AU2004242178A1 - A method for identifying a subset of components of a system - Google Patents

Description

WO 2004/104856 PCT/AU2004/000696 A METHOD FOR IDENTIFYING A SUBSET OF COMPONENTS OF A SYSTEM FIELD OF THE INVENTION 5 The present invention relates to a method and apparatus for identifying components of a system from data generated from samples from the system, which components are capable of predicting a feature of the sample within the system and, particularly, but not exclusively, the present invention 10 relates to a method and apparatus for identifying components of a biological system from data generated by a biological method, which components are capable of predicting a feature of interest associated with a sample applied to the biological system. 15 BACKGROUND OF THE INVENTION There are any number of systems in existence that can be classified according to one or more features thereof. The 20 term "system" as used throughout this specification is considered to include all types of systems from which data (e.g. statistical data) can be obtained. Examples of such systems include chemical systems, financial systems and geological systems. It is desirable to be able to utilise 25 data obtained from the systems to identify particular features of samples from the system; for instance, to assist with analysis of financial system to identify groups such as those who have good credit and those who are a credit risk. Often the data obtained from the systems is relatively large 30 and therefore it is desirable to identify components of the systems from the data, the components being predictive of the particular features of the samples from the system. However, when the data is relatively large it can be difficult to identify the components because there is a 35 large amount of data to process, the majority of which may not provide any indication or little indication of the features of a particular sample from which the data is WO2004/104856 PCT/AU2004/000696 - 2 taken. Furthermore, components that are identified using a training sample are often ineffective at identifying features on test sample data when the test sample data has a high degree of variability relative to the training sample 5 data. This is often the case in situations when, for example, data is obtained from many different sources, as it is often difficult to control the conditions under which the data is collected from each individual source. 10 An example of a type of system where these problems are particularly pertinent, is a biological system, in which the components could include, for example, particular genes or proteins. Recent advances in biotechnology have resulted in the development of biological methods for large scale 15 screening of systems and analysis of samples. Such methods include, for example, microarray analysis using DNA or RNA, proteomics analysis, proteomics electrophoresis gel analysis, and high throughput screening techniques. These types of methods often result in the generation of data that 20 can have up to 30,000 or more components for each sample that is tested. It is highly desirable to be able identify features of interest in samples from biological systems. For example, 25 to classify groups such as "diseased" and "non-diseased". Many of these biological methods would be useful as diagnostic tools predicting features of a sample in the biological systems. For example, identifying diseases by screening tissues or body fluids, or as tools for 30 determining, for example, the efficacy of pharmaceutical compounds. Use of biological methods such as biotechnology arrays in such applications to date has been limited due to the large 35 amount of data that is generated from these types of methods, and the lack of efficient methods for screening the data for meaningful results. Consequently, analysis of WO2004/104856 PCT/AU2004/000696 - 3 biological data using existing methods is time consuming, prone to false results and requires large amounts of computer memory if a meaningful result is to be obtained from the data. This is problematic in large scale screening 5 scenarios where rapid and accurate screening is required. It is therefore desirable to have a method, in particular for analysis of biological data, and more generally, for an improved method of analysing data from a system in order to 10 predict a feature of interest for a sample from the system. SUMMARY OF THE INVENTION According to a first aspect of the present invention, there 15 is provided a method of identifying a subset of components of a system based on data obtained from the system using at least one training sample from the system, the method comprising the steps of: obtaining a linear combination of components of the 20 system and weightings of the linear combination of components, the weightings having values based on the data obtained from the system using the at least one training sample, the at least one training sample having a known feature; 25 obtaining a model of a probability distribution of the known feature, wherein the model is conditional on the linear combination of components; obtaining a prior distribution for the weighting of the linear combination of the components, the prior 30 distribution comprising a hyperprior having a high probability density close to zero, the hyperprior being such that it is not a Jeffreys hyperprior; combining the prior distribution and the model to generate a posterior distribution; and 35 identifying the subset of components based on a set of the weightings that maximise the posterior distribution.

WO 2004/104856 PCT/AU2004/000696 - 4 The method utilises training samples having the known feature in order to identify the subset of components which can predict a feature for a training sample. Subsequently, knowledge of the subset of components can be used for tests, 5 for example clinical tests, to predict a feature such as whether a tissue sample is malignant or benign, or what is the weight of a tumour, or provide an estimated time for survival of a patient having a particular condition. 10 The term "feature" as used throughout this specification refers to any response or identifiable trait or character that is associated with a sample. For example, a feature may be a particular time to an event for a particular sample, or the size or quantity of a sample, or the class or 15 group into which a sample can be classified. Preferably, the step of obtaining the linear combination comprises the step of using a Bayesian statistical method to estimate the weightings. 20 Preferably, the method further comprises the step of making an apriori assumption that a majority of the components are unlikely to be components that will form part of the subset of components. 25 The apriori assumption has particular application when there are a large amount of components obtained from the system. The apriori assumption is essentially that the majority of the weightings are likely to be zero. The model is 30 constructed such that with the apriori assumption in mind, the weightings are such that the posterior probability of the weightings given the observed data is maximised. Components having a weighting below a pre-determined threshold (which will be the majority of them in accordance 35 with the apriori assumption) are ignored. The process is iterated until the correct diagnostic components are identified. Thus, the method has the potential to be quick, WO2004/104856 PCT/AU2004/000696 5 mainly because of the apriori assumption, which results in rapid elimination of the majority of components. Preferably, the hyperprior comprises one or more adjustable 5 parameter that enable the prior distribution near zero to be varied. Most features of a system typically exhibit a probability distribution, and the probability distribution of a feature 10 can be modelled using statistical models that are based on the data generated from the training samples. The present invention utilises statistical models that model the probability distribution for a feature of interest or a series of features of interest. Thus, for a feature of 15 interest having a particular probability distribution, an appropriate model is defined that models that distribution. Preferably, the method comprise a mathematical equation in the form of a likelihood function that provides the 20 probability distribution based on data obtained from the at least one training sample. Preferably, the likelihood function is based on a previously described model for describing some probability 25 distribution. Preferably, the step of obtaining the model comprises the step of selecting the model from a group comprising a multinomial or binomial logistic regression, generalised 30 linear model, Cox's proportional hazards model, accelerated failure model and parametric survival model. In a first embodiment, the likelihood function is based on the multinomial or binomial logistic regression. The 35 binomial or multinomial logistic regression preferably models a feature having a multinomial or binomial distribution. A binomial distribution is a statistical WO 2004/104856 PCT/AU2004/000696 - 6 distribution having two possible classes or groups such as an on/off state. Examples of such groups include dead/alive, improved/not improved, depressed/not depressed. A multinomial distribution is a generalisation of the 5 binomial distribution in which a plurality of classes or groups are possible for each of a plurality of samples, or in other words, a sample may be classified into one of a plurality of classes or groups. Thus, by defining a likelihood function based on a multinomial or binomial 10 logistic regression, it is possible to identify subsets of components that are capable of classifying a sample into one of a plurality of pre-defined groups or classes. To do this, training samples are grouped into a plurality of sample groups (or "classes") based on a predetermined 15 feature of the training samples in which the members of each sample group have a common feature and are assigned a common group identifier. A likelihood function is formulated based on a multinomial or binomial logistic regression conditional on the linear combination (which incorporates the data 20 generated from the grouped training samples). The feature may be any desired classification by which the training samples are to be grouped. For example, the features for classifying tissue samples may be that the tissue is normal, malignant, benign, a leukemia cell, a healthy cell, that the 25 training samples are obtained from the blood of patients having or not having a certain condition, or that the training samples are from a cell from one of several types of cancer as compared to a normal cell. 30 In the first embodiment, the likelihood function based on the multinomial or binomial logistic regression is of the form: eik eiG n G-1 x fO L = I G-1 1-G-1 i=1 g= g1 + G e }J' 1 + h e=1 h g=1 h=1 WO 2004/104856 PCT/AU2004/000696 - 7 wherein x,g is a linear combination generated from input data from training sample i with component weights fig; 5 x, is the components for the i th Row of X and Pg is a set of component weights for sample class g; and X is data from n training samples comprising p components and the eik are defined further in this specification. 10 In a second embodiment, the likelihood function is based on the ordered categorical logistic regression. The ordered categorical logistic regression models a binomial or multinomial distribution in which the classes are in a particular order (ordered classes such as for example, 15 classes of increasing or decreasing disease severity). By defining a likelihood function based on an ordered categorical logistic regression, it is possible to identify a subset of components that is capable of classifying a sample into a class wherein the class is one of a plurality 20 of predefined ordered classes. By defining a series of group indentifiers in which each group identifier corresponds to a member of an ordered class, and grouping the training samples into one of the ordered classes based on predetermined features of the training samples, a 25 likelihood function can be formulated based on a categorical ordered logistic regression which is conditional on the linear combination (which incorporates the data generated from the grouped training samples). 30 In the second embodiment, the likelihood function based on the categorical ordered logistic regression is of the form: N G -1 (' ri i, \r k 1 i=1 k=1 ik+1 ik+1l Wherein Yik is the probability that training sample i belongs to a 35 class with identifier less than or equal to k (where the WO2004/104856 PCT/AU2004/000696 - 8 total of ordered classes is G).The ri is defined further in the document. In a third embodiment of the present invention, the 5 likelihood function is based on the generalised linear model. The generalised linear model preferably models a feature that is distributed as a regular exponential family of distributions. Examples of regular exponential family of distributions include normal distribution, guassian 10 distribution, poisson distribution, gamma distribution and inverse gaussian distribution. Thus, in another embodiment of the method of the invention, a subset of components is identified that is capable of predicting a predefined characteristic of a sample which has a distribution 15 belonging to a regular exponential family of distributions. In particular by defining a generalised linear model which models the characteristic to be predicted. Examples of a characteristic that may be predicted using a generalised linear model include any quantity of a sample that exhibits 20 the specified distribution such as, for example, the weight, size or other dimensions or quantities of a sample. In the third embodiment, the generalised linear model is of the form: N L=logp(y I, +) {y, -b() + c(yi,) } ,= ai(O) 25 where y = (yl,..., Yn)T and ai(4) = 4 /wi with the wi being a fixed set of known weights and 4 a single scale parameter. The other terms in this expression are defined later in this document. 30 In a fourth embodiment, the method of the present invention may be used to predict the time to an event for a sample by utilising the likelihood function that is based on a hazard model, which preferably estimates the probability of a time 35 to an event given that the event has not taken place at the WO2004/104856 PCT/AU2004/000696 9 time of obtaining the data. In the fourth embodiment, the likelihood function is selected from the group comprising a Cox's proportional hazards model, parametric survival model and accelerated failure times model. Cox's proportional 5 hazards model permits the time to an event to be modelled on a set of components and component weights without making restrictive assumptions about time. The accelerated failure model is a general model for data consisting of survival times in which the component measurements are assumed to act 10 multiplicatively on the time-scale, and so affect the rate at which an individual proceeds along the time axis. Thus, the accelerated survival model can be interpreted in terms of the speed of progression of, for example, disease. The parametric survival model is one in which the distribution 15 function for the time to an event (eg survival time) is modelled by a known distribution or has a specified parametric formulation. Among the commonly used survival distributions are the Weibull, exponential and extreme value distributions. 20 In the fourth embodiment, a subset of components capable of predicting the time to an event for a sample is identified by defining a likelihood based on Cox's proportional standards model, a parametric survival model or an 25 accelerated survival times model, which comprises measuring the time elapsed for a plurality of samples from the time the sample is obtained to the time of the event. In the fourth embodiment, the likelihood function for 30 predicting the time to an event is of the form: N Log (Partial) Likelihood = gj ( ,; , ;,y,c) i=1 where 8' =(,,fl2,---,flp) and p' =(p1,p 2 ,'",pq)are the model parameters, y is a vector of observed times and c is an indicator vector which indicates whether a time is a true WO 2004/104856 PCT/AU2004/000696 - 10 survival time or a censored survival time. In the fourth embodiment, the likelihood function based on Cox's proportional hazards model is of the form: 5 ' 'd; N exp (Z p ) f exp Zi/) where the observed times are be ordered in increasing magnitude denoted as t=(t(1),t(2),-..t(N)), t(i+ ) >t(i). and Z denotes the Nxp matrix that is the re-arrangement of the rows of X 10 where the ordering of the rows of Zcorresponds to the ordering induced by the ordering of t. Also PT =(#1 02,.... /n), Zj = the jth row of Z, and 9ij= {i:i=j,j+,...---,N}= the risk set at the jth ordered event time t). 15 In the fourth embodiment, wherein the likelihood function is based on the Parametric Survival model it is of the form: L = cilog(p i)- pi+c i log (Yi) i=1 A (yi,;_) where pi=Ayi;;)exp(XfJ) and A denotes the integrated 20 parametric hazard function. For any defined models, the weightings are typically estimated using a Bayesian statistical model (Kotz and Johnson, 1983) in which a posterior distribution of the 25 component weights is formulated which combines the likelihood function and a prior distribution. The component weightings are estimated by maximising the posterior WO 2004/104856 PCT/AU2004/000696 - 11 distribution of the weightings given the data generated for the at least one training sample. Thus, the objective function to be maximised consists of the likelihood function based on a model for the feature as discussed above and a 5 prior distribution for the weightings. Preferably, the prior distribution is of the form: p ( )= p v2)p (V2)dv2 V2 10 wherein v is a p x 1 vector of hyperparameters, and where pvfl IV) is N0(O,diagjv) and p(v m ) is some hyperprior distribution for v 2 Preferably, the hyperprior comprises a gamma distribution 15 with a specified shape and scale parameter. This hyperprior distribution (which is preferably the same for all embodiments of the method) may be expressed using different notational conventions, and in the detailed 20 description of the embodiments (see below), the following notational conventions are adopted merely for convenience for the particular embodiment: As used herein, when the likelihood function for the 25 probability distribution is based on a multinomial or binomial logistic regression, the notation for the prior distribution is: G-1 72 g=1 30 where '=(/ ,..J_-1 and r=( 'I G-) WO 2004/104856 PCT/AU2004/000696 - 12 and p(fg 12) is N(0,diag{ r,2}) and P('r) is some hyperprior distribution for rg. As used herein, when the likelihood function for the 5 probability distribution is based on a categorical ordered logistic regression, the notation for the prior distribution is: N P(f1,62," , n p (' i 2)p(V 2) d V 2 r i=1 10 where p,, -2 ,, are component weights, P( vi) isN(0,vt)and P(v i ) some hyperprior distribution for v 1 . As used herein, when the likelihood function for the distribution is based on a generalised linear model, the 15 notation for the prior distribution is: p(f)= Jp8 v2)p(V 2)dv2 f2 wherein v is a p x 1 vector of hyperparameters, and where p(P V2) is N(O,diag{v2}) and p(V2) is some prior distribution 20 for v 2 . As used herein, when the likelihood function for the distribution is based on a hazard model, the notation for the prior distribution is: 25 where p(p*j ) is N(O,diag{ 2}) and p r ) some hyperprior distribution for r .

WO2004/104856 PCT/AU2004/000696 - 13 The prior distribution comprises a hyperprior that ensures that zero weightings are used whenever possible. P(p*)= p(fl*lr )P(-r )dr In an alternative embodiment, the hyperprior is an inverse 22 5 gamma distribution in which each t i =I/v7 has an independent gamma distribution. In a further alternative embodiment, the hyperprior is a 22 gamma distribution in which each vi, ,Ty or z (depending on the 10 context) has an independent gamma distribution. As discussed previously , the prior distribution and the likelihood function are combined to generate a posterior distribution. The posterior distribution is preferably of 15 the form: p(fp vly)a L(yI (p)p( IV)p(v) or P (8, ,) aL Pr)P 1)P(T) 20 wherein L f,T) is the likelihood function. Preferably, the step of identifying the subset of components comprises the step of using an iterative procedure such that 25 the probability density of the posterior distribution is maximised. During the iterative procedure, component weightings having a value less than a pre-determined threshold are eliminated, 30 preferably by setting those component weights to zero. This results in the substantially elimination of the WO2004/104856 PCT/AU2004/000696 - 14 corresponding component. Preferably, the iterative procedure is an EM algorithm. 5 The EM algorithm produces a sequence of component weighting estimates that converge to give component the weightings that maximise the probability density of the posterior distribution. The EM algorithm consists of two steps, known as the E or Expectation step and the M, or Maximisation 10 step. In the E step, the expected value of the log posterior function conditional on the observed data is determined. In the M step, the expected log-posterior function is maximised to give updated component weight estimates that increase the posterior. The two steps are 15 alternated until convergence of the E step and the M step is achieved, or in other words, until the expected value and the maximised value of the expected log-posterior function converges. 20 It is envisaged that the method of the present invention may be applied to any system from which measurements can be obtained, and preferably systems from which very large amounts of data are generated. Examples of systems to which the method of the present invention may be applied include 25 biological systems, chemical systems, agricultural systems, weather systems, financial systems including, for example, credit risk assessment systems, insurance systems, marketing systems or company record systems, electronic systems, physical systems, astrophysics systems and mechanical 30 systems. For example, in a financial system, the samples may be particular stock and the components may be measurements made on any number of factors which may affect stock prices such as company profits, employee numbers, rainfall values in various cities, number of shareholders 35 etc.

WO2004/104856 PCT/AU2004/000696 - 15 The method of the present invention is particularly suitable for use in analysis of biological systems. The method of the present invention may be used to identify subsets of components for classifying samples from any biological 5 system which produces measurable values for the components and in which the components can be uniquely labelled. In other words, the components are labelled or organised in a manner which allows data from one component to be distinguished from data from another component. For 10 example, the components may be spatially organised in, for example, an array which allows data from each component to be distinguished from another by spatial position, or each component may have some unique identification associated with it such as an identification signal or tag. For 15 example, the components may be bound to individual carriers, each carrier having a detectable identification signature such as quantum dots (see for example, Rosenthal, 2001, Nature Biotech 19: 621-622; Han et al. (2001) Nature Biotechnology 19: 631-635), fluorescent markers (see for 20 example, Fu et al, (1999) Nature Biotechnology 17: 1109 1111), bar-coded tags (see for example, Lockhart and trulson (2001) Nature Biotechnology 19: 1122-1123). In a particularly preferred embodiment, the biological 25 system is a biotechnology array. Examples of biotechnology arrays include oligonucleotide arrays, DNA arrays, DNA microarrays, RNA arrays, RNA microarrays, DNA microchips, RNA microchips, protein arrays, protein microchips, antibody arrays, chemical arrays, carbohydrate arrays, proteomics 30 arrays, lipid arrays. In another embodiment, the biological system may be selected from the group including, for example, DNA or RNA electrophoresis gels, protein or proteomics electrophoresis gels, biomolecular interaction analysis such as Biacore analysis, amino acid analysis, 35 ADMETox screening (see for example High-throughput ADMETox estimation: In Vitro and In Silico approaches (2002), Ferenc Darvas and Gyorgy Dorman (Eds), Biotechniques Press), WO2004/104856 PCT/AU2004/000696 - 16 protein electrophoresis gels and proteomics electrophoresis gels. The components may be any measurable component of the 5 system. In the case of a biological system, the components may be, for example, genes or portions thereof, DNA sequences, RNA sequences, peptides, proteins, carbohydrate molecules, lipids or mixtures thereof, physiological components, anatomical components, epidemiological 10 components or chemical components. The training samples may be any data obtained from a system in which the feature of the sample is known. For example, training samples may be data generated from a sample applied 15 to a biological system. For example, when the biological system is a DNA microarray, the training sample may be data obtained from the array following hybridisation of the array with RNA extracted from cells having a known feature, or cDNA synthesised from the RNA extracted from cells, or if 20 the biological system is a proteomics electrophoresis gel, the training sample may be generated from a protein or cell extract applied to the system. It is envisaged that an embodiment of a method of the 25 present invention may be used in re-evaluating or evaluating test data from subjects who have presented mixed results in response to a test treatment. Thus, there is a second aspect to the present invention. 30 The second aspect provides a method for identifying a subset of components of a subject which are capable of classifying the subject into one of a plurality of predefined groups, wherein each group is defined by a response to a test treatment, the method comprising the steps of: 35 exposing a plurality of subjects to the test treatment and grouping the subjects into response groups based on responses to the treatment; WO2004/104856 PCT/AU2004/000696 - 17 measuring components of the subjects; and identifying a subset of components that is capable of classifying the subjects into response groups using a statistical analysis method. 5 Preferably, the statistical analysis method comprises the method according to the first aspect of the present invention. 10 Once a subset of components has been identified, that subset can be used to classify subjects into groups such as those that are likely to respond to the test treatment and those that are not. In this manner, the method of the present invention permits treatments to be identified which may be 15 effective for a fraction of the population, and permits identification of that fraction of the population that will be responsive to the test treatment. According to a third aspect of the present invention, there 20 is provided an apparatus for identifying a subset of components of a subject, the subset being capable of being used to classify the subject into one of a plurality of predefined response groups wherein each response group, is formed by exposing a plurality of subjects to a test 25 treatment and grouping the subjects into response groups based on the response to the treatment, the apparatus comprising: an input for receiving measured components of the subjects; and 30 processing means operable to identify a subset of components that is capable of being used to classify the subjects into response groups using a statistical analysis method. 35 Preferably, the statistical analysis method comprises the method according to the first or second aspect.

WO2004/104856 PCT/AU2004/000696 - 18 According to a fourth aspect of the present invention, there is provided a method for identifying a subset of components of a subject that is capable of classifying the subject as being responsive or non-responsive to treatment with a test 5 compound, the method comprising the steps of: exposing a plurality of subjects to the test compound and grouping the subjects into response groups based on each subjects response to the test compound; measuring components of the subjects; and 10 identifying a subset of components that is capable of being used to classify the subjects into response groups using a statistical analysis method. Preferably, the statistical analysis method comprises the 15 method according to the first aspect. According to a fifth aspect of the present invention, there is provided an apparatus for identifying a subset of components of a subject, the subset being capable of being 20 used to classify the subject into one of a plurality of predefined response groups wherein each response group is formed by exposing a plurality of subjects to a compound and grouping the subjects into response groups based on the response to the compound, the apparatus comprising: 25 an input operable to receive measured components of the subjects; processing means operable to identify a subset of components that is capable of classifying the subjects into response groups using a statistical analysis method. 30 Preferably, the statistical analysis method comprises the method according to the first or second aspect of the present invention. 35 The components that are measured in the second to fifth aspects of the invention may be, for example, genes or small nucleotide polymorphisms (SNPs), proteins, antibodies, WO2004/104856 PCT/AU2004/000696 - 19 carbohydrates, lipids or any other measurable component of the subject. In a particularly embodiment of the fifth aspect, the 5 compound is a pharmaceutical compound or a composition comprising a pharmaceutical compound and a pharmaceutically acceptable carrier. The identification method of the present invention may be 10 implemented by appropriate computer software and hardware. According to a sixth aspect of the present invention, there is provided an apparatus for identifying a subset of components of a system from data generated from the system 15 from a plurality of samples from the system, the subset being capable of being used to predict a feature of a test sample, the apparatus comprising: a processing means operable to: obtain a linear combination of components of the system 20 and obtain weightings of the linear combination of components, each of the weightings having a value based on data obtained from at least one training sample, the at least one training sample having a known feature; obtaining a model of a probability distribution of a 25 second feature, wherein the model is conditional on the linear combination of components; obtaining a prior distribution for the weightings of the linear combination of the components, the prior distribution comprising an adjustable hyperprior which 30 allows the prior probability mass close to zero to be varied wherein the hyperprior is not a Jeffrey's hyperprior; combining the prior distribution and the model to generate a posterior distribution; and identifying the subset of components having component 35 weights that maximize the posterior distribution.

WO2004/104856 PCT/AU2004/000696 - 20 Preferably, the processing means comprises a computer arranged to execute software. According to a seventh aspect of the present invention, 5 there is provided a computer program which, when executed by a computing apparatus, allows the computing apparatus to carry out the method according to the first aspect of the present invention. 10 The computer program may implement any of the preferred algorithms and method steps of the first or second aspect of the present invention which are discussed above. According to an eighth aspect of the present invention, 15 there is provided a computer readable medium comprising the computer program according with the seventh aspect of the present invention. According to a ninth aspect of the present invention, there 20 is provided a method of testing a sample from a system to identify a feature of the sample, the method comprising the steps of testing for a subset of components that are diagnostic of the feature, the subset of components having been determined by using the method according to the first 25 or second aspect of the present invention. Preferably, the system is a biological system. According to a tenth aspect of the present invention, there 30 is provided an apparatus for testing a sample from a system to determine a feature of the sample, the apparatus comprising means for testing for components identified in accordance with the method of the first or second aspect of the present invention. 35 According to an eleventh aspect of the present invention, there is provided a computer program which, when executed by WO2004/104856 PCT/AU2004/000696 - 21 on a computing device, allows the computing device to carry out a method of identifying components from a system that are capable of being used to predict a feature of a test sample from the system, and wherein a linear combination of 5 components and component weights is generated from data generated from a plurality of training samples, each training sample having a known feature, and a posterior distribution is generated by combining a prior distribution for the component weights comprising an adjustable 10 hyperprior which allows the probability mass close to zero to be varied wherein the hyperprior is not a Jeffrey's hyperprior, and a model that is conditional on the linear combination, to estimate component weights which maximise the posterior distribution. 15 Where aspects of the present invention are implemented by way of a computing device, it will be appreciated that any appropriate computer hardware e.g. a PC or a mainframe or a networked computing infrastructure, may be used. 20 According to a twelfth aspect of the present invention, there is provided a method of identifying a subset of components of a biological system, the subset being capable of predicting a feature of a test sample from the biological 25 system, the method comprising the steps of: obtaining a linear combination of components of the system and weightings of the linear combination of components, each of the weightings having a value based on data obtained from at least one training sample, the at 30 least one training sample having a known first feature; obtaining a model of a probability distribution of a second feature, wherein the model is conditional on the linear combination of components; obtaining a prior distribution for the weightings of 35 the linear combination of the components, the prior distribution comprising an adjustable hyperprior which allows the probability mass close to zero to be varied; WO2004/104856 PCT/AU2004/000696 - 22 combining the prior distribution and the model to generate a posterior distribution; and identifying the subset of components based on the weightings that maximize the posterior distribution. 5 BRIEF DESCRIPTION OF THE DRAWINGS Notwithstanding any other embodiments that may fall within the scope of the present invention, an embodiment of the 10 present invention will now be described, by way of example only, with reference to the accompanying figures, in which: figure 1 provides a flow chart of a method according to the embodiment of the present invention; 15 figure 2 provides a flow chart of another method according to the embodiment of the present invention; figure 3 provides a block diagram of an apparatus 20 according to the embodiment of the present invention; figure 4 provides a flow chart of a further method according to the embodiment of the present invention; 25 figure 5 provides a flow chart of an additional method according to the embodiment of the present invention; and figure 6 provides a flow chart of yet another method according to the embodiment of the present invention. 30 DETAILED DESCRIPTION OF AN EMBODIMENT The embodiment of the present invention identifies a relatively small number of components which can be used to 35 identify whether a particular training sample has a feature. The components are "diagnostic" of that feature, or enable discrimination between samples having a different feature.

WO2004/104856 PCT/AU2004/000696 - 23 The number of components selected by the method can be controlled by the choice of parameters in the hyperprior. It is noted that the hyperprior is a gamma distribution with a specified shape and scale parameter. Essentially, from all 5 the data which is generated from the system, the method of the present invention enables identification of a relatively small number of components which can be used to test for a particular feature. Once those components have been identified by this method, the components can be used in 10 future to assess new samples. The method of the present invention utilises a statistical method to eliminate components that are not required t 9 correctly predict the feature. 15 The inventors have found that component weightings of a linear combination of components of data generated from the training samples can be estimated in such a way as to eliminate the components that are not required to correctly predict the feature of the training sample. The result is 20 that a subset of components are identified which can correctly predict the feature of the training sample. The method of the present invention thus permits identification from a large amount of data a relatively small and controllable number of components which are capable of 25 correctly predicting a feature. The method of the present invention also has the advantage that it requires usage of less computer memory than prior art methods. Accordingly, the method of the present 30 invention can be performed rapidly on computers such as, for example, laptop machines. By using less memory, the method of the present invention also allows the method to be performed more quickly than other methods which use joint (rather than marginal) information on components for 35 analysis of, for example, biological data.

WO2004/104856 PCT/AU2004/000696 - 24 The method of the present invention also has the advantage that it uses joint rather than marginal information on components for analysis. 5 A first embodiment relating to a multiclass logistic regression model will now be described. A. Multi Class Logistic regression model 10 The method of this embodiment utilises the training samples in order to identify a subset of components which can classify the training samples into pre-defined groups. Subsequently, knowledge of the subset of components can be used for tests, for example clinical tests, to classify 15 samples into groups such as disease classes. For example, a subset of components of a DNA microarray may be used to group clinical samples into clinically relevant classes such as, for example, healthy or diseased. 20 In this way, the present invention identifies preferably a small and controllable number of components which can be used to identify whether a particular training sample belongs to a particular group. The selected components are "diagnostic" of that group, or enable discrimination between 25 groups. Essentially, from all the data which is generated from the system, the method of the present invention enables identification of a small number of components which can be used to test for a particular group. Once those components have been identified by this method, the components can be 30 used in future to classify new samples into the groups. The method of the present invention preferably utilises a statistical method to eliminate components that are not required to correctly identify the group the sample belongs to. 35 The samples are grouped into sample groups (or "classes") based on a pre-determined classification. The classification WO2004/104856 PCT/AU2004/000696 - 25 may be any desired classification by which the training samples are to be grouped. For example, the classification may be whether the training samples are from a leukemia cell or a healthy cell, or that the training samples are obtained 5 from the blood of patients having or not having a certain condition, or that the training samples are from a cell from one of several types of cancer as compared to a normal cell. In one embodiment, the input data is organised into an 10 nxpdata matrix X=(xi) with n training samples and p components. Typically, p will be much greater than n. In another embodiment, data matrix X may be replaced by an n x n kernel matrix K to obtain smooth functions of X as 15 predictors instead of linear predictors. An example of the kernel matrix K is kij=exp(-0.5*(x,-xj)t(xi-xj)/a 2 ) where the subscript on x refers to a row number in the matrix X. Ideally, subsets of the columns of K are selected which give sparse representations of these smooth functions. 20 Associated with each sample class (group) may be a class label yi, where yi=k,ke{1,..,G}, which indicates which of G sample classes a training sample belongs to. We write the nxl vector with elements yi as y.. Given the vector y we 25 can define indicator variables 1, Yi = 9 eig = 0, otherwise (Al) In one embodiment, the component weights are estimated using a Bayesian statistical model (see Kotz and Johnson, 1983). 30 Preferably, the weights are estimated by maximising the posterior distribution of the weights given the data generated from each training sample. This results in an objective function to be maximised consisting of two parts. The first part a likelihood function and the second a prior 35 distribution for the weights which ensures that zero weights are preferred whenever possible. In a preferred embodiment, WO 2004/104856 PCT/AU2004/000696 - 26 the likelihood function is derived from a multiclass logistic model. Preferably, the likelihood function is computed from the probabilities: XiI8g Pig = 1,,G - (A2) 1+ 1 e Xi J; h=1 5 and 1 PiG + G- (A3)

~

1 + @exIf jh h=1 wherein 10 Pig is the probability that the training sample with input data X, will be in sample class g; x TPg is a linear combination generated from input data from training sample i with component weights fig; T th orteth xis the components for the i Row of X and P. is a set of 15 component weights for sample class g; Typically, as discussed above, the component weights are estimated in a manner which takes into account the apriori assumption that most of the component weights are zero. 20 In one embodiment, components weights fig in equation (A2) are estimated in a manner whereby most of the values are zero, yet the samples can still be accurately classified. 25 In one embodiment, the component weights are estimated by maximising the posterior distribution of the weights given the data in the Bayesian model referred to above'. Preferably, the component weights are estimated by 30 WO 2004/104856 PCT/AU2004/000696 - 27 (a) specifying a hierarchical prior for the component weights A,..-,fG-1; and (b) specifying a likelihood function for the input data; (c) determining the posterior distribution of the weights 5 given the data using (A5); and (d) determining component weights which maximise the posterior distribution. In one embodiment, the hierarchical prior specified for the 10 parameters A,...,flG-i is of the form: G-1 P(A...,f,_-)= f tPT & p(2)dr2 (A4) T2 g=1 where /3 =( f',...f_1 , = (.

T ) p/glr) is N0(,diag{ 7 and 15 pr is a suitable prior. n In one embodiment, p(r)= p(ri) where p(;) is a prior i=1 wherein t 2=1/rF has an independent gamma distribution. I2~ 2 20 In another embodiment, pig) is a prior wherein rig has an independent gamma distribution. In one embodiment, the likelihood function is L(yfll,...,fG of the form in equation (8) and the posterior distribution of 25 /3 and r given y is p(f rIy) a L(yl )p(/8 I2)p( 2) (A5) In one embodiment, the likelihood function has a first and 30 second derivative. In one embodiment, the first derivative is determined from the following algorithm: WO 2004/104856 PCT/AU2004/000696 - 28 alogL a=g -xr eg-pg), g 1,...,G-1 (A6) d/g wherein =ei=1,...,n)p =(pii=...,n) are vectors indicating 5 membership of sample class g and probability of class g respectively. In one embodiment, the second derivative is determined from the following algorithm: 10

D

2 1ogL= -XT d i ag {6hgpg - PhPg }X (A7) where hg is 1 if h equals g and zero otherwise. 15 Equation A6 and equation A7 may be derived as follows: (a) Using equations (Al), (A2) and (A3), the likelihood function of the data can be written as: ' 9 eiG n G-1 xe 20 L= G-1 G (AB) i=1 g=1 1+ eP_ 1 + ex g=1 } h=1 (b) Taking logs of equation (A6) and using the fact that G eih=1 for all i gives: h=1 n= G-1 G- 1 T 25 logL= ex j-log 1+

Y

e g= (A9) c) Differentiating equation (A) with respect to g gives (c) Differentiating equation (A8) with respect to fig gives WO 2004/104856 PCT/AU2004/000696 - 29 al=gL xe- pg), g = ,...,G -1 (A1O) aMg whereby '= (eig,i=1,n), p" =(pig,i=1,n) are vectors indicating 5 membership of sample class g and probability of class g respectively. (d) The second derivative of equation (9) has elements 10 a =lgL -XTdiag {45hgpg - phPg}X (All) where 1, h=g 0hg , otherwise 15 Component weights which maximise the posterior distribution of the likelihood function may be specified using an EM algorithm comprising an E step and an M step. 20 In conducting the EM algorithm, the E step preferably comprises the step computing a term of the form: G-1 n P=E ig/rig I g} g=1 i=1 (Alla) G-1 n = I-"3j/dg g=l i=1 25 where d,g=E{1/,| fig}-s idg=(dlg,d 2 g,..., dpg)T and dig =l/dg=Oiff,=O. Preferably, equation (11a) is computed by calculating the 2 =1 2 IrL) i conditional expected value of tg =1/r when p(/9gTg) is N(0,f;) and p(r) has a specified prior distribution.

WO 2004/104856 PCT/AU2004/000696 - 30 Explicit formulae for the conditional expectation will be presented later. 5 Typically, the EM algorithm comprises the steps: (a) performing an E step by calculating the conditional expected value of the posterior distribution of component weights using the 10 function: 1 Q=Q(rly,) =logL- 2 diag{d(? )}- Y (A12) 2=1 where xfg=x[P, Pgyg in equation (8), d(fg)=Pgd,, and 15 d is defined as in equation (1la) evaluated at = -Pgg. Here Pg is a matrix of zeroes and ones derived from the identity matrix such that PgTflg selects non-zero elements of fig which are denoted by r. 20 (b) performing an M step by applying an iterative procedure to maximise Q as a function of y whereby: 25 Y =7 r ty 2 (d (A13) where a' is a step length such that 0<a' _; and y = (79, g=l ... , G-1). 30 Equation (A12) may be derived as follows: Calculate the conditional expected value of (A5) given the observed data y and a set of parameter estimates 6.

WO 2004/104856 PCT/AU2004/000696 - 31 Q=Q(fl y, fl)= E{logp(3,ry) y,) Consider the case when components of 8 (and / ) are set to zero i.e for g=1,...,G-l,1fg=Pgg and #g=Pgjg. 5 Ignoring terms not involving yand using (A4), (A5), (A9) we get: 1G-1 n 2 Q= log L--EE '" y, 2 72 2 g=1 i=1 ig 1 G-1 n 10 Q= logL 2 G i E 2 - Y, 2 g=1 i=1 ig G-1 log L - T diag dg(f) 7, (A14) 2g=1 where xf,g=x[Pgg in (A8), dg(?g)=P

T

dg where d is defined 15 as in equation (Alla) evaluated at ig =Pig. Note that the conditional expectation can be evaluated from first principles given (A4). Some explicit expressions are given later. 20 The iterative procedure may be derived as follows: To obtain the derivatives required in (11), first note that from (A8), (A9) and (A10) writing d(f)={dg(fg),g=l,...,G-l}, we get 25 DQ_ 8f@Io l g L -Q _ (- Lo diag {d()} - 2 7 diay(d(f } a7 ar ap8 X(e - pi) = e -diag{d()} 7 (A15) -XG_ (eG-1 PG-1) WO 2004/104856 PCT/AU2004/000696 - 32 and

)

2 Q (, g2 log L 1,., -2 T - diag {d (f ) 5 10 and XI A X ... XIAG-AXG In a preferred embodiment, the iterative procedure may be 15 simplified by using only the block diagonals of equation (Al6) in equation (A13). For g=,...G-1, this gives: =+ - . .diag Id )} (A16) (A18) 20 Rearranging equation (AI8) leads to 7 XG-1 G-Illi XG-1AG-lG-11XG-1, A+ = diag 3d( g ( - p ,h (AI9) 5 where =1, g S0, otherwisediag{dgg)X d(?) )= (d(?g), g = ,.,G - 1) 10 and XgT = Pg'X', g = 1,...G -1I (A17) In a preferred embodiment, the iterative procedure may be 15 simplified by using only the block diagonals of equation (A16) in equation (A13) . For g= 1,...G -1, this gives: y"' = 7 a fX gA.Xg + diag (d,(,) T- -1Xeg - pg) - diag fdg (f,)j y (A18) 20 Rearranging equation (A18) leads to y9 = y +a'diag dg(f,)) (YgTAggY +I)-1 Y" (e - pg)-diag dg(g)I 7gt (A19) 25 where Yr = diag Idg (fg)j X WO 2004/104856 PCT/AU2004/000696 - 33 Writing p(g) for the number of columns of Yg, (A19) requires the inversion of a p(g)xp(g) matrix which may be quite large. This can be reduced to an nxn matrix for p(g)>n by 5 noting that: ( Y f 9 ( + )1 -I (A20) = I - ZgZ I Zg where Zg =A Y. Preferably, (A19) is used when p(g)>n and 10 (A19) with (A20) substituted into equation (A19) is used when p(g) n. Note that when r has a Jeffreys prior we have: 15 E{t2 I }ig} = 1/2 In one embodiment, t 2 =1/r 2 has an independent gamma In ne mboimet, ig r19 distribution with scale parameter b>0 and shape parameter 2 k>0 so that the density of tg is: 20 *"t, (t2/ b)k-leXp(-t2 /b)/I(k) Omitting subscripts to simplify the notation, it can be shown that 25 E{ t 2 I 3 } = (2k+1)/(2/b + ) (A21) as follows: 30 Define I(p,b,k)= J(t 2 )P t exp(-0.5fl 2 t 2 )t 2 ,b,k)dt 2 0 then WO 2004/104856 PCT/AU2004/000696 - 34 I(p,b,k)=b p0 5 {r(p+k+0.5)/F(k) } (1+0.5b 2 )-(p+k+0.5) Proof 5 Let s=-/ 2 /2 then I(p,b,k)=becO s (t2/b)p+0.

5 exp(-st 2 2, b, k)dt2 0 Now using the substitution u = t 2 /b we get I(p,b,k)=b p+ o

'

s f(u) p+ 0 s exp(-sub)M(u,1,k)du 0 Now let s'=bs and substitute the expression for (u,1,k). This 10 gives I(p,bk)=bP°s5 exp(-(s'+1)u)uP+k+S'-ldu / F(k) 0 Looking up a table of Laplace transforms, eg Abramowitz and Stegun, then gives the result. 15 The conditional expectation follows from E{ t 2 I 3 } = I(1,b,k)/I(0,b,k) = (2k+1)/(2/b + 02) As k tends to zero and b tends to infinity we get the 20 equivalent result using Jeffreys prior. For example, for k=0.005 and b=2*10 s E{ t 2 I 0 } = (1.01)/(10-s +o2) Hence we can get arbitrarily close to the Jeffreys prior 25 with this proper prior. The algorithm for this model has WO 2004/104856 PCT/AU2004/000696 - 35 d=E{t 2 1 -0.5 =E{- I )

-

} where the expectation is calculated as above. 2 In another embodiment, Tig has an independent gamma 5 distribution with scale parameter b>0 and shape parameter k>0. It can be shown that fuk-3/2-1exp(-(y ./u+u))du E { rTig -2 Iig }= o b fuk-1/2-1exp(-( i/u +u))du 0 f 1 K3/2-k (2Xi) (A22) = I I K 2 (A2 2) b ,| l/-k (2 ) 1 (2Fie)K 3

/

2 -k (2 F) I O 3 ig 12 K1/2-k (2j )ig 10 where yig=Og/2b and K denotes a modified Bessel function. For k=1 in equation (A22) E{-r, -0 2

}

= V2/b (1/|4f, |) 15 For K=0.5 in equation (A22) E{rg I3pig}= 42/b(1/3ig ) {K, (24-,ig)/Ko (2 7,ig)} or equivalently 20 E { z|g If3,}= (1/3ji12 ) {21-i K, (2 jg4)/K 0 (24Vy)} Proof of (A.1) From the definition of the conditional expectation, writing 25 =03 2 /2b, we get WO 2004/104856 PCT/AU2004/000696 - 36 r-2 -lexp(-yr-2)b-'(r-2/b)k-lexp(r- 2 /b)dr 2 E{ r- 2 103}= r lexp(-x-2)b-'(r- 2 / kexp(r- 2 /b)d r 2 0 Rearranging, simplifying and making the substitution u=r 2 /b gives the first equation in (A22). 5 The integrals in (22) can be evaluated by using the result x

-

b - exp - x+- = Kb(2a) where K denotes a modified Bessel function, see Watson(1966). 10 Examples of members of this class are k=l in which case E{ V-ig -21 3ig}= 21b(1/13igI) 15 which corresponds to the prior used in the Lasso technique, Tibshirani(1996). See also Figueiredo(2001). The case k=0.5 gives 20 E{ r - Ag}= M(1/|ig)f{K,(2F)/K o (2 )} or equivalently E{ r Iflig} = (1/1f l i 2 ){24iK (2 F )/Ko(2 J)} 25 where K 0 and K3 are modified Bessel functions, see Abramowitz and Stegun(1970). Polynomial approximations for evaluating these Bessel functions can be found in Abramowitz and Stegun(1970, p 379 ). The expressions above demonstrate the 30 connection with the Lasso model and the Jeffreys prior model.

WO2004/104856 PCT/AU2004/000696 - 37 It will be appreciated by those skilled in the art that as k tends to zero and b tends to infinity the prior tends to a Jeffreys improper prior. 5 In one embodiment, the priors with 0< k 51 and b>0 form a class of priors which might be interpreted as penalising non zero coefficients in a manner which is between the Lasso prior and the specification using Jeffreys hyper prior. 10 The hyperparameters b and k can be varied to control the number of components selected by the method. As k tends to zero for fixed b the number of components selected can be decreased and conversely as k tends to 1 the number of 15 selected components can be increased. In a preferred embodiment, the EM algorithm is performed as follows: 20 1. Set n=0 ,Pg =I and choose an initial value for fo.Choose a value for b and k in equation (A22). For example b=1e7 and k=0 gives the Jeffreys prior model to a good degree of approximation. This is done by ridge regression of log(Pig/PiG ) on Xi where Pig is chosen to be near one for 25 observations in group g and a small quantity >0 otherwise - subject to the constraint of all probabilities summing to one. 2.Do the E step i.e evaluate Q=Q(ryy,"). Note that this 30 also depends on the values of k and b. 3. Set t=0. For g=1,...,G-1 calculate: a) 8 = -- using (A19) with (A20) substituted into (A19) when p(g)2n. 35 (b) Writing '=(3,g=,...,G-1) Do a line search to find the value of a' in y+' = y' +a' ' which maximises (or simply increases) (12) as a function of a'.

WO2004/104856 PCT/AU2004/000696 - 38 c) set t+'=y and t=t+l Repeat steps (a) and (b) until convergence. 5 This produces y*n+ say which maximises the current Q function as a function of y r .n+ *i+11 For g=1,...G-1 determine S = j:y jg E maxy kg I k I J1 Where E<<1 , say 10 -s. Define P so that flig =0 for ieSgand 10 #n+1 = rf*n+I , = g , gIj, This step eliminates variables with small coefficients from the model. 15 4.Set n=n+l and go to 2 until convergence. A second embodiment relating to an ordered cat logistic regression will now be described. 20 B. Ordered categories model The method of this embodiment may utilise the training samples in order to identify a subset of components which 25 can be used to determine whether a test sample belongs to a particular class. For example, to identify genes for assessing a tissue biopsy sample using microarray analysis, microarray data from a series of samples from tissue that has been previously ordered into classes of increasing or 30 decreasing disease severity such as normal tissue, benign tissue, localised tumour and metastasised tumour tissue are used as training samples to identify a subset of components which is capable of indicating the severity of disease associated with the training samples. The subset of 35 components can then be subsequently used to determine whether previously unclassified test samples can be WO 2004/104856 PCT/AU2004/000696 - 39 classified as normal, benign, localised tumour or metastasised tumour. Thus, the subset of components is diagnostic of whether a test sample belongs to a particular class within an ordered set of classes. It will be apparent 5 that once the subset of components have been identified, only the subset of components need be tested in future diagnostic procedures to determine to what ordered class a sample belongs. 10 The method of the invention is particularly suited for the analysis of very large amounts of data. Typically, large data sets obtained from test samples is highly variable and often differs significantly from that obtained from the training samples. The method of the present invention is 15 able to identify subsets of components from a very large amount of data generated from training samples, and the subset of components identified by the method can then be used to classifying test samples even when the data generated from the test sample is highly variable compared 20 to the data generated from training samples belonging to the same class. Thus, the method of the invention is able to identify a subset of components that are more likely to classify a sample correctly even when the data is of poor quality and/or there is high variability between samples of 25 the same ordered class. The components are "predictive" for that particular ordered class. Essentially, from all the data which is generated from the system, the method of the present invention enables 30 identification of a relatively small number of components which can be used to classify the training data. Once those components have been identified by this method, the components can be used in future to classify test samples. The method of the present invention preferably utilises a 35 statistical method to eliminate components that are not required to correctly classify the sample into a class that is a member of an ordered class.

WO 2004/104856 PCT/AU2004/000696 - 40 In the following there are N samples, and vectors such as y, z and 9 have components yi, zi and pi for i = 1, ..., N. Vector multiplication and division is defined componentwise and 5 diag{ } denotes a diagonal matrix whose diagonals are equal to the argument. We also use to denote Euclidean norm. Preferably, there are N observations y*i where yi takes 10 integer values 1,...,G. The values denote classes which are ordered in some way such as for example severity of disease. Associated with each observation there is a set of covariates (variables, e.g gene expression values) arranged into a matrix X* with n row and p columns wherein n is the *T 15 samples and p the components. The notation xi denotes the i t h row of X'. Individual (sample) i has probabilities of belonging to class k given by &k=7rk(X,). Define cumulative probabilities k Yik IZik , k = 1,G g= 1 20 Note that Y'ik is just the probability that observation i belongs to a class with index less than or equal to k. Let C be a N by p matrix with elements cy given by 1, if observation i in classj C - 0, otherwise 25 and let R be an n by P matrix with elements ry given by i = g=1 These are the cumulative sums of the columns of C within rows. 30 For independent observations (samples) the likelihood of the data may be written as: WO 2004/104856 PCT/AU2004/000696 - 41 N G-1 rar , - -r. I = k_ , i k+1 ik i=1 k=1 ik+1 Yik+l and the log likelihood L may be written as: NV G-I 5 L= EEr log +( -rk) log rik+I Yik i=1 j=1 ik+1 Yik+1 l The continuation ratio model may be adopted here as follows: logit Yik+l Yik l=ogit C ik = + x T ik+1 ik+1 10 for k = 2,...,G , see McCullagh and Nelder(1989) and McCullagh(1980) and the discussion therein. Note that logit +~ ik+ =- k logit k ( Yik+l ( k ik+l 15 The likelihood is equivalent to a logistic regression likelihood with response vector yand covariate matrix X y =vec{R} X =BITB2... Bx TT T]T

B

i = o[I-, lo-Ix r wherel_l is the G-1 by G-1 identity matrix and 1

I

G-1 is a G-1 20 by 1 vector of ones. Here vec{ } takes the matrix and forms a vector row by row. Typically, as discussed above, the component weights are estimated in a manner which takes into account the a priori 25 assumption that most of the component weights are zero. Following Figueiredo(2001), in order to eliminate redundant variables (covariates), a prior is specified for the WO 2004/104856 PCT/AU2004/000696 - 42 parameters f* by introducing a p x 1 vector of hyperparameters. Preferably, the prior specified for the component weights is 5 of the form p (/*)= Jp v p 2)p(v2)dv2 2 (al) where p(fl*v2) is N(0,diag{v2}) and p(v 2 ) is a suitably chosen hyperprior. For example, p(v 2 )rjp(v7) is a suitable form of i=1 10 Jeffreys prior. In another embodiment, p(v ) is a prior wherein t i =1/v i has an independent gamma distribution. 15 In another embodiment, p(V7) is a prior wherein Vi/ has an independent gamma distribution. The elements of theta have a non informative prior. 20 Writing L(y f38*) for the likelihood function, in a Bayesian framework the posterior distribution of , and v given y is p f'p vy) aL (y 13*fp)p (f* v)p(v) (2) 25 By treating v as a vector of missing data, an iterative algorithm such as an EM algorithm (Dempster et al, 1977) can be used to maximise (2) to produce maximum a posteriori estimates of P and 0. The prior above is such that the WO2004/104856 PCT/AU2004/000696 - 43 maximum a posteriori estimates will tend to be sparse i.e. if a large number of parameters are redundant, many components of P* will be zero. 5 Preferably f8T=(OT,/*T) in the following: For the ordered categories model above it can be shown that aL =Xt (y -A) (11) E{ }= - X t diag {u(1-p)}X (12) 10 Where p= exp(xTf8)/(1+exp(xTf8))and fl'=( 0 2,...,0,,) . The iterative procedure for maximising the posterior distribution of the components and component weights is an EM algorithm, such as, for example, that described in 15 Dempster et al, 1977. Preferably, the EM algorithm is performed as follows: 1. Chose a hyperprior and values b and k for its parameters. Set n=0, So = {1,2,..., p } , o ( ) , and E =10 -s (say). Set the 20 regularisation parameter K at a value much greater than 1, say 100. This corresponds to adding 1/K 2 to the first G-1 diagonal elements of the second derivative matrix in the M step below. 25 If p N compute initial values P* by

*=(X

t

X+M),

1 XTg(y+ ) (B2) and if p > N compute initial values P* by 30 1 #*= (I -X T

(XX

T +M)-1X)X

T

g(y+ ) (B3) where the ridge parameter X satisfies 0 < X 1 and 4 is small and chosen so that the link function g(z)=log(z/(1-z)) is well defined at z = y+ .

WO 2004/104856 PCT/AU2004/000696 - 44 2. Define 5 p n { ¢, iC S. n 0, otherwise and let Pn be a matrix of zeroes and ones such that the nonzero elements y " (n) of (n) satisfy 10 ) = p.T 13 (n) , =() p , fn) 7 = P.TI , = Pn7 Define w,=(w,i=l,p), such that 1, i >G wpi 0, otherwise and let wy =Pw6 15 3. Perform the E step by calculating Q(n ), I (") = E{ logp(p, p , v I y) I y, 0('), "n) (15) (15) = L(y 10, 4") )-0.5 (|I (P*w6)/ (1) 2 ) where L is the log likelihood function of y and 20 dig =E{lrg I i8}-0'5 dg=(dl2g " ,dpg) and for convenience we define d=1/dig=O,= if& =0. . Using f=Pnyand #(n) = p Yn) (15) can be written as Q(7 I Yn ), 4() = L(y I Pny, 4(n))-0.5 (Il(y*wr)/d(yn))I ) (B4) 25 with d(r"))=prd( ") evaluated at "Y(n). 4. Do the M step. This can be done with Newton Raphson iterations as follows. Set Yo = Y(n) and for r=0,1,2,... Tr+I1 WO 2004/104856 PCT/AU2004/000696 - 45 = Yr + ar 8r where ar is chosen by a line search algorithm to ensure Q(y,,Q | n) q"') >Q(y , I ), 0(n)) For p < N use 8, = A(d*(7"))[YnV; Y.+I]'(Y.zr- ,)) (BS) 5 where d(y()), i _ G d* (y("')=( ic, otherwise YJ= A(d (y(n)))PnTX T 10 V,-'=diag {up,(1-pr)} zr =(Y-) and ILr = exp( XPn yr)/(1+exp( XP.,y)). 15 For p > N use Tr wyYr B,= A(d*(y(")))[I-YT(yYY+V,)'Yn](Y/z, wr ) (B6) with Vr and Zr defined as before. 20 Let y* be the value of yr when some convergence criterion is satisfied e.g II Yr - Yr+l| I < s (for example 10- 5 ) 25 5. Define P* = PnY* , Sn+ 1 ={iG: fli >max(I jl*Ei)} where s is a jaG small constant, say le-5. Set n=n+l . 6. Check convergence. If I Iy* - Yn)I I < 82 where 92 is suitably small then stop, else go to step 2 above. 30 Recovering the probabilities Once we have obtained estimates of the parameters 1 are obtained, calculate WO 2004/104856 PCT/AU2004/000696 - 46 aik = 'i ik agk= Yik for i=l,...,N and k = 2,...,G. Preferably, to obtain the probabilities we use the recursion ciG aiG 5r - ai-1 _ ik );ik aik and the fact that the probabilities sum to one, for i = 1,- ,N. In one embodiment the covariate matrix X 10 with rows xi T can be replaced by a matrix K with ijt h element kj and kij = c( x i - xj ) for some kernel function K. This matrix can also be augmented with a vector of ones. Some example kernels are given in Table 1 below, see Evgeniou et al(1999). 15 Kernel function Formula for K( x - y ) Gaussian radial basis function exp( - I x - y / a) , a>0 Inverse multiquadric ( x y I2 2 -1/2 multiquadric ( x - y 1+ C 2 I Thin plate splines x - y 2n+1 Ix - y I12nln(1 x - y 1) Multi layer perceptron tanh( x'y-0 ) , for suitable 0 Ploynomial of degree d (1 + xy ) B splines B 2 n+1 (x - y) Trigonometric polynomials sin(( d +1/2 )(x-y))/sin((x y)/2) Table 1: Examples of kernel functions In Table 1 the last two kernels are preferably one dimensional i.e. for the case when X has only one column. 20 Multivariate versions can be derived from products of these kernel functions. The definition of B 2 n.+ 1 can be found in De Boor(1978). Use of a kernel function results in mean values WO 2004/104856 PCT/AU2004/000696 - 47 which are smooth (as opposed to transforms of linear) functions of the covariates X. Such models may give a substantially better fit to the data. 5 A third embodiment relating to generalised linear models will now be described. C. Generalised Linear Models 10 The method of this embodiment utilises the training samples in order to identify a subset of components which can predict the characteristic of a sample. Subsequently, knowledge of the subset of components can be used for tests, for example clinical tests to predict unknown values of the 15 characteristic of interest. For example, a subset of components of a DNA microarray may be used to predict a clinically relevant characteristic such as, for example, a blood glucose level, a white blood cell count, the size of a tumour, tumour growth rate or survival time. 20 In this way, the present invention identifies preferably a relatively small number of components which can be used to predict a characteristic for a particular sample. The selected components are "predictive" for that 25 characteristic. By appropriate choice of hyperparameters in the hyper prior the algorithm can be made to select subsets of varying sizes. Essentially, from all the data which is generated from the system, the method of the present invention enables identification of a small number of 30 components which can be used to predict a particular characteristic. Once those components have been identified by this method, the components can be used in future to predict the characteristic for new samples. The method of the present invention preferably utilises a statistical 35 method to eliminate components that are not required to correctly predict the characteristic for the sample.

WO2004/104856 PCT/AU2004/000696 - 48 The inventors have found that component weights of a linear combination of components of data generated from the training samples can be estimated in such a way as to eliminate the components that are not required to predict a 5 characteristic for a training sample. The result is that a subset of components are identified which can correctly predict the characteristic for samples in the training set. The method of the present invention thus permits identification from a large amount of data a relatively 10 small number of components which are capable of correctly predicting a characteristic for a training sample, for example, a quantity of interest. The characteristic may be any characteristic of interest. In 15 one embodiment, the characteristic is a quantity or measure. In another embodiment, they may be the index number of a group, where the samples are grouped into two sample groups (or "classes") based on a pre-determined classification. The classification may be any desired classification by which 20 the training samples are to be grouped. For example, the classification may be whether the training samples are from a leukemia cell or a healthy cell, or that the training samples are obtained from the blood of patients having or not having a certain condition, or that the training samples 25 are from a cell from one of several types of cancer as compared to a normal cell. In another embodiment the characteristic may be a censored survival time, indicating that particular patients have survived for at least a given number of days. In other embodiments the quantity may be any 30 continuously variable characteristic of the sample which is capable of measurement, for example blood pressure. In one embodiment, the data may be a quantity yi, where ie {1,...,N}. We write the nxl vector with elements yi as y. We 35 define a p x 1 parameter vector 0 of component weights (many of which are expected to be zero), and a q x 1 vector of parameters T (not expected to be zero). Note that q could be WO2004/104856 PCT/AU2004/000696 - 49 zero (i.e. the set of parameters not expected to be zero may be empty). In one embodiment, the input data is organised into an 5 nxpdata matrix X=(x) with n test training samples and p components. Typically, p will be much greater than n. In another embodiment, data matrix X may be replaced by an n x n kernel matrix K to obtain smooth functions of X as 10 predictors instead of linear predictors. An example of the kernel matrix K is kij=exp(-0.5*(xi-xj)t(x--xj)/ 2 ) where the subscript on x refers to a row number in the matrix X. Ideally, subsets of the columns of K are selected which give sparse representations of these smooth functions. 15 Typically, as discussed above, the component weights are estimated in a manner which takes into account the apriori assumption that most of the component weights are zero. 20 In one embodiment, the prior specified for the component weights is of the form: p(f)= fp(f 8v2)p(v2)dv2 V2 (Cl) wherein v is a p x 1 vector of hyperparameters, and where 25 p('8 V2) is N(O,diag{v2}) and p(V2) is some hyperprior distribution for v 2. P A suitable form of hyperprior is, p (v2)alp(v7). Jeffreys i=1 In another embodiment, the hyperprior p(v 2) is such that 30 each t i 2 =1/v/ 2 has an independent gamma distribution. In another embodiment, the hyperprior p(V 2 ) is such that each vi'2 has an independent gamma distribution.

WO2004/104856 PCT/AU2004/000696 - 50 Preferably, an uninformative prior for T is specified. The likelihood function is defined from a model for the 5 distribution of the data. Preferably, in general, the likelihood function is any suitable likelihood function. For example, the likelihood function L(y|}fl) may be, but not restricted to, of the form appropriate for a generalised linear model (GLM), such as for example, that described by 10 Nelder and Wedderburn (1972). Preferably, in this case, the likelihood function is of the form: L =log p(y |4,)= -) { y -()+ c(y,,#) } (C2) i=1 ai(o) where y = (Y-..., yn)T and ai (4) = q /wi with the wi being a 15 fixed set of known weights and k a single scale parameter. Preferably, the likelihood function is specified as follows: We have E{yi} = b'(0) 20 Var{y) = b"(04)ai( = ai( (C3) Each observation has a set of covariates xi and a linear predictor 7± = xi T j. The relationship between the mean of the i th observation and its linear predictor is given by the link function ni = g(Ai) = g( b' (i) ). The inverse of the 25 link is denoted by h, i.e Ai = b'(0) = h(Wi). In addition to the scale parameter, a generalised linear model may be specified by four components: 30 * the likelihood or (scaled) deviance function, * the link function * the derivative of the link function * the variance function.

WO2004/104856 PCT/AU2004/000696 - 51 Some common examples of generalised linear models are given in the table below. Distribution Link function Derivative Variance Scale g(y) of link function parame function ter Gaussian 4 1 1 Yes Binomial log(A/(l- A)) 1/( A(l- p)) d(1- No i) /n Poisson log (A) 1/ A A No Gamma 1/ A -1/ ;2 2 Yes Inverse 1/ 42 -2/ p3 3 Yes Gaussian 5 In another embodiment, a quasi likelihood model is specified wherein only the link function and variance function are defined. In some instances, such specification results in the models in the table above. In other instances, no distribution is specified. 10 In one embodiment, the posterior distribution of 8 op and v given y is estimated using: p(fpvly) a L(yfl(p)p(fl/v)p(v) (C4) 15 wherein L(y/58) is the likelihood function. In one embodiment, v may be treated as a vector of missing data and an iterative procedure used to maximise equation 20 (C4) to produce maximum a posteriori estimates of 0. The prior of equation (Cl) is such that the maximum a posteriori estimates will tend to be sparse i.e. if a large number of parameters are redundant, many components of P will be zero. 25 As stated above, the component weights which maximise the posterior distribution may be determined using an iterative procedure. Preferable, the iterative procedure for WO 2004/104856 PCT/AU2004/000696 - 52 maximising the posterior distribution of the components and component weights is an EM algorithm comprising an E step and an M step, such as, for example, that described in Dempster et al, 1977. 5 In conducting the EM algorithm, the E step preferably comprises the step of computing terms of the form P=.E{ ?/vf I i} i=1 (C4a) p i=1I where dj=d,(j,)=E{11vi ;}~O5 and for convenience we define 10 di=1/di= OifiA=0. In the following we write d=d(fl)=(d,d 2 ,...,d,)

T

. In a similar manner we define for example d(fl (n) ) and d(y"))=P, Td(Py

"

)) where fl("=y(,) and P, is obtained from the p by p identity matrix by omitting columns j for which ")=0. 15 Preferably, equation (C4a) is computed by calculating the conditional expected value of ti 2 =1/Vi 2 when p(i3v7) is N(O,v) and p(v) has a specified prior distribution. Specific examples and formulae will be given later. 20 In a general embodiment, appropriate for any suitable likelihood function, the EM algorithm comprises the steps: (a) Selecting a hyperprior and values for its parameters. Initialising the algorithm by setting n=0, So = {1,2,..., p } , initialise 9(O) 25 , P* and applying a value for s, such as for example c = 10-s5; (b) Defining , {if S. (C5) 0, otherwise and let Pn be a matrix of zeroes and ones such 30 that the nonzero elements y (n) of p satisfy WO 2004/104856 PCT/AU2004/000696 - 53 n) pT~ 3 (n) 3(n) -- Pn n) 'y=P 1 T , 8 = ., (c) performing an estimation (E) step by calculating the conditional expected value of the posterior 5 distribution of component weights using the function: Q(P I

,

(p() , ()) = E{ logp(p, v I v y) I y, ("), O ")} ' (C6) = L(yl f, on) -0.5 fi T A(d(9",)))- 2 , 10 where L is the log likelihood function of y. Using #=P.y and d(

")

) as defined in (C4a), (C6) can be written as Q(YI 7 n, e(") = L(y I P,, (nC))-0-5 YT A(d('"))-2 C7) 15 (d) performing a maximisation (M) step by applying an iterative procedure to maximise Q as a function of y whereby yo = y( n ) and for r=0,1,2, Yr4l = Yr + ar br and where ar is chosen by a 20 line search algorithm to ensure Q(,Yl In), O(n)) > Q(Yr I7", I "n) ), and a 2 L L(n) (C8) 6,= A(d(y (n )) [-A(d(y (n

))

) A(d(y " ))+I-'1 (A(d(y ))( L 8) 'Y' o d(70 ' ) ) 25 where: d(yn)=pTd(p.(n))as in (C4a); and L =aL 8 L T a 2 L =P-P , = P Pn yr "8 ' 0 2Y% n 2r for , = P. •Y 30 WO 2004/104856 PCT/AU2004/000696 - 54 (e) Let y* be the value of yr when some convergence criterion is satisfied, for example, |I yr yr+11 < s (for example 10 - s ); 5 (f) Defining B* = Pny* , Sn+,={i: |i I >max(3j I*e 1 ) } where ex is a small constant, for example le-5. (g) Set n=n+l and choose 9(n+i) = (n)+Kn ( 9 - 9(Pn) where 9q satisfies L(y1Pai4,)=O and Kn is a 10 damping factor such that 0< Kn - 1; and (h) Check convergence. If I y* - Y(n) < E2 where 62 is suitably small then stop, else go to step (b) above. 15 In another embodiment, t =1/v 2 has an independent gamma distribution with scale parameter b>0 and shape parameter k>0 so that the density of t2 is 2 0 )(t2,b,k)=b- (t2/b)k-lexp(-t2/b)/I(k) It can be shown that E{ t 2 I 3 } = (2k+1)/(2/b + 2) 25 as follows: Define 30 I(p,b,k)= J(t2)P t exp(-0.53 2 t 2 )'(t 2 ,b,k)dt 2 0 then I(p,b,k)=b 0 °s {r(p+k+0.5)/rF(k)}(1+0.5b/3 2) - (p

+

k+0.5) WO 2004/104856 PCT/AU2004/000696 - 55 Proof Let s=l2/2 then I(p,b,k)=b

P

ro

-

s f(t2 p+

'

0.

5 exp(-st2)_t 2 ,b,k)dt 2 0 5 Now using the substitution u = t 2 /b we get I(p,b,k)=b"" (u)P+o s exp(-sub)-(u, 1,k)du 0 o Now let s'=bs and substitute the expression for -(u,1,k). This gives 10 I(p,b,k)=bPwas exp(-(s'+1)u)uPk+0.

5 -ldu / I(k) 0 Looking up a table of Laplace transforms, eg Abramowitz and Stegun, then gives the result. 15 The conditional expectation follows from E{ t 2 i } = I(1,b,k)/I(0,b,k) = (2k+ 1)/(2/b + #2) As k tends to zero and b tends to infinity we get the 20 equivalent result using Jeffreys prior. For example, for k=0.005 and b=2*10s E{ t 2 13 } = (1.01)/(10 -s + 3 2 ) Hence we can get arbitrarily close to the algorithm with a 25 Jeffreys hyperprior with this proper prior. In another embodiment, v7 has an independent gamma distribution with scale parameter b>0 and shape parameter k>0. It can be shown that WO 2004/104856 PCT/AU2004/000696 - 56 fuk-3/2-lexp(-(A/u +u)) du E{vj-2j i3,} 0 b uk-/2-1exp(-(4/u +u))du 0 [ 1 K3/2-k (2/' -) b I I I K 11 2-k (2j,) ()) 1 (2 )K 3

/

2 -k (2 rj7) I i | Kl/2-k (2j ) where =8 i/2b and K denotes a modified Bessel function, which can be shown as follows: For k=1 in equation (c9) 5 E { Vi-2 Ifi 2j ( 1i1 For K=0.5 in equation (C9) 0 E{VI-| } = (1/I |I|) {K, (2 , -()/Ko (2j )} or equivalently E{i -2 Ii} =(1/1i 2 ){2j. K, (2-)/Ko (2j)} 15 Proof From the definition of the conditional expectation, writing A=fl2/2b, we get SVi2 il exp(-AV -2)b' ( i /b)k-exp(z i /b)dv i 2 202 20 E{i-2 i I 0 o Svi- exp(-, V -2)b' (Vi-2 / kb)-exp(Vi- 2 /b)dv 2 0 Rearranging, simplifying and making the substitution u=pi/b gives A.1 The integrals in A.1 can be evaluated by using the result 25 WO 2004/104856 PCT/AU2004/000696 - 57 X p - exp[ X + =b K (2a) 0a where K denotes a modified Bessel function, see Watson(1966). 5 Examples of members of this class are k=1 in which case Elpv-2 10i) = 2(/1fI I) which corresponds to the prior used in the Lasso technique, 10 Tibshirani(1996). See also Figueiredo(2001). The case k=0.5 gives E{pi-2 10j} = 1/ (1/|l|){K, (2 )/Ko(2 ,) } 15 or equivalently E {i-2 i }= ( 2/l {2 )2jK(2j)/Ko(2,)} 20 where K 0 and K, are modified Bessel functions, see Abramowitz and Stegun(1970). Polynomial approximations for evaluating these Bessel functions can be found in Abramowitz and Stegun(1970, p379). Details of the above calculations are given in the Appendix. 25 The expressions above demonstrate the connection with the Lasso model and the Jeffreys prior model. It will be appreciated by those skilled in the art that as k 30 tends to zero and b tends to infinity the prior tends to a Jeffreys improper prior. In one embodiment, the priors with 0< k 51 and b>0 form a class of priors which might be interpreted as penalising non 35 zero coefficients in a manner which is between the Lasso prior and the original specification using Jeffreys prior.

WO 2004/104856 PCT/AU2004/000696 - 58 In another embodiment, in the case of generalised linear models, step (d) in the maximisation step may be estimated 8'L82 by replacing with its expectation E{ .a2 This is 82 2, 5 preferred when the model of the data is a generalised linear model. 8 2 L For generalised linear models the expected value E{ l may a2 71 be calculated as follows. Beginning with 10 a= XT {A(1 ay' X )}(010 (' =X r . ... ri (c10) where X is the N by p matrix with it h row xiT and a2L aL aLr 15 E{ } = -E{(-)(-) } (Cll) we get

E{

2 } =-X T A(a(P)r 2 (_ ) 2 ) x 20 Equations (Cl0) and (ClI) can be written as -=X'V-( ()(y-y) (C12) aL E{ }= -X'V'X (C13) where V=A(ai(q)[)(

-

0 ) 2 ) 25 WO 2004/104856 PCT/AU2004/000696 - 59 Preferably, for generalised linear models, the EM algorithm comprises the steps: (a) Choose a hyper prior and its parameters. Initialising the algorithm by setting n=0, So = 5 {1,2,..., p } , () , applying a value for 8, such as for example s = 10s , and If p < N compute initial values P* by g* =(XtX+XI)-X

T

g(y+

"

) (C14) and if p > N compute initial values P* by , 1 10 0 1x (I-X(XX +M)"X)XTg(y+) (C15) where the ridge parameter X satisfies 0 < X < 1 and is small and chosen so that the link function is well defined at y+ . (b) Defining 15 S{, ie S. 0, otherwise and let Pn be a matrix of zeroes and ones such that the nonzero elements y(n) of P(n) satisfy 20 Yn) = (n) (n) = V n) Y= Pn3 , 3 =P.'Y (c) performing an estimation (E) step by calculating 25 the conditional expected value of the posterior distribution of component weights using the function: Q(13 pn), (pn)) = E { logp(13, <p , v | y) I y, 3(n

"

), (n)}6) (c16) = L(y I 0, 0 ") ) - 0 . 5 '6T A(d(f()))-2p 30 WO 2004/104856 PCT/AU2004/000696 - 60 where L is the log likelihood function of y. Using $=P.7 and 3(n)=p=n,) (C16) can be written as 5 Q(yI _/n), (n)= L(yI Psy, 0")-0.5 A(d(r )))- 2 r (17) (d) performing a maximisation (M) step by applying an iterative procedure, for example a Newton Raphson iteration, to maximise Q as a function of r whereby 10 70 =

Y

(n) and for r=0,1,2,... 7r,,i = Yr + ar 8r where r, is chosen by a line search algorithm to ensure Q(Yr+, I n), 4 n) > Q(Yr I 'n) , 4')n), and for p < N use 15 6r= A(d(y(n)))Yi V Yn i- (i(YTVIZ r ) (C18) d( nq) where Y = A(d(y"))PnX 20 V=A(ai P)T2 ( i )2) z o aA, and the subscript r denotes that these quantities are evaluated at = h(XPn-T,) For p > N use 25 8, = A(d(7(n))) [I -Y (Yn Y+V,)-'Yn](YT V z, ) (C19) d(y" )) with Vr and Zr defined as before. (e) Let y* be the value of Yr when some convergence 30 criterion is satisfied e.g I1 7Yr - 7r+.I < 8 (for example 10- ) WO2004/104856 PCT/AU2004/000696 - 61 (f) Define * = Py* , S.+ 1 ={i: Ii |>max(loj I*E )} where ex is a small constant, say le-5. Set n=n+l and choose (p' = qn + Kn( * - pn) where * satisfies SL(y I P,,4)= 0 and Kn is a damping factor such that 5 0< Kn 1. Note that in some cases the scale parameter is known or this equation can be solve explicitly to get an updating equation for (p. The above embodiments may be extended to incorporate quasi 10 likelihood methods Wedderburn (1974) and McCullagh and Nelder (1983)). In such an embodiment, the same iterative procedure as detailed above will be appropriate, but with L replaced by a quasi likelihood as shown above and, for example, Table 8.1 in McCullagh and Nelder (1983). In one 15 embodiment there is a modified updating method for the scale parameter T. To define these models requires specification of the variance function T 2 , the link function g and the derivative of the link function -. Once these are defined the above algorithm can be applied. 20 In one embodiment for quasi likelihood models, step 5 of the above algorithm is modified so that the scale parameter is updated by calculating 25 " N_ 1 -(yi-Vi) 2 25 _ where L and T are evaluated at 3* = Py* . Preferably, this updating is performed when the number of parameters s in the model is less than N. A divisor of N -s can be used when s 30 is much less than N.

WO 2004/104856 PCT/AU2004/000696 - 62 In another embodiment, for both generalised linear models and Quasi likelihood models the covariate matrix X with rows xi T can be replaced by a matrix K with ijth element kLj and kij = K(xi-xj) for some kernel function X . This matrix can 5 also be augmented with a vector of ones. Some example kernels are given in Table 2 below, see Evgeniou et al(1999). Kernel function Formula for K( x - y ) Gaussian radial basis exp( - I x - y 112 / a) , function a>0 Inverse multiquadric ( I x - y 12 + c 2 -1/2 Multiquadric ( I x - y 112+ c 2 ) Thin plate splines | x - y 2n+1 x - Iy "ln( x - y II) Multi layer perceptron tanh( x'y-O ) , for suitable 0 Ploynomial of degree d (1 + x'y ) B splines B 2 n+ 1 (x - y) Trigonometric polynomials sin(( d +1/2 )(x-y))/sin((x y)/2) 10 Table 2: Examples of kernel functions In Table 2 the last two kernels are one dimensional i.e. for the case when X has only one column. Multivariate versions can be derived from products of these kernel functions. The 15 definition of B 2

,+

1 can be found in De Boor(1978 ). Use of a kernel function in either a generalised linear model or a quasi likelihood model results in mean values which are smooth (as opposed to transforms of linear) functions of the covariates X. Such models may give a substantially better 20 fit to the data.

WO2004/104856 PCT/AU2004/000696 - 63 A fourth embodiment relating to a proportional hazards model will now be described. 5 D. Proportional Hazard Models The method of this embodiment may utilise training samples in order to identify a subset of components which are capable of affecting the probability that a defined event 10 (eg death, recovery) will occur within a certain time period. Training samples are obtained from a system and the time measured from when the training sample is obtained to when the event has occurred. Using a statistical method to associate the time to the event with the data obtained from 15 a plurality of training samples, a subset of components may be identified that are capable of predicting the distribution of the time to the event. Subsequently, knowledge of the subset of components can be used for tests, for example clinical tests to predict for example, 20 statistical features of the time to death or time to relapse of a disease. For example, the data from a subset of components of a system may be obtained from a DNA microarray. This data may be used to predict a clinically relevant event such as, for example, expected or median 25 patient survival times, or to predict onset of certain symptoms, or relapse of a disease. In this way, the present invention identifies preferably a relatively small number of components which can be used to 30 predict the distribution of the time to an event of a system. The selected components are "predictive" for that time to an event. Essentially, from all the data which is generated from the system, the method of the present invention enables identification of a small number of 35 components which can be used to predict time to an event. Once those components have been identified by this method, the components can be used in future to predict statistical WO2004/104856 PCT/AU2004/000696 - 64 features of the time to an event of a system from new samples. The method of the present invention preferably utilises a statistical method to eliminate components that are not required to correctly predict the time to an event 5 of a system. By appropriate selection of the hyperparameters in the model some control over the size of the selected subset can be achieved. As used herein, "time to an event" refers to a measure of 10 the time from obtaining the sample to which the method of the invention is applied to the time of an event. An event may be any observable event. When the system is a biological system, the event may be, for example, time till failure of a system, time till death, onset of a particular 15 symptom or symptoms, onset or relapse of a condition or disease, change in phenotype or genotype, change in biochemistry, change in morphology of an organism or tissue, change in behaviour. 20 The samples are associated with a particular time to an event from previous times to an event. The times to an event may be times determined from data obtained from, for example, patients in which the time from sampling to death is known, or in other words, "genuine" survival times, and 25 patients in which the only information is that the patients were alive when samples were last obtained, or in other words, "censored" survival times indicating that the particular patient has survived for at least a given number of days. 30 In one embodiment, the input data is organised into an nxpdata matrix X=(x) with n test training samples and p components. Typically, p will be much greater than n. 35 For example, consider an Nxp data matrix X=(x,) from, for example, a microarray experiment, with N individuals (or samples) and the same p genes for each individual.

WO2004/104856 PCT/AU2004/000696 - 65 Preferably, there is associated with each individual i(i=1,2,--...,N) a variabley, (yO) denoting the time to an event, for example, survival time. For each individual there may also be defined a variable that indicates whether 5 that individual's survival time is a genuine survival time or a censored survival time. Denote the censor indicators as ciwhere c , if yi is uncensored Ci= [0, if yi is censored 10 The Nxl vector with survival times y, may be written as y and the Nxl vector with censor indicators cias c. Typically, as discussed above, the component weights are 15 estimated in a manner which takes into account the a priori assumption that most of the component weights are zero. Preferably, the prior specified for the component weights is of the form 20 N P (A3,/8 2 ,...,/3P)(AI IP7- ) P(ri ) dv (Dl) T i=1 where ,l,---,', fl are component weights, P(Alri) is N(0O,rf) and P(ri) is some hyperprior distribution 25 P(r)= P ft) i=1 that is not a Jeffrey's hyperprior. In one embodiment, the prior distribution is an inverse 30 gamma prior for T in which t i =1/r, has an independent gamma distribution with scale parameter b>0 and shape parameter k>0 so that the density of t2 is WO 2004/104856 PCT/AU2004/000696 - 66 ft',b,k)=b

-

' (t /b)k-1eXp(-tf/b)/I(k). It can be shown that: 5 E{ t 2 | } = (2k+1)/(2/b + 02) (A) Equation A can be shown as follows: Define 10 I(p,b,k)= f(t2)p t exp(-0.5 2 t 2 )'(t 2 ,b,k)dt 2 0 then I(p,b,k)=b {I'{(p+k+.5)/F(k)} (1+0.5b 2 )4p+k+0.5) 15 Proof Let s =62 /2 then I(p,b,k)=b °5 J(t2/b)+05 exp(-st 2 ) , (t 2 ,b,k)dt 2 0 Now using the substitution u = t 2 /b we get 20 I(p,b,k)=b P+ ' 5 (u) p+

.

s exp(-sub)Mu,1,k)du 0 Now let s'=bs and substitute the expression for )(u,1,k). This gives I(p,b,k)=bP+o 5 s exp(-(s'+ 1)u)up+k '5- Idu / F(k) 0 25 Looking up a table of Laplace transforms, eg Abramowitz and Stegun, then gives the result. The conditional expectation follows from 30 E{ t' 2 I } = I(1,b,k)/I(0,b,k) = (2k+l)/(2/b + 32) WO 2004/104856 PCT/AU2004/000696 - 67 As k tends to zero and b tends to infinity, a result equivalent to using Jeffreys prior is obtained. For example, for k=0.005 and b=2*10 s 5 E{ t 2 1 3 } = (1.01)/(1 0- 5 + 3) Hence we can get arbitrarily close to a Jeffery's prior with this proper prior. 10 The modified algorithm for this model has b) =E{t 2 (n))}-0.5 where the expectation is calculated as above. 15 In yet another embodiment, the prior distribution is a gamma distribution for 7 . Preferably, the gamma distribution has scale parameter b>0 and shape parameter k>0. 20 It can be shown, that uk-3/2-exp(-(yi/u +u)) du E{ i-2 I ) 0 b uk-1/2-lexp(-("i/U +u))du 0 2 1 K 3 /2-k(2 -i) =b I I K/2-k (2Ji) 1 (2 j)K 3

/

2 k (2 F) I i 12 Ki/2-k (2f -V ) where Yi=i/2b and K denotes a modified Bessel function. 25 Some special members of this class are k=l in which case E{r-2 I i - = / 1 1 ) WO 2004/104856 PCT/AU2004/000696 - 68 which corresponds to the prior used in the Lasso technique, Tibshirani(1996). See also Figueiredo(2001). The case k=0.5 gives 5 Efr I@}= Ifi47(I/1/|) {K, (25 )/Ko (25)} or equivalently 10 E{ ri - 2 I}= (1 ){2 11li 12 )2 K, (2J )/Ko (2[-)} where Ko and K3 are modified Bessel functions, see Abramowitz and Stegun(1970). Polynomial approximations for evaluating these Bessel functions can be found in Abramowitz and 15 Stegun(1970, p379). The expressions above demonstrate the connection with the Lasso model and the Jeffreys prior model. 20 Details of the above calculations are as follows: For the gamma prior above and y=fl3 2 /2b uk3/2-lexp(-(7/u+u)) du E{q-2 j3m}= b fuk-/2"-exp(-(yi/u +u))du 0 S1 K 3

/

2 -k (2 i ) = (D2) bI| Al K 1 /2-k (2 fW) 1 (2 '-)K 3

/

2 k (2 7 ) I i 12K /2-k(2 25 where K denotes a modified Bessel function. For k=1 in (D2) E }- -42-{(1/-2ij) WO 2004/104856 PCT/AU2004/000696 - 69 For K=0.5 in (D2) E{i2 Ijti i )(1/Il1) {K, (2- )/K o (2 j)} 5 or equivalently E{ 'ji- 2 I}= (1/i 2) {2 K, (2 ' )/Ko (2 - )} 10 Proof From the definition of the conditional expectation, writing _yi=0i2/2b, we get 'ri -2' i -1 exp(-y i -2)b-' (fzi-2/b) k-exp(r i -2/b)drti2 E { ri 2 Ifi} = rti-1exp(-yiTi-2 )b-' (- 2 /b)k-exp( i -2/b)dri 2 0 15 Rearranging, simplifying and making the substitution u=i/b gives A.1 The integrals in A.1 can be evaluated by using the result 20 b-I 2 20 x-b-1eXp- X+ = Kb(2a) 0a where K denotes a modified Bessel function, see Watson(1966) . In a particularly preferred embodiment, p(Ti) a1/,r is a 25 Jeffreys prior, Kotz and Johnson(1983). The likelihood function defines a model which fits the data based on the distribution of the data. Preferably, the likelihood function is of the form: 30 N Log (Partial) Likelihood = g, (flT,;X,y,c) i=1 WO2004/104856 PCT/AU2004/000696 - 70 where PT = (,882 ... p) and pT = (PV 2 ,",Pq )are the model parameters. The model defined by the likelihood function may be any model for predicting the time to an event of a system. 5 In one embodiment, the model defined by the likelihood is Cox's proportional hazards model. Cox's proportional hazards model was introduced by Cox (1972) and may preferably be used as a regression model for survival data. In Cox's 10 proportional hazards model, 61 is a vector of (explanatory) parameters associated with the components. Preferably, the method of the present invention provides for the parsimonious selection (and estimation) from the parameters 1' =(,012 .. "' p) for Cox's proportional hazards model given 15 the data X, y and c. Application of Cox's proportional hazards model can be problematic in the circumstance where different data is obtained from a system for the same survival times, or in 20 other words, tied survival times. Tied survival times may be subjected to a pre-processing step that leads to unique survival times. The pre-processing proposed simplifies the ensuing code as it avoids concerns about tied survival times in the subsequent application of Cox's proportional hazards 25 model. The pre-processing of the survival times applies by adding an extremely small amount of insignificant random noise. Preferably, the procedure is to take sets of tied times and 30 add to each tied time within a set of tied times a random amount that is drawn from a normal distribution that has zero mean and variance proportional to the smallest non-zero distance between sorted survival times. Such pre-processing achieves an elimination of tied times without imposing a 35 draconian perturbation of the survival times.

WO 2004/104856 PCT/AU2004/000696 - 71 The pre-processing generates distinct survival times. Preferably, these times may be ordered in increasing magnitude denoted as t=(t1),t( 2 ),*..*t(N)) t(i+l) >t). Denote by Zthe Nxp matrix that is the re-arrangement 5 of the rows of X where the ordering of the rows of Zcorresponds to the ordering induced by the ordering of t; also denote by Z the jeb row of the matrix Z. Let d be the result of ordering c with the same permutation required to order t. 10 After pre-processing for tied survival times is taken into account and reference is made to standard texts on survival data analysis (eg Cox and Oakes, 1984), the likelihood function for the proportional hazards model may 15 preferably be written as N exp (Z ) I(t | # = flI(D3) j I exp(ZI) M) ~iE91. where 'B=(T, 1 2 ,..., ), Zy = the jth row of Z, and 9ij= {i:i=j,j+1,...---,N}= the risk set at the jth ordered event 20 time t W. The logarithm of the likelihood (ie L=Ilog(l)) may preferably be written as N L~t |p) =Ydi Z p -logE exp (Zjp i= je91i NV N = 'd i Zi-log ji exp Zj )1 , (D4) i=1 j= where 0, ifj <i ij = 1, ifj > i WO2004/104856 PCT/AU2004/000696 - 72 Notice that the model is non-parametric in that the parametric form of the survival distribution is not specified - preferably only the ordinal property of the 5 survival times are used (in the determination of the risk sets). As this is a non-parametric case 4 is not required (ie q=0). In another embodiment of the method of the invention, the 10 model defined by the likelihood function is a Parametric survival model. Preferably, in a parametric survival model, /3 is a vector of (explanatory) parameters associated with the components, and qpi is a vector of parameters associated with the functional form of the survival density function. 15 Preferably, the method of the invention provides for the parsimonious selection (and estimation) from the parameters 81 and the estimation of ip' =1,2,,pq) for parametric survival models given the data X, y and c. 20 In applying a parametric survival model, the survival times do not require pre-processing and are denoted as y. The parametic survival model is applied as follows: Denote by f(y;q,I,X)the parametric density function of the 25 survival time, denote its survival function by (y;<p,fX = Jf u; ,,/Xu where p are the parameters relevant Y to the parametric form of the density function and 8,X are as defined above. The hazard function is defined as h (yg;p,f,X = f (y;(p,f,X)/ (y; p,f/,X). 30 Preferably, the generic formulation of the log-likelihood function, taking censored data into account, is WO 2004/104856 PCT/AU2004/000696 - 73 N L= {cilog( f(yi; of ,X)) +( - ci)log(S(Yi;Pfl'X))} i=1 Reference to standard texts on analysis of survival time data via parametric regression survival models reveals a collection of survival time distributions that may be used. 5 Survival distributions that may be used include, for example, the Weibull, Exponential or Extreme Value distributions. If the hazard function may be written as 10 h(yi;(P,fLX) = A(yi,p)exp(Xif) then S(yi;q,#,X) = exp(-A(yi;p)eXif) and f(yi;p,f,X)= A(yi;,p)exp(XiPf-A(yi)eXi) where A(y;) fi A (u;p)du dA yg;P) is the integrated hazard function and A y;P ; X is the i t h row of X. 15 The Weibull, Exponential and Extreme Value distributions have density and hazard functions that may be written in the form of those presented in the paragraph immediately above. 20 The application detailed relies in part on an algorithm of Aitken and Clayton (1980) however it permits the user to specify any parametric underlying hazard function. Following from Aitkin and Clayton (1980) a preferred 25 likelihood function which models a parametic survival model is: L= Zcilog(u)-p+ci log (Yi) (D5) i=1 Ay;p) where pi=Ayi;p)exp(Xf/). Aitkin and Clayton (1980) note that a consequence of equation (11) is that the ci's may be WO2004/104856 PCT/AU2004/000696 - 74 treatedas Poisson variates with means puand that the last term in equation (11) does not depend on 6 (although it depends on ). 5 Preferably, the posterior distribution of p, and r given y is P #,?,Ej yg al f J,T,Cg ejP(E) (D6) wherein I,) is the likelihood function. 10 In one embodiment, T may be treated as a vector of missing data and an iterative procedure used to maximise equation (D6) to produce a posteriori estimates of f . The prior of equation (D1) is such that the maximum a posteriori 15 estimates will tend to be sparse i.e. if a large number of parameters are redundant, many components of 8 will be zero. Because a prior expectation exists that many components of 20 /' are zero, the estimation may be performed in such a way that most of the estimated fly's are zero and the remaining non-zero estimates provide an adequate explanation of the survival times. 25 In the context of microarray data this exercise translates to identifying a parsimonious set of genes that provide an adequate explanation for the event times. As stated above, the component weights which maximise the 30 posterior distribution may be determined using an iterative procedure. Preferable, the iterative procedure for maximising the posterior distribution of the components and component weights is an EM algorithm, such as, for example, that described in Dempster et al, 1977. 35 WO 2004/104856 PCT/AU2004/000696 - 75 If the E step of the EM algorithm is examined, from (D6) ignoring terms not involving beta, it is necessary to compute ~E tg'/Ir- 2 1 j} ijl (D7) n i=1 5 where d,=d,( ,)=E{1/v 2 j-,}.

5 and for convenience we define d2i=1d= Oif/3i=O. In the following we write d=d(^)=(d 2

,-

2 ... , ,d). In a similar manner we define for example d(fl

(

'

"

) and d(y("))=Pgd(Py ( )) where ( )=Py(") and P, is obtained from the p by p identity matrix by omitting columns j for which 6(

")

=0. 10 Hence to do the E step requires the calculation of the conditional expected value of ti2 =1/ 2 when p(i 72) is N(O,,r2) and p ()i2has a specified prior distribution as discussed above. 15 In one embodiment, the EM algorithm comprises the steps: 1. Choose the hyperprior and values for its parameters, namely b and k. Initialising the algorithm by setting n=O, 20 So = (1,2,, p I , initialise fl(0)=P*, () 2. Defining (n_ { ,K , i s. 0, otherwise 25 and let P be a matrix of zeroes and ones such that the nonzero elements 2 A") of Pr) satisfy (n)_ = pT p(n) = (n) Y(n) (D8) r PI= P ,r WO 2004/104856 PCT/AU2004/000696 - 76 3. Performing an estimation step by calculating the expected value of the posterior distribution of component weights. This may be performed using the function: 5 Q(6I ",enp") = E log(P(,ry ly)) Iyp,"(,p ( A 2 (D9) 2j=1 di,(")) where L is the log likelihood function of y. Using P = PIy and f(n) = Py") we have 10 Q(r I P("), (n)) = L(t I P.2, " ) 1 (A(d(r ")), (D10) 2 4. Performing the maximisation step. This may be performed using Newton Raphson iterations as follows: Set 70 Y(r) and for r=0,1,2,... Yr+1 = Yr + ar -r where ar is chosen by a line search algorithm to ensure Q(Yr+, I Y( M,()) > 15 Q (Yr I r("),p")) and 82L aL 71 q, =A(d(r("'))[-A(d( " ))) A(d(r" )))+I]-'( 2r ~~, d (7 " ) aL ,L 82 L 82L where P ,- =P for fir =ny r (Dli) where ar =P ar 8 2 yr 8~ ~3, Let r be the value of Yr when some convergence criterion is satisfied e.g ||Yr-Yr+11 < c (for example 6

=

1 0

-

S) 20 5. Define /=Pny , Sn= i:1fi | > -maxj where c 1 is a small constant, say J Set n=n+l, choose (n+)(n) constant, say 10 - .

Set n=n+1, choose 9 ~l=9 n+rCn 9*-9() WO 2004/104856 PCT/AU2004/000696 - 77 . BL Py Pn/9) where q satisfies =0 and rn is a damping factor such that 0<n <1. 6. Check convergence. If |ry -y" '<e2 where e2 is suitably small then stop, else go to step 2 above. 5 In another embodiment, step (D11) in the maximisation step may be estimated by replacing 2 with its expectation E{ ) •a2L71 10 In one embodiment, the EM algorithm is applied to maximise the posterior distribution when the model is Cox's proportional hazard's model. To aid in the exposition of the application of the EM 15 algorithm when the model is Cox's proportional hazards model, it is preferred to define "dynamic weights" and matrices based on these weights. The weights are WO 4i'l exp (Zll6 Wil= N E ij exp (Z A) j=1 N w = diwi,l , i=1 -;= d, -w 1 . 20 Matrices based on these weights are - WO 2004/104856 PCT/AU2004/000696 - 78 Wi, I Wi,2 Wi= <Wi,N , w 2 W ='- . ' 0) N,, N w** d w,.wj r i=1 In terms of the matrices of weights the first and second derivatives of L may be written as OL= zrTWk 02L zT( W ** A (W*)) z=ZTKZ (D12) where K=W -AW . Note therefore from the transformation matrix P described as part of Step (2) of the EM algorithm (Equation DS) (see also Equations (D11)) it follows that OL _PTLPZf L 2 L (D3) 0= 2L Ta2 =PTZT(W**-A(W*))ZP =p

T

Z

T

KZpD y -

-P

WO 2004/104856 PCT/AU2004/000696 - 79 Preferably, when the model is Cox's proportional hazards model the E step and M step of the EM algorithm are as follows: 1. Choose the hyperprior and its parameters b and k. Set 5 n=O, So = {1,2,.., p}. Let v be the vector with components V 1 fl-e, if ci=1 E if ci =O 10 for some small E , say .001. Define f to be log(v/t). If p N compute initial values 6* by fl = (Z

T

Z + I)' Z

T

f 15 If p > N compute initial values 8*by *1 S=-(I -Z T

(ZZ

T + 2I)"'Z)Z

T

f where the ridge parameter A satisfies 0 < A _ 1. 20 2. Define ,?,= #,, i S 0, otherwise Let Pnbe a matrix of zeroes and ones such that the nonzero elements yr(n)of 6(n) satisfy Pr f hpe(n) E t(n) p c la(n) 25 Y PTfl /P 7 r 3. Perform the E step by calculating WO 2004/104856 PCT/AU2004/000696 - 80 Q(f I ,8n)) = E 1og(p(p it | tp(")} [2 L 1 )____ d L(t--)- Y,'(\ where Lis the log likelihood function of t given by Equation (8). Using f = P,y and fl") = Pry("') we have 5 Q(r l

(

")) = L(t I1P.y)- I "A(d(r7"))7 2 4. Do the M step. This can be done with Newton Raphson iterations as follows. Set y 0 = Y' and for r=0, 1, 2,... Yr+l

=

1 Yr + ar r where ar is chosen by a line search algorithm to ensure Q7Iyr+i i y",">Q y, . 10 For p :5N use Jr =A (d(y (n)))(YTKY+I,'' (yTk - A (1/d(r(n) where Y = ZPnA (d(7(n) For p > N use & =A (d(y (n)))( YT (yyT +K-YY -A 1d((n) Let y* be the value of yr when some convergence 15 criterion is satisfied e.g Tr- Yr+I] < E (for example 5. Define * =Pny , Sn= i:1fi >6jmaxjljI where el is a 1.1 small constant, say 10 -5 . This step eliminates variables with very small coefficients. 20 6. Check convergence. If jy*-y(n) j<2 where c 2 is suitably small then stop, else set n=n+l, go to step 2 above and repeat procedure until convergence occurs.

WO 2004/104856 PCT/AU2004/000696 - 81 In another embodiment the EM algorithm is applied to maximise the posterior distribution when the model is a parametric survival model. 5 In applying the EM algorithm to the parametic survival model, a consequence of equation (11) is that the ci's may be treated as Poisson variates with means ui and that the last term in equation (11) does not depend on 8 (although it depends on p) . Note that log(,p)=logAy;(p))+Xi@ and so 10 it is possible to couch the problem in terms of a log-linear model for the Poisson-like mean. Preferably, an iterative maximization of the log-likelihood function is performed where given initial estimates of p the estimates of f are obtained. Then given these estimates of f, updated 15 estimates of p are obtained. The procedure is continued until convergence occurs. Applying the posterior distribution described above, we note that (for fixed q) Olog(p) 1 8p 8p 8log(pu) 8log(pi) = : "-- = p and Xi (D14) a- jaf a a,6 aj6 20 Consequently from Equations (11) and (12) it follows that aL a 2 L" aL = X ( -,p) and a2=XA(e)X. The versions of Equation (12) relevant to the parametric survival models are _ =_ .T L = l T1 F~ X T(C _.u)1 BL rL - X' -p --=P = P P( = -X'A]p) a2 2. ar j r ap2 To solve for p after each M step of the EM algorithm (see step 5 below) preferably put (n+l)= (n+l) n * (n) where* step 5 below) preferably put qi =9 +Kn' -9 where q' WO 2004/104856 PCT/AU2004/000696 - 82 satisfies aL-=0 for 0<Kn_<1 and P is fixed at the value a ( obtained from the previous M step. It is possible to provide an EM algorithm for parameter selection in the context of parametric survival models and 5 microarray data. Preferably, the EM algorithm is as follows: 1. Choose a hyper prior and its parameters b and k eg b=1e7 and k=0.5. Set n=O, So = {1,2,-., p} p(initial)= ( 0 O). Let v be 10 the vector with components -= IE, if ci=1 i e , if ci=0 for some small s , say for example .001. Define f to be 15 log(v/A(y,9)). If p N compute initial values f* by f =(XTX+M)'IXTf. If p > N compute initial values f* by 1 /3' =-(I -X (XY T " + AI)-'X)X

T

f 20 where the ridge parameter A satisfies 0 < A E 1. 2. Define 8(n)= f. , i E S 0, otherwise Let Pnbe a matrix of zeroes and ones such that the nonzero 25 elements y(n)of f(n) satisfy (n) = pTf(n) p(n) - P t r =P P. , 8 =P, WO 2004/104856 PCT/AU2004/000696 - 83 3. Perform the E step by calculating Q(8 fl ,". ,cp )) E 10lg ( p Tr Iy) I y," " , 2 LY,m> )1-E A, = L(yj= (_ d(")) 5 where L is the log likelihood function of Y and (n) Using 8 = P,y and (") = P(n)" we have Q(y Iy("), T

("

)) = L(y 1 Py,

(

p ")) - yr A ( d ( y (

"

)) ) y 2 4. Do the M step. This can be done with Newton Raphson 10 iterations as follows. Set yo=y(r)and for r=0,1,2,... Yr+1 = r + Gr 8,r where a r is chosen by a line search algorithm to ensure Q(Yr+ I y"n),(n)) >Q(r Y("),P(n)) For p ! N use -, -A(d(7'"')))[(Y) YV'T(Y T (c-p-A(1/d(/f )))) where Y= XP A (d(Y"))). 15 For p > N use .r =-A d(7(n)) I-Y T(YYT + (1)) y

)

(Y

1 (Ync)- (l/d(y(n)))r) Let y* be the value of yr when some convergence criterion is satisfied e.g Yr - Yr+i < s (for example 10s ). 20 5. Define 8 =Pny , Sn= i:| i >elmax1jI where el is a small constant, say 10 - s . Set n=n+l, choose rp(n+l) = q(n)+ n (_(n) WO 2004/104856 PCT/AU2004/000696 - 84 where p satisfies ~ ~0 and K n is a damping factor such that 0< c n<l. 6. Check convergence. If jjy- )jj<e2 where £2 is suitably 5 small then stop, else go to step 2. In another embodiment, survival times are described by a Weibull survival density function. For the Weibull case ( is preferably one dimensional and 10 A(y;p) = ya, A (y;p =aya-1, Preferably, -= -+(cN-pi)1og(yN)=O is solved a a a .= i=1 after each M step so as to provide an updated value of a. 15 Following the steps applied for Cox's proportional hazards model, one may estimate aand select a parsimonious subset of parameters from f that can provide an adequate explanation for the survival times if the survival times follow a Weibull distribution. A numerical example is now 20 given. The preferred embodiment of the present invention will now be described by way of reference only to the following non limiting example. It should be understood, however, that 25 the example is illustrative only, should not be taken in any way as a restriction on the generality of the invention described above. Example: WO2004/104856 PCT/AU2004/000696 - 85 Full normal regression example 201 data points 41 basis functions k=0 and b=1e7 5 the correct four basis functions are identified namely 2 12 24 34 estimated variance is 0.67. With k=0.2 and b=1e7 10 eight basis functions are identified, namely 2 8 12 16 19 24 34 estimated variance is 0.63. Note that the correct set of basis functions is included in this set. 15 The results of the iterations for k=0.2 and b=1e7 are given below. EM Iteration: 0 expected post: 2 basis fns 41 20 sigma squared 0.6004567 EM Iteration: 1 expected post: -63.91024 basis fns 41 sigma squared 0.6037467 EM Iteration: 2 expected post: -52.76575 basis fns 41 25 sigma squared 0.6081233 EM Iteration: 3 expected post: -53.10084 basis fns 30 sigma squared 0.6118665 30 EM Iteration: 4 expected post: -53.55141 basis fns 22 sigma squared 0.6143482 EM Iteration: 5 expected post: -53.79887 basis fns 18 35 sigma squared 0.6155 EM Iteration: 6 expected post: -53.91096 basis fns 18 WO2004/104856 PCT/AU2004/000696 - 86 sigma squared 0.6159484 EM Iteration: 7 expected post: -53.94735 basis fns 16 sigma squared 0.6160951 5 EM Iteration: 8 expected post: -53.92469 basis fns 14 sigma squared 0.615873 EM Iteration: 9 expected post: -53.83668 basis fns 13 10 sigma squared 0.6156233 EM Iteration: 10 expected post: -53.71836 basis fns 13 sigma squared 0.6156616 EM Iteration: 11 expected post: -53.61035 basis fns 12 15 sigma squared 0.6157966 EM Iteration: 12 expected post: -53.52386 basis fns 12 sigma squared 0.6159524 20 EM Iteration: 13 expected post: -53.47354 basis fns 12 sigma squared 0.6163736 EM Iteration: 14 expected post: -53.47986 basis fns 12 25 sigma squared 0.6171314 EM Iteration: 15 expected post: -53.53784 basis fns 11 sigma squared 0.6182353 EM Iteration: 16 expected post: -53.63423 basis fns 11 30 sigma squared 0.6196385 EM Iteration: 17 expected post: -53.75112 basis fns 11 sigma squared 0.621111 35 EM Iteration: 18 expected post: -53.86309 basis fns 11 sigma squared 0.6224584 WO2004/104856 PCT/AU2004/000696 - 87 EM Iteration: 19 expected post: -53.96314 basis fns 11 sigma squared 0.6236203 EM Iteration: 20 expected post: -54.05662 basis fns 11 5 sigma squared 0.6245656 EM Iteration: 21 expected post: -54.1382 basis fns 10 sigma squared 0.6254182 10 EM Iteration: 22 expected post: -54.21169 basis fns 10 sigma squared 0.6259266 EM Iteration: 23 expected post: -54.25395 basis fns 9 15 sigma squared 0.6259266 EM Iteration: 24 expected post: -54.26136 basis fns 9 sigma squared 0.6260238 EM Iteration: 25 expected post: -54.25962 basis fns 9 20 sigma squared 0.6260203 EM Iteration: 26 expected post: -54.25875 basis fns 8 sigma squared 0.6260179 25 EM Iteration: 27 expected post: -54.25836 basis fns 8 sigma squared 0.626017 EM Iteration: 28 expected post: -54.2582 basis fns 8 30 sigma squared 0.6260166 For the reduced data set with 201 observations and 10 variables, k=0 and b=1e7 Gives the correct basis functions, namely 1 2 3 4. With 35 k=0.25 and b=1e7, 7 basis functions were chosen, namely 1 2 3 4 6 8 9. A record of the iterations is given below. Note that this set also includes the correct set.

WO 2004/104856 PCT/AU2004/000696 - 88 EM Iteration: 0 expected post: 2 basis fns 10 sigma squared 0.6511711 5 EM Iteration: 1 expected post: -66.18153 basis fns 10 sigma squared 0.6516289 EM Iteration: 2 expected post: -57.69118 basis fns 10 10 sigma squared 0.6518373 EM Iteration: 3 expected post: -57.72295 basis fns 9 sigma squared 0.6518373 EM Iteration: 4 expected post: -57.74616 basis fns 8 15 sigma squared 0.65188 EM Iteration: 5 expected post: -57.75293 basis fns 7 sigma squared 0.6518781 20 Ordered categories examples Luo prostate data 15 samples 9605 genes. For k=0 and b=1e7 we get the following results misclassification table 25 pred y 1 2 3 4 1 4 0 0 0 2 0 2 1 0 3 0 0 4 0 30 4 0 0 0 4 Identifiers of variables left in ordered categories model 6611 35 For k=0.25 and b=1e7 we get the following results misclassification table WO 2004/104856 PCT/AU2004/000696 - 89 pred y 1 2 3 4 14000 20300 1 4 0 0 0 2 0 3 0 0 5 3 0 0 4 0 40004 4 0 0 0 4 Identifiers of variables left in ordered categories model 6611 7188 10 Note that we now have perfect discrimination on the training data with the addition of one extra variable. A record of the iterations of the algorithm is given below. 15 *********************************************** Iteration 1 : 11 cycles, criterion -4.661957 misclassification matrix fhat 20 f 1 2 1 23 0 2 0 22 row =true class 25 Class 1 Number of basis functions in model : 9608 WWWWWWWWWWWWWWWWW*eettWWWWWWWWWWWW**ttttWWWWWWW Iteration 2 : 5 cycles, criterion -9.536942 misclassification matrix 30 fhat f 1 2 1 23 0 2 1 21 row =true class 35 Class 1 Number of basis functions in model : 6431 AtttttttttkWWWWW*kttttttteetweetWW*etttttttettt WO2004/104856 PCT/AU2004/000696 - 90 Iteration 3 : 4 cycles, criterion -9.007843 misclassification matrix fhat 5 f 1 2 1 23 0 2 0 22 row =true class 10 Class 1 Number of basis functions in model : 508 *****************************w****w*w******* Iteration 4 : 5 cycles, criterion -6.47555 misclassification matrix 15 fhat f 1 2 1 23 0 2 0 22 row =true class 20 Class 1 Number of basis functions in model : 62 *********************************************** Iteration 5 : 6 cycles, criterion -4.126996 25 misclassification matrix fhat f 1 2 1 23 0 2 1 21 30 row =true class Class 1 Number of basis functions in model : 20 Iteration 6 : 6 cycles, criterion -3.047699 35 misclassification matrix fhat WO2004/104856 PCT/AU2004/000696 - 91 f 1 2 1 23 0 2 1 21 row =true class 5 Class 1 Number of basis functions in model : 12 ***********************************www********** Iteration 7 : 5 cycles, criterion -2.610974 10 misclassification matrix fhat f 1 2 1 23 0 2 1 21 15 row =true class Class 1 : Variables left in model 1 2 3 408 846 6614 7191 8077 regression coefficients 20 28.81413 14.27784 7.025863 -1.086501e-06 4.553004e-09 16.25844 0.1412991 -0.04101412 *********************************************** Iteration 8 : 5 cycles, criterion -2.307441 25 misclassification matrix fhat f 1 2 1 23 0 30 2 1 21 row =true class Class 1 : Variables left in model 1 2 3 6614 7191 8077 35 regression coefficients 32.66699 15.80614 7.86011 -18.53527 0.1808061 -0.006728619 WO2004/104856 PCT/AU2004/000696 - 92 Iteration 9 : 5 cycles, criterion -2.028043 misclassification matrix 5 fhat f 1 2 1 23 0 2 0 22 row =true class 10 Class 1 : Variables left in model 1 2 3 6614 7191 8077 regression coefficients 36.11990 17.21351 8.599812 -20.52450 0.2232955 -0.0001630341 15 *********************************************** Iteration 10 : 6 cycles, criterion -1.808861 misclassification matrix 20 fhat f 1 2 1 23 0 2 0 22 row =true class 25 Class 1 : Variables left in model 1 2 3 6614 7191 8077 regression coefficients 39.29053 18.55341 9.292612 -22.33653 0.260273 -8.696388e-08 30 ****** ***** *** **** *************** **** **** *** *** Iteration 11 : 6 cycles, criterion -1.656129 misclassification matrix 35 fhat f 1 2 1 23 0 WO2004/104856 PCT/AU2004/000696 - 93 2 0 22 row =true class Class 1 : Variables left in model 5 1 2 3 6614 7191 regression coefficients 42.01569 19.73626 9.90312 -23.89147 0.2882204 * *** ***** *** **** *** *** **** *********** *****w* 10 Iteration 12 : 6 cycles, criterion -1.554494 misclassification matrix fhat f 1 2 15 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 20 1 2 3 6614 7191 regression coefficients 44.19405 20.69926 10.40117 -25.1328 0.3077712 ************** ************ ***** ****** ***** Iteration 13 : 6 cycles, criterion -1.487778 25 misclassification matrix fhat f 1 2 1 23 0 30 2 0 22 row =true class Class 1 : Variables left in model 1 2 3 6614 7191 35 regression coefficients 45.84032 21.43537 10.78268 -26.07003 0.3209974 WO2004/104856 PCT/AU2004/000696 - 94 * *** ***** ***** *** **** **** **** *** ** **** ******* Iteration 14 : 6 cycles, criterion -1.443949 misclassification matrix 5 fhat f 1 2 1 23 0 2 0 22 row =true class 10 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 47.03702 21.97416 11.06231 -26.75088 0.3298526 15 *********************************************** Iteration 15 : 6 cycles, criterion -1.415 misclassification matrix 20 fhat f 1 2 1 23 0 2 0 22 row =true class 25 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 47.88472 22.35743 11.26136 -27.23297 0.3357765 30 ********************************ww******ww***** Iteration 16 : 6 cycles, criterion -1.395770 misclassification matrix 35 fhat f 1 2 1 23 0 WO 2004/104856 PCT/AU2004/000696 - 95 2 0 22 row =true class Class 1 : Variables left in model 5 1 2 3 6614 7191 regression coefficients 48.47516 22.62508 11.40040 -27.56866 0.3397475 ************************** ****************ne 10 Iteration 17 : 5 cycles, criterion -1.382936 misclassification matrix fhat f 1 2 15 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 20 1 2 3 6614 7191 regression coefficients 48.88196 22.80978 11.49636 -27.79991 0.3424153 25 Iteration 18 : 5 cycles, criterion -1.374340 misclassification matrix fhat f 1 2 30 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 35 1 2 3 6614 7191 regression coefficients 49.16029 22.93629 11.56209 -27.95811 0.3442109 WO2004/104856 PCT/AU2004/000696 - 96 **********************************wwwwwwwwwwwww Iteration 19 : 5 cycles, criterion -1.368567 5 misclassification matrix fhat f 1 2 1 23 0 2 0 22 10 row =true class Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 15 49.34987 23.02251 11.60689 -28.06586 0.3454208 Iteration 20 : 5 cycles, criterion -1.364684 20 misclassification matrix fhat f 1 2 1 23 0 2 0 22 25 row =true class Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 30 49.47861 23.08109 11.63732 -28.13903 0.3462368 ***************************w**w******w****w** Iteration 21 : 5 cycles, criterion -1.362068 35 misclassification matrix fhat f 1 2 WO2004/104856 PCT/AU2004/000696 - 97 1 23 0 2 0 22 row =true class 5 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 49.56588 23.12080 11.65796 -28.18862 0.3467873 10 *********************************************** Iteration 22 : 5 cycles, criterion -1.360305 misclassification matrix fhat 15 f 1 2 1 23 0 2 0 22 row =true class 20 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 49.62496 23.14769 11.67193 -28.22219 0.3471588 25 *********************************************** Iteration 23 : 4 cycles, criterion -1.359116 misclassification matrix fhat 30 f 1 2 1 23 0 2 0 22 row =true class 35 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients WO2004/104856 PCT/AU2004/000696 - 98 49.6649 23.16588 11.68137 -28.2449 0.3474096 Iteration 24 : 4 cycles, criterion -1.358314 5 misclassification matrix fhat f 1 2 1 23 0 10 2 0 22 row =true class Class 1 : Variables left in model 1 2 3 6614 7191 15 regression coefficients 49.69192 23.17818 11.68776 -28.26025 0.3475789 * w****************************************** Iteration 25 : 4 cycles, criterion -1.357772 20 misclassification matrix fhat f 1 2 1 23 0 25 2 0 22 row =true class Class 1 : Variables left in model 1 2 3 6614 7191 30 regression coefficients 49.71017 23.18649 11.69208 -28.27062 0.3476932 wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww**************** Iteration 26 : 4 cycles, criterion -1.357407 35 misclassification matrix fhat WO2004/104856 PCT/AU2004/000696 - 99 f 1 2 1 23 0 2 0 22 row =true class 5 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 49.72251 23.19211 11.695 -28.27763 0.3477704 10 Iteration 27 : 4 cycles, criterion -1.35716 misclassification matrix 15 fhat f 1 2 1 23 0 2 0 22 row =true class 20 Class 1 : Variables left in model 1 2 3 6614 7191 regression coefficients 49.73084 23.19590 11.69697 -28.28237 0.3478225 25 WWWWWW*tttttWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW Iteration 28 : 3 cycles, criterion -1.356993 misclassification matrix 30 fhat f 1 2 1 23 0 2 0 22 row =true class 35 Class 1 : Variables left in model 1 2 3 6614 7191 WO2004/104856 PCT/AU2004/000696 - 100 regression coefficients 49.73646 23.19846 11.6983 -28.28556 0.3478577 wwwwwww*************wwwwwwwwwwww**************** 5 Iteration 29 : 3 cycles, criterion -1.356881 misclassification matrix fhat f 1 2 10 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 15 1 2 3 6614 7191 regression coefficients 49.74026 23.20019 11.6992 -28.28772 0.3478814 ********************************************www 20 Iteration 30 : 3 cycles, criterion -1.356805 misclassification matrix fhat f 1 2 25 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 30 1 2 3 6614 7191 regression coefficients 49.74283 23.20136 11.69981 -28.28918 0.3478975 1 35 misclassification table pred WO2004/104856 PCT/AU2004/000696 - 101 y 12 3 4 1 4 0 0 0 2 0 3 0 0 3 0 0 4 0 5 4 0 0 0 4 Identifiers of variables left in ordered categories model 6611 7188 10 Ordered categories example Luo prostate data 15 samples 50 genes. For k=0 and b=le7 we get the following results misclassification table 15 pred y 12 3 4 1 4 0 0 0 2 0 2 1 0 3 0 0 4 0 20 4 0 0 0 4 Identifiers of variables left in ordered categories model 1 25 For k=0.25 and b=1e7 we get the following results misclassification table pred y 1 2 3 4 30 14 0 0 0 2 0 3 0 0 3 0 04 0 4 0 0 0 4 35 Identifiers of variables left in ordered categories model 1 42 WO2004/104856 PCT/AU2004/000696 - 102 A record of the iterations for k=0.25 and b=1e7 is given below 5 Iteration 1 : 19 cycles, criterion -0.4708706 misclassification matrix fhat f 1 2 10 1 23 0 2 0 22 row =true class Class 1 Number of basis functions in model : 53 15 *********************************************** Iteration 2 : 7 cycles, criterion -1.536822 misclassification matrix fhat 20 f 1 2 1 23 0 2 0 22 row =true class 25 Class 1 Number of basis functions in model : 53 WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW*kttttttttt Iteration 3 : 5 cycles, criterion -2.032919 misclassification matrix 30 fhat f 1 2 1 23 0 2 0 22 row =true class 35 Class 1 Number of basis functions in model : 42 * * *********** ***** **************** WO2004/104856 PCT/AU2004/000696 - 103 Iteration 4 : 5 cycles, criterion -2.132546 misclassification matrix fhat 5 f 1 2 1 23 0 2 0 22 row =true class 10 Class 1 Number of basis functions in model : 18 ********************************************www Iteration 5 : 5 cycles, criterion -1.978462 misclassification matrix 15 fhat f 1 2 1 23 0 2 0 22 row =true class 20 Class 1 Number of basis functions in model : 13 *********************************************** Iteration 6 : 5 cycles, criterion -1.668212 25 misclassification matrix fhat f 1 2 1 23 0 2 0 22 30 row =true class Class 1 : Variables left in model 1 2 3 4 10 41 43 45 regression coefficients 35 34.13253 22.30781 13.04342 -16.23506 0.003213167 0.006582334 -0.0005943874 -3.557023 WO2004/104856 PCT/AU2004/000696 - 104 Iteration 7 : 5 cycles, criterion -1.407871 misclassification matrix 5 fhat f 1 2 1 23 0 2 0 22 row =true class 10 Class 1 : Variables left in model 1 2 3 4 10 41 43 45 regression coefficients 36.90726 24.69518 14.61792 -17.16723 1.112172e-05 5.949931e 15 06 -3.892181e-08 -4.2906 *********************************************** Iteration 8 : 5 cycles, criterion -1.244166 20 misclassification matrix fhat f 1 2 1 23 0 2 0 22 25 row =true class Class 1 : Variables left in model 1 2 3 4 10 45 regression coefficients 30 39.15038 26.51011 15.78594 -17.99800 1.125451e-10 -4.799167 **********************************w*wwwww Iteration 9 : 5 cycles, criterion -1.147754 35 misclassification matrix fhat f 1 2 WO2004/104856 PCT/AU2004/000696 - 105 1 23 0 2 0 22 row =true class 5 Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 40.72797 27.73318 16.56101 -18.61816 -5.115492 10 *********************************************** Iteration 10 : 5 cycles, criterion -1.09211 misclassification matrix fhat 15 f 1 2 1 23 0 2 0 22 row =true class 20 Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 41.74539 28.49967 17.04204 -19.03293 -5.302421 25 *********************************************** Iteration 11 : 5 cycles, criterion -1.060238 misclassification matrix fhat 30 f 1 2 1 23 0 2 0 22 row =true class 35 Class 1 : Variables left in model 1 2 3 4 45 regression coefficients WO 2004/104856 PCT/AU2004/000696 - 106 42.36866 28.96076 17.32967 -19.29261 -5.410496 t*****et***W*WW*W***WWWWWWW**t*etW**W********** Iteration 12 : 5 cycles, criterion -1.042037 5 misclassification matrix fhat f 1 2 1 23 0 10 2 0 22 row =true class Class 1 : Variables left in model 1 2 3 4 45 15 regression coefficients 42.73908 29.23176 17.49811 -19.44894 -5.472426 *********************************************** Iteration 13 : 5 cycles, criterion -1.031656 20 misclassification matrix fhat f 1 2 1 23 0 25 2 0 22 row =true class Class 1 : Variables left in model 1 2 3 4 45 30 regression coefficients 42.95536 29.38894 17.59560 -19.54090 -5.507787 * ** ***** ********** **** ************ ****** Iteration 14 : 4 cycles, criterion -1.025738 35 misclassification matrix fhat WO 2004/104856 PCT/AU2004/000696 - 107 f 1 2 1 23 0 2 0 22 row =true class 5 Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 43.08034 29.47941 17.65163 -19.59428 -5.527948 10 ********************************************www Iteration 15 : 4 cycles, criterion -1.022366 misclassification matrix 15 fhat f 1 2 1 23 0 2 0 22 row =true class 20 Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 43.15213 29.53125 17.68372 -19.62502 -5.539438 25 ********************************************www Iteration 16 : 4 cycles, criterion -1.020444 misclassification matrix 30 fhat f 1 2 1 23 0 2 0 22 row =true class 35 Class 1 : Variables left in model 1 2 3 4 45 WO2004/104856 PCT/AU2004/000696 - 108 regression coefficients 43.19322 29.56089 17.70206 -19.64265 -5.545984 *********************************************** 5 Iteration 17 : 4 cycles, criterion -1.019349 misclassification matrix fhat f 1 2 10 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 15 1 2 3 4 45 regression coefficients 43.21670 29.57780 17.71252 -19.65272 -5.549713 *********************************************** 20 Iteration 18 : 3 cycles, criterion -1.018725 misclassification matrix fhat f 1 2 25 1 23 0 2 0 22 row =true class Class 1 : Variables left in model 30 1 2 3 4 45 regression coefficients 43.23008 29.58745 17.71848 -19.65847 -5.551837 35 Iteration 19 : 3 cycles, criterion -1.01837 misclassification matrix WO2004/104856 PCT/AU2004/000696 - 109 fhat f 1 2 1 23 0 2 0 22 5 row =true class Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 10 43.23772 29.59295 17.72188 -19.66176 -5.553047 *********************************************** Iteration 20 : 3 cycles, criterion -1.018167 15 misclassification matrix fhat f 1 2 1 23 0 2 0 22 20 row =true class Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 25 43.24208 29.59608 17.72382 -19.66363 -5.553737 WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW*ettttttttttt Iteration 21 : 3 cycles, criterion -1.018052 30 misclassification matrix fhat f 1 2 1 23 0 2 0 22 35 row =true class Class 1 : Variables left in model WO2004/104856 PCT/AU2004/000696 - 110 1 2 3 4 45 regression coefficients 43.24456 29.59787 17.72493 -19.66469 -5.55413 5 *********************************************** Iteration 22 : 3 cycles, criterion -1.017986 misclassification matrix fhat 10 f 1 2 1 23 0 2 0 22 row =true class 15 Class 1 : Variables left in model 1 2 3 4 45 regression coefficients 43.24598 29.59889 17.72556 -19.6653 -5.554354 20 1 misclassification table pred y 1 2 3 4 25 1 4 0 0 0 2 0 3 0 0 3 0 0 4 0 4 0 0 0 4 Identifiers of variables left in ordered categories model 30 1 42

Claims (30)

1. A method of identifying a subset of components of a system based on data obtained from the system using at least 5 one training sample from the system, the method comprising the steps of: obtaining a linear combination of components of the system and weightings of the linear combination of components, the weightings having values based on data 10 obtained from the at least one training sample, the at least one training sample having a known feature; obtaining a model of a probability distribution of the known feature, wherein the model is conditional on the linear combination of components; 15 obtaining a prior distribution for the weighting of the linear combination of the components, the prior distribution comprising a hyperprior having a high probability density close to zero, the hyperprior being such that it is not a Jeffreys hyperprior; 20 combining the prior distribution and the model to generate a posterior distribution; and identifying the subset of components based on a set of the weightings that maximise the posterior distribution. 25

2. The method as claimed in claim 1, wherein the step of obtaining the linear combination comprises the step of using a Bayesian statistical method to estimate the weightings.

3. The method as claimed in claim 1 or 2, further 30 comprising the step of making an apriori assumption that a majority of the components are unlikely to be components that will form part of the subset of components.

4. The method as claimed in any one of the preceding 35 claims, wherein the hyperprior comprises one or more adjustable parameters that enable the prior distribution near zero to be varied. WO 2004/104856 PCT/AU2004/000696 - 112

5. The method as claimed in any one of the preceding claims, wherein the model comprise a mathematical equation in the form of a likelihood function that provides the 5 probability distribution based on data obtained from the at least one training sample.

6. The method as claimed in claim 5, wherein the likelihood function is based on a previously described model 10 for describing some probability distribution.

7. The method as claimed in any one of the preceding claims, wherein the step of obtaining the model comprises the step of selecting the model from a group comprising a 15 multinomial or binomial logistic regression, generalised linear model, Cox's proportional hazards model, accelerated failure model and parametric survival model.

8. The method as claimed in claim 7, wherein the model 20 based on the multinomial or binomial logistical regression is in the form of: eik e n G-1 x g S= G-1 G-1 g=1 )h=1 25

9. The method as claimed in claim 7, wherein the model based on the generalised linear model is in the form of: L= log p(y ,) y0i ,-b(O) + c(yi,o)} i=1 a/4O)

10. The method as claimed in claim 7, wherein the model 30 based on the Cox's proportional hazards model is in the form WO2004/104856 PCT/AU2004/000696 - 113 of: rdI N exp Z i ) j =1ex p (Z i,6

11. The method as claimed in claim 7, wherein the model based on the Parametric Survival model is in the form of: 5 L=Z cilog(pi)-pi+Ci log [ i) i=1 A (yi;

12. The method as claimed in any one of the preceding claims, wherein the step of identifying the subset of components comprises the step of using an iterative 10 procedure such that the probability density of the posterior distribution is maximised.

13. The method as claimed in claim 12, wherein the iterative procedure is an EM algorithm. 15

14. A method for identifying a subset of components of a subject which are capable of classifying the subject into one of a plurality of predefined groups, wherein each group is defined by a response to a test treatment, the method 20 comprising the steps of: exposing a plurality of subjects to the test treatment and grouping the subjects into response groups based on responses to the treatment; measuring components of the subjects; and 25 identifying a subset of components that is capable of classifying the subjects into response groups using a statistical analysis method. WO2004/104856 PCT/AU2004/000696 - 114

15. The method as claimed in claim 14, wherein the statistical analysis method comprises the method as claimed in any one of claims 1 to 13. 5

16. An apparatus for identifying a subset of components of a subject, the subset being capable of being used to classify the subject into one of a plurality of predefined response groups wherein each response group, is formed by exposing a plurality of subjects to a test treatment and 10 grouping the subjects into response groups based on the response to the treatment, the apparatus comprising: an input for receiving measured components of the subjects; and processing means operable to identify a subset of 15 components that is capable of being used to classify the subjects into response groups using a statistical analysis method.

17. The apparatus as claimed in claim 16, wherein the 20 statistical analysis method comprises the method as claimed in any one of claims 1 to 15.

18. A method for identifying a subset of components of a subject that is capable of classifying the subject as being 25 responsive or non-responsive to treatment with a test compound, the method comprising the steps of: exposing a plurality of subjects to the test compound and grouping the subjects into response groups based on each subjects response to the test compound; 30 measuring components of the subjects; and identifying a subset of components that is capable of being used to classify the subjects into response groups using a statistical analysis method. 35

19. The method as claimed in claim 18, wherein the statistical analysis method comprises the method as claimed in any one of claims 1 to 13. WO2004/104856 PCT/AU2004/000696 - 115

20. An apparatus for identifying a subset of components of a subject, the subset being capable of being used to classify the subject into one of a plurality of predefined 5 response groups wherein each response group is formed by exposing a plurality of subjects to a compound and grouping the subjects into response groups based on the response to the compound, the apparatus comprising; an input operable to receive measured components of 10 the subjects; processing means operable to identify a subset of components that is capable of classifying the subjects into response groups using a statistical analysis method. 15

21. The apparatus as claimed in claim 20, wherein the statistical analysis method comprises the method as claimed in any one of claims 1 to 15.

22. An apparatus for identifying a subset of components of 20 a system from data generated from the system from a plurality of samples from the system, the subset being capable of being used to predict a feature of a test sample, the apparatus comprising: a processing means operable to: 25 obtain a linear combination of components of the system and obtain weightings of the linear combination of components, each of the weightings having a value based on data obtained from at least one training sample, the at least one training sample having a known feature; 30 obtaining a model of a probability distribution of a second feature, wherein the model is conditional on the linear combination of components; obtaining a prior distribution for the weightings of the linear combination of the components, the prior 35 distribution comprising an adjustable hyperprior which allows the prior probability mass close to zero to be varied wherein the hyperprior is not a Jeffrey's hyperprior; WO2004/104856 PCT/AU2004/000696 - 116 combining the prior distribution and the model to generate a posterior distribution; and identifying the subset of components having component weights that maximize the posterior distribution. 5

23. The apparatus as claimed in claim 22, wherein the processing means comprises a computer arranged to execute software. 10

24. A computer program which, when executed by a computing apparatus, allows the computing apparatus to carry out the method as claimed in any one of claims 1 to 13.

25. A computer readable medium comprising the computer 15 program as claimed in claim 24.

26. A method of testing a sample from a system to identify a feature of the sample, the method comprising the steps of testing for a subset of components that are diagnostic of 20 the feature, the subset of components having been determined by using the method as claimed in any one of claims 1 to 15.

27. The method as claimed in claim 26, wherein the system is a biological system. 25

28. An apparatus for testing a sample from a system to determine a feature of the sample, the apparatus comprising means for testing for components identified in accordance with the method as claimed in any one of claims 1 to 15. 30

29. A computer program which, when executed by on a computing device, allows the computing device to carry out a method of identifying components from a system that are capable of being used to predict a feature of a test sample 35 from the system, and wherein a linear combination of components and component weights is generated from data generated from a plurality of training samples, each WO2004/104856 PCT/AU2004/000696 - 117 training sample having a known feature, and a posterior distribution is generated by combining a prior distribution for the component weights comprising an adjustable hyperprior which allows the probability mass close to zero 5 to be varied wherein the hyperprior is not a Jeffrey's hyperprior, and a model that is conditional on the linear combination, to estimate component weights which maximise the posterior distribution. 10 30. A method of identifying a subset of components of a biological system, the subset being capable of predicting a feature of a test sample from the biological system, the method comprising the steps of: obtaining a linear combination of components of the 15 system and weightings of the linear combination of components, each of the weightings having a value based on data obtained from at least one training sample, the at least one training sample having a known feature; obtaining a model of a probability distribution of the 20 known feature, wherein the model is conditional on the linear combination of components; obtaining a prior distribution for the weightings of the linear combination of the components, the prior distribution comprising an adjustable hyperprior which 25 allows the probability mass close to zero to be varied; combining the prior distribution and the model to generate a posterior distribution; and identifying the subset of components based on the weightings that maximize the posterior distribution.

30 DATED this 2 6 th day of May 2004 CSIRO By their Patent Attorneys GRIFFITH HACK 35