Respondents of contingent valuation surveys may place a null value on the public good, for reasons that differ from a genuine indifference to the good, but that can be interpreted as a "protest": either against the interview, or the public management, or both. A good survey design can effectively reduce them, but protest votes can hardly be completely removed from the dataset, and, if there is sample selection bias, they lead to biased estimates for the wtp measure. We propose a survey design, and a sample selection model, that allows taking into account, and correcting, the possible bias due to protest votes. Since the asymptotic standard errors estimated by means of the inverse of the information matrix containing the sample selection parameter are not reliable, we use an alternative procedure based on the likelihood profile. It will be seen that sample selection models may present estimation problems because of the flatness of the likelihood function: in some cases confidence intervals around the sample selection coefficient are too wide to give evidence of presence or absence of sample selection bias. We maintain that even in these circumstances the sample selection model with the protest votes should be preferred to the model without protest votes, since it takes into account the uncertainty about the estimates of the willingness to pay for the public good.