A majority of the literature on the net seems to support that non-anonymous and anonymous-surveys give very similar results.
The paper "A comparison of confidential versus anonymous survey procedures: Effects on reporting of drug use and related attitudes and beliefs in a national study of students." studies anonymous vs. confidential surveys in a specific case (drug usage reported in 8th and 10th graders), and concluded that there was a very modest or possibly no effect on the data.
From the abstract:
This study presents a comparison of
reporting of drug use and related
attitudes and beliefs by national
samples of 8th and 10th grade students
under two different modes of
administration conditions:
confidential and anonymous. Data come
from the Monitoring the Future project
for the year 1998. The results show
that there were clearly no differences
between the conditions in 10th
graders' reports of drug use and
related attitudes and beliefs. With
8th graders, the results show, at
most, only a very modest mode of
administration effect and quite
possibly no effect at all.
The same conclusion is drawn in more countries in the paper "Adolescent Substance Abuse in Mexico, Puerto Rico and the United States: Effect of Anonymous versus Confidential Survey Formats".
Similarly another study "The Effect of Anonymous Vs. Nonanonymous Rating Conditions on Patient Satisfaction and Motivation Ratings in a Population of Substance Abuse Patients" concluded that
Anonymity had either no effect on
ratings or accounted for <1% of the
variance.
In a study about collecting data regarding bullying, "The Efficacy of Non-Anonymous Measures of Bullying" concluded that
The findings supported the hypotheses
that the respondents did not differ in
their report of the incidence of
either bullying or victimization,
regardless of whether they were
required to identify themselves by
writing down their names on the
questionnaire forms.
Another study, "Differences between ‘talking about' and ‘admitting' sensitive behaviour in anonymous and non-anonymous web-based interviews" which (among other things) compared difference between two questionnaire modes- anonymous web-based forms or dektop-based video interviews concluded that:
Nevertheless, the expected differences
between the interview modes were not
observed.
On the other hand:
A study called "High risk behaviour and fertility desires among heterosexual HIV-positive patients with a serodiscordant partner – two challenging issues"(PDF) says that
Non-anonymous data collection on
condom use may underestimate high risk
behaviour
However, there are several caveats to these papers, some issues which may need exploring:
- The surveys sometimes seems to be
carried out through a trusted
authority.
- The surveys seem to
target some specific portion of the
population. The validity for general
polls like opinion polls etc. may be
different.
- Non-anonymous may mean
several things, perhaps each with
it's own effect on results.