  Many data mining and statistical machine learning algorithms have been
developed to select a subset of covariates to associate with a response
variable. Spurious discoveries can easily arise in high-dimensional data
analysis due to enormous possibilities of such selections. How can we know
statistically our discoveries better than those by chance? In this paper, we
define a measure of goodness of spurious fit, which shows how good a response
variable can be fitted by an optimally selected subset of covariates under the
null model, and propose a simple and effective LAMM algorithm to compute it. It
coincides with the maximum spurious correlation for linear models and can be
regarded as a generalized maximum spurious correlation. We derive the
asymptotic distribution of such goodness of spurious fit for generalized linear
models and $L_1$ regression. Such an asymptotic distribution depends on the
sample size, ambient dimension, the number of variables used in the fit, and
the covariance information. It can be consistently estimated by multiplier
bootstrapping and used as a benchmark to guard against spurious discoveries. It
can also be applied to model selection, which considers only candidate models
with goodness of fits better than those by spurious fits. The theory and method
are convincingly illustrated by simulated examples and an application to the
binary outcomes from German Neuroblastoma Trials.
