Web surveys are usually characterized by massive non response and undercoverage. A popular way to deal with these problem in the sampling literature is by means of weighting observations in the estimation process. If the population totals of some auxiliary variables are accurately known poststratification or, more in general, calibration methods can be used to reduce both non-response and undercoverage bias (Sarndal and Lundstrom, 2oo5). They are based on the correction of a set of ‘basic weights’, i.e. the inverse of the inclusion probabilities when sampling is random. Alternatively an estimated propensity score is sometimes used to create a weight adjustment to account for nonresponse in surveys where some variables are known for both respondents and nonrespondents (Little and Rubin, 1987).

When correction of basic weights is substantial the resulting weights may be characterized by high variability and a skewed distribution with extreme values. This may have a bad impact of the mean square error of estimators. In fact the reduction in the bias may be overcome by an increase in the variance. This is true especially when non response is noninformative, that is the probability of being a respondent is independent of the study variables.

Compromise solutions may be obtained by means of weight trimming or smoothing (Elliot and Little, 2000). In the case of calibration we may also modify the distance function for the same purposes.

In this paper we deal with several problems related to weighting, such as variable selection in building propensity scores and calibration, comparison of different trimming threshold and smoothing methods. We are also interested in evaluating the impact of ‘basic’ or ‘initial’ weights on final weights. This problem is relevant for those web surveys in which it is not possible to obtain basic weights as inverse inclusion probabilities. We illustrate these problems using data from a web survey conducted on a population of university students. More detail we consider a survey on a stratified random sample of students from the University of Bergamo, contacted via e-mail and asked to complete a customer satisfaction questionnaire on the University library services on the internet.