This dissertation reports on the development of the mathematical and statistical framework that was necessary for the analysis of data from a novel single-cell assay designed to address questions in fundamental biology. Many biological systems function by generating new cells from activated ancestors through cellular division. To investigate such systems, a high throughput experimental protocol was recently developed that marks initial cells so that their cellular offspring, the number of rounds of division from their ancestor, and their phenotype can be determined. The clonal data that result from this technique, however, are characterised by familial associations that impede their analysis using classical quantitative tools, necessitating the development of a new mathematical framework where suitable statistics are formulated that take these complex dependencies into account. The design, development and implementation of that framework, as well as inferences made from its use, are the subject of the present thesis.