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Publication date: 11/10/2021

Singularity Details

The linear regression model is formulated as Y = Xβ + ε. Here X is a matrix whose first column consists of 1s, and whose remaining columns are the values of the non-intercept terms in the model. If the model consists of p terms, including the intercept, then X is an n by p matrix, where n is the number of observations. The parameter estimates, denoted by the vector b, are typically given by the formula:

Equation shown here

However, this formula presumes that XX1 exists, in other words, that the p × p matrix XX is invertible, or equivalently, of full rank. Situations often arise when XX is not invertible because there are linear dependencies among the columns of X.

In such cases, the matrix XX is singular, and the Fit Least Squares report contains the Singularity Details report. This report contains a table of expressions that describe the linear dependencies. The terms involved in these linear dependencies are aliased (confounded).

Figure 3.64 shows reports for the Reactor 8 sample data table. To obtain these reports, fit a model with Percent Reacted as Y. Enter Feed Rate, Catalyst, Stir Rate, Temperature, Concentration, Catalyst*Stir Rate, Catalyst*Concentration, and Feed Rate*Catalyst as model effects.

Figure 3.64 Singularity and Parameter Estimates Report for Model with Linear Dependencies 

Singularity and Parameter Estimates Report for Model with Linear Dependencies

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