Today personalized medicine is one of the most popular interdisciplinary research eld, risk group identi cation being one of its most important tasks. Even though the rst attempts to estimate the effect of patient's characteristics on the outcome were proposed in statistics in the middle of the twentieth century, it is still an open question how to explore such e ects properly. In this paper we propose a trial version of the approach to risk group speci cation based on pattern structures and competing risk estimation, and discuss further steps of research on its performance and speci city.
Risk group identi cation is one of the task of personalized medicine. Often clinicians are required to de ne or to specify risk groups of patients to change treatment protocols properly. The task becomes more complicated if treatment outcome variable is not binary or numerical, but censored with several possible events. Such outcome is typical for most of the oncological diseases. Medical statistics calls the risk of occurrence of such event as competing risk and has a collection of techniques for their estimation and comparison [1]. However, there are almost no good techniques for risk groups speci cation when we deal with this type of outcome. To solve this problem, we presented an idea of approach to risk group identi cation when patients are described by nominal and/or numerical features with censored multi-event outcome. The mining process is based on pattern structures construction [2{5] with several adjustments which make it possible to apply the general approach to the dataset on acute lymphoblastic leukemia from ALL-MB-2008 trial [6] to obtain some promising results demonstrating the potential of the proposed approach.
The paper is organized as follows. In section 2 the theoretical basics of pattern structures and competing risks essential for the proposed approach are presented. Section 3 is devoted to the proposed approach. In section 4 we brie y describe results of application of the proposed approach to the data on acute lymphoblastic leukemia. Section 5 concludes the paper.
In this section we give an introduction to pattern structures and discuss some of their applications.
Let G be a set (of objects), (D; u) be a meet-semilattice (of all possible
object descriptions), and : G ! D be a mapping. Then (G; (D; u); ) is called
a pattern structure, provided that the set (G) = f (g) j g 2 Gg generates a
complete subsemilattice (D ; u) of (D; u), i.e. every subset X of (G) has an
in mum FX in (D; u). Elements of D are called patterns and are naturally
ordered by subsumption relation v: c v d , c u d = c, where c; d 2 D. If
(G; (D; u); ) is a pattern structure we de ne the derivation operators which
form a Galois connection between the powerset of G and (D; u) as:
A = gF2A (g)
d = fg 2 G j d v (g)g
for A G
for d 2 D
(
If objects are described by binary attributes from set M , then D = }(M ), the powerset of M , and (g) is prime operator ( )0 in the context (G; M; I): (g) = fm 2 M j gImg, and d1 u d2 = d1 \ d2 where d1; d2 2 D. So, subsumption corresponds to set inclusion: d1 v d2 , d1 u d2 = d1 , d1 \ d2 = d1 , d1 d2.
If objects are described by k nominal features f 1; :::; kg, we assume that i takes values from f1; :::; lig, where li 2 N, for i 2 f1; :::; kg. Nominal features can be transformed into binary attributes in many di erent ways. In this paper we will assume just one variant of transformation. For each i we construct li binary attributes f i1; :::; ili g such that ij : G ! f0; 1g and ij (g) : g ! i(g) 6= j where g 2 G; j = 1; :::; li. As a result we get Pi=1;:::;n li binary attributes. Patterns on such binary attributes de ne subsets of values of nominal features. For instance, if a pattern contains binary attributes i1 and i4 and no other ij then in terms of nominal feature it will look like i 2 f2; 3; 5; :::; lig.
To operate with numerical features interval pattern structures [3{5] can be
applied. If objects are described by n numerical features, we can represent them
as a set of functions f'1; '2; :::; 'ng, such that 'i : G ! R for i = 1; :::; n. For
each feature 'i we construct an interval attribute i : G ! [R; R] such that
if 'i(g) = x for g 2 G, then i(g) = [x; x], where x 2 R. Then each object
is described by a n-dimensional tuple of intervals. Let a and b be tuples of n
intervals, so a = h[vi; wi]ii=1;:::;n and b = h[xi; yi]ii=1;:::;n, where vi; wi; xi; yi 2
R 8i = 1; :::; n. In this case operator u is de ned as follows:
a u b = h[vi; wi]ii=1;:::;n u h[xi; yi]ii=1;:::;n = h[vi; wi] u [xi; yi]ii=1;:::;n
(
In a case when objects have both nominal and numerical features, we
associate the set of binary attributes with each nominal feature and an
interval attribute with each numerical one. Assume there are k binary and n
interval attributes, then every d 2 D can be represented as d = h ; i where
is an interval tuple of length n, and is a subset of binary attributes.
If d1; d2 2 D, d1 = h 1; 1i, and d2 = h 2; 2i operator u can be set as
d1 u d2 = h 1 u 2; 1 u 2i where operator u for interval tuples and the sets
of binary attributes is de ned above. The subsumption is also de ned by
subsumption on interval tuples and the sets of binary attributes:
d1 v d2 () d1 u d2 = d1 () h 1; 1i u h 2; 2i = h 1; 1i ()
() 1 u 2 = 1; 1 u 2 = 1 () 1 v 2; 1 v 2
(
In this paper the outcome variable is represented by a pair (T; C), where C 2 f0; 1; :::; pg and T 2 R+. Here C = i; i 2 f1; :::; pg corresponds to a type of event (for instance, relapse or death in case of cancer-study data), and in this case T is the time from the beginning of observation to the event occurrence. When C = 0 corresponds to censorship (observation had ended before any event occurred), T is the time from the beginning of observation to the moment of censorship. Then the survival function is de ned as S(t) = P(T > t), the probability of being free from any event at time t [1, Chapter 2]. However, if we aim at estimation of the risk of a speci c event i (event of interest) in the presence of other competing events we are more interested in the cumulative incidence function (CIF), de ned as Fi(t) = P(T t; C = i), the probability that event i occurs before time t [1, Chapter 4, p. 55]. Let t1 < t2 < ::: < tr be observed unique uncensored time points, dij be the number of events of type i observed at time tj , nj be the number of patients for whom T is not less than tj . Then, rst, we nonparametrically estimate the survival function through Kaplan-Meier estimator [7, 8]: Second, we estimate the CIF of the event of interest i as [1, Chapter 4, p. 56]: S^(t) =
Y h1 tj t
Pp
i=1 dij i
nj
F^i(t) =
tj t nj
X dij S^(tj 1)
(
To test the con dence of the di erence in risk of event of interest occurrence between M non-overlapping groups of patients, we use Gray's test for multiple groups [9] [1, Chapter 5, p. 74]. The main subject of it is to test the null hypothesis H0 : Fim(t) = Fi(t); m = 1; :::; M , where Fim(t) is the CIF of event i in group m, and Fi is the CIF of event i without groups speci cation. The general form of the score for group m is
Z tr
0 zm =
Wim(t)f im(t) i(t)gdt where tr is the maximum observed time, im(t) = 1FimFim(t)(0t) is the hazard of event i in group m, i(t) = 1FiF(ti)(0t) is the hazard of event i without groups speci cation, and Wim(t) is a weight function. The test statistics is a quadratic form Z 1Zt where Z = (z1; :::; zM 1) and is the corresponding covariance matrix.
For M = 2 only z1 needs to be computed. Frequently, Wi1(t) is assumed to be equal to Ri1(t), an adjusted number of individuals at risk. Let nj1 be the number of patients in group 1 for whom T is not less than tj . We put Ri1j = R^i1(tj ) = nj1 1 S^F1^(i1t(jtj1)1) , where S^1(tj 1) is Kaplan-Meier estimation of survival function in group 1 and F^i1(tj 1) is the estimation of the CIF of event i in group 1. Then the score is where di1j and di2j are the number of events i at time tj in groups 1 and 2, respectively. 3
In this section the procedure of identi cation of groups of patients with high
or low risk of the event of interest is proposed. Here we assume that patients
are described by numerical and/or nominal features, which can be transformed
into ordinal and binary attributes, respectively. The most obvious way of group
identi cation is the exhaustive search among all possible relevant descriptions.
However, several group descriptions may correspond to the same subset of
patients, therefore it is reasonable to search them only among pattern intents, which
we also call closed descriptions, as all descriptions of any subset of patients are
subsumed by the corresponding pattern intent. Hence, the search process can
be considerably reduced. Further we will see that the reduction of the search
is also important in terms of multiple testing. To construct all closed
descriptions object-wise version of Close-by-One (CbO) algorithm [11] can be applied.
(
As the number of closed descriptions is theoretically exponential in the number of patients and the number of unique values of all features, it may be still necessary to curtail the search. The possible solution is to construct closed descriptions only from data on patients who experienced exactly the event of interest, but to perform all further steps on the whole dataset.
We may also consider a closed description as a rule which splits the whole set of patients into two parts: those who satisfy and not satisfy this description. It allows us to perform Gray's test for every closed description with two parts of the corresponding split as groups. However, it may be also reasonable to perform additional pre-selection steps before.
The idea of pre-selection is to de ne one or several measures of di erence between CIFs of two parts of a split di erent from Gray's test statistics. If Fcd(t) and ncd are the CIF of the event of interest and the number of patients satisfying a closed description and Fr(t) and nr are the CIF of the event of interest and the number of the remaining patients, you can nd two examples of di erence measures below: { Absolute di erence of total CIFs
Entropy:
F^cd(
F^r(
ncd ncd + nr log
ncd ncd + nr { Absolute t-test statistics:
F^cd(
F^r(
ncd nr
As we want to obtain groups with high or low risk of the event of interest we attempt to maximize di erence measures chosen for pre-selection. The main problem here is choosing the cuto of pre-selection. A possible solution would be computing the value of di erence measure for every closed description, sorting computed values in ascending order and setting cuto at the point from which the values of measure increase faster than before it. So, for further analysis we retain only closed descriptions with the value of di erence measure not less than the chosen cuto value.
After all pre-selection steps we compute Gray's test p-value for all remaining closed descriptions. Taking into account multiple testing correction we select only closed descriptions with con dent di erence in CIF between two parts of the corresponding splits. Let us call them con dent descriptions. Then from all con dent descriptions we retain only those which are not subsumed by other con dent descriptions.
To present the obtained descriptions in a better way, one can remove uninformative attributes from the descriptions, such as interval attributes taking the whole attribute range, and transform binary attributes to the subsets of values of nominal features. Finally, when several similar splits (i.e. closed descriptions) are obtained we try to combine them through intersection and uni cation. nr ncd + nr log
nr ncd + cr
i
(
The proposed procedure was applied to the data of MB-ALL-2008 study [6]. We tried to specify relapse risk groups separately for patients from standard risk group (SRG) and intermediate risk group (ImRG) (more than 1000 patients each). Patients are described by 5 nominal and 4 numerical features. The outcome is represented by the censored variable with 5 possible events, one of which is a relapse (the event of interest).
As for SRG, closed descriptions were constructed on the relapse patient data. Closed descriptions with the small inside or outside number of patients (less than 50) were excluded beforehand as the corresponding splits are too unbalanced, which may badly a ect the value of Gray's test statistics. So, we started from 8383 closed descriptions for SRG. After that they were pre-selected with the use of two di erence measures provided in section 4 as di erence measure examples. The cuto vales were set manually after looking at the graphs in Figure 1.
Applying both measures with chosen cuto s only 18 closed descriptions were selected and corresponding splits were tested with Gray's test. Among 18 performed Grey's tests p-value of 14 tests was less than corrected level of con dence 2:8 10 3. Here and further Bonferroni correction [12] was applied. Subsumption removing resulted in 3 quite similar descriptions, which were nally transformed into one split of all SRG patients into two groups with Gray's test p-value 7:8 10 11: 1. High risk (201 patients): (age is not less than 11 years) OR (age 2 [4; 11) in years AND spleen enlargement is more than 2 cm) 2. Low risk (959 patients): others.
CIF estimations with 95% con dence intervals are shown in Figure 2. The pvalue of Gray's test is very small and may satisfy even more strict multiplicity corrections. For instance, if we apply Bonferroni correction for all closed descriptions the di erence between CIFs will remain con dent. This de nitely looks like a promising result.
As far as ImRG concerns, 588178 closed descriptions, for which the inner and outer number of patients were not less than 100, were constructed. After preselection with two measures mentioned above, the number of candidate closed descriptions decreased to 4599. Performing Gray's test with Bonferroni corrected p-value 1:1 10 5 we cut down the number of descriptions to 61, and after subsumption removal we resulted in 12 con dent closed descriptions. Most of the descriptions did not di er a lot, hence after their transformation and combination we resulted in the split of ImRG into 3 groups: 1. High risk (97 patients): (age is not less than 6 years old) AND (neuroleukemia
OR white blood cell count is larger than 100/nl) 2. Low risk (575 patients): no neuroleukemia AND white blood cell count is not larger than 100/nl AND age is less than 6 years old 3. Medium risk (388 patients): others.
You can nd the illustration of CIF estimations with 95% con dence intervals in Figure 3. Gray's test p-value for 3 groups is 3 10 4, pairwise Gray's test pvalues: low-medium { 1:1 10 2, medium-high { 1:3 10 3, low-high { 1:8 10 8. So, the signi cance of the di erence in relapse risk between found groups of ImRG patients is questionable, except the di erence between the low- and highrisk groups. 5
In this paper we proposed the rst version of the procedure which realizes the approach to risk group speci cation based on pattern structures. This approach was applied to data with nominal and/or numerical features and censored outcome with several possible events and one event of interest. The reason for using pattern structures and closed descriptions comes from the idea that, rst, de nitions of risk groups should be interpretable and, second, not a greedy optimization, but global one should be realized. However, the general procedure still employs several heuristics that allow us to reduce the global search. The open questions that remain are the following: what is the best decision for multiple testing problem in terms of the proposed approach? What should be done with multiple solutions given by overlapping groups? Although some promising results were obtained, a comparison to other approaches has to be performed.
The article was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and supported within the framework of a subsidy by the Russian Academic Excellence Project '5-100'.