The method of least squares will fit any model to a data set, but is the resulting model "good"?One criterion is that the model should fit the data significantly better than a simpler model with fewer predictors. After all, if the fit is not significantly better, then the model with fewer predictors is almost as good. For linear models, this approach is equivalent to testing if selected regression parameters vanish. This chapter discusses procedures for testing such hypotheses. In interpreting such hypotheses, it is important to recognize that a regression parameter for a given predictor quantifies the expected rate of change of the predict and while holding the other predictors constant. Equivalently, the regression parameter quantifies the dependence between two variables after controlling or regressing out other predictors. These concepts are important for identifying a confounding variable, which is a third variable that influences two variables to produce a correlation between those two variables. This chapter also discusses how detection and attribution of climate change can be framed in a regression model framework.

We explain linear regression models with multiple predictors, including an overview of partial regression coefficients. The related concept of partial correlation is discussed in a separate section. We also contrast the overall model test using the F-ratio statistic and the t tests of partial effects of individual predictors. The adjusted coefficient of determination is presented as a more accurate way of conveying the explanatory power of a regression model. Finally, we characterise the family of general linear models, focusing specifically on analysis of covariance (ANCOVA). We provide examples of ANCOVA models and demonstrate their usefulness when applied to the analysis of biological experiments. The methods described in this chapter are accompanied by a carefully-explained guide to the R code needed for their use, including the effects and ppcor packages.