We have a self-selected sample when the units of the frame population are not chosen by the researcher, but these units choose to participate in the survey process by self-decision. We can nd self-selected samples very often in web surveys, but the reasons of self selection are very dierent. In fact, suppose you have a mailing list from a frame population i.e. the students of a university course and suppose you send the questionnaire to all students, our intention is therefore to plan a census, however only a part of the students replies to the questionnaire. Or suppose you post a questionnaire in a website of a rm and ask the hypothetical customers to ll in the questionnaire. In both cases we will have a self selected sample. However the survey backgrounds are very dierent. In the rst case we know the frame population and we can know auxiliary variables for each unit. On the contrary, in the second case we do not know the frame therefore we cannot have auxiliary information we do not know the population size. Moreover, some web surfers that ll in the questionnaire could not be customers and many customers are not web surfers.

It is well known that the self selected sample is not probabilistic, therefore we would not have estimators according to the design-based approach.

Heckman (1979) proposed a method to estimate the values of the study variable for the unobserved units, in order to overcome the problem of sample selection bias. The method is performed in two steps and requires the use of some auxiliary variables. Then it can be employed when the frame population is known. The contribution of this paper is to use the Heckman method in a customer satisfaction survey where the study variable is not quantitative but categorical.