Bionformatics, SRA, ITC-irst & Dept. of Information and Communication Technology, University of Trento, Trento, Italy

Department of Pathology, Brigham and Women's Hospital, Boston, MA, USA

Harvard Medical School, Boston, MA, USA

Dipartimento di Informatica e Sistemistica, Università di Pavia, Pavia, Italy

Dana Farber Harvard Cancer Center, Boston, MA, USA

Abstract

Background

Uncertainty often affects molecular biology experiments and data for different reasons. Heterogeneity of gene or protein expression within the same tumor tissue is an example of biological uncertainty which should be taken into account when molecular markers are used in decision making. Tissue Microarray (TMA) experiments allow for large scale profiling of tissue biopsies, investigating protein patterns characterizing specific disease states. TMA studies deal with multiple sampling of the same patient, and therefore with multiple measurements of same protein target, to account for possible biological heterogeneity. The aim of this paper is to provide and validate a classification model taking into consideration the uncertainty associated with measuring replicate samples.

Results

We propose an extension of the well-known Naïve Bayes classifier, which accounts for biological heterogeneity in a probabilistic framework, relying on Bayesian hierarchical models. The model, which can be efficiently learned from the training dataset, exploits a closed-form of classification equation, thus providing no additional computational cost with respect to the standard Naïve Bayes classifier. We validated the approach on several simulated datasets comparing its performances with the Naïve Bayes classifier. Moreover, we demonstrated that explicitly dealing with heterogeneity can improve classification accuracy on a TMA prostate cancer dataset.

Conclusion

The proposed Hierarchical Naïve Bayes classifier can be conveniently applied in problems where within sample heterogeneity must be taken into account, such as TMA experiments and biological contexts where several measurements (replicates) are available for the same biological sample. The performance of the new approach is better than the standard Naïve Bayes model, in particular when the within sample heterogeneity is different in the different classes.

Background

The biomedical sciences are fraught with uncertainty. The sources of this uncertainty are manifold. Devices used to monitor biological processes vary in terms of resolutions. Gaps in the full understanding of basic biology compound this problem. Biological diversity or heterogeneity may make predictions difficult. Finally, uncertainty may be due to the unpredictable sources of noise, which can be inside or outside the biological system itself.

In molecular biology uncertainty is ubiquitous; for example, tissue heterogeneity makes it difficult to compare a tissue sample composed of pure tumor cell populations with one composed of tumor and other non-tumoral elements such as supporting structural tissues (i.e. stroma) and vessels. However, in molecular biology, one rarely can examine an entire tumor and biopsies are taken with the assumption that they represent a portion of the whole tumor.

This paper addresses the uncertainty associated with measuring replicate samples. Understanding such kind of uncertainty would help guide decision making and allow for alternate strategies to be explored. Usually, the measurements of replicate samples are averaged to derive a single measurement. This value is then used for example when building a classification system which may play a critical role in the decision making process. Unfortunately, an average measurement (or median value) hides the uncertainty or heterogeneity present in the replicates, and may thus lead to decision making rules which are too reliant on this pooled data. This process may lead to a model that is not sufficiently robust to work in an independent dataset.

TMA studies represent a context where the issue of biological heterogeneity is particularly relevant. Where gene expression microarray experiments provide researchers with quantitative evaluation of transcripts, TMA evaluate DNA, RNA or protein targets through

TMA datasets usually include replicate core biopsies of the same tissue from the same individual to ensure that enough representative tissue is available in each experiment and to better represent the biological variability of the tissue itself and of the protein activity (i.e. accurate sampling). Most TMA datasets are evaluated using straightforward pooling of the data from replicates, thus ignoring variations among biopsies from the same patient (the so called within sample variability). The mean, the maximum or the minimum is usually adopted and the strategy may be based on biological knowledge or on known protein associations. However, it has been found that different choices can lead to covariates with different significance levels in Cox regression

However the degree of heterogeneity of the tumor tissue may be an important biological parameter. In a probabilistic framework, for example, accounting for the within sample variability caused by the tumor tissue heterogeneity, could alter the probability of a case belonging to a certain class (even changing the predicted class), providing insight into the particular case study. When measurement occurs at different levels, i.e. different biopsies of the same tumor or different tumors, standard statistical techniques are not appropriate because they either assume that groups belong to entirely different populations or ignore the aggregate information entirely.

Hierarchical models (multilevel models) provide a way of pooling the information for the different groups without assuming that they belong to precisely the same population

Herein we propose a classification model, which accounts for the tumor within sample variability in a probabilistic framework, relying on Bayesian hierarchical models. Hierarchical Bayes models have been used for modeling individual effects in several experimental contexts, ranging from toxicology to tumor risk

The paper is structured as follows: we first provide relevant background on Bayesian classifiers (specifically on the Naïve Bayes classifiers) and on Bayesian hierarchical models. Then we describe the proposed method and compare its performances to a Naive Bayesian classifier, in which we applied standard pooling strategies. The results will be shown on simulated datasets characterized by different ratios of within and between samples variability and on a real classification problem based on TMA data we generated in our laboratory from a prostate cancer progression array (TMA Core of Dana Farber Harvard Cancer Center, Boston, MA) developed to identify proteins that can distinguish aggressive from indolent forms of this common tumor type.

Bayesian classifiers and the Naïve Bayes

In this paper we focus on classification problems, where, given the data coming from a target case, we must decide to which class the case belongs. For example, given the set of tumor marker values measured on biopsies of a tissue of a given patient, we must decide if the patient is affected by a particular kind of tumor.

From a Bayesian viewpoint, a classification problem can be written as the problem of finding the class with maximum probability given a set of observed attribute values. Such probability is seen as the posterior probability of the class given the data, and is usually computed using the Bayes theorem, as ** X**) =

Bayesian classifiers are known to be the optimal classifiers, since they minimize the risk of misclassification. However, they require defining ** X**|

Conversely, in the framework of Naïve Bayes classifiers, the attributes are assumed to be independent from each other given the class. This allows us to write, following Bayes theorem, the posterior probability of the class ** X**) = ∏

Bayesian Hierarchical Models

Bayesian hierarchical models (BHM)

Bayesian hierarchical models provide a natural way to represent this relationship by specifying a suitable conditional independence structure and a suitable set of conditional probability distributions.

The simpler structure of a BHM can be summarized as follows: let us suppose that a certain variable _{rep }times in _{i}| _{i}). The assumption that the individuals are "related" to each other can be then represented by introducing the conditional probability _{i}|_{1}, _{2}, ..., _{m}}. Once a data set ** X **= {

BHMs have been applied in a variety of contexts, ranging from signal processing

Results

The Hierarchical Naïve Bayes approach

From a probabilistic perspective, a classification problem can be viewed as the selection of the class which has the highest (posterior) probability given the available data. Here we explicitly handle data with multiple replicate values.

In the case of TMA, we can think of one case as being the tumor tissue of one patient and replicates being the multiple biopsies from that patient. Therefore, the evaluation of a target protein (feature) on a TMA section will provide the pathologist with multiple evaluations of protein expression for each patient (case).

Let _{ij}, _{rep}, _{Ck}, where _{Ck }is the number of cases in the class _{k}.

Let us assume that the values of replicates of the generic case _{i }with a variance ^{2 }(independent from _{ij}~_{i}, ^{2}). The mean values _{i }are, at their turn, normally distributed around a "population" mean value ^{2}, i.e. _{i}~^{2}). The assumption that the variance is the same for all patients belonging to the same class reflects the intuitive notion that the variability over replicates, due for example to the tissue heterogeneity, is a property of the disease. Such an assumption, which is realistic in TMA data, turns out to be convenient when estimating the variance from the data: the reliability of the estimate is increased by the higher number of measurements exploited. The resulting hierarchical model is presented in Fig.

Structure of a hierarchical model

**Structure of a hierarchical model**. The replicates _{i }with a within sample variance ^{2}, i.e. _{ij}~_{i}, ^{2}). The mean values _{i }are normally distributed around a "population" mean value ^{2}, i.e. _{i}~^{2}).

Given this probabilistic model, we here describe how to classify a new case, supposing that the class model parameters ^{2}, ^{2 }are known for each class. In the Methods section we detail how to learn the model parameters from a training dataset. Moreover, the same section reports the classification and the learning phase of Standard Naïve Bayes (StNB) classifier, in order to highlight the differences.

To classify a new case in the Bayesian framework it is necessary to evaluate the posterior probability of each class given the case data. Let us define the vector **X**_{i }= (_{i1}, _{i2},..., _{inrepi}), which represents the replicate measurements of the

By applying the Bayes' theorem, the posterior probability for the class C_{k }given the set of data is

To evaluate the posterior probability, the marginal likelihood ** X**|

The marginal likelihood can be written as (for sake of readability the subscript ^{2}, ^{2 }have been omitted in the left hand side of the equation):

Applying simple algebra (details are reported

The marginal likelihood for the Hierarchical Naïve Bayes Model.

Click here for file

Finally, given the model parameters, the new case ** X **can be classified into the class that maximize the posterior probability, which is proportional to the marginal likelihood if the classes are

It is interesting to note that the main novelty of the method is that the classification rule (through the marginal likelihood) includes the information on the within sample heterogeneity. Such information is expressed by the parameter ^{2}, which may therefore guide decisions when there is a clear difference between the within sample heterogeneity of cases belonging to the different classes. Let us note that standard approaches, such as the StNB or the quadratic discriminant analysis, can take into account only between samples variability, expressed in our model by the parameter ^{2 }. Moreover, since the classification rule can be calculated in closed-form, it can be used in real-time applications, such as the StNB classifier.

The generalization for the multivariate case, i.e. **X**^{1}, **X**^{2}**,...,X**^{Nfeature}), can be obtained by assuming, as in the StNB classifier, the conditional independence of the features given the class, i.e.

In this case, the posterior probability of class

Results on simulated and real data

We present the results we obtained using both computationally generated datasets and a real TMA protein expression dataset.

A first set of simulated data was generated to represent the best scenario, by incrementally varying the within sample variance ^{2 }for one class only. A second set of normally distributed data was generated using a variety of parameter values (see

Parameter values used to generate simulated data.

Click here for file

In both studies, being the Hierarchical Naïve Bayes (HierNB) classifier an extension of the StNB classifier to cope with replicate measurements, results are compared with the StNB classifier to highlight, without introducing additional bias due to the different classification techniques, the advantage of the new approach. However, the real data were analyzed by several other classification methods and the results are reported in the

Comparison of the proposed approach with other classification strategies on the TMA protein expression dataset.

Click here for file

The classifiers are compared on the basis of two indexes: the accuracy (defined as the ratio of the properly classified and the total classified cases) and the Brier score

The data analysis reported in this paper has been implemented in the R statistical package

Simulated data

Data description

We generated simulated datasets with 4000 patients (2000 for class 1 and 2000 for class 2), 5 replicates each and a number of independent features ranging from 1 to 10. For each feature, the values of the five replicates were randomly extracted from a normal distribution with fixed variance ^{2 }(dependent on the class and on the feature) and mean randomly generated from a normal distribution with fixed mean M and variance ^{2}, again dependent on the class and on the feature.

A first set of experiments was run for the univariate case (one feature), so that the two classes basically had the same parameter values, but for the within sample variance parameter ^{2 }that is assumed bigger in the second class. In particular they had a similar population mean M and exactly the same class variance ^{2 }(M_{1 }= 100, M_{2 }= 105, _{1}^{2 }= _{2}^{2 }= 300, _{1}^{2 }= 10). The parameter _{2}^{2 }varied from 15 to 75.

A second set of experiments was run simulating both univariate case studies and multivariate case studies (here we report results for two, three and ten features). The values of the first feature were generated for all these experiments using 90, 300 and 100 as M, ^{2 }and ^{2 }for class 1 and 140, 600 and 400 as M, ^{2 }and ^{2 }for class 2.

The complete set of parameter values used for the model in the multivariate set of experiment is reported in the

We assessed the performances of the two classifiers by equally dividing the dataset into training and test set.

Data results

The results of the first experiment are shown in Tables

Results on simulated data for 1 feature with different level of within sample heterogeneity in the different classes.

HierNB Classifier

StNB Classifier

Exp

_{2}
^{2}

Acc

Brier

Acc

Brier

1

15

0.620 [0.604 0.636]

0.449

0.559 [0.539 0.580]

0.490

2

30

0.762 [0.740 0.790]

0.307

0.560 [0.540 0.580]

0.490

3

45

0.830 [0.813 0.849]

0.228

0.554 [0.537 0.570]

0.490

4

60

0.878 [0.866 0.888]

0.176

0.556 [0.531 0.582]

0.490

5

75

0.899 [0.883 0.914]

0.147

0.560 [0.534 0.586]

0.490

Exp = experiment number, Acc = Accuracy, Brier = Brier Score. In brackets the 95% confidence intervals for the estimate of the accuracy.

Results for the second set of four experiments are presented in Table

Results on simulated data: experiments were done using increasing number of features.

HierNB Classifier

StNB Classifier

Exp

N Feat.

Acc

Brier

Acc

Brier

1

1

0.925 [0.917, 0.933]

0.112

0.874 [0.864, 0.884]

0.184

2

2

0.966 [0.960, 0.971]

0.052

0.921 [0.912, 0.929]

0.118

3

3

0.987 [0.983, 0.990]

0.020

0.946 [0.938, 0.952]

0.082

4

10

0.998 [0.997, 0.999]

0.002

0.985 [0.981, 0.989]

0.023

Exp = number of experiment, N. Feat = Number of features, Acc = Accuracy, Brier = Brier Score. In brackets the 95% confidence intervals for the estimate of the accuracy.

The HierNB classifier performs better in all the experiments, showing higher accuracy and lower Brier score.

In Figure

Posterior probabilities of the three feature simulated experiment

**Posterior probabilities of the three feature simulated experiment**. Histograms of posterior probabilities of the three feature experiment (Exp.3, Table 2) on a simulated dataset. Panels A and B show the results obtained with the HierNB classifier for class 1 and 2 respectively; panels C and D show results obtained with the StNB classifier. In the upper right corner of each panel the frequency of the bin corresponding to the highest posterior probability range is reported.

The HierNB classifier shows a better separation between the two classes not only in term of accuracy but also in term of credibility of the classification (as highlighted by the Brier Score). The confidence intervals of the estimated accuracy confirm that in all the experiments the proposed method outperforms the standard classifier. This second experiment shows that, even if the experimental context is complex since the classes show an overlap due to the values of the within and between sample variability, the method is able to perform equal or better than the StNB.

Real data

Data description

We used a research dataset obtained from a recently constructed prostate progression TMA, as previously described

The TMA was constructed to test molecular differences between localized and metastatic prostate cancer samples, on a total of 288 core biopsies. In this paper, we explore the expression of two proteins, i.e. EZH2 and AMACR known to be differentially expressed in non aggressive or localized tumors (class 1 or negative class) versus aggressive or metastatic prostate cancers (class 2 or positive class). The Polycomb Group protein, EZH2, is over-expressed in hormone-refractory and metastatic prostate cancer

The TMA dataset includes 72 patients (samples), 36 for each class, each case having four replicates. After the processing of the TMA slides, 35 and 34 cases were suitable for analysis for class 1 and class 2 respectively (69 cases) and each case was characterized measuring from 1 to 4 times for two proteins. The assumption that the data have a Gaussian distribution given the class has been verified by applying the Kolmogorov-Smirnov test. Table ^{2 }estimated according to the StNB and ^{2 }estimated as the average within case variance for the two classes. Since the estimate of ^{2 }is different in the two classes, this classification problem may benefit from the use of the HierNB approach.

TMA data description: model parameters of localized (class 1) and metastatic prostate cancer tumors (class 2) for two proteins.

M

^{2}

^{2}

Class

1

2

1

2

1

2

AMACR

155.2

148.8

201.2

208.8

49.1

85.7

EZH2Int

146.2

141.6

86.7

107.9

135.7

53.9

M = class mean; ^{2 }= class variance; ^{2 }= averaged within sample variance.

We assessed the performances of the two classifiers by applying one hundred times a 10 fold cross-validation procedure with different fold randomization and then by computing the average results.

Data results

Results obtained for the prostate cancer dataset are presented in Table

Results on TMA dataset for the two proteins.

**Acc**

**Spec**

**Sens**

**AUC**

**Brier**

**HierNB Model**

0.65 [0.62–0.68]

0.71 [0.66–0.74]

0.60 [0.56–0.62]

0.69 [0.689–0.693]

0.41 [0.39–0.42]

**StNB Model**

0.58 [0.54–0.61]

0.58 [0.54–0.62]

0.57 [0.53–0.61]

0.62 [0.617–0.622]

0.47 [0.46–0.48]

Acc = Accuracy, Spec = specificity, Sens = Sensitivity, Brier = Brier Score. In brackets the 95% confidence intervals for the estimates, AUC = area under the ROC.

ROC Curves for the TMA protein expression dataset as calculated by running 100 times 10-fold cross validation (Table

Click here for file

Posterior probabilities of prostate cancer cases

**Posterior probabilities of prostate cancer cases**. Histograms of the posterior probabilities of prostate cancer cases. Panels A and B show the results evaluated with the HierNB classifier for class 1 (localized tumors) and 2 (aggressive tumors) respectively; panels C and D show results evaluated with the StNB classifier.

The classification performance of the HierNB model clearly outperforms the StNB one, for what concerns all the evaluation parameters considered. In particular, both accuracy and the Brier score are significantly better in the HierNB case than in the StNB one, also considering the 95% confidence interval of the estimates. The fact that classification accuracy in distinguishing localized prostate cancer from metastases is only about 60% is not surprising. The complexity of this classification problem has been recently discussed in Bismar, Demichelis et al.

The HierNB classifier also shows a significantly higher specificity, and a similar sensitivity. We note from the histogram of posterior probabilities (Figure

Discussion

Few studies have dealt with the problem of uncertain data in classification

In this paper we have proposed a classifier based on Bayesian hierarchical models and have applied it on TMA datasets. The approach permits embedding in the classification model the tumor variability (heterogeneity of protein levels across tumor tissue), using the tuple of protein level measurements of each case instead of unique representative value as done by conventional approaches.

Bayesian hierarchical models have two main advantages with respect to other methods: i) they coherently manage uncertainty in the framework of probability theory; ii) they make explicit the assumptions which the model relies on. The implementation of the Bayesian classifier presented in this paper is an extension of the well known Naïve Bayes classifier. It assumes that all the attributes are independent among each other given the class. Moreover, we have also assumed that the probability distributions are conditionally Gaussian.

Preliminary performance tests on simulated data give us some clues about the applicability of the proposed model. With respect to classification, we observed that when classes have similar within sample variances no differences in terms of classification accuracy are obtained, as expected. However, increasing differences in the posterior distributions are detected as the difference of the within sample variability increases, e.g. _{1}^{2}<<_{2}^{2}. In this case the HierNB model outperforms the standard approach.

On TMA real data, we saw that the hierarchical model may improve specificity, which is part of the clinical question, and emphasizes the information available at every level, accounting for the spread of the replicate measures and thus may provide interesting insights into the biology of the tumor samples being analyzed. Rather interestingly, in this case the classification model is able to improve the data comprehension, highlighting if the heterogeneity of the tumor tissue sample is critical or not in the decision making process. Moreover, hierarchical models are also able to exploit the information on the lack of heterogeneity.

Heterogeneous and homogeneous protein expression may reflect different biological processes occurring in tumors. Exploiting this data may be critical in understanding the underlying biology.

Finally, the HierNB presents interesting robustness properties when comparing the results obtained in the data-rich case of the simulation study (4000 samples) with the relatively data-poor real one (69 samples). The real case is much more difficult than the simulated one, due to the lower number of samples and the smaller difference of the mean values of the markers in the two classes. Such a difficulty results in more spread posterior distributions and in lower accuracy and higher Brier score values of all tested classification models. However, in both the simulated and the real cases the HierNB shows nearly the same gain in accuracy with respect to the StNB, taking advantage from the within sample variability information to better separate classes.

From a practical point of view, in TMA experiments in which hundreds of cases are evaluated and only a fraction do not fit well into one class or another, one can imagine that by using the hierarchical Naïve Bayes model, cases with a posterior probability within a certain window around 0.5 would be classified as ambiguous and would require re-review.

From a methodological point of view, in order to generalize the proposed approach, we are now working on the following aspects:

1) Learning: while the classification step fully follows the Bayesian approach, the learning phase of the proposed method is not fully Bayesian. This choice was motivated by the need to perform fast learning from potentially large datasets for the needed probability distributions. However, it is also possible to resort to a more rigorous learning procedure by paying the price of implementing iterative procedures, such as Expectation Maximization (EM) or Monte Carlo Markov chain (MCMC) approaches

2) Non Gaussian distributions: we have also implemented a version of the hierarchical Naïve Bayes approach for discrete variables, relying on multinomial and Dirichlet probability distributions

Conclusion

We have proposed a novel approach for dealing with uncertain data in classification, with applications to TMA microarrays. The proposed model has, as its unique property, the capability of handling data heterogeneity in a sound probabilistic way, without requiring additional computational burden with respect to the standard Naïve Bayes approach. Based on the results obtained on simulated and real data, we can conclude that the proposed approach is particularly useful when the within sample heterogeneity differs between classes. Its application to TMA data has been shown to provide more insight into the information available in the database and to improve the decision making process also in presence of a very limited number of features. The proposed model can be conveniently applied and extended to deal with other application domains in Bioinformatics.

Methods

Tissue microarray technology

TMAs were recently developed to facilitate tissue-based research

Cylindrical tissue biopsies are transferred with a biopsy needle from carefully selected morphologically representative areas from original paraffin blocks (donor blocks), each containing tumor tissue from a patient. Core tissue biopsies are then arrayed into a new "recipient" paraffin block by using a tissue arrayer using a precise spacing pattern along x and y axis, which generates a regular matrix of cores. Typically, more than one biopsy from each patient is included in a TMA block; replicates allow for good representation of the patient's tumor and to potentially detect heterogeneous expression of markers (e.g. proteins) of interest within the tumor. How well TMA samples represent entire tumors has been the focus of several recent studies

Learning the Hierarchical Naïve Bayes Model

The classification algorithm described in the Results section assumes that the model parameters (^{2}, ^{2}) have been estimated from a training dataset. In the implementation of the method presented in this paper, we have adopted an approximation of the maximum likelihood estimation approach, called empirical learning

Following such approach, the within sample means and variances are estimated as:

The estimate of the population variance includes two terms, representing

Classification by standard Naïve Bayes Model

Also in this case the classification of a new case requires the evaluation of the posterior probability of each class given the case data. However, the standard Naïve Bayes classifier does not consider the replicate measurements, but only their aggregate value (standard pooling strategy, e.g. mean value). Let X be the mean value of a given feature (univariate case) of the case to classify. The likelihood of X, X ∈ R, is

Learning of standard Naïve Bayes Model

Also in this case, the model parameters (M, ^{2}) have to be estimated from a training dataset. They can be computed as:

Authors' contributions

FD, PM and RB developed and implemented the Hierarchical Naïve Bayes Model presented in the paper. MAR was involved in the discussions about the suitability of the model to face protein expression heterogeneity in human tumors and generated the TMA protein expression datasets. PP helped in the evaluation of the model performances. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank Robert Kim for his technical expertise in performing the TMA experiments, Juan- Miguel Mosquera and Kirsten D. Mertz for their pathology evaluation of the tissue microarray experiments and Andrea Sboner and Rossana Dell'Anna for critical discussion on the paper. RB and PM acknowledge the FIRB project "Learning theory and engineering applications", funded by the Italian Ministry of University and scientific research. FD was supported by a Prostate Cancer Foundation award.