Zero Pivot Element Detected During Model Fitting

During parameter estimation in Generalized Linear/Nonlinear Models, you may occasionally encounter a warning message that a "zero pivot element" was detected. This message generally means that during parameter estimation a redundancy was detected in the design matrix. In other words, one of the variables or columns in the design matrix (predictors) is redundant with other columns (predictors).  

This of course will usually happen when categorical predictors are coded using the overparameterized model (see GLM Introductory Overview - Sigma-Restricted and Overparameterized Model). For example, if you have a categorical predictor, "Gender," with two values, "Male" and "Female," in the overparameterized model, the program would create two indicator columns in the design matrix: One for males (1=yes, 0=no), and one for females (1=yes, 0=no). Obviously, these two columns are redundant and, hence, during estimation the program will drop one of them.

This mechanism is equivalent to the application of the generalized inverse in GLM to automatically remove redundant columns from the design matrix.

Note: in some cases this message may occur with continuous predictors as well, or when the overparameterized model is not being used. In this case, it simply means that some predictors are entirely redundant with other predictors and, thus, unique parameter estimates for those predictors cannot be computed (and they are zeroed instead).