We present a design for a (fragment of) Breast Cancer ontology encoded in the probabilistic description logic P-SROIQ which supports determining the consistency of distinct statistical experimental results which may be described in diverse ways. The key contribution is a method for approximating sampling distributions such that the inconsistency of the approximation implies the statistical inconsistency of the continuous distributions.
The current amount of knowledge about breast cancer is overwhelming. For example, a meta-study conducted in 2006 by Key et al. [4] covered 98 unique studies focused only on the impact of a single risk factor, alcohol consumption. At the same time there are no common knowledge bases which would combine and formally represent ndings produced by the multitude of studies.3 This makes it di cult to have a global view of breast cancer risk factors and, consequently, develop tools like risk assessment calculators.
The probabilistic description logic P-SROIQ can be used to represent general knowledge about breast cancer in the form of a probabilistic ontology (the BRC ontology) [5]. However, a general knowledge ontology need not support risk entailments for various combinations of risk factors | that is, compete (poorly) with narrowly speci c risk calculators4 which have a direct implementation of simple equations derived from statistical risk models (such as the Gail model [2]). Instead, its main goal is to formally and unambiguously describe the background theory of breast cancer embracing as many reliable ndings as possible and serving as a common knowledge base for more speci c tools, such as risk assessment calculators or decision support systems. This sort of task seems to be a better t for a probabilistic logic.
The set of use cases for the general knowledge ontology is wider than for the BRC ontology. In addition to maintaining a birds-eye view of breast cancer, it may be used for nding and analyzing inconsistencies in outcomes of di erent 3 There are some lower level databases, such as ROCK (http://rock.icr.ac.uk/)|a cancer speci c functional genomic database. However, they do not explicitly represent case study ndings and do not support such services as risk assessment. 4 Such as http://www.cancer.gov/bcrisktool studies. It can support studying mechanisms of interactions between risk factors, for example, how alcohol consumption a ects estrogen level. Finally, it may play a useful role in planning and coordination of future medical studies by helping to identify the most controversial or insu ciently studied risk factors or exposures.
In this paper, we present a design of general P-SROIQ ontology about breast cancer (i.e., the BRC ontology) which incorporates a substantial amount of statistical knowledge. While we do not present a fully eshed out instance of this design, we do tackle a major representational challenge, namely, the representation of the statistical results of experiments. We present a method for approximate representations of di erent sampling distributions and their use in determining consistency between experimental data. 2
Preliminaries of P-SROI Q P-SROIQ [8] is a probabilistic extension of the DL SROIQ [3]. It provides means for expressing probabilistic relationships between arbitrary SROIQ concepts and a certain class of probabilistic relationships between classes and individuals. Any SROIQ, and thus OWL 2 DL (as it can be seen as a notational variant of SROIQ), ontology can be used as a basis for a P-SROIQ ontology, which facilitates transition from classical to probabilistic ontologies. We presume the reader is reasonbly familiar with class/object oriented description logics such as SROIQ, though very little in this paper turns on speci c details.
The only syntactic construct in P-SROIQ (in addition to all of the SROIQ syntax) is the conditional constraint.
De nition 1 (Conditional Constraint). A conditional constraint is an expression of the form (DjC)[l; u], where C and D are concept expressions in SRIQ (i.e., SROIQ without nominals) called evidence and conclusion, respectively, and [l; u] [0; 1] is a closed real-valued interval. In the case where C is > the constraint is called unconditional.
Ontologies in P-SROIQ are separated into a classical and a probabilistic part. It is assumed that the set of individual names NI is partitioned onto two sets: classical individuals NCI and probabilistic individuals NP I . De nition 2 (PTBox, PABox, and Probabilistic Knowledge Base). A probabilistic TBox (PTBox) is a pair P T = (T ; P) where T is a classical ( nite) SROIQ TBox and P is a nite set of conditional constraints. A probabilistic ABox (PABox) is a nite set of conditional constraints associated with a probabilistic individual op 2 NP I . A probabilistic knowledge base (or a probabilistic ontology) is a triple P O = (T ; P; fPop gop2NP I ), where the rst two components de ne a PTBox and the last is a a set of PABoxes.
Informally, a PTBox constraint (DjC)[l; u] expresses a conditional statement of the form \if a randomly chosen individual is an instance of C, the probability of it being an instance of D is in [l; u]". A PABox constraint, which we write as (DjC)o[l; u] where o is a probabilistic individual, states that \if a speci c individual (that is, o) is an instance of C, the probability of it being an instance of D is in [l; u]". For more details we refer the reader to [8]. 3
The classical part of a P-SROIQ ontology (or OWL part) provides a medical
vocabulary which can be used on its own in a variety of applications or used in the
representation of probabilistic knowledge. In this paper we focus on providing
an OWL terminology for probabilistic statements. The ontology contains the
following main class hierarchies (taxonomies):
Taxonomy of breast cancers Breast cancer is a heterogeneous disease. Some
risk factors can be associated with increase in risk of developing one
particular type of breast cancer and not the other. Thus it is important to classify
types of breast cancer. In particular, our ontology distinguishes breast
cancers by hormone receptor status. Estrogen and progesterone positive breast
cancers are modeled using concepts ERPositiveBRC and PRPositiveBRC
while their complements are modeled using ERNegativeBRC and
PRNegativeBRC (we use shorthands ER+/- and PR+/- with obvious
meaning.). Another important classi cation is based on histology. The ontology
distinguishes between invasive and non-invasive (e.g. in situ) cancers.
Taxonomy of risk factors Dozens of risk factors are known so far. Some are
established and strongly associate with increased risks, such as BRCA1(
The last two taxonomies induce the corresponding classi cations of women, i.e., classes of women w.r.t. risk factors and w.r.t. risk. For example, any risk factors RF gives rise to a class of women Woman u 9hasRiskFactor.RF. Women having various combinations of risk factors are modeled as conjunctive concept expressions. Analogously, given a certain kind of risk R the expression Woman u 9hasRisk.R models those women who are in the risk group R, for example, have moderately increased risk of developing ER+ breast cancer. These taxonomies of women may or may not be explicitly present in the ontology. In other words, it is possible, but not essential, to generate a concept name for each interesting class of women since P-SROIQ (and our reasoner Pronto) allows for complex concept expressions in conditional constraints.
A future, more complete version of the ontology would certainly make use of existing bio-medical ontologies which cover substantial portions of the domain either by direct reuse or by ontology alignment techniques. 4
The probabilistic part of the ontology captures statistical background knowledge about breast cancer. We distinguish between knowledge which explicitly associates quanti es speci c risk factors and more general statistical relationships which are not necessarily risk related. The distinction could be useful for importing knowledge from other medical ontologies. We begin with the latter.
General statistical knowledge mostly includes relationships between various risk factors. For example, Ashkenazi Jew women are more likely to develop BRCA gene mutations, while early menarche, late rst child (or no live births), lack of breastfeeding and alcohol consumption all increase levels of estrogen in blood.5 Such relationships are important because they can help to infer the presence of some risk factors given the set of known factors. They are typically easy to represent by using conditional constraints of the form (Woman u 9hasRiskFactor.RFYjWoman u 9hasRiskFactor.RFX)[l,u] which says that the chances of having risk factor RF Y given RF X are between l and u. One possible source of complications is continuous variables, e.g. the level of estrogen, which are discussed below.
Most of statistical ndings available in medical literature quantitatively describe risk increase for categories of women with speci c risk factors. Such ndings are presented by giving estimated parameters of a probability distribution where the random variable represents the relative risk of a random woman in the population. Such parameters include the estimated mean value and the estimated variance. Table 1 presents an example of the reported association between alcohol intake and the risk increase among postmenopausal women taken from [10]. There are two main di culties with representing this kind of data in PSROIQ. First, the risk increase is a continuous random variable so it needs to be discretized. Second, the available language supports only conditional constraints so a straightforward encoding of probability distributions is not possible.
Discretization of a continuous variable is technically straightforward. We introduce a set of disjoint concept names each of which models women in the corresponding group of risk. Speci cally, we de ne concepts WomenAtWeakRisk, WomenAtModerateRisk and WomenAtHighRisk with the obvious meanings described using OWL 2 datatype support to describe the exact boundaries. We have chosen ranges (1, 1.5], (1.5, 3.0] and (3.0, + inf) respectively.6
The inability to represent distributions is a more severe limitation. It leaves the modeler with the only option of approximating the continuous distribution using a nite set of points. In other words, each distribution, for example, risk increase for women consuming a certain amount of alcohol, can be approximated by specifying the probability that a randomly taken woman with the given exposure belongs to a speci c group of risk, i.e. WomenAtWeakRisk, WomenAtModerateRisk or WomenAtHighRisk. This is the semantics of P-SROIQ conditional constraints.
Assuming that the random variable is real-valued, a standard way of approximating a continuous distribution is to take each interval and compute the probability that the variable takes on a value in that interval. Then the approximation of a distribution P r(x) w.r.t. a nite set of intervals U is simply a function P^r such that P^r(Ui) = R P r(x)dx.
Ui
Unfortunately, this approximation of results of statistical experiments is unsatisfactory because it maps every interval to a single point. The problem is that any arbitrarily small di erence between two or more sampling distributions will results in con icting probabilistic statements for every interval (because the point-valued probabilities will be di erent) even though the results can con rm each other from a purely statistical point of view. Consequently this approach does not support working with results reported by multiple studies.
Our goal is to approximate sampling distributions in P-SROIQ in a statistically coherent way. Informally it means that satis ability of probabilistic formulas representing two or more sampling distributions must agree with their mutual statistical consistency, i.e., whether they support a common statistical hypothesis. The hypothesis, in this case, is that there exists a distribution (not necessarily a unique one) over G with parameters ; such that it is supported by all sampling distributions with the required level of con dence.
We assume a ( nite) population G of size NG and a random variable X which is normally distributed across G. We also make the realistic assumption that G is large enough so that evaluating X for all members of G is not feasible. A common approach is to take one or more random samples from G, evaluate X for them and estimate the actual distribution over G based on the sampling distributions. We use ; to denote the mean and the variance of the actual distribution and X(i); S(i) for the mean and the variance of the sample X(i). For simplicity we nally assume that the population distribution is normal.
The mainstream approach for comparing two or more sampling distributions
is based on statistical hypothesis tests. For example, given two normal
distribu6 The choice of intervals is obviously ambiguous but this issue is orthogonal to the
approximation method presented in this paper.
tions X(
Our approach is slightly di erent from the outlined above. It is not based on
tests but on con dence regions for sampling distributions. The approach, which
generalizes con dence intervals and dates back to Mood [9], is to estimate a
region R in the parameter space for ( ; 2) such that it will contain the ; 2
pair of the actual distribution 100(
P (( ; 2) 2 R ) = 1
; for all ( ; 2)
(
Informally, the con dence region speci es how far sampling distributions can
deviate from the population distribution while supporting it with 100(
Consider the sample X1; : : : ; Xn where all Xi are independent random variables with the normal distribution (N ( ; 2)). Then X = n1 Pn i=1 Xi and S2 = n1 1 Pin=1 (Xi X)2, i.e., the sample mean and the sample variance, are ran2 dom variables. It is well known that X has the normal distribution N ( ; n ) (or, equivalently, X=pn N (0; 1)) while (n 1)S2= 2 has the chi-square distribution with n 1 degrees of freedom [9].
The standard tables for N (0; 1) and 2n 1 provide numbers a; b; c such that
for xed p1; p2 the following equalities hold [1]:
7 The null hypothesis is a default position which, in this case, could be that the
population mean is di erent from at least one of X(
The crucial fact is that the two random variables are independent (see [9] for a proof) which implies that:
P (X
a pn <
< X + a pn ;
=pn < a; b <
Thus, the 100(p1)(p2)% con dence region for ( ; 2) takes the following form:
Rp1;p2 (X; S) =
( ; 2) : X
(n
pn <
1)S2
<
< X +
2 < (n
pn ;
1)S2
(
De nition 3. Let P r(X(
Now we can return to the issue of approximating a continuous sampling
distribution by a discrete set of points. Assume that the domain E of a continuous
real-valued random variable X is a disjoint union of a nite number of intervals
U = f( 1; r1]; (r1; r2]; : : : ; (rl 1; rl]; (rl; +1)g. Then the approximation of the
sampling distribution P r(X) with mean and variance (X; S2) is the function P^r
which maps each interval Ui to the following real-valued set:
De nition 4. Two sampling distributions P r(X(
As with any approximation, the utility of approximations of sampling
distributions depends on what conclusions they help to draw about the distributions
themselves. Given that we are interested in the matter of consistency, it is
important to understand the relationships between the notions of consistency and
approximate consistency of sampling distributions. Fortunately, consistency
implies approximate consistency regardless of partitioning of the real line:
Theorem 1. If two sampling distributions P r(X(
The following corollary from the above theorem is at heart of our method. As we demonstrate below, the inconsistency of approximations can be proved by logical reasoning in P-SROIQ (i.e., by solving the probabilistic satis ability problem), which means that the result enables approximate reasoning about sampling distributions in a purely logical way. Even though the power of such reasoning is currently limited to consistency checking, its integration with OWL/DL reasoning and the ability to use common, formally de ned terminology for representation of statistical experiments is promising.
Corollary 1. If sampling distributions P r(X(
Now we present an example of approximate representation of sampling distributions in P-SROIQ. The task is to take two results of statistical experiments aimed at investigating associations between alcohol consumption and the increased risk of breast cancer among postmenopausal women. Unfortunately it is common for medical papers to not explicitly present all parameters that characterize results of their statistical analyses. Typically, only the estimated mean and the con dence interval are presented while, for example, the kind of distribution is left to the reader to infer from other information. Due to that fact and because the approach above has only been developed for normal distributions, we illustrate it on an arti cial example. The information given in the example is analogous to that given in medical literature, e.g. [10, 11], but is complete in the sense that all parameters and the type of sampling distributions are known. Example 1. Consider two hypothetical papers which report results of independent studies of associations between alcohol consumption among postmenopausal women and their relative risk of developing breast cancer. According to study A the mean relative risk (RR) of ER+ breast cancer for women drinking 4g of ethanol a day is 1.8 and has variance of 0.5. Study B has reported that the mean RR of ER+ breast cancer for the same level of drinking is 2.2 (variance 0.7). The number of cases in the studies was 230 and 150 respectively.
We propose the following four step procedure for an approximate representation of statistical results, similar to those in the example above, in P-SROIQ: 1. Preparing concepts The rst step is to de ne the concepts/roles used to describe the distribution. In our case evidence concepts should describe categories of women with respect to speci c risk factors, e.g. alcohol intake, while conclusion concepts describe groups of women strati ed by risk increase. For instance, the concept expression C Woman u 9hasRiskFactor:(Postmenopause u ModerateConsumption) is used to model postmenopausal women with moderate level of alcohol intake.9 On the other hand the expression:
D
Woman u 9hasRisk:(ModeratelyIncreasedRisk u 9riskOf.ERPositiveBRC) 9 The level of intake is a continuous variable which we also split onto categories LimitedConsumption, ModerateConsumption and HeavyConsumption which correspond to 4, 4 9:9 and 10g of ethanol per day. models women who are at moderately increased risk of developing ER-positive breast cancer. Using these expressions the modeler can specify the probability than a random women the class C also belong the risk group D as (D|C)[l,u].
2. Determining parameters of sampling distributions (if required) Sometimes parameters of sampling distributions can be determined from other information. For example, knowing the kind of distribution, sample mean, sample size, con dence interval and the methodology of its estimation, one can calculate the sample variance.10 In our case it is not needed as the distributions are normal and the parameters are known.
3. Choosing intervals Choice of intervals for an approximation of a continuous random variable is driven by balancing the quality of the approximation (i.e., how closely it models the continuous distribution) and the number of statements required. The latter has a direct impact on performance. For Example 1 we use three concepts WomenAtWeakRisk, WomenAtModerateRisk and WomenAtHighRisk which correspond to relative risk intervals of (1,1.5], (1.5, 3.0] and (3.0,+1) respectively.
4. Computing the approximation The nal (and the central) step is to
compute probability intervals for the statements that approximate the
continuous distribution. Each statement speci es the lower and upper probabilities
that the continuous random variable X will fall into an interval Ui given that
parameters of the distribution can vary within the con dence region (
Now the optimization problem (
f(W u 9hR:(WeaklyIncreasedRisk u 9riskOf.ERPositiveBRC)jC)[0:219; 0:298]; (W u 9hR:(ModeratelyIncreasedRisk u 9riskOf.ERPositiveBRC)jC)[0:655; 0:878]; (W u 9hR:(StronglyIncreasedRisk u 9riskOf.ERPositiveBRC)jC)[0:239; 0:586]g and f(W u 9hR:(WeaklyIncreasedRisk u 9riskOf.ERPositiveBRC)jC)[0:116; 0:224]; (W u 9hR:(ModeratelyIncreasedRisk u 9riskOf.ERPositiveBRC)jC)[0:562; 0:769]; (W u 9hR:(StronglyIncreasedRisk u 9riskOf.ERPositiveBRC)jC)[0:189; 0:568]g where W and hR abbreviate Woman and hasRisk, respectively and C
W u 9hasRiskFactor:(Postmenopause u ModerateConsumption)
Probabilistic consistency of the above set of statements can be proved by solving the probabilistic satis ability problem (PSAT). Modern algorithms can decide PSAT for over a thousand of P-SROIQ statements (in addition to thousands of OWL axioms), so the method could be computationally practical [6]. 6
Checking consistency of sampling distributions in P-SROIQ may well appear cumbersome and pointless given that the same task can be done in a much simpler way and without any logical reasoning, e.g. via testing or by analyzing con dence regions. However, our aim is not to reduce statistical testing to logical reasoning (that aim is indeed pointless). Our aim is to represent results of statistical experiments using common, unambiguously de ned logical vocabulary and be able to reason about them. Even though probabilistic reasoning about statistical results is currently limited to approximate consistency checking, the 11 We use Wolfram Mathematica for this purpose. potential bene ts are in combining it with reasoning about the classical knowledge. For example, the BCRA ontology contains a little taxonomy of breast cancers by hormone receptor status. This enables us to combine results of the studies which are of di erent levels of granularity. For instance, Sellers et al. [10] report associations between alcohol intake and ER(+/-) breast cancer risk, while Suzuki et al. [11] divide it further to ER(+/-)PR(+/-) risks. In that simple case non-logical reasoning about the reported results becomes much less straightforward, while studies can also distinguish histologic types of breast cancer (see [7]). In such complex situations reasoning about ndings does involve reasoning about background knowledge, e.g. the taxonomy of breast cancers, so a combination of OWL and probabilistic reasoning is potentially bene cial.