### Appendix 12:

### [Cluster Analysis For Grouping of Food Categories]{.smallcaps}

Appendix 12: Cluster Analysis for Grouping of Food Categories

The results of the uncertainty analysis of the risk assessment were
summarized by a cluster analysis of food categories. The similarity
between categories was evaluated for the predicted number of cases of
listeriosis expressed as the risk per serving and per annum. Cluster
analysis is a descriptive statistical technique by which a set of
objects are partitioned or classified into subsets according to some
measure of similarity between objects[^1]. Typically, this partitioning
is defined to generate hierarchical subsets of the objects to be
classified. A single level of disjoint partitioning, without any
sub-partitioning of the objects within the primary clusters, is a
special case of the more general objective of obtaining a hierarchical
classification.

The use of a cluster analysis to summarize the results of the *L.
monocytogenes* risk assessment provides a means to convey the
implications of the uncertainty analysis of the rankings of food
categories which is, in some sense, more informative than statistical
point null hypothesis tests of differences in the location of the
distribution of ranks across food categories (e.g., as provided by
Kruskal-Wallis test or sign test). Testing for differences in location
(e.g., the median) of the uncertainty distributions of risk rankings,
according to either risk per serving or cases per annum, does not
incorporate any consideration of whether or not the differences obtained
are meaningful on a practical level.

Although the possibility exits that the elicitation and specification of
the variability and uncertainty of the model could result in two or more
pairings of food categories with identical distributions for either risk
per serving or expected cases per annum, this is very unlikely and small
differences in the location of rank distributions are expected. In this
event, statistical analysis of the output of the simulation based on use
of point null hypothesis tests to define differences between food
categories is likely to result in categorizing all such (small)
differences as significant (i.e., provided that the output of the
simulation is sufficiently large). While composite rather than point
null hypotheses could be used to define practical or meaningful
differences between the risk rankings of different food categories
(e.g., by equivalence testing methods), the application of these methods
is not readily available. Consequently, a cluster analysis approach was
adopted as an alternative.

Central to any cluster analysis is the specification of a definition of
similarity, or conversely dissimilarity, between the objects to be
classified[^2]. With respect to a cluster analysis of risk ranking of
the food categories, the "objects" to classified are the uncertainty
distributions (of risk per serving and expected cases per annum) and
thus a classification requires a definition of the "distance" or
dissimilarity between any two such distributions. The measure of
similarity adopted here for the cluster analysis was defined by the
degree to which any two uncertainty distributions overlap. If, for two
food categories, the uncertainty distribution of their risk rankings
were identical then the distributions would overlap maximally and it
would be reasonable to infer that they are two food categories that
should be judged to be similar in risk ranking. Conversely, if the risk
rank distributions of two food categories did not overlap at all then it
would be reasonable to infer that they are very dissimilar foods in
regard to risk ranking.

Based on this intuitive notion of distance between two distributions the
following measure of dissimilarity was used:

$\text{distance}(A,B)\  = \ \text{Pr}(\text{rank}(A) > \text{rank}(B))$

where A and B denote any two food categories, and rank() denotes their
rank distributions (according to either risk per serving or expected
number of cases per annum). Thus, if the rank of food category A is
higher than that of food category B with a high probability of belief
(i.e., according to their uncertainty distributions) then A and B would
be considered sufficiently dissimilar to belong in different clusters. A
level of 90% probability of belief that the rank of one food category
was higher than another was chosen as a cut-off value for classifying
any two distributions as dissimilar. That is to say, any two food
categories A and B were considered to be of different risk category (or
cluster) if:

$\text{distance}(A,B)\  > \ 0\text{.}\text{90}$

Obviously, both the definition of distance used and what constitutes a
"significant" distance based on the definition are subjective. With
respect to the latter, this is not intrinsically different from the
specification of confidence levels in frequentist-based hypothesis
testing. A level of 0.05 is common by convention but it is a subjective
choice nonetheless and other significance levels can and often have been
advocated. With respect to the former, we note that the chosen measure
of distance is not the only one that could be made. Also, it is a
pseudo-distance measure because it does not satisfy all properties of
distance measure proper; specifically it is not a symmetric function of
the argument. However, other more sophisticated information-theoretic
measures of the distance between two distributions such as the
Kullback-Leibler divergence are computationally difficult and also do
not satisfy all of the properties of a distance measure per se (i.e.,
they are quasi- or pseudo-distances).

Given the chosen definition of distance between two distributions and
the cut-off probability value for significant distance, all food
categories were compared in a pairwise fashion. Based on these
comparisons a partitioning of the food categories into disjoint subsets
of similar risk (either by risk per serving or cases per annum) was
obtained by defining clusters in the ordering of food categories from
highest median rank to lowest median rank. Specifically, the food
categories were ranked according their median rank and then partitions
where formed by taking the first cluster as being the largest set of
ordered food categories (starting from the first) for which all pairwise
comparisons of food categories within the set were equivalent based on
the definition of significant distance between their respective
uncertainty distributions. This process was repeated with all of the
remaining food categories until each food category was assigned to one
(and only one) cluster. If, for any given food category, there was no
other food category that was similar, based on the definition, then that
single food category was taken to form a cluster of one.

The results of the calculations of dissimilarity (or distance) between
the twenty-three food categories are shown in Tables A12-1 and A12-2
based on the simulation output of the uncertainty distributions of mean
risk per serving and expected number of cases per annum, respectively (n
= 4,000 uncertainty samples or iterations). Based on these calculations
the results of clustering the food categories according to either per
serving risk or cases per annum are shown in Table A13-3. The
sensitivity of the results to different specification of cut-off values
for belief that one food category ranks higher than another, and is
therefore dissimilar, is shown in Table A12-4. A level of 90%
probability was chosen here as a reasonable summarization in order to
obtain a relatively small number of clusters. At the 90% cut-off value
there is a high degree of belief that, based on the uncertainty
distributions, the foods in one cluster are of appreciably higher risk
than those foods in any lower ranked cluster. While there are
differences in risk rankings of food categories within any given cluster
we are not "confident at a 90% level" that the differences are
practically significant given all the attendant uncertainties that have
been incorporated into the assessment.

**Table A12-1. Probabilities^1^ (over uncertainty) that food categories
rank higher (or lower) than other food categories based on the mean risk
per serving.**

  ----- -------- --------- ------- -------- -------- -------- --------- --------- -------- --------- -------- -------- -------- ------- --------- --------- --------- -------- -------- -------- -------- -------- --------
        **DM**   **FNR**   **P**   **UM**   **SS**   **CR**   **HFD**   **SUC**   **PM**   **FSC**   **FR**   **PF**   **RS**   **F**   **DFS**   **SSC**   **SRC**   **V**    **DS**   **IC**   **PC**   **CD**   **HC**

  DM    0.0%     50.6%     65.8%   84.9%    82.8%    94.7%    98.5%     95.1%     97.4%    100.0%    100.0%   99.1%    100.0%   95.8%   99.6%     99.9%     99.9%     100.0%   100.0%   100.0%   100.0%   100.0%   100.0%

  FNR   49.5%    0.0%      71.8%   86.5%    84.7%    96.3%    98.3%     96.8%     97.8%    99.9%     100.0%   99.1%    100.0%   96.1%   99.8%     100.0%    99.9%     100.0%   100.0%   100.0%   100.0%   100.0%   100.0%

  P     34.2%    28.2%     0.0%    77.4%    76.6%    90.1%    96.0%     91.0%     96.0%    99.9%     100.0%   98.7%    100.0%   93.1%   99.2%     100.0%    99.9%     100.0%   100.0%   100.0%   100.0%   100.0%   100.0%

  UM    15.1%    13.6%     22.6%   0.0%     49.9%    56.0%    68.9%     69.2%     80.1%    92.0%     94.9%    91.0%    96.5%    84.4%   92.2%     97.0%     95.3%     98.4%    98.3%    100.0%   99.9%    99.5%    99.8%

  SS    17.3%    15.3%     23.5%   50.2%    0.0%     52.8%    66.6%     69.1%     81.7%    95.1%     99.8%    92.0%    99.5%    84.8%   93.6%     98.8%     97.0%     100.0%   99.3%    100.0%   100.0%   100.0%   100.0%

  CR    5.4%     3.7%      10.0%   44.0%    47.2%    0.0%     71.0%     68.0%     84.6%    97.6%     99.8%    93.3%    99.7%    83.5%   94.4%     99.4%     97.3%     100.0%   99.6%    100.0%   100.0%   100.0%   100.0%

  HFD   1.5%     1.8%      4.0%    31.2%    33.5%    29.0%    0.0%      57.9%     74.9%    94.6%     98.8%    87.9%    99.3%    79.5%   91.1%     98.3%     95.6%     99.9%    99.1%    100.0%   100.0%   99.9%    100.0%

  SUC   4.9%     3.2%      9.0%    30.8%    30.9%    32.0%    42.1%     0.0%      57.2%    78.1%     85.2%    78.5%    87.3%    73.4%   81.6%     87.9%     84.8%     90.7%    91.7%    97.2%    97.0%    96.7%    98.3%

  PM    2.6%     2.2%      4.0%    19.9%    18.4%    15.5%    25.1%     42.8%     0.0%     80.0%     93.2%    78.3%    96.6%    75.0%   84.1%     95.8%     88.9%     99.2%    97.0%    99.8%    100.0%   99.4%    100.0%

  FSC   0.0%     0.2%      0.1%    8.1%     5.0%     2.5%     5.4%      21.9%     20.0%    0.0%      66.6%    63.1%    78.2%    61.8%   68.4%     83.2%     75.1%     88.6%    89.2%    98.0%    98.4%    95.7%    98.5%

  FR    0.0%     0.0%      0.0%    5.2%     0.2%     0.3%     1.2%      14.9%     6.8%     33.4%     0.0%     57.4%    77.1%    58.0%   62.8%     82.3%     70.2%     86.5%    86.6%    99.2%    99.5%    96.0%    99.0%

  PF    1.0%     0.9%      1.4%    9.1%     8.0%     6.7%     12.1%     21.5%     21.7%    36.9%     42.6%    0.0%     50.6%    48.4%   50.7%     57.5%     57.6%     62.6%    69.8%    84.5%    83.7%    83.3%    91.1%

  RS    0.0%     0.0%      0.0%    3.6%     0.5%     0.4%     0.7%      12.7%     3.4%     21.8%     23.0%    49.5%    0.0%     50.3%   51.8%     65.9%     60.2%     73.1%    78.0%    96.8%    97.8%    91.7%    96.7%

  F     4.2%     3.9%      6.9%    15.6%    15.2%    16.6%    20.5%     26.6%     25.0%    38.2%     42.0%    51.6%    49.7%    0.0%    50.5%     57.2%     57.8%     60.9%    69.5%    84.5%    84.3%    83.2%    91.3%

  DFS   0.4%     0.2%      0.8%    7.8%     6.4%     5.6%     8.9%      18.4%     15.9%    31.6%     37.2%    49.3%    48.2%    49.5%   0.0%      58.1%     58.3%     64.4%    71.8%    88.8%    89.1%    85.8%    92.7%

  SSC   0.1%     0.0%      0.0%    3.0%     1.2%     0.6%     1.7%      12.1%     4.2%     16.9%     17.7%    42.5%    34.2%    42.9%   42.0%     0.0%      50.5%     58.5%    69.0%    89.7%    90.4%    84.7%    92.3%

  SRC   0.1%     0.1%      0.1%    4.7%     3.0%     2.8%     4.5%      15.2%     11.2%    24.9%     29.8%    42.4%    39.8%    42.2%   41.7%     49.5%     0.0%      55.3%    63.1%    80.7%    80.9%    79.3%    88.5%

  V     0.0%     0.0%      0.0%    1.6%     0.0%     0.0%     0.1%      9.3%      0.9%     11.4%     13.5%    37.4%    26.9%    39.1%   35.7%     41.5%     44.7%     0.0%     63.0%    86.1%    85.8%    81.1%    91.8%

  DS    0.0%     0.0%      0.1%    1.8%     0.7%     0.5%     0.9%      8.3%      3.1%     10.9%     13.4%    30.2%    22.0%    30.6%   28.2%     31.0%     36.9%     37.0%    0.0%     72.7%    72.3%    71.2%    85.1%

  IC    0.0%     0.0%      0.0%    0.1%     0.0%     0.0%     0.1%      2.9%      0.2%     2.0%      0.8%     15.5%    3.3%     15.5%   11.3%     10.4%     19.3%     14.0%    27.4%    0.0%     50.9%    53.0%    70.5%

  PC    0.0%     0.0%      0.0%    0.2%     0.0%     0.0%     0.0%      3.1%      0.0%     1.6%      0.5%     16.3%    2.2%     15.7%   11.0%     9.6%      19.1%     14.2%    27.8%    49.1%    0.0%     51.9%    69.4%

  CD    0.0%     0.0%      0.0%    0.6%     0.1%     0.1%     0.2%      3.3%      0.6%     4.3%      4.0%     16.7%    8.3%     16.8%   14.2%     15.3%     20.8%     18.9%    28.8%    47.1%    48.2%    0.0%     65.9%

  HC    0.0%     0.0%      0.0%    0.2%     0.0%     0.0%     0.0%      1.7%      0.1%     1.5%      1.1%     8.9%     3.3%     8.7%    7.3%      7.7%      11.5%     8.2%     14.9%    29.5%    30.7%    34.1%    0.0%
  ----- -------- --------- ------- -------- -------- -------- --------- --------- -------- --------- -------- -------- -------- ------- --------- --------- --------- -------- -------- -------- -------- -------- --------

1.  Probabilities are defined as Prob(rank(A) \> rank(B)) where A is the
    food category identified in the row labels and B is the food
    category identified in the column labels (based on 4,000 uncertainty
    iterations of the model).

+-----+---------------------------+-----+-----------------------------+
| ### |                           |     |                             |
| ### |                           |     |                             |
| ##  |                           |     |                             |
| LEG |                           |     |                             |
| END |                           |     |                             |
+-----+---------------------------+-----+-----------------------------+
| DM  | Deli Meats                | RS  | Raw Seafood                 |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| FNR | Frankfurters (not         | F = | Fruits                      |
| =   | reheated)                 |     |                             |
+-----+---------------------------+-----+-----------------------------+
| P = | Pâté and Meat Spreads     | DFS | Dry/Semi-Dry Fermented      |
|     |                           | =   | Sausages                    |
+-----+---------------------------+-----+-----------------------------+
| UM  | Unpasteurized Fluid Milk  | SSC | Semi-soft Cheese            |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| SS  | Smoked Seafood            | SRC | Soft Ripened Cheese         |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| CR  | Cooked Ready-To-Eat       | V = | Vegetables                  |
| =   | Crustaceans               |     |                             |
+-----+---------------------------+-----+-----------------------------+
| HFD | High Fat and Other Dairy  | DS  | Deli-type Salads            |
|     | Products                  | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| SUC | Soft Unripened Cheese     | IC  | Ice Cream and Frozen Dairy  |
| =   |                           | =   | Products                    |
+-----+---------------------------+-----+-----------------------------+
| PM  | Pasteurized Fluid Milk    | PC  | Processed Cheese            |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| FSC | Fresh Soft Cheese         | CD= | Cultured Milk Products      |
| =   |                           |     |                             |
+-----+---------------------------+-----+-----------------------------+
| FR  | Frankfurters (reheated)   | HC  | Hard Cheese                 |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| PF  | Preserved Fish            |     |                             |
| =   |                           |     |                             |
+-----+---------------------------+-----+-----------------------------+

**Table A12-2. Probabilities ^1^ (over uncertainty) that food categories
rank higher (or lower) than other food categories based on the number of
cases per annum.**

  ----- -------- -------- --------- --------- --------- -------- -------- -------- -------- ------- -------- -------- --------- --------- --------- --------- -------- -------- -------- -------- -------- -------- --------
        **DM**   **PM**   **HFD**   **FNR**   **SUC**   **P**    **CR**   **UM**   **SS**   **F**   **FR**   **V**    **DFS**   **FSC**   **SSC**   **SRC**   **DS**   **RS**   **PF**   **IC**   **PC**   **CD**   **HC**

  DM    0.0%     91.9%    98.5%     99.6%     99.8%     100.0%   100.0%   99.8%    99.6%    92.4%   100.0%   100.0%   100.0%    100.0%    100.0%    100.0%    100.0%   100.0%   100.0%   100.0%   100.0%   100.0%   100.0%

  PM    8.1%     0.0%     60.3%     75.1%     83.8%     96.0%    98.0%    93.8%    94.5%    77.9%   100.0%   99.2%    99.1%     100.0%    99.9%     99.6%     99.5%    100.0%   99.9%    100.0%   100.0%   100.0%   100.0%

  HFD   1.5%     39.7%    0.0%      69.7%     80.4%     95.3%    97.9%    93.1%    94.0%    75.4%   99.8%    99.1%    99.0%     100.0%    99.8%     99.8%     99.4%    100.0%   100.0%   100.0%   100.0%   100.0%   100.0%

  FNR   0.4%     24.9%    30.3%     0.0%      70.2%     92.0%    95.8%    87.2%    91.0%    72.5%   99.3%    98.5%    98.0%     100.0%    99.7%     99.6%     98.9%    100.0%   99.8%    100.0%   100.0%   100.0%   100.0%

  SUC   0.2%     16.2%    19.7%     29.9%     0.0%      60.6%    64.9%    60.8%    69.8%    59.5%   83.8%    78.8%    85.8%     91.1%     90.3%     88.4%     88.2%    92.8%    92.9%    95.6%    95.7%    96.0%    97.7%

  P     0.1%     4.0%     4.7%      8.0%      39.4%     0.0%     57.2%    54.0%    69.8%    59.6%   94.0%    82.2%    90.1%     100.0%    98.2%     93.6%     93.2%    100.0%   98.9%    99.7%    100.0%   99.6%    100.0%

  CR    0.0%     2.0%     2.2%      4.2%      35.1%     42.8%    0.0%     48.7%    65.8%    57.0%   91.8%    79.3%    89.2%     99.9%     98.0%     92.2%     92.5%    100.0%   98.5%    99.6%    100.0%   99.2%    99.9%

  UM    0.3%     6.2%     6.9%      12.8%     39.2%     46.0%    51.3%    0.0%     61.3%    56.5%   80.5%    76.2%    85.7%     96.6%     93.9%     89.3%     89.4%    97.7%    95.3%    98.4%    98.7%    97.4%    99.0%

  SS    0.4%     5.5%     6.0%      9.0%      30.2%     30.2%    34.3%    38.7%    0.0%     54.0%   72.7%    71.0%    82.2%     98.7%     94.7%     87.0%     88.8%    99.5%    95.4%    99.5%    99.8%    97.6%    99.3%

  F     7.6%     22.1%    24.7%     27.6%     40.5%     40.5%    43.0%    43.5%    46.0%    0.0%    54.7%    57.6%    69.7%     75.9%     74.6%     74.5%     76.8%    82.3%    79.9%    89.7%    89.4%    89.1%    94.2%

  FR    0.0%     0.1%     0.2%      0.7%      16.2%     6.0%     8.2%     19.5%    27.3%    45.4%   0.0%     58.5%    75.2%     89.8%     88.9%     78.4%     82.4%    98.4%    88.4%    98.7%    99.3%    95.7%    98.5%

  V     0.0%     0.9%     0.9%      1.6%      21.3%     17.8%    20.7%    23.8%    29.0%    42.5%   41.6%    0.0%     66.4%     78.6%     77.6%     72.9%     76.7%    89.5%    81.2%    92.8%    94.7%    91.0%    96.4%

  DFS   0.0%     1.0%     1.0%      2.0%      14.2%     9.9%     10.8%    14.4%    17.9%    30.3%   24.9%    33.7%    0.0%      59.2%     58.0%     57.9%     58.9%    67.6%    66.2%    78.2%    79.6%    79.7%    88.3%

  FSC   0.0%     0.0%     0.0%      0.0%      8.9%      0.0%     0.1%     3.4%     1.3%     24.1%   10.2%    21.5%    40.8%     0.0%      50.1%     50.1%     51.4%    67.1%    60.1%    76.2%    78.7%    77.1%    87.2%

  SSC   0.0%     0.2%     0.2%      0.3%      9.7%      1.8%     2.0%     6.2%     5.3%     25.4%   11.1%    22.4%    42.0%     49.9%     0.0%      50.1%     53.1%    66.7%    60.4%    77.5%    78.5%    76.8%    86.8%

  SRC   0.0%     0.4%     0.2%      0.4%      11.6%     6.4%     7.8%     10.7%    13.1%    25.6%   21.6%    27.1%    42.1%     49.9%     49.9%     0.0%      50.6%    58.6%    60.1%    69.0%    70.5%    72.6%    81.3%

  DS    0.0%     0.5%     0.6%      1.1%      11.8%     6.8%     7.5%     10.6%    11.2%    23.2%   17.6%    23.3%    41.1%     48.6%     47.0%     49.4%     0.0%     53.3%    58.8%    71.9%    72.8%    74.5%    86.3%

  RS    0.0%     0.0%     0.0%      0.0%      7.2%      0.0%     0.0%     2.3%     0.5%     17.7%   1.6%     10.5%    32.5%     33.0%     33.3%     41.5%     46.7%    0.0%     52.6%    69.8%    72.0%    72.8%    84.9%

  PF    0.0%     0.1%     0.0%      0.2%      7.1%      1.1%     1.5%     4.7%     4.6%     20.1%   11.7%    18.8%    33.8%     39.9%     39.6%     39.9%     41.2%    47.5%    0.0%     59.0%    60.7%    62.0%    71.4%

  IC    0.0%     0.0%     0.0%      0.1%      4.4%      0.3%     0.4%     1.7%     0.5%     10.3%   1.3%     7.2%     21.8%     23.8%     22.5%     31.0%     28.2%    30.2%    41.1%    0.0%     53.0%    59.8%    74.9%

  PC    0.0%     0.0%     0.0%      0.0%      4.3%      0.0%     0.0%     1.3%     0.2%     10.6%   0.7%     5.3%     20.4%     21.3%     21.5%     29.5%     27.3%    28.0%    39.3%    47.0%    0.0%     57.0%    72.6%

  CD    0.0%     0.0%     0.0%      0.0%      4.0%      0.4%     0.9%     2.7%     2.4%     10.9%   4.3%     9.0%     20.3%     22.9%     23.2%     27.4%     25.6%    27.2%    38.1%    40.2%    43.0%    0.0%     62.0%

  HC    0.0%     0.0%     0.0%      0.0%      2.3%      0.0%     0.2%     1.1%     0.7%     5.8%    1.6%     3.6%     11.7%     12.9%     13.3%     18.7%     13.8%    15.1%    28.6%    25.1%    27.5%    38.0%    0.0%
  ----- -------- -------- --------- --------- --------- -------- -------- -------- -------- ------- -------- -------- --------- --------- --------- --------- -------- -------- -------- -------- -------- -------- --------

^1^ Probabilities are defined as Prob(rank(A) \> rank(B)) where A is the
food category identified in the row labels and B is the food category
identified in the column labels (based on 4,000 uncertainty iterations
of the model).

+-----+---------------------------+-----+-----------------------------+
| ### |                           |     |                             |
| ### |                           |     |                             |
| ##  |                           |     |                             |
| LEG |                           |     |                             |
| END |                           |     |                             |
+-----+---------------------------+-----+-----------------------------+
| DM  | Deli Meats                | DFS | Dry/Semi-Dry Fermented      |
| =   |                           | =   | Sausages                    |
+-----+---------------------------+-----+-----------------------------+
| PM  | Pasteurized Fluid Milk    | FSC | Fresh Soft Cheese           |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| HFD | High Fat and Other Dairy  | SSC | Semi-soft Cheese            |
|     | Products                  | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| FNR | Frankfurters (not         | SRC | Soft Ripened Cheese         |
| =   | reheated)                 | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| SUC | Soft Unripened Cheese     | DS  | Deli-type Salads            |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| P = | Pâté and Meat Spreads     | RS  | Raw Seafood                 |
|     |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| CR  | Cooked Ready-To-Eat       | PF  | Preserved Fish              |
| =   | Crustaceans               | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| UM  | Unpasteurized Fluid Milk  | IC  | Ice Cream and Frozen Dairy  |
| =   |                           | =   | Products                    |
+-----+---------------------------+-----+-----------------------------+
| SS  | Smoked Seafood            | PC  | Processed Cheese            |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| F = | Fruits                    | CD= | Cultured Milk Products      |
+-----+---------------------------+-----+-----------------------------+
| FR  | Frankfurters (reheated)   | HC  | Hard Cheese                 |
| =   |                           | =   |                             |
+-----+---------------------------+-----+-----------------------------+
| V = | Vegetables                |     |                             |
+-----+---------------------------+-----+-----------------------------+

Table A12-3. Clustering of Similar Food Categories Based on the
Uncertainty Distribution of Relative Risk Ranking on Per Serving and Per
Annum Basis.

+-------------+--------------------------------+-----------------------+
| **Cluster** | **Risk per Serving**           | **Risk per Annum**    |
+-------------+--------------------------------+-----------------------+
| Cluster 1   | Deli Meats                     | Deli Meats            |
|             |                                |                       |
|             | Frankfurters, not reheated     |                       |
|             |                                |                       |
|             | Pâté and Meat Spreads          |                       |
|             |                                |                       |
|             | Unpasteurized Fluid Milk       |                       |
|             |                                |                       |
|             | Smoked Seafood                 |                       |
+-------------+--------------------------------+-----------------------+
| Cluster 2   | Cooked RTE Crustaceans         | High Fat and Other    |
|             |                                | Dairy Products        |
|             | High Fat and Other Dairy       |                       |
|             | Products                       | Frankfurters, not     |
|             |                                | reheated              |
|             | Pasteurized Fluid Milk         |                       |
|             |                                | Pasteurized Fluid     |
|             | Soft Unripened Cheese          | Milk                  |
|             |                                |                       |
|             |                                | Soft Unripened Cheese |
+-------------+--------------------------------+-----------------------+
| Cluster 3   | Deli-type Salads               | Cooked RTE            |
|             |                                | Crustaceans           |
|             | Dry/Semi-dry Fermented         |                       |
|             | Sausages                       | Fruits                |
|             |                                |                       |
|             | Fresh Soft Cheese              | Pâté and Meat Spreads |
|             |                                |                       |
|             | Frankfurters, reheated         | Unpasteurized Fluid   |
|             |                                | Milk                  |
|             | Fruits                         |                       |
|             |                                | Smoked Seafood        |
|             | Preserved Fish                 |                       |
|             |                                |                       |
|             | Raw Seafood                    |                       |
|             |                                |                       |
|             | Semi-soft Cheese               |                       |
|             |                                |                       |
|             | Soft Ripened Cheese            |                       |
|             |                                |                       |
|             | Vegetables                     |                       |
+-------------+--------------------------------+-----------------------+
| Cluster 4   | Cultured Milk Products         | Deli-type Salads      |
|             |                                |                       |
|             | Ice Cream and Frozen Dairy     | Dry/Semi-dry          |
|             | Products                       | Fermented Sausages    |
|             |                                |                       |
|             | Processed Cheese               | Frankfurters,         |
|             |                                | reheated              |
|             | Hard Cheese                    |                       |
|             |                                | Fresh Soft Cheese     |
|             |                                |                       |
|             |                                | Semi-Soft Cheese      |
|             |                                |                       |
|             |                                | Soft Ripened Cheese   |
|             |                                |                       |
|             |                                | Vegetables            |
+-------------+--------------------------------+-----------------------+
| Cluster 5   | Not Applicable                 | Cultured Milk         |
|             |                                | Products              |
|             |                                |                       |
|             |                                | Hard Cheese           |
|             |                                |                       |
|             |                                | Ice Cream and Frozen  |
|             |                                | Dairy Products        |
|             |                                |                       |
|             |                                | Preserved Fish        |
|             |                                |                       |
|             |                                | Processed Cheese      |
|             |                                |                       |
|             |                                | Raw Seafood           |
+-------------+--------------------------------+-----------------------+

Table A12-4. Sensitivity of clustering procedure to the cut-off
probability used to define similar versus dissimilar food categories.

  ---------------- ------------------ ------------------ ------------------
  Measure for      Cut-off            Total \# of        \# of distinct
  ranking          probability        pairwise           disjoint clusters
                   (distance) for     comparisons for    ^2^ of similarly
                   defining any two   which food         ranked food
                   categories as      categories are not categories
                   dissimilar         judged dissimilar  
                                      ^1^                

                                                         

  Risk per serving 0.95               139                4

  Risk per serving 0.90               116                4

  Risk per serving 0.75               61                 7

                                                         

  Cases per annum  0.95               149                4

  Cases per annum  0.90               124                5

  Cases per annum  0.75               69                 7

                                                         
  ---------------- ------------------ ------------------ ------------------

> ^1^There are a total of 276 pairwise comparisons of 23 food types; two
> food categories where considered dissimilar if Pr(rank(A) \> rank(B))
> \> the cut-off probability value where A is the food with higher mean
> rank and B is the food with lower mean rank

^2^A cluster is defined here as a collection of food categories for
which Pr(rank(A) \> rank(B)) \< cut-off probability value for any pair
(A,B) in the cluster; each food is assigned to only one cluster and
therefore clusters are disjoint.

[^1]: Jain A.K., Murty M.N. and Flynn P.J. (1999). Data Clustering: A
    review. ACM Computing Surveys 31(3), pg 264-323.

[^2]: ^1^