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Handling Misconduct - Technical Assistance
Statistical Forensics: 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 (The paper was retracted, and in a related Department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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