  Opinion dynamics models have been developed to study and predict the
evolution of public opinion. Intensive research has been carried out on these
models, especially exploring the different rules and topologies, which can be
considered two degrees of freedom of these models. In this paper we introduce
what can be considered a third degree of freedom.
  Since it is not possible to directly access someone's opinions without
measuring them, we always need to choose a way to transform real world opinions
(e.g. being anti-Trump) into numbers. However, the properties of this
transformation are usually not discussed in opinion dynamics literature. For
example, it would be fundamental to know if this transformation of opinions
into numbers should be unique, or if several are possible; and in the latter
case, how the choice of the scale would affect the model dynamics.
  In this article we explore this question by using the knowledge developed in
psychometrics. This field has been studying how to transform psychological
constructs (such as opinions) into numbers for more than 100 years.
  We start by introducing this phenomenon by looking at a simple example in
opinion dynamics. Then we provide the necessary mathematical background and
analyze three opinion dynamics models introduced by Hegselmann and Krause.
Finally, we test the models using agent-based simulations both in the case of
perfect scales (infinite precision) and in the case of real world scales.
  Both in the theoretical analysis and in the simulations, we show how the
choice of the scale (even in the case of perfect accuracy and precision) can
strongly change the model's dynamics. Indeed, by choosing a different scale it
is possible to (1) find different numbers of final opinion clusters, (2) change
the mean value of the final opinion distribution up to a change of $\pm 100 \%$
and (3) even transform one model into another.
