tailieunhanh - Statistics for Environmental Engineers Second Edition phần 4

Các phương pháp đồng bằng-lognormal ước tính trung bình của một mẫu kích thước n bình quân của nc thay thế kiểm duyệt giá trị và n - nc kiểm duyệt lognormally phân phối giá trị. Các phương pháp Aitchison (1955, 1969) | The transformation equations to convert these into estimates of the mean and variance of the untransformed y s are n y exp n x 72 Ởy 77y exp 72 - 1 Substituting the parameter estimates f x and 72 gives n y exp exp jlg L 7y2 2 exp - 1 jUg L 2 ờy Hg L The Delta-Lognormal Distribution The delta-lognormal method estimates the mean of a sample of size n as a weighted average of nc replaced censored values and n - nc uncensored lognormally distributed values. The Aitchison method 1955 1969 assumes that all censored values are replaced by zeros D 0 and the noncensored values have a lognormal distribution. Another approach is to replace censored values by the detection limit D MDL or by some value between zero and the MDL . EPA 1989 Owen and DeRouen 1980 . The estimated mean is a weighted average of the mean of nc values that are assigned value D and the mean of the n - nc fully measured values that are assumed to have a lognormal distribution with mean _ _ _2 nx and variance 7 . ỈỊy Dn 1 - nj exp nx - 72 where 17 x and ờx are the estimated mean and variance of the log-transformed noncensored values. This method gives results that agree well with Cohen s method but it is not consistently better than Cohen s method. One reason is that the user is required to assume that all censored values are located at a single value which may be zero the limit of detection or something in between. Comments The problem of censored data starts when an analyst decides not to report a numerical value and instead reports not detected. It would be better to have numbers even if the measurement error is large relative to the value itself as long as a statement of the measurement s precision is provided. Even if this were practiced universally in the future there remain many important data sets that have already been censored and must be analyzed. Simply replacing and deleting censored values gives biased estimates of both the .

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