# Statistics for Environmental Engineers

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The true difference in survival proportions (p) that is to be detected with a given degree of confidence must be specified. That difference (8 = pe -pc) should be an amount that is deemed scientifically or environmentally important. The subscript e indicates the exposed group and c indicates the control group.

The variance of a binomial response is Var(p) = p(1 —p)/n. In the experimental design problem, the variances of the two groups are not equal. For example, using n = 20, pc = 0.95 and pe = 0.8, gives:

Var( pe) = pe( 1 — pe)/n = 0.8( 1 — 0.8)/20 = 0.008

and

Var( pc) = pc( 1 — pc)/n = 0.95( 1 — 0.95)/20 = 0.0024

As the difference increases, the variances become more unequal (for p = 0.99, Var( p) = 0.0005). This distortion must be expected in the bioassay problem because the survival proportion in the control group should approach 1.00. If it does not, the bioassay is probably invalid on biological grounds.

The transformation x = arcsin Vp will “stretch” the scale near p = 1.00 and make the variances more nearly equal (Mowery et al., 1985). In the following equations, x is the transformed survival proportion and the difference to be detected is:

8 = xc xe = arcsin л/р — arcsin J~pe

For a binomial process, 8 is approximately normally distributed. The difference of the two proportions is also normally distributed. When x is measured in radians, Var(x) = 1/4n. Thus, Var(8) = Var(x1 — x2) = 1/4n + 1/4n = 1/2n. These results are used below.

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