Latin Hypercube Sensitivity analysis (LHS; Blower and Dowlatabadi 1994) is a type of stratified Monte Carlo sampling. It is an extremely efficient sampling design because each value of a parameter is only used once in the analysis. The estimation of uncertainty for each parameter is modeled by treating each parameter as a random variable. Probability distribution functions (pdfs) are defined for each parameter. We used normal, gamma, or beta distributions (see Table 1). For a sample size of N, we broke each of these distributions into N intervals, each of equal probability. We then chose the midpoint of each interval and generated an LHS table as an N × K matrix, where N is the number of simulations and K is the number of sampled input parameters. The matrix is generated as follows: N sampling indices of the first variable are paired randomly with N sampling indices of the second variable, these N pairs are randomly paired with the N values of the third variable, and so on until all K input variable are included. This N × K matrix gives N sets of parameter values, which are used for simulations. The N results are basis for the sensitivity conclusions. In particular, sensitivity to a given parameter is judged by partial correlation between the parameter and the simulation results. This partial rank correlation is based on ranks of the results and of the parameter values within their columns, rather than on the raw values.
An exact formula to calculate an appropriate sample size does not exist (1994), but it has been established empirically that N > 4/3K (McKay et al. 1979). In our case K = 44, and we simulated each of the four scenarios with 150 different parameter combinations (N = 150) or more than three times the suggested minimum. We repeated each parameter combination 20 times because the model is stochastic. Therefore the whole sensitivity analysis is based on 12000 simulations (4 × 150 × 20). See Fig. D1 for more details on the LHS.
As only a few of the parameter distribution functions were normal distributions the output variable is frequently a nonlinear function of the input variables. Therefore, we normalized all input variables and then compared the relative effect of changing a parameter value one standard deviation above or below the mean value on . We used a partial rank correlation coefficient (PRCC) to evaluate statistical relationships. PRCC calculates the statistical relationship between each input parameter and while keeping all other input parameters constant at their expected value (Conover 1980). This procedure enables us to determine the independent effects of each parameter, even if the parameters are correlated. We examined the monotonicity between a specific input variable and by examining scatter plots. The sign of the PRCC indicates the qualitative relationship between the input variable and, and the magnitude of PRCC indicates the importance of the input variable in contributing to the imprecision in predicting the difference in the proportion of size genes, . The relative importance of the input variables can be directly evaluated by comparing the PRCC values. The calculation of PRCC is described in Blower and Dowlatabadi (1994).
|FIG. D1. Schematic Latin Hypercube Sampling technique for a hypothetical two-parameter model, modified from (Blower and Dowlatabadi 1994). Probability density functions (pdfs) of our model were divided into 150 equi-probable intervals. For each simulation a value for each parameter combination is selected from one of these intervals at random, and without replacement.|
Blower, S. M., and H. Dowlatabadi. 1994. Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. International Statistical Review 62:229243.
Conover, W. J. 1980. Practical nonparametric statistics, Second edition. John Wiley and Sons, New York, New York, USA.
McKay, M. D., W. J. Conover, and R. J. Beckman. 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239245.