Training artificial neural networks directly on the concordance index for censored data using genetic algorithms.

OBJECTIVE: The concordance index (c-index) is the standard way of evaluating the performance of prognostic models in the presence of censored data. Constructing prognostic models using artificial neural networks (ANNs) is commonly done by training on error functions which are modified versions of the c-index. Our objective was to demonstrate the capability of training directly on the c-index and to evaluate our approach compared to the Cox proportional hazards model. METHOD: We constructed a prognostic model using an ensemble of ANNs which were trained using a genetic algorithm. The individual networks were trained on a non-linear artificial data set divided into a training and test set both of size 2000, where 50% of the data was censored. The ANNs were also trained on a data set consisting of 4042 patients treated for breast cancer spread over five different medical studies, 2/3 used for training and 1/3 used as a test set. A Cox model was also constructed on the same data in both cases. The two models' c-indices on the test sets were then compared. The ranking performance of the models is additionally presented visually using modified scatter plots. RESULTS: Cross validation on the cancer training set did not indicate any non-linear effects between the covariates. An ensemble of 30 ANNs with one hidden neuron was therefore used. The ANN model had almost the same c-index score as the Cox model (c-index=0.70 and 0.71, respectively) on the cancer test set. Both models identified similarly sized low risk groups with at most 10% false positives, 49 for the ANN model and 60 for the Cox model, but repeated bootstrap runs indicate that the difference was not significant. A significant difference could however be seen when applied on the non-linear synthetic data set. In that case the ANN ensemble managed to achieve a c-index score of 0.90 whereas the Cox model failed to distinguish itself from the random case (c-index=0.49). CONCLUSIONS: We have found empirical evidence that ensembles of ANN models can be optimized directly on the c-index. Comparison with a Cox model indicates that near identical performance is achieved on a real cancer data set while on a non-linear data set the ANN model is clearly superior

Neural Network Approaches To Survival Analysis

Predicting the probable survival for a patient can be very challenging for many diseases. In many forms of cancer, the choice of treatment can be directly impacted by the estimated risk for the patient. This thesis explores different methods to predict the patient's survival chances using artificial neural networks (ANN). ANN is a machine learning technique inspired by how neurons in the brain function. It is capable of learning to recognize patterns by looking at labeled examples, so-called supervised learning. Certain characteristics of medical data make it difficult to use ANN methods and the articles in this thesis investigates different methods of overcoming those difficulties. One of the most prominent difficulties is the missing data known as censoring. Survival data usually originates from medical studies, which only are conducted during a limited time period for example during five years. During this time, some patients will leave the study for various reasons like death by unrelated causes. Some patients will also survive the study without experiencing cancer recurrence or death. These patients provide partial information about the survival characteristics of the disease but are challenging to include in statistical models. Articles 1-3, and 5 utilize a genetic algorithm to train ANN models to maximize (or minimize) non-differentiable functions, which are impossible to combine with traditional ANN training techniques which rely on gradient information. One of these functions is the concordance index, which compares survival predictions in a pair-wise fashion. This function is often used to compare prognostic models in survival analysis, and is maximized directly using the genetic algorithm approach. In contrast, Article 5 tries to produce the best grouping of the patients into low, intermediate, or high risk by maximizing, or minimizing the area under the survival curve. Article 4 does not use a genetic algorithm approach but instead takes the approach to modify the underlying data. Regular gradient methods are used to train ANNs on survival data where censored times are estimated in a maximum likelihood framework

Finding Risk Groups by Optimizing Artificial Neural Networks on the Area under the Survival Curve Using Genetic Algorithms

We investigate a new method to place patients into risk groups in censored survival data. Properties such as median survival time, and end survival rate, are implicitly improved by optimizing the area under the survival curve. Artificial neural networks (ANN) are trained to either maximize or minimize this area using a genetic algorithm, and combined into an ensemble to predict one of low, intermediate, or high risk groups. Estimated patient risk can influence treatment choices, and is important for study stratification. A common approach is to sort the patients according to a prognostic index and then group them along the quartile limits. The Cox proportional hazards model (Cox) is one example of this approach. Another method of doing risk grouping is recursive partitioning (Rpart), which constructs a decision tree where each branch point maximizes the statistical separation between the groups. ANN, Cox, and Rpart are compared on five publicly available data sets with varying properties. Cross-validation, as well as separate test sets, are used to validate the models. Results on the test sets show comparable performance, except for the smallest data set where Rpart's predicted risk groups turn out to be inverted, an example of crossing survival curves. Cross-validation shows that all three models exhibit crossing of some survival curves on this small data set but that the ANN model manages the best separation of groups in terms of median survival time before such crossings. The conclusion is that optimizing the area under the survival curve is a viable approach to identify risk groups. Training ANNs to optimize this area combines two key strengths from both prognostic indices and Rpart. First, a desired minimum group size can be specified, as for a prognostic index. Second, the ability to utilize non-linear effects among the covariates, which Rpart is also able to do

Ensembles of genetically trained artificial neural networks for survival analysis

We have developed a prognostic index model for survival data based on an ensemble of artificial neural networks that optimizes directly on the concordance index. Approximations of the c-index are avoided with the use of a genetic algorithm, which does not require gradient information. The model is compared with Cox proportional hazards (COX) and three support vector machine (SVM) models by Van Belle et al. [10] on two clinical data sets, and only with COX on one artificial data set. Results indicate comparable performance to COX and SVM models on clinical data and superior performance compared to COX on non-linear data

Analysis of regional bone scan index measurements for the survival of patients with prostate cancer

A bone scan is a common method for monitoring bone metastases in patients with advanced prostate cancer. The Bone Scan Index (BSI) measures the tumor burden on the skeleton, expressed as a percentage of the total skeletal mass. Previous studies have shown that BSI is associated with survival of prostate cancer patients. The objective in this study was to investigate to what extent regional BSI measurements, as obtained by an automated method, can improve the survival analysis for advanced prostate cancer

Predictions for the test set.

<p>Overall the results for the three models are quite similar. One notable exception is for <i>lung</i> where the low risk group predicted by Rpart actually has the worst survival.</p

Group sizes for the test set.

<p>Group sizes for the test set.</p

Median survival time in 10 × 3 cross-validation.

<p>This is only possible to compute if a group’s survival rate reaches 0.5, which no grouping in <i>nwtco</i> did. Groupings in <i>lung</i> had so poor survival that even the low-risk groups could be included.</p

Schematic of the training procedure.

<p>It is important to have similar group sizes so that the properties of the survival curves can be compared. To be able to include Rpart in the comparison, it is necessary to compensate for its inability to pre-determine suitable group sizes. The predicted group sizes on the training data (only sizes of groups, not the actual predictions) are set as parameters on Cox and ANN to generate similarly sized risk groups later on the test/validation data.</p