When is Latin Hypercube Sampling (LHS) a good idea? In this paper: http://salserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/pdf-files/emat08.pdf [1] equation 6 shows that if $\operatorname{ cov} \left(f\left(x_1\right),f\left(x_2\right)\right)$ is positive LHS does not reduce random sampling error. Is there a rule of thumb for determining if LHS sampling will actually have less error than random sampling? I do not want to try this out across all points of interest, it would be much more useful to have a general method for qualitatively determining LHS efficiency.
[1] Anna Matala (2008), "Sample Size Requierement for Monte Carlo - simulations using Latin Hypercube Sampling", 60968U (20.5.2008) Mat-2.4108 Independent Research Projects in Applied Mathematics, Helsinki University of Technology       
 A: In practical terms (i.e. excluding very small sample sizes) Latin Hypercube Sampling is always a good idea, it's more a question of 'how good'. 
Why? Well, the report you reference is interesting, but it leaves open the question as to how $\operatorname{ cov} \left(f\left(x_1\right),f\left(x_2\right)\right)$ behaves for finite sample sizes i.e. they only present asymptotic results as the sample size $N\to\infty$. I'm surprised that there is no reference to the work of Art Owen as he has worked extensively on Latin Hypercube Sampling and related sampling methods. If you look at his book chapter then Proposition 10.4 states that the variance of the LHS estimate $\mathrm{Var}(\hat{\mu}_{LHS})$ is at most $\frac{n}{n-1}\mathrm{Var}(\hat{\mu}_{IID})$ where $n$ is the sample size (the proof is in: Owen, 1997, Monte Carlo variance of scrambled net quadrature, SIAM Journal of Numerical Analysis 34:1884-1910). In other words, LHS can never be worse than IID sampling on a sample with one less point. Often, LHS is much better than this. Proposition 10.1 in the book chapter shows that the extent to which LHS is an improvement depends on the degree to which your function $f$ is additive.
