  We use p-values to identify the threshold level at which a regression
function takes off from its baseline value, a problem motivated by applications
in toxicological and pharmacological dose-response studies and environmental
statistics. We study the problem in two sampling settings: one where multiple
responses can be obtained at a number of different covariate-levels and the
other the standard regression setting involving limited number of response
values at each covariate. Our procedure involves testing the hypothesis that
the regression function is at its baseline at each covariate value and then
computing the potentially approximate p-value of the test. An estimate of the
threshold is obtained by fitting a piecewise constant function with a single
jump discontinuity, otherwise known as a stump, to these observed p-values, as
they behave in markedly different ways on the two sides of the threshold. The
estimate is shown to be consistent and its finite sample properties are studied
through simulations. Our approach is computationally simple and extends to the
estimation of the baseline value of the regression function, heteroscedastic
errors and to time-series. It is illustrated on some real data applications.
