure 1. Stream temperature vs. elevation in Oregon. Solid black line is a simple linear regression fit to the data. Blue lines are the 95% confidence intervals on the estimated mean, and green lines are the 95% prediction intervals.

It is often useful to plot your data and superimpose the estimated regression line with confidence or prediction intervals (Figure 1). Confidence intervals provide an estimate of the range of possible values for the estimated mean response for any given values of explanatory variables, while prediction intervals provide an estimate of the range of possible values of the response in individual samples. In general, confidence and predictions intervals are only meaningful in cases in which regression assumptions are satisfied (see Regression Analysis: Details for more information).

#### How do I use regression analysis in causal analysis?

Linear regression is the underlying, basic analysis on which many more advanced techniques are based.  Some examples of these advanced analyses include:

Linear regression is also a key technique for describing stressor-response relationships.

Technical details for linear regression are available here.