  Computer experiments with both qualitative and quantitative input variables
occur frequently in many scientific and engineering applications. How to choose
input settings for such experiments is an important issue for accurate
statistical analysis, uncertainty quantification and decision making. Sliced
Latin hypercube designs are the first systematic approach to address this
issue. However, it comes with the increasing cost associated with an increasing
large number of level combinations of the qualitative factors. For the reason
of run size economy, marginally coupled designs were proposed in which the
design for the quantitative factors is a sliced Latin hypercube design with
respect to each qualitative factor. The drawback of such designs is that the
corresponding data may not be able to capture the effects between any two (and
more) qualitative factors and quantitative factors. To balance the run size and
design efficiency, we propose a new type of designs, doubly coupled designs,
where the design points for the quantitative factors form a sliced Latin
hypercube design with respect to the levels of any qualitative factor and with
respect to the level combinations of any two qualitative factors, respectively.
The proposed designs have the better stratification property between the
qualitative and quantitative factors compared with marginally coupled designs.
The existence of the proposed designs is established. Several construction
methods are introduced, and the properties of the resulting designs are also
studied.
