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4
Statistical Considerations
Abstract The Cumulative Sum (CUSUM), a widely accepted method for maintaining
process control in manufacturing, is a feasible approach to statistical evaluation of the
slaughter process. It can help ensure that procedural or policy changes, personnel
turnovers, inadvertent mistakes, poor training, or mechanical deficiencies do not
establish an upward trend in defects that are detectable by inspection. CUSUM is a
good choice to help control the process and ensure uniformity of processing. However,
many people erroneously believe CUSUM as used in SIS-C is intended to ensure that
no contaminated or otherwise defective meat reaches the consumer. That is not its
purpose, and it is inadequate to provide that assurance in cattle slaughtering operations.
Therefore, CUSUM in SIS-C will not directly ensure a pleasing, clean,
wholesome, and toxin- or pathogen-free product. As one part of the processing and
inspection effort, it monitors system performance by checking that the process is
consistent in its ability to detect and remove visible defects and that processing
modifications do not establish an upward trend in these defects.
The limitations of CUSUM as an inspection tool for assessing a
nonuniform product must be clearly recognized. Because of these limitations, greater
flexibility must be built into the sampling scheme along with an emphasis on optimizing
the quality of the final product rather than simply obtaining acceptable CUSUM scores.
CUSUM is appropriate for maintaining process control provided
appropriate tolerance limits are set and provided all lots are adequately sampled.
However, CUSUM cannot eliminate the need for adequate inspection and for
microbiologic and toxicologic testing.
The primary statistical considerations related to SIS-C involve three points: (~)
the rationale supporting a change from Acceptable Quality Level (AQL) scores to
CUSUM for STS-C; (2) the use of CUSUM for process control; and (3) the method of
sampling for CUSUM determinations.
Rationale Supporting the Change from AQI, to SIS
The rationale for favoring SIS-C over traditional inspection came from AQL,
scores (see definitions in Appendix G), based on both traditional and STS-C procedures.
These data showed that one of the five plants sampled (plant #5) had lower AQL
scores when STS-C was in effect (FSIS, 1989; Wesson, 1983~. These data are consistent
with the hypothesis that STS-C might produce lower AQL scores in that specific plant
(plant #5), but are an inadequate basis for accepting this hypothesis or inferring that
SIS-C produces lower AQL scores in other plants. Furthermore, these conclusions
should not be drawn from data on this one plant, because there was a downward trend
in AQL scores in that plant when traditional inspection was still in place. Other
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variables, such as management changes, that are unrelated to inspection procedures
could have affected the results. After the initial improvements, little consistent further
improvement was observed.
Use of CUSUM for Process Control
MUM IS a widely accepted method tor maintaining process control In
manufacturing products that must be kept within certain tolerances. At appropriate
stages of production, product samples are examined for deviations from specified limits
in composition, dimensions, or other features. CUSUM is a gooc] method for
monitoring manufacturing processes to detect changes that cause failure of products to
meet required specifications. CUSUM is an acceptable method of monitoring the
process to discover problems introduced by factors such as personnel turnover,
inadequately trained personnel, mechanical deficiencies, plant policy, and changes in
procedures. This is the purpose for which FSIS uses CUSUM. However, many people
believe erroneously that CUSUM as used in SIS-C is intended to ensure that no
contaminated or otherwise defective meat reaches the consumer. CUSUM sampling
and calculations are not designed to detect defects that are missed by inspectors or
trimmers.
~1
In SIS-C, CUSUM is used to monitor slaughtering of cattle; this is similar to
process control in manufacturing facilities. However, unlike manufacturing processes
that merge raw materials into some product, in SIS-C the incoming material is modified
by separating or removing certain parts and by detecting and removing defects. In STS-
C, a nonuniform product, i.e., cattle with various defects and from different sources,
enters the processing plant. In addition, certain defects can be introduced by the
processing. CUSUM attempts to monitor the system's performance in detecting and
removing both types of defects and producing a uniform (with respect to defects) final
product.
According to "FSTS Data Presentation to the Committee," CUSUM involves use
of a tolerance level "set at the national average for the total of the weighted non-
conformances for all of the plants tested" (FSIS, 1989b). Data from a randomly
selected sample of plants were used to determine the average quality (cleanliness and
processing defects) of products produced in the United States. This average is applied
to SIS-C plants whose initial performance should have been much better than the
national average because of the uniform age, size, and weight of the cattle they
slaughter. In CUSUM, as in SIS-C, the tolerance limits might more appropriately be
based on what is acceptable for every plant rather than on an average reflecting what
plants were doing at any particular time. It would be surprising if these measures were
identical. The issue is not whether CUSUM is appropriate; it can be a valuable part
of SIS-C. Rather, the important factors are (~) the tolerance limits within which plants
can operate and be labeled "in control" or "out of control", (2) the appropriate actions
that require more conservative processing practices such as slower line speeds and
reprocessing, and (3) the way in which CUSUM is utilized and interpreted within the
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entire framework of SIS-C.
Cattle arrive at the processing plant with various defects. Defects may be
clustered in herds or lots. They may be related to traumas during transport, to
diseases, or to nutritional influences at the feediot. In addition, defects are also
introduced by processing. Therefore, defects measured in STS-C can come from two
sources: the cattle entering the plant and the process itself. If cattle entering the
plant had standard defects, then monitoring the defects would monitor the process. A
goal of SIS-C monitoring is to remove defects from both sources without distinguishing
between them. One question to be answered is whether the effectiveness of CUSUM
monitoring is influenced by the condition of the lot entering the plant. Is process
control affected by the number of defects, the number of cattle with defects, the
clustering of defects, and the kinds and locations of defects? Is it possible for some
lots with high numbers of defects to enter the plant and not be included in the
sampling for CUSUM?
If the processing of cattle is the same for all lots, the nature and location of
defects do not influence the ability of inspectors to discover them, and if all lots are
adequately represented in CUSUM, then the clustering of defects within lots might be
less important. However, if defects of particular kinds or in certain locations are more
difficlit to observe, then clustering of defects in lots is important, and the monitoring
method needs to ensure that there is adequate sampling within lots (possibly
representing all herds). The feasibility of representing all sources in the sampling is at
best questionable. On the other hand, sampling from all lots may be feasible and
could be incorporated into the plan for CUSUM in the same way that sampling during
certain time intervals is planned.
Finally, CUSUM in SIS-C plays an indirect, not a direct, role in ensuring
pleasing, clean, wholesome, and toxin- or pathogen-free products as part of the entire
processing and inspection effort. It is a check on the process to ensure that processing
modifications, inadvertent mistakes, and poor training do not establish an upward trend
in the number of defects undetected by the inspection. It does not guarantee that the
product is free from defects, but should ensure that the process is consistent in its
ability to detect and remove defects.
The Sampling Basis for CUSUM
If there is substantial variation in the average number of defects among lots,
then the sampling for CUSUM needs to include an adequate number of subgroups
from each lot. The plan for CUSUM monitoring of carcass and viscera includes the
plant's sampling subgroups of three consecutive carcass sides with a frequency of one
subgroup sample per hour and FSIS sampling subgroups of three consecutive sides
every ~ hours; other products are sampled with different criteria. If carcasses from a
lot composed of several herds are distributed randomly along the production line, the
probabilities of detecting defects in a particular lot should be determined to understand
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how well sampling will represent the lot.
The probability that defects in a given lot will be detected can be studied by
examining the converse probability that the sampled units will not include any defects
in a lot, when in fact they exist. Figure 4-1 depicts these probabilities based on the
following assumptions:
I. The distribution of defects in lots is Poisson.
2. The average number of defects per carcass side is 0.~.
3. Three carcass sides are observed each hour.
4. Carcass sides are independent. (This is, of course, an oversimplification.)
The probabilities have been calculated and plotted for line speeds from 200 to
400 per hour in increments of 50, and for lot sizes of 250 to 3,000 in increments of
250.
The probabilities of observing no defects among sampled units from single lots
having an average of 0.1 defects per carcass are very high for the smaller lot sizes and
decrease as the lot sizes increase. This is to be expected, since more carcass sides
would be sampled from the larger lots for any given line speed. The probabilities are
consistently higher for faster line speeds, because a smaller proportion of carcass sides
is sampled for faster line speeds, regardless of the lot size. By looking at the points
where the lines intersect with a particular probability on the graph, it can be seen that
the lot size must increase for each increment in line speed to achieve that probability.
1
0.8
m 0.6
6
m
0 0.4
0.2
i_
O 1
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000
LOT SIZE
Lumber of
Cable per Hour
200
~ 250
· -a--- 300
- - 350
_ 400
Figure 4-! Probability of Observing No Defects in Various Lot Sizes and
Line Speeds When the Average Number of Defects/Sample Unit
(Carcass Side) is 0.l
3
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For example, the following line speeds and lot sizes were all associated with the same
probability (0.223) of observing no defects in the lot (line speed/lot size): 200/1,000,
250/1,250, 300/1,500, 350/1,750, and 400/2,000. Higher line speeds can be used with
larger lots to achieve probabilities comparable with those of lower line speeds and
smaller lots. However, these probabilities relate only to the discovery of at least one
defect in a lot (~.0 minus the probability of not detecting a defect) and do not address
adequate numbers to assess the level of defects in a lot. Of course, when faster line
speeds are used with small lots, there is a chance that some small lots will not be
represented in the sample.
In summary, a simple Poisson mode] was used to compute probabilities that
some defective product will be accepted using CUSUM. These computations showed
that defects among cattle in small lots are more likely to be missed than those in large
lots and that faster line speeds are associated with higher probabilities of accepting
defective meat. This demonstrated that a CUSUM-based inspection scheme will allow-
-with disturbingly high probability--some defective meat to pass undetected in small lots
and in plants with high line speeds, and reinforced the position that its use should be
limited to process control.
A second approach to studying the potential for defective cattle to pass the
CUSUM monitoring process is presented in Figure 4-2. This figure presents the
operation characteristic curve for CUSUM. When the fraction nonconforming is 0.] or
lower, the probability of accepting the lot is 0.73 or higher. Disturbingly high
probabilities of acceptance (0.5 i, 0.34, 0.21) remain even when the fraction of
nonconforming products is high (e.g., probabilities of 0.51, 0.34, and 0.21 are associated
with nonconforming fractions of 0.2, 0.3, and 0.4, respectively).
These probability computations demonstrate the need to clearly define the
objectives of the inspection program. If the objective is to control the process and
ensure uniformity of processing, CUSUM is a good choice. However, if the objective is
to ensure that no contaminated or otherwise defective meat reaches the consumer,
CUSUM is inadequate. This distinction is crucial.
Discussion
In a broad sense, the question of the usefulness of the CUSUM is not a
statistical one but relates more to the objectives of inspection and quality control in
plants. The materials upon which this document is based suggest that the purpose of
quality control and the role of meat inspectors in processing plants are viewed
differently, depending on whose interests are being served. At least four different
interests are involved in the processing of cattle:
1. Industry wants to produce an appealing finished product as efficiently,
inexpensively, and quickly as possible.
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1
LU
~7
<: 0.8
111
o 0.6
lo
o
~ 0.4
m
~ 0.2
o
a:
1~
0L ' ~
o
-
-
-
-
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
FRACTION OF NONCONFORMING PRODUCTS
Figure 4-2 Evaluation of CUSUM for SIS-C, Assuming n=3, c=0
2. USDA wants to fulfill its mandate to ensure wholesomeness, protect
public health, prevent adulteration, and protect against economic
fraud.
3. Consumer groups want a safe final product that has a good
appearance.
4. Health organizations want to ensure that the product is not
contaminated with disease-causing organisms, toxins, or chemicals.
These goals may be an oversimplification of the issues, but they are important in
evaluating inspection procedures and designing sampling schemes.
There are probably no economically feasible sampling schemes that could satisfy
all these concerns. An attempt to sample adequately is built into SIS-C (see Chapter
3) by having plants institute their own quality control programs that are monitored by
USDA. According to this plan, USDA samples only for finished product standards and
continues 100% inspection of head, carcass, and viscera. After this inspection, the
head and viscera are separated from the carcass and company personnel cut and trim
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for the cooler. At another company station, the cuts are again viewed and three
carcass sides are sampled once or twice each shift and the products are examined for
processing defects only. The inspectors maintain their CUSUM, which should be
similar to the plant's score.
This plan could be expected to accomplish the following in the best of
circumstances:
1.
The company quality control program (QC) could be designed to detect visual
defects exceeding acceptable limits among the cattle being processed and
remove the defects or condemn the defective meat. This would require a
sampling scheme that could take into account cattle lots from different
sources and other potentially important factors.
2. The 100% inspection of heads, viscera, and carcasses would accomplish
exactly what traditional inspection has accomplished, assuming that the
inspection is performed with the same thoroughness as in the traditional
program.
^. Documentation and monitoring of plant QC would help ensure that deviations
from the plant norm would be followed by appropriate actions.
4. CUSUM would allow inspectors to monitor the plant process so that the
defects introduced and missed during processing could be monitored, and
trends toward increasing numbers of defects could be detected and corrected.
However, the method alone could not be expected to ensure that carcasses
leaving the plant meet the expectations mentioned above.
CUSUM is appropriate for maintaining process control, provided the tolerance
levels are set for maintaining an acceptable process for all plants and are not at the
level observed in a sampling of plants, and provided al] Jots are adequately sampled.
However, CUSUM cannot eliminate the need for the other parts of SIS-C, including
inspection and other testing (e.g., microbiological and toxicologic).
It is important to distinguish between two objectives that might be part of the
SIS-C in order to realize the impact of the probabilities presented in this chapter. One
objective might be process control (i.e., ensuring that the process remains consistent in
its detection of defects). CUSUM wall be helpful in attaining this objective. The other,
and primary, objective should be preventing contaminated or defective meat from being
accepted. CUSUM wall not ensure that the final product is free from defects or from
microbial and chemical contamination.
Recommendations
The theoretical and statistical considerations underlying CUSUM should be
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reevaluated with the following points in mind:
The limitations of CUSUM as an inspection too] for assessing a nonuniform
product must be clearly recognized. Because of these limitations, greater
flexibility must be built into the sampling process along with an emphasis on
optimizing the quality of the final product rather than simply obtaining
acceptable CUSUM scores.
For CUSUM, or any comparable sampling system, the statistical basis for setting
initial sample sizes and tolerance levels must be validated in a series of ongoing
controlled studies.
USDA should evaluate CUSIJM's performance with respect to accepting
`defective lots of meat. This should be done in two ways: (1) Operating
characteristic curves should be developed for various schemes by using CUSUM.
These can be used to evaluate performance under various conditions and
assumptions by providing probabilities of accepting defective meat. Ideally, the
probability of accepting defective meat should be zero or very close to zero. (2)
CUSUM should be tested in plants to determine how often it actually accepts
defective meat. This evaluation should also include observations on microbial or
chemical contamination and should be designed with adequate sample numbers
to facilitate valid interpretations of the results.
41
Representative terms from entire chapter:
process control
