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FAPPENDIx Cons traction of Confidence Intervals for Mathematical Combinations of Random Variables Many of the variables central to risk and benefit analyses will generally be measured with imprecision. Accordingly, the Committee has recom- mended in this report that the estimates for such variables be reported as 90 percent confidence intervals rather than as single point estimates. (Of course, inadequacies with the data will require most of the ranges to be subjectively determined, on the basis of the analyst's judgment.) In such instances, variables are estimated by mathematically combin- ing (e.g., adding, multiplying) estimates of other variables. For instance, total benefits forgone due to the withdrawal of chlorobenzilate are measured by the sum of benefits forgone from the citrus and noncitrus uses. If the mathematical manipulations involve two or more variables measured as intervals, caution must be exercised in forming the confidence interval for the derived estimate. This appendix presents the correct procedure for combining estimates of random variables to derive estimates of and confidence intervals for other random variables.) The following discussion is framed largely in terms of the two-variable case. The randomly distributed variables are denoted by x and y. Further, their expected values and variances are denoted by E(x) and E(y) and by V(x) and V(y), respectively. The covariance between x end y is denoted by C(x, y). 283
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284 SUMS AND DIFFERENCES OF RANDOM VARIABLES Appendix F If the variables x end y are to be combined to form a new variable z = x + y, then E(z) = E(x + y) = E(x) + Em. Thus, the best estimate of z is simply the sum (or difference) of the best estimates of x andy. The variance of z is V(z) = V(x) + V(v] + 2C(x. Y). Clearly, if the ~ ~ ~ , .' ~ ,~ variables are independently distributed, the variance of their sum or difference is simply the sum of their variances: V(x) + V(y). These results easily generalize to the case involving three or more random variables (see Mood et al. 1974~. PRODUCT OF RANDOM VARIABLES Suppose that x and y are to be combined to form a product, z = xy. In this case, E(z) = E(xy) = E(x)E(y) + C(x, y). Of course, if x and y are uncorrelated, then E(z) = E(x)E(y). The variance of the product of two random variables assumes a rather complex form when x and y are correlated. In general, the data available to oPP would not permit the use of this formula, so it is not shown here. (The interested reader may refer to Mood et al. 1974.) If x and y are independently distributed, the variance of the product is relatively straightforward: V(xy) = E(x)2 Vim + EQ)2 V(x) + V(x) Vim. These results also generalize to cases involving three or more variables (Goodman 1960~. QUOTIENT OF TWO RANDOM VARIABLES In general, there are no simple exact formulas for the mean and variance of the quotient of two random variables, although there are some approximate formulas (Mood et al. 1974~. The formulas used by the Committee are E ~ x ~ E(x) _ C(x, y) + E(x jV(y) y E(y) E~y'2 E(y) and V ax ~ E(x) )2 ~ V(X) + V(y) _ 2C(x, y) By ~ E(y) J [(x,2 E(y'2 E(x)E`y'
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Appendix F Clearly, both of these formulas simplify somewhat when x and y are uncorrelated. 285 APPLYING THE FORMULAS In applying the formulas described above, the Committee found it necessary to adopt the following assumptions. 1. Both x end y have normal distributions. 2. The interval estimates for x and y represent 90 percent confidence intervals. 3. The midpoints of the ranges estimate the expected values for the variable. 4. The product and quotient of x and y create distributions that can be reasonably approximated by the normal distribution. 5. x end y are uncorrelated (unless stated otherwise). The application of these formulas can be illustrated with an example from Chapter 7. The benefits of chlorobenzilate to the Florida IPM program were estimated in Chapter 7 to range from $0 to $3 million/year. The non-~PM benefits to Florida citrus growers were estimated to fall between $0.6 and $6.6 million annually. What is the appropriate interval for the sum of these two benefits? In accordance with the above-mentioned assumptions, we can restate, say, the non-~PM benefits as equalling $3.6 million ~ + $3.0 million). The upper limit for the 90 percent confidence interval is presumed to be $6.6 million in this instance. Thus, $6.6 million = $3.6 million + 1.64sx,~where so represents the estimated standard deviation around the estimated mean value for the non-~PM benefits. This equation clearly implies that the variance around the estimated mean is she = $3.33 x 10~2. Similar reasoning applied to the IPM benefit estimates yields an estimated variance of $8.31 x 10~. The square root of the sum of these variances provides a correct estimate of the standard deviation around the sum of the mean values for the two variables, namely so+ = $2.04 million. Thus, the sum of the midpoints of the two benefit measures yields an estimate of the aggregate benefits in Florida equal to $5.1 million (+~1.64~$2.04 million] = $3.35 million). Alternatively, the aggregate Florida benefits are estimated to range from $1.75 to $8.45 million. It is interesting to note that this estimated range is quite different from the one obtained from simple additions of the lower and upper limits for the individual benefit estimates. This "naive" approach implies a much larger range: $0.6 to $9.6 million.
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286 NOTE 1. This discussion draws heavily on Mood et al. (1974). REFERENCES Appendix F Goodman, L.A. (1960) On the exact variance of products. Journal of the American Statistical Association 55:70~713. Mood, A.M., F.A. Graybill, and D.C. Does (1974) Introduction to the Theory of Statistics. New York: McGraw-Hill.