Applied Biomathematics Reconstructing Scattergram Data from Regression Statistics. S. Ferson, H. M. Regan, and D. S. Myers; Applied Biomathematics and NCEAS, University of California-Santa Barbara
Risk analysts frequently must look to the scientific and engineering literature to obtain estimates of relevant distributions. In many cases, the empirical information recorded there has been encoded and summarized into statistical regressions. How should a risk analyst interpret regression statistics to reconstruct the distributions of interest? Some authors have used the following formulation: the dependent variate Y is simulated as aX+b where X is the independent variable and a and b are random slope and intercept values selected as independent normally distributed random numbers with means and standard deviations taken respectively to be the reported regression statistics and their standard errors. This approach is flawed, however, because it does not reconstruct the original scatter of Y variable. In fact, it can grossly underestimate the true scatter of the distribution of the Y variable. The correct procedure to use in this context untangles the underlying basic linear regression model to simulate Y as aX+b+e where a and b are the simple constant (mean) regression statistics for the slope and intercept, and e is an independent normally distributed random variable with mean zero and standard deviation taken to be the standard error s of the regression as a whole (rather than of either regression coefficient). We discuss the consequences for a risk analysis of using the two approaches. Because the value of s may not be explicitly reported in a regression summary, we describe several convenient formulas that permit the computation of s from incomplete descriptions of regressions that often appear in the literature.
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