In this paper we formulate a discrete choice model that incorporates thresholds in the perception of changes in attribute values. The model considers multiple options and allows for changes in several attributes. We postulate that if thresholds exist they could be random, differ between individuals, and even be a function of socio-economic characteristics and choice conditions. Our formulation allows estimation of the parameters of the threshold probability distribution starting from information about choices. The model is applied to synthetic data and also to real data from a stated preference survey. We found that where perception thresholds exist in the population, the use of models without them leads to errors in estimation and prediction. Clearly, the effect is more relevant when the typical size of change in the attribute value is comparable with the threshold, and when the contribution of this attribute in the utility function is substantial. Finally, we discuss the implications of the threshold model for estimation of the benefits of transport investments, and show that where thresholds exist, models that do not represent them can overestimate benefits substantially.