**3.2.2 Relations between concepts**

We define the primary relations between concepts.

1. Relations of patients and events: The relations are defined between the [patient] and all event concepts. For example, the following relation denotes the relations between patients and their hospital stays.

⟨subject (of an event)⟩⊆ [patient]×[hospital stay].

Note that these relations share the same name "subject (of an event)". We omit the explanation of the relations between patients and other events.

Representation System for Quality Indicators by Ontology 201

a. concept *C*,

ii. R(�):=\**0*,

v. C(�):=*C*.

iv. L(�)(\**0*):=C,

i. N(�):={\**0*, …, \**n*},

b. attributes *A1*,…, *An* of *C*, and

 L(�)(\**i*):=*ai* for *i=1,…, n*, and, L(�)(*fi*):=*Ai* for *i=1,…, n*,

a. an integer *n* with *n*≧*1*,

the empty set).

i. N(�):= {\**0*, …, \**n*},

iv. L(�)(\**i*):=�*i* (*i=0,…, n*), L(�)(*f i*):=*Ri* (*i=0,…, n*) and,

ii. R(�):= \**0*,

v. C(�):= C(�*0*).

�1, and �2, as follows.

c. values *a1*,…, *an* of *A1*,…, *An*, respectively.

Then, we define an objective graph �, as follows.

**Case 2**. Assume that the following data are given:

Then, we define an objective graph �, as follows.

L(�)(*f j,ik*):= *Ri,jk* (*i,j=0,…, n* and *k=1,…, n(i, j)*).

b. a set of objective graphs {�*0*, …, �*n*},

d. a set of integers {*n(i,j)*}*<sup>0</sup>*≦*i*≦*n, <sup>0</sup>*≦*j*≦*<sup>n</sup>* and,

each *f i,jk* is an edge from \**i* to \**j*.

**4.2 Example of an objective graph** 

iii. E(�):={*f1*,…,*fn*}, where each *f i* is an edge from \**0* to \**i*.

Note that if n=0, then N(�) is the singleton set {\**0*} and E(�) is the empty set.

c. a set of relations {*R1*,…, *Rn*}, where each *Ri* is a relation between C(�*i*) and C(�*0*),

e. for each *i* with *0*≦*i*≦*n* and *j* with *0*≦*j*≦*n*, the set of relations is {*Ri,j1,…, Ri,jn(i,j)*}, where each *Ri,jk* is a relation between C(�*i*) and C(�*j*). (Note: if *n(i,j)=0*, the set {*Ri,j1,…, Ri,jn(i,j)*} is

iii. E(�):={*f 1,…, f n*}∪(∪*<sup>0</sup>*≦*i*≦*n*, *0*≦*j*≦*<sup>n</sup>*{*f i,j1*,…, *f i,jn(i,j)*}), where each *f i* is an edge from \**i* to \**0* and

We give an example of an objective graph. For example, let us consider the quality indicator "5-year stomach cancer survival rate". The definition of the quality indicator is the ratio of the number of 5-year surviving patients to all stomach cancer patients, where a "stomach cancer patient" is a patient who had a diagnosis whose result was stomach cancer, and a "5 year surviving patient" is a patient who had a diagnosis whose result was stomach cancer but who is alive 5 years after that medical examination. Thus, we will first express the set of 5-year surviving patients in Figure 6. To this end, we construct three objective graphs �0,

Each *f i* is called a main edge of � and each *f i,jk* is called an optional edge of �.

(1) �*0* = ({\*}, \*, ∅ (the empty set), *L0*, [patient]), where *L0*(\*)=[patient].

2. **Relations of patients and states**: The relations are defined between the [patient] and all state concepts. For example, the following relation denotes the relationship between patients and their states of diseases.

〈subject (of a state)⟩⊆ [patient]×[state of disease].

Note that these relations also share the same name "subject (of an state)" and that all concepts of states have the attributes of starting time points and terminating time points. We omit the explanation of the relations between patients and other states.

3. **Relations of time ordering**: The relations are defined between the concepts of events and the states. For example, the following relations denote the relationships between operations.

> 〈more than <p> before〉 ⊆ [operation]×[operation], 〈less than <p> befor〉 ⊆ [operation]×[operation], 〈less than <p> after〉 ⊆ [operation]×[operation] and 〈more than <p> after〉 ⊆ [operation]×[operation].

Here, "<p>" denotes a parameter. For example, the relation 〈before more than <2 weeks>〉 consists of a pair <op*1*, op*2*> if op*1* and op*2* are performed and if op*1* is performed more than two weeks before op*2*.

4. **Belonging relations of events**: The relations are defined between concepts of events with no term and events with terms. For example, the following relation denotes the relations between operations and hospital stays that have operations.

〈belonging〉 ⊆ [operation]×[hospital stay].

The relation contains a pair (op, sty) of an event of an operation op and that of a hospital stay sty if op is performed in the duration of sty.
