  Increasing nonresponse rates and the cost of data collection are two pressing
problems encountered in traditional probability surveys. The proliferation of
inexpensive data from web surveys stimulates interest in statistical techniques
for valid inferences from web samples. We consider estimation of population and
domain means in the two-sample setup, where the web sample contains variables
of interest and covariates that are shared with an auxiliary probability survey
sample. First, we propose an estimator of population mean, based on the
estimated propensity of response to a web survey. This makes inferences from
web samples that are similar to well-established techniques used for
observational studies and missing data problems. Second, we propose an
'implicit' logistic regression for estimating parameters of the web response
model in the two-sample setup. Implicit logistic regression uses selection
probabilities, nominally defined for web sample units, and the size of the
hypothetic population of responders to a web survey. A simulation study
confirms the validity of implicit logistic regression and its higher efficiency
comparing to alternative estimators of web response propensity.
