Statistics for Environmental Engineers

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Regression on Rankits

It is possible to replace the probabilities with rankits (also called normal order scores or order statistics) and then to use regression to fit a line to the probability plot (Gilliom and Helsel, 1986; Hashimoto and Trussell, 1983; Travis and Land, 1990). This is equivalent to rescaling the graph in terms of standard deviations instead of probabilities.

If the data are normally distributed, or have been transformed to make them normal, the probabili-ties(p) are converted to rankits (normal order scores), Rt = F ‘(pi) where F-1 is the inverse cumulative normal probability distribution and pt is the plotting position of the ith ranked observation. The rankits can be calculated (using the NORMSINV function in EXCEL) or looked up in standard statistical tables (for n < 50). The analysis could also be done using probits, which are obtained by adding five to the rankits to eliminate the negative values.

A straight line yi = b0 + b1 Rt is fitted to the rankits of the noncensored portion of the data. The rankits( R) are treated as the independent variable and the data are the dependent variable. For normally distributed data, the b0 and b1 are estimates of the mean and standard deviation of the noncensored distribution. The intercept b0 estimates the median because rankits are symmetrical about zero and the 50th percentile corresponds to Rt = 0. For the normal distribution, the median equals the mean, so the 50th percentile also estimates the mean. The slope b1 estimates the standard deviation because the rankits are scaled so that one rankit is one standard deviation on the original scale.

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