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Statistical Forensics

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Check Rightmost Digits for Uniform Distribution

Numbers are often recorded beyond the repeatability of the experimental procedure. When counts or measurements are recorded to higher precision than can be repeated in replications of an experiment, the rightmost digits of the recorded numbers have little biological meaning. Consider a count of radioactivity for a biological preparation, for example, 5179. In a recount of the sample, or in a replication of the assay, it is unlikely that the rightmost digits will be the same. Thus, with three repetitions, 5179, 5118 and 5134 could be expected.



The rightmost digits of these three numbers differ. Thus xx 79 differs from xx 18, and, in turn, both differ from xx 34.



In large samples of numbers, such rightmost digits often occur with the same frequency, like lottery digits where each of the digits 0, 1, 2, . . . , 9 has the same expectation. Statistically speaking, rightmost digits are approximately uniformly distributed in many circumstances.



In one ORI case, the respondent's notebook contained fabricated counts as well as un-fabricated counts. For the fabricated counts the radioactive spots on the experimental sheets had not been excised and hence could not have been counted in the scintillation counter. The un-fabricated counts were supported by counter tapes.



Investigators from ORI's Division of Investigative Oversight (DIO) compared rightmost digits of fabricated and un-fabricated counts. The fabricated digits differed significantly from uniform. The un-fabricated digits did not so differ. (The respondent accepted voluntary exclusion from receiving Federal funds for 3 years.)



In another case, one column of a published table of numbers was not supported by notebook data. DIO investigators found that the rightmost digits of the unsupported column differed significantly from uniform. The rightmost digits of the supported columns did not so differ. (The paper was retracted, and in a related Department of Justice settlement, the Government recovered over $1 million from two universities.)



To succeed in fabricating data, the fabricator must make the leftmost digits exhibit the desired biological magnitudes. Rightmost digits, given little thought, may be subject to personal preferences of the moment, and hence not uniform. Even when instructed to "make up" numbers with uniform digits, many subjects appear unable to do so. (See "Data Fabrication: Can people generate Random Digits?" J.E. Mosimann, C.V. Wiseman and R.E. Edelman, Accountability in Research, 4, 31-55, 1995).



In cases of scientific misconduct, un-scientific details, like rightmost digits, are worthy of attention.

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