AbstractTaylor’s power law (TPL) from empirical ecological theory has had many explanations proposed for its widespread observation in data. We show that the class of reduced-rank vector generalized linear models (RR-VGLMs) for coupling two parameters from a statistical distribution linearly together creates hybrid models that satisfy TPL or very similar. These include the RR-negative binomial, RR-inverse Gaussian and RR-generalized Poisson distributions. Some advantages of RR-VGLMs include the handling of covariates and an implementation exists in the form of the VGAM package. The software is demonstrated to show how these models may be fitted conveniently.