  The paper considers variable selection in linear regression models where the
number of covariates is possibly much larger than the number of observations.
High dimensionality of the data brings in many complications, such as (possibly
spurious) high correlations between the variables, which result in marginal
correlation being unreliable as a measure of association between the variables
and the response. We propose a new way of measuring the contribution of each
variable to the response which takes into account high correlations between the
variables in a data-driven way. The proposed tilting procedure provides an
adaptive choice between the use of marginal correlation and tilted correlation
for each variable, where the choice is made depending on the values of the hard
thresholded sample correlation of the design matrix. We study the conditions
under which this measure can successfully discriminate between the relevant and
the irrelevant variables and thus be used as a tool for variable selection.
Finally, an iterative variable screening algorithm is constructed to exploit
the theoretical properties of tilted correlation, and its good practical
performance is demonstrated in a comparative simulation study.
