Appendix A. Problems with estimating and using correlations in vital rates.
When developing a stochastic simulation model to make predictions about future population growth and extinction risk, we need to use the full set of estimated correlations to repeatedly pick sets of random, but correlated, vital rate values. Doing so involves the use of the full correlation matrix (or equivalently, the covariance matrix) for the vital rates. The basic procedure, as outlined in several sources (Burgman et al. 1993, Gross et al. 1998, Todd and Ng 2001, Morris and Doak 2002) is to generate sets of correlated vital rates by initially choosing uncorrelated, normally-distributed values and then multiplying these values by a matrix of values derived from an eigenvalue /eigenvector decomposition of the estimated correlation matrix, thereby generating correlated normal variables. These correlated, normally-distributed variables can then be used to create sets of correlated values with the appropriate distributions, means, and variances to match each vital rate. This can be done in several ways, one of which is to pick values for each vital rate that have the same cumulative distribution function values as do the corresponding, correlated, normal values. This procedure only preserves the rank correlations between non-normal variables, rather than the Pearson product-moment correlations, but since rank correlations are more appropriate for variables such as survival rates that can be highly non-normal, this is not too worrisome.
Unfortunately, there is a hidden problem in using matrices of estimated correlations to perform these simulations. Any real correlation matrix must be "positive semi-definite," with only positive or zero eigenvalues, and this feature is necessary in order for the decomposition mentioned above to work. While any legitimate correlation matrix is positive semi-definite, limited data can result in an estimated correlation matrix that is not. This can happen in two ways. First, if all vital rates were not estimated in all years of a study, the missing data can result in an estimated correlation matrix that includes impossible sets of pair-wise correlations, as we mention in the main text. Second, if the number of years of data is less than the number of vital rates +1, many of the eigenvalues for the correlation matrix should be zero, but rounding errors during estimation may give them small negative or imaginary values. This is a smaller problem than the first one, but must still be corrected in order to use an estimated correlation structure.
It is easy to test whether a correlation matrix meets the positive semi-definite criteria by checking for negative eigenvalues. If some eigenvalues are negative but nearly zero, setting those eigenvalues equal to zero will make the matrix positive semi-definite with minimal changes to the correlation matrix; the steps involved in making these corrections are described in detail in Morris and Doak (2002), Chapter 8. This approach will usually resolve any problems if every vital rate was observed in all years of the study. Another approach is to make an assumption about the form of the joint probability distribution for the full set of vital rates, and estimate the parameters of this distribution simultaneously. In this case, the resulting correlation matrix is guaranteed to be positive semi-definite. For variables that lie between 0 and 1, such joint distributions exist, but have unsatisfying restrictions on the possible variance-covariance matrices (Todd and Ng. 2001). An alternative option is to transform the variables to resemble normal random variable, and fit a multivariate normal distribution to the transformed variables. One of us (K. Gross) has written an R package (mvnmle; http://cran.r-project.org) that automates this fitting procedure.
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