  We study a new privacy model where users belong to certain sensitive groups
and we would like to conduct statistical inference on whether there is
significant differences in outcomes between the various groups. In particular
we do not consider the outcome of users to be sensitive, rather only the
membership to certain groups. This is in contrast to previous work that has
considered locally private statistical tests, where outcomes and groups are
jointly privatized, as well as private A/B testing where the groups are
considered public (control and treatment groups) while the outcomes are
privatized. We cover several different settings of hypothesis tests after group
membership has been privatized amongst the samples, including binary and real
valued outcomes. We adopt the generalized $\chi^2$ testing framework used in
other works on hypothesis testing in different privacy models, which allows us
to cover $Z$-tests, $\chi^2$ tests for independence, t-tests, and ANOVA tests
with a single unified approach. When considering two groups, we derive
confidence intervals for the true difference in means and show traditional
approaches for computing confidence intervals miss the true difference when
privacy is introduced. For more than two groups, we consider several mechanisms
for privatizing the group membership, showing that we can improve statistical
power over the traditional tests that ignore the noise due to privacy. We also
consider the application to private A/B testing to determine whether there is a
significant change in the difference in means across sensitive groups between
the control and treatment.
