Nested Latin hypercube designs (LHDs) are proposed for conducting multiple computer experiments with different levels of accuracy. Orthogonality is shown to be an important feature. However, little is known about the construction of nested orthogonal LHDs. In this paper, we present methods to construct nested orthogonal LHDs with two or more layers, making use of orthogonal designs. The constructed designs possess the property that the sum of the elementwise products of any three columns is zero, which is shown to be desirable for factor screening.

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Latin hypercube designs (LHDs), widely used for computer experiments, are a very large class of designs with desirable properties. Recently, a number of methods have been proposed to construct orthogonal LHDs. In this paper, we introduce an approach to constructing 2 r-order orthogonal designs. The methods are simple and easy to implement. Using orthogonal designs, we propose some methods for constructing orthogonal and nearly orthogonal LHDs so that the elementwise square of each column and the elementwise product of any two distinct columns are orthogonal to all columns. Further, the resulting nearly orthogonal LHDs with 2 r+1 +2 runs and 2 r factors have the minimum correlation between any two distinct columns.

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