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Appendix C
Committee Evaluation of
Statistical Analysis Report
The Statistical Analysis Report (B2M10) was submitted in response to a
contract with the FBI, to analyze the results of the assays on the 1,070 FBI
Repository (FBIR) samples and determine whether the results of the assays could
be related to those obtained on the evidentiary material. If such a relationship
could be identified, a secondary issue was the development of a measure of
its “statistical strength.” Two of the attack letters assayed positive for the four
mutations A1, A3, D, and E. Results on 1,059 of the 1,070 samples were tabu-
lated in the Statistical Analysis Report. Eight samples tested positive for all four
mutations; seven of these eight samples came from one institution (USAMRIID)
and the remaining sample came from a different institution (Battelle Memorial
Institute [BMI]). A table of documented transfers of samples from one institu -
tion to another showed a transfer of sample material from the first institution
(USAMRIID) to the second institution (BMI). This Appendix discusses the
validity of the inferences and calculations in the Statistical Analysis Report sub-
mitted to the FBI.
As noted in Chapter 6, the statistical analyses used in the report (e.g.,
95 percent confidence interval for the proportion of samples with four muta -
tions, chi-squared tests of independence) require two key assumptions to be
valid:
1. Representativeness: The 1,059 samples are assumed to be a representa-
tive and random collection of samples from some well-defined popula-
tion of samples.
2. Independence: The 1,059 samples are assumed to be independent of
one another (i.e., have no connection with each other, beyond that they
all come from the same population).
The Statistical Analysis Report acknowledges that neither assumption can
be validated from these data. The committee agrees with this assessment. As a
consequence, many of the statistical methods applied to these data cannot be
185

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186 APPENDIX C
validated. The consequences of the violation of these assumptions and their
impacts are listed below.
1. FBIR is not a representative and random collection of samples from a
well-defined population of B. anthracis samples.
The 1,059 samples do not appear to satisfy assumption 1. They were
obtained in response to a request from the FBI. No information is
available on samples in the population that were not submitted. In fact,
the “target population’’ seems not to have been defined. It could be
the population of all unique preparations of B. anthracis Ames in the
United States, or in the world, or from selected institutions. The absence
of a definition of “well-defined population” makes it difficult to assess
representativeness of the collection. The elimination of samples that had
“inconclusive” results on assays also appears to be nonrandom, as some
institutions had many more “inconclusive” assays than others.
2. The 1,059 samples in the FBIR are not independent.
The FBI submitted to the committee a table of known transfers of
samples between institutions. Hence, the second assumption is violated.
Thus, the results of the chi-squared tests for independence of the muta -
tions that are calculated in the report are not meaningful. Further, the
confidence interval for the proportion 8/947 is not appropriate. The
correct denominator for this proportion is likely not 947. A more accu -
rate numerator and denominator might refer to the number of known
independent preparations rather than the number of samples, but such
information may not be possible to obtain.
3. Violation of assumptions renders invalid the inferences from the statistical
analyses.
Because the FBIR is not a representative and random collection of
independent samples, the results on the assays from the repository may
be biased. Virtually all statistical procedures assume that the units on
which measurements are made comprise a random, representative col-
lection from the target population. (The effects of biased sampling on
inferences have been well documented; see, e.g., Freedman et al., 2007).
Without an appropriate model that characterizes the nonrepresentative-
ness and the degree of dependence among the samples, it is not possible
to calculate a meaningful measure of “statistical significance” in the
results.

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187
APPENDIX C
4. Results on 112 samples beyond the 947 samples
The Statistical Analysis Report eliminated from most of its tables the
results of the assays on 112 samples that showed “inconclusive” for A1,
A3, MRI-D, or E. Twenty-one of these 112 sample that were eliminated
from the statistical analysis assayed positive for 1, 2, or 3 mutations.
Table C-1 lists these samples. (Five samples—05-022, 49-014, 53-014,
53-068, 54-008—are listed twice because they were reported as “incon -
clusive” or “variant” on two assays.)
TABLE C-1 Samples with Positive and “Inconclusive” or “Variant” Assays
FBIR Number A1 A3 MRI-D IITRI-D E +Mutations
039-010 inc + - - - A3
044-034 var - - + - IITRI-D
049-014 inc var + + - MRI-D, IITRI-D
053-004 var + - - + A3, E
053-010 var + + + + A3, MRI-D, IITRI-D, E
053-014 var inc - - + E
053-068 inc inc + - - MRI-D
054-008 inc + + inc + A3, MRI-D, E
061-030 inc - + + - MRI-D, IITRI-D
066-015 inc inc + + - MRI-D, IITRI-D
005-022 + var - inc + A1, E
017-006 - var + + - MRI-D, IITRI-D
049-014 inc var + + - MRI-D, IITRI-D
049-018 - var + + - MRI-D, ITRI-D
053-014 var inc - - + E
053-068 inc inc + - - MRI-D
054-066 + inc + + + A1, MRI-D, IITRI-D, E
054-068 - var + - + MRI-D, E
005-020 - - + inc - MRI-D
005-022 + var - inc + A1, E
043-016 - - + inc - MRI-D
044-020 - + - inc + A3, E
052-026 + + + inc - A1, A3, MRI-D
054-008 inc + + inc + A3, MRI-D, E
054-022 - - + inc - MRI-D
057-036 - - + inc - MRI-D

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188 APPENDIX C
inc = inconclusive
IITRI = Illinois Institute for Technology Research Institute
MRI = Midwest Research Institute
var = variant
In addition to the two 3-positive samples (+++) among the 947 samples, the
four samples below also tested positive for 3 mutations (ordered by FBIR
number):
052-026 + + + inc - A1, A3, MRI-D
053-010 var + + + + A3, MRI-D, IITRI-D, E
054-008 inc + + inc + A3, MRI-D, E
054-066 + Inc + + + A1, MRI-D, IITRI-D, E
The following four samples revealed positive assays for 2 of the 4 mutations,
in addition to the 11 samples noted among the 947 samples (ordered by FBIR
number):
005-022 + var - inc + A1, E
044-020 - + - inc + A3, E
053-004 var + - - + A3, E
054-068 - var + - + MRI-D, E
DILUTION EXPERIMENTS
Dilution experiments were conducted to assess the sensitivity of the assays
to various concentrations. Thirty samples were prepared from RMR-1029 at
dilution 10.0. As with the other samples, some of the assays were “inconclu -
sive.” Genotype E tested positive in all 30 samples; all 4 mutations tested posi -
tive for 16 samples. But in the remaining 14 samples, assays for one or more
of the genotypes were negative. In fact, one sample tested negative for A1, A3,
and D; it was positive for only E. Five samples were positive for two mutations
only (A3 and E), and eight samples were positive for only three of the four
mutations (7 for A3, D, E; 1 for A1, A3, E). Thus, 6 of the 30 replicate samples
(20 percent) tested positive for only 1 or 2 of the mutations. Given that 50 of
the 947 FBIR samples showed only 1 positive, and 11 of the 947 showed only 2
positives, this variation indicates that some of the samples may have harbored
mutations that went undetected. Absent any repeat testing of these samples,
however, it is difficult to know how such false negatives might have affected
the inferences.
Additional experiments were conducted on RMR-1029 and another sample,
“SPS.266 Tube#5,” at 10 dilutions levels (10.1, . . ., 10.10). The results of the

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189
APPENDIX C
three replicates at each dilution level, for each of the five genotypes, for samples
from both RMR-1029 and SPS.266 Tube#5 were reported in Chapter 6. Vari-
ability in the results on replicates, even from the same sample at the same dilu-
tion level, demonstrates the value, and need for, replicate testing. For example,
the results on the three replicates from RMR-1029 at dilution 10.1, ordered as
A1, A3, MRI-D, IITRI-D, E, were: (- + + + +), (- + + + -), (+ + + + -). Clearly,
dilution affects the assay result: the greater the dilution, the more likely the assay
is negative. Moreover, it is perhaps unexpected that greater dilutions sometimes
give positive results when not all replicates at lesser dilutions did so.
CONCORDANCE OF TESTS FROM IITRI-D AND MRI-D
The FBI retained both the Illinois Institute for Technology Research Insti -
tute (IITRI) and Midwest Research Institute (MRI) to conduct the D assays.
Because the assays on the 1,059 samples can be considered to be independent
between IITRI and MRI, the Statistical Analysis Report (Table 3, p. 7, as pre-
sented below) tabulates the results of the D assays from the two facilities:
IITRI-D
MRI-D Inconclusive Negative No growth Pending Positive Total
Inconclusive 0 22 12 0 0 34
Negative 17 909 1 1 12 940
Negative-u 1 20 0 0 0 21
Positive 6 12 0 0 46 64
TOTAL 24 963 13 1 58 1,059
The Statistical Analysis Report combined the “negative-u” results with the
“negative” results, and eliminated the 12 samples that showed “no-growth” by
IITRI-D and “inconclusive” by MRI-D as well as the one “pending” sample,
to yield the following table:
IITRI-D
MRI-D Inconclusive Negative Positive Total
Inconclusive 0 22 0 22
Negative 18 929 12 959
Positive 6 12 46 64
TOTAL 24 963 58 1,045
Eliminating the 14 “no-growth” and “pending” samples, the concordance
rate is 975/1045 = 0.933, with a 95 percent confidence interval (0.916, 0.947).
Thus, the agreement between the facilities is unlikely to be lower than 91.6 per-

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190 APPENDIX C
cent and likely does not exceed 94.7 percent. Of greater interest, however, are the
12 samples that were positive by IITRI-D but negative for MRI-D, the 12 samples
that were negative by IITRI-D but positive by MRI-D, and the six samples that
were positive by MRI-D but inconclusive by IITRI-D. While concordance is
informative, these 30 samples with discordant results might provide increased
information about the samples and the assay process. On the other hand, we also
know from the repeated assays of the dilution series that some discordance
also arises owing to variation even when using the same assay procedure.
In any case, because genotype D is the only one of the four genotypes that
was subjected to independent testing by a second organization one cannot say
whether the results on the other genotypes might have been different if they
also had been subjected to independent testing.
“SIGNIFICANCE” OF SEVEN (++++) SAMPLES FROM
INSTITUTION F
The Statistical Analysis Report notes in its conclusions:
“In summation, though the random chance of occurrence of the sample type
(++++) is 8 out of 947 (i.e., 0.84%) with exact 95% confidence interval of
0.0037 to 0.0166 (I.e., from 1 in 270 to 1 in 60), this sample type has been
found in only two institutions thus far sampled (USAMRIID and BMI), and
its occurrence in BMI is explained by a recent sample transfer from USAM
to BMI, since there is no documented record of sample transfers in the other
direction.’’ (p. 2)
As noted in Chapter 6, 598 of the 947 samples (63 percent) came from
Institution F. (Twelve of the institutions submitted 6 or fewer samples; 4 insti -
tutions submitted 15-31 samples, and 4 institutions submitted 49-74 samples.)
Therefore, one would not be surprised to find more “mutation-positive” sam -
ples from Institution F than, say, from Institution B (which contributed only
one sample). One might naturally ask: How unusual is the occurrence of seven
“4-mutation” samples—or even all eight—from Institution F? Given that Insti-
tution F contributed almost 2/3 of the 947 samples, how many of the 4-positive
(++++) samples would Institution F receive if the 4-mutation samples were
distributed completely at random?
The answer to this question is given by the probabilities of observing 0
or 1 or 2 or ... or 8 of the eight (++++) samples from Institution F, given that
Institution F submitted 598 of the 947 samples that yielded definitive results on
the A1, A3, MRI-D, and E assays. These probabilities (from the hypergeometric
probability distribution) (Johnson et al., 2005) are shown in Table C-2.

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191
APPENDIX C
TABLE C-2 Probabilities of k 4-Mutation Samples in Institution F
k= 0 1 2 3 4 5 6 7 8
Probability 0.0003 0.0045 0.0276 0.0955 0.2058 0.2826 0.2415 0.1174 0.0248
This table shows that the chance of Institution F having ended up with
seven or eight of the eight (++++) samples is (0.1174 + 0.0248) = 0.1422, or
about 1/7. Therefore, while the observed data showing that seven of the eight
(++++) samples appeared in Institution F is not completely typical, it also could
hardly be considered extreme.

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