Liver fibrosis is the most important factor that determines the prognosis in chronic liver disease patients. Liver cirrhosis which is the extreme of liver fibrosis is an independent risk factor for liver cancer, chronic hepatitis B is the most important cause (62.5-73%) in Korea [9]. Liver biopsy is the gold standard for assessing liver fibrosis, but its invasive method can be risk of complications such as pain and bleeding.

Transient elastography (Fibroscan) is a rapid and non-invasive method to measure liver stiffness and this allow the assessment of liver fibrosis [8]. Researcher is under way to learn more and physicians are to assess the diagnostic accuracy of measuring the liver stiffness.

In this study, the liver stiffness data is taken from 228 HBsAg positive patients whose liver stiffness was measured by Fibroscan between March 2005 and September 2005. Liver biopsy examinations were performed in 34 patients. The fibrosis (F) was staged on a 0-4 scale according to the Ludwig classification. Kim et al. [8] concluded that Fibroscan is a reliable and rapid noninvasive method to diagnose the severity of chronic liver disease and to predict fibrosis in patients with chronic hepatitis B in addition to using the aspartate aminotransferase-to-platelet ratio index (APRI) and aspartate aminotransferase/aspartate aminotransferase (AST/ALT) ratio.

In this subsection, we briefly review the ME architecture [4] and the EM algorithm [3]. The ME architecture is shown in (Figure 1). The architecture is composed of a gating network sit at the nonterminals of a tree and several expert networks sit at leaves of a tree. The gating network receives the input vector x as input and produces scalar outputs that are partition of unity at each point in the input space. Each expert network produces an output vector for each input vector. These output vectors proceed up the tree, being mixture of the gating network outputs. The gating network provides linear combination coefficients as vertical probabilities for expert networks and, therefore, the final output of the ME architecture is a convex weighted sum of all the output vectors produced by expert networks. Suppose that there are N expert networks in the ME architecture. All the expert networks are linear with a single output nonlinearity that is also referred to as 'generalized linear' [10]. The i-th expert network produces its output o_i(x) as a generalized linear function of the input x:

o_i(x) = f(W_ix)

where W_i is a weight matrix and f(·) is generally taken to be the logistic function or the identity function. These models are smoothed piecewise analogs of the corresponding generalized log-linear model (GLIM) models [3]. The gating network is also generalized linear function, and its i-th output, g(x,v_i), is the multinomial logit or softmax function of intermediate variables ξ_i [10]:

where and v_i is a weight vector. The overall output o(x) of the ME architecture is

The ME architecture can be given a probabilistic interpretation. For an input-output pair (x,y), the values of g(v_i,x) are interpreted as the multinomial probabilities associated with the decision that terminates in a regressive process that maps x to y. Once the decision has been made, resulting in a choice of regressive process i, the output y is then chosen from a probability density P(y∣x,W_i), where W_i denotes the set of parameters or weight matrix of the i-th expert network in the model. Therefore, the total probability of generating y from x is the mixture of the probabilities of generating y from each component densities, where the mixing proportions are multinomial probabilities:

where Φ includes the expert network parameters as well as the gating network parameters. Moreover, the probabilistic component of the model is generally assumed to be a Bernoulli distribution in the case of binary classification, a multinomial distribution in the case of multiclass classification and a Gaussian distribution in the case of regression. Based on the probabilistic model, a learning algorithm for the ME architecture is treated as a maximum likelihood estimation problem. Jordan and Jacobs [3] have developed a learning algorithm for the ME architecture based on EM framework.

Suppose that the training set is given as . The EM algorithm consists of two steps. In E-step, the posterior probabilities h_i^(t)(i=1,…,N), which can be interpreated as the probabilities P(i∣x_t,y_t) are computed for the s-th epoch:

The M-step solves the following maximization problems:

and

where V is the set of all the parameters in the gating network. Therefore, the EM algorithm is summarized as the following [3]:

For each data pair (x_t,y_t), compute the posterior probabilities h_i^(t) using the current values of the parameters.

For each expert network i, solve the a maximization problem in W_i^(s+1) with observations and observation weights .

For the gating network, solve the maximization problem in V^(s+1) with observations .

Iterate by using the updated parameter values.

The ME architecture used for the diagnosis of liver cirrhosis is shown in (Figure 2). Since two-group classification was investigated exclusively, the ME was configured with two local experts and a gating network.

There are a total of 228 HBsAg positive patients whose liver stiffness was measured by Fibroscan between March 2005 and September 2005. The diagnosis of liver cirrhosis (LC) were considered such as LC with compensated, LC with decompensated and hepatocellular carcinoma (HCC) cases.

Statistical analysis was performed by R 2.13.0 (software available at http://www.r-project.org). The characteristics of LC and non-LC patients were compared using two-sample t-test and chi-square test. Classical logistic regression model and ME model was described, and classifying of the hidden subgroups.

Receiver operating characteristic (ROC) curves were shown for comparing classical logistic regression and ME model from this data for the validation of model.

A total of 228 patients were enrolled in this study. The mean age of the patients was 45.9 years and 177 patients were male. In diagnosis, 29 (12.7%) patients were inactive carrier, 106 (46.5%) patients had chronic hepatitis, 63 (27.6%) patients had LC with compensated, 26 patients had LC with decompensated and 4 (1.8%) patients were HCC. The mean age of the liver biopsy patients was 39.4 years and fibrosis stage 1 (portal fibrosis) was 4 (11.8%) patients, stage 2 (periportal fibrosis) was 12 (35.3%) patients, stage 3 (septal fibrosis) was 2 (5.9%) patients and stage 4 (cirrhosis) was 16 (47.1%) patients (Table 1).

According to the clinical diagnosis, the median values of liver stiffness were 7.0 ± 2.7 kPa for inactive carriers (n = 29), 8.3 ± 5.3 kPa for chronic hepatitis patients (n = 106), 15.9 ± 8.3 kPa for compensated cirrhosis patients (n = 63), 31.8 ± 20.3 kPa for decompensated cirrhosis patients (n = 26), and 45.1 ± 34.5 kPa for HCC patients (n = 4). The degree of liver stiffness was statistically significant (p < 0.001) (Table 2).

In classification, the aim is to assign the input patterns to one of several classes, usually represented by outputs restricted to lie in the range from 0 to 1, so that they represent the probability of class membership [1]. While the classification is carried out, a specific pattern is assigned to a specific class according to the characteristic features selected for it. In this application, there were two classes: low risk and high risk. Classification results of ME were displayed by a confusion matrix. The confusion matrix showing the classification results of the ME is given as following (Table 3).

According to the confusion matrix, 8 patients were classified incorrectly by ME as non-LC patient in low risk group and no patient was classified as non-LC in high risk group.

The values of statistical parameters are given in (Table 4). The ME classified expert 1 (low risk) is lower parameter than expert 2 (high risk) except AST. Estimated gating networks are expert 1 (49.76%) and expert 2 (50.24%).

The performance of the ME model can be evaluated by plotting a ROC curve for the test (Figure 3). For a given result obtained by a classifier system, four possible alternatives exist that describe the nature of the result: 1) true positive (TP), 2) false positive (FP), 3) true negative (TN), and 4) false negative (FN) [11]. ROC curves which are shown in (Figure 3) represent performances of the stand-alone logistic regression and ME structure in liver stiffness data. This shows that the performance of the ME is higher than that of the stand-alone logistic regression.