Prediction and Monitoring Method for Breast Cancer: A Case Study for Data from the University Hospital Centre of Coimbra

Introduction

Breast cancer is a malignant tumor (a collection of cancer cells) arising from the cells of the breast.¹ Although breast cancer predominantly occurs in women, it can also affect men. This article addresses breast cancer in women. Breast cancer and its complications can affect nearly every part of the body. Breast cancer screening is an important strategy for early detection and to ensure a greater probability of having a good treatment outcome. Robust predictive models based on data that may be collected in routine consultations and blood analysis are sought to provide an important contribution by offering more screening tools, and are important for detecting whether there is a change in the breast cancer index.^2,3

Statistical process control (SPC) has been frequently used for fault detection.^4–6 One major concern of SPC is whether there has been a change of distribution from the target in the process, that is, the process has gone out of control. Many researchers have discussed and proposed useful charts for detecting whether there is a change in a process. The most commonly used control schemes include the Shewhart chart,⁷ the CUSUM chart⁸ and the EWMA chart.⁹ These proposed control schemes are efficient for fault diagnosis in practice. Statistical properties of a control chart are usually evaluated in terms of the average run length (ARL), that is, the average number of observations required to signal a change for a particular size of the shift. When a process stays IC, control charts with larger ARL (ARL₀) are considered performing better. Otherwise, when the process is OC, these charts with smaller ARL (ARL₁) are considered better.

Control chart schemes attach increasing importance to biosurveillance studies. For example, Rogerson and Yamada¹⁰ proposed a multivariate cumulative sum approach to detect changes in spatial patterns and applied it to county-level breast cancer data in the Northeastern United States. The comparison results suggested that the multivariate chart performed well. Abdollahian and Hayati Rezvan¹¹ applied a multivariate EWMA control chart to monitor patients’ progress after cardiac surgery, in which the multivariate EWMA chart can detect an out-of-control signal that was missed by the univariate EWMA charts. Yue et al¹² proposed a new combined risk-adjusted EWMA and Variable life-adjusted display (VLAD) chart for detecting Surgical Outcome Monitoring and Improvement Program (SOMIP) data. In addition, Various kinds of control charts have been used to monitor surgical outcomes.^13–15 However, in most surgical contexts, the risk of mortality estimated pre-operatively would vary from patient to patient. Considering this fact, Cook et al¹⁶ proposed a risk-adjusted chart to track outcomes in intensive care. Steiner¹⁷ proposed a new CUSUM chart to monitor surgical performance in which the risk is adjusted to reflect the surgical risk of each patient. Moreover, many researchers have studied the application of risk-adjusted control charts to assess surgical outcomes.^18–20

Otherwise, most control charts require that the monitoring observations be univariate and usually assume that the observations follows a normal distribution. With data becoming complex and high dimensional, the monitoring of multivariate data has become increasingly important in quality control. The classical chart includes the T² control chart, which was proposed by Hotelling and assumes that the dataset distributions are multivariate normal.²¹ That is, both the mean vector and variance matrix are known. In addition, a multivariate CUSUM chart based on T² statistics was proposed by Lowry et al.²² These methods perform well under the multivariate normal distribution assumption. When the underlying distribution and the magnitude of the shifts are both unknown, Yue and Liu²³ used the Mahalanobis data depth method to propose a chart for monitoring processes with multivariate quality measurements. In addition, Liu et al²⁴ proposed a new multivariate EWMA chart based on ranks. Their method performs well for detecting a range of changes.

In this paper, based on Liu et al²⁴ we provide a nonparametric statistical method to predict and detect breast cancer occur. The remainder of this paper is organized as follows: in Section 2, we review the existing proposed rank-based control chart. In Section 3, the Breast Cancer Coimbra data are studied to illustrate the performance of the proposed chart. Finally, several remarks conclude the article in Section 4.

Review

Rank-Based Methods

Liu et al²⁵ introduced the rank-based method and assumed that observations , which are independent, follow the model below:

where and are the IC location parameter and the OC location parameter, respectively. represents the unknown change point. is an unknown continuous distribution function. Let denote the th sequential rank; the formula for the rank of among is as follows²⁵

The standardized sequential rank is given by

where

Therefore,

Therefore, the distribution of is defined in the interval

The asymptotic distribution of is as ²⁵.

EWMA Chart for a Multivariate Process

An EWMA control scheme begins with a time series graph. It is based on the statistic

together with UCL’s and LCL’s. is a smoothing parameter. The sequentially recorded observations, , can be individually observed values from the process. The process is considered OC and action should be taken whenever falls outside the range of the control limits. The EWMA chart performs well for small shifts with an appropriate smoothing parameter.²⁶

We cite this method proposed by Liu et al²⁷ in the context of a multivariate process, and they supposed that there are independent observations from an unknown multivariate continuous distribution with dimensionality . That is, where There are characteristics of interest to be examined. For a set of variables, which represents the th characteristic with observations, the rank-based method can be used to construct statistics. When the observations are -dimensional, the th observation is For the th component, denotes the th standardized sequential rank with the arrival of the th component Therefore, the vectors can be obtained. In addition, each component follows the same uniform distribution as Then, the EWMA statistics can be constructed, which are based on T² statistics. We cite the the method proposed by Liu et al0,²⁷ and the EWMA statistics are given by

where represents the smoothing parameter. represents the -dimensional identity matrix. If there is no a priori information given, different smoothing parameters are needed for different components; then, are used, and the starting value is The process is considered to be out of control, and action should be taken whenever where is the control limit. We cite the method proposed by Liu et al²⁷ the covariance matrix of is as follows:

In particular, when is a fixed value. Usually, we take the limit form, , the covariance matrix of , is estimated from samples in practice. We use this method for detection in the Breast Cancer Coimbra dataset.

Beast Cancer Coimbra Data

Data Source

For each of the 116 participants several clinical features were observed or measured. Clinical features were observed or measured for 64 patients with breast cancer and 52 healthy controls. Quantitative attributes including age (years), BMI (kg/m2), glucose (mg/dL), insulin (U/mL), HOMA, leptin (ng/mL), adiponectin (g/mL), resistin (ng/mL), and MCP-1 (pg/dL).²⁸ The characteristics are anthropometric data and parameters which can be gathered in routine blood analysis. The characteristics can potentially be used as a biomarker of breast cancer. The data are publicly available in the “ Breast Cancer Coimbra Data Set” from the UCI Machine Learning Repository and can be downloaded from the web site http://archive.ics.uci.edu/mL/datasets/Breast+Cancer+Coimbra. In this work, we aim to monitor the Beast Cancer Coimbra data and identify whether there are changes.

Figure 1 The corresponding normal Q-Q plots.

A quantile-quantile (Q-Q) plot of each region, which includes 116 historical observations, is presented in Figure 1. Figure 1 suggests that the normality assumption for the data are invalid, which leads us to reject the null hypothesis that the data are normally distributed. Therefore, a nonparametric control chart might be more suitable for this dataset. The correlation of nine attributes is shown in Figure 2 for a total of lines. Figure 2 shows that the cross-correlation is not stable. Therefore, we update the covariance matrix with the arrival of new observations. It should be noted that the covariance matrix is updated, as presented in Section 3.2.

Figure 2 Correlations of the nine attributes.

Data Analysis

The proposed multivariate EWMA control chart is used to monitor the Breast Cancer Coimbra data, which may have a certain correlation. We cite the spectral analysis,^29,30 which is used to identify interepidemic periods. Based on the spectral analysis, the trend in the incidence data are determined. Inspired by Liu et al²⁷ the procedure comprises the following 3 steps. First, the Breast Cancer Coimbra data are preprocessed. In step II, the temporal behavior of the period is investigated. Second, nonlinear least squares fitting is used for the fitting analysis. This trend is then removed by subtracting the nonlinear least squares fitted curve from the data, thereby yielding the residual time-series data. Third, the obtained residual time-series datasets are monitored. The vertical coordinates of Figure 3 represent the power spectral density (PSD). Figure 3 indicates the numbers of the maximum entropy method (MEM) spectral periods.

Figure 3 Spectral analysis of the breast cancer coimbra data series.

The Breast Cancer Coimbra data indicate changes after observation 52. Therefore, we use the 1~50 IC data to find the control limits. These control charts have the same IC zero-state ARL. Then, we use the control limits to monitor the remainder of the process. The EWMA control chart of the residual data are presented in Figure 4. Figure 4 shows that the EWMA statistics fall outside the range of the control limits at observation 53, suggesting that the proposed method can provide relatively early detection in a process.

Figure 4 EWMA control chart for the breast cancer coimbra data.

Conclusions

In this paper, the Breast Cancer Coimbra data are provided for analysis. We use a nonparametric statistical process control chart to monitor them. Spectral analysis is also reviewed and conducted to investigate the periodicities of shorter time series, and then nonlinear least squares fitting is used for the fitting analysis. Finally, the residual data series are obtained and monitored. The Breast Cancer Coimbra data show that the statistics fall outside the control limit at observation 53. It means there is a significant sign at this point to show one has high risk to get the Breast Cancer. Future diagnosis should be done by relevant medicine specialist.