Statistical models are often used as the defining basis of measurement in social science. However, important issues such as uncertainty, model criticism and the impact of model assumptions are often neglected. These issues are explored in the context of three leading cases of model-based measurement. The relative index of inequality (RII) is used to measure socioeconomic inequality in health outcomes. A new definition of the RII is introduced, in place of the standard formulation based on linear regression. Non-linear outcome rates are accommodated, and estimation using cubic splines fitted by maximum penalized likelihood is developed. The modeling approach taken naturally handles standardizing variables such as age. Bootstrap estimation of variability is pursued. Simulations and a real data example demonstrate a reduction in bias, at the cost of some increase in variance, when incidence rates are non-linear. Social mobility can be measured using an index of mobility, which summarizes a mobility table in a single number. A certain index, based on the supposition of an underlying continuous-time Markov process, satisfies criteria set out in the literature. The 'embedding problem' is explored to assess when it is valid to apply this index. A sufficient condition for the distribution of a likelihood ratio test statistic is derived, and used to develop confidence intervals. In an example using French data, the index is used to caution against the conclusions of the popular UNIDIFF model. Electoral transitions between parties can be measured via a transition table of votes. The aggregated compound multinomial model for such a table appears rigid. Flexibility is introduced in two ways: the log-linear dependence on covariates is relaxed to one involving thin plate splines, and a new model with a general covariate structure is introduced. However, little improvement is offered in the example of describing the dependence of the 2005 British general election results on the 2001 results. This is illustrated using novel plots of residuals.