Information-Theory Analysis of Cell Characteristics in Breast Cancer Patients
David Blokh · International Journal on Bioinformatics & Biosciences · 2013
International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 DOI : 10.5121/ijbb.2013.3101 1 Information-Theory Analysis of Cell Characteristics in Breast Cancer Patients David Blokh C. D. Technologies Ltd., Israel [email protected] Abstract A problem of selecting a subset of parameters containing a maximum amount of information on all parameters of a given set is considered. The proposed method of selection is based on the informationtheory analysis and rank statistics. The uncertainty coefficient (normalized mutual information) is used as a measure of information about one parameter contained in another parameter. The most informative characteristics are selected from the set of cytological characteristics of breast cancer patients. Keywords Normalized mutual information, Newman-Keuls test, Cytological characteristics, Breast cancer 1. Introduction The selection of a subgroup of parameters containing a maximum amount of information on all parameters of a given group of parameters is an important problem of medical informatics. This problem has applications in analyzing oncology patients data [1], mathematical modeling of tumor growth [2] and developing intellectual medical systems [3, 4]. In the present article, a subgroup of parameters describing cytological characteristics of breast cancer patients and containing the maximum amount of information on all parameters of this group is selected. This selection is required for constructing models of cancer diagnosis and prognosis. The problem under consideration is rather difficult. First, this group includes both discrete and continuous parameters; second, the distributions of continuous parameters are non-Gaussian and, third, the interrelations between parameters are nonlinear. Any method of selecting information-intensive parameters is based on the use of a measure of parameters correlation. In the majority of methods, a correlation coefficient is used as such a measure. However, the application of the correlation coefficient suggests that parameters distributions are Gaussian, and the correlations of parameters are linear. Therefore, in the present article, the uncertainty coefficient (normalized mutual information) is used as a measure of parameters correlation. This coefficient evaluates nonlinear correlations between parameters with arbitrary distributions. Thus, this article presents a method of selecting the most informative parameters, which has no restrictions on the distribution of the parameters and on the correlations between the parameters. Application of the uncertainty coefficient has made it possible to obtain interesting results in medicine [5] and, in particular, in oncology [6, 7, 8]. The approach proposed in [6] is presented in monograph [9]. International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 2 2. Materials and Methods 2.1. Materials To illustrate the method, we used data from the Wisconsin Breast Cancer Database [10]. These data cover 663 patients, each patient being presented by 10 cytological parameters, 9 of which are continuous (clump thickness, uniformity of cell size, uniformity of cell shape, marginal adhesion, single epithelial cell size, bare nuclei, bland chromatin, normal nucleoli, and mitoses) and one is discrete (Class). 2.2. Data Analysis 2.2.1. Statement of the problem Assume that the initial data on n objects are presented in the form of a n m× array [ ] kj a , where each row k is an object described by m parameters. It is needed to find a parameter or a subgroup of parameters containing the greatest amount of information about all m parameters. 2.2.2. Description of the algorithm The algorithm of selecting a subgroup of the most informative parameters from the entire group of parameters includes four procedures. A short description of each procedure is as follows. A more complete description of the application of information-theory analysis to the selection problem is presented in [11]. 1. Discretization. This procedure transforms parameters having continuous values into parameters having discrete values. If an acceptable method of a continuous parameter discretization is unavailable, use a formal approach to the discretization [12]. 2. Construction of the uncertainty coefficient matrix. For i-th and j-th parameters 1 , ≤ ≤ i j m, calculate the uncertainty coefficient ij c [5] and construct m m× uncertainty coefficient matrix [ ]ij c . 3. Construction of the rank matrix. For each column of the matrix [ ]ij c , we rank its elements and assign rank 1 to the smallest element of the column. We obtain the matrix m m× of ranks [ ]ij r , where each column of the matrix contains ranks from 1 to m. We estimate the amount of information about all m parameters contained in the i-th parameter by the sum of all the entries of the i-th row of the matrix [ ]ij r . 4. Application of the multiple comparison method. Apply the multiple comparison method to the sums of [ ]ij r matrix rows [13]. This gives a clustering of parameters that contains the desired subgroup of parameters. Remark. Programs for the algorithm of uncertainty coefficient calculation and the Friedman test are implemented in the package SPSS [14]. International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 3 3. Results We consecutively perform the four procedures of the selection algorithm. 1. Categories of parameters contained in the Wisconsin Breast Cancer Database are given below. Parameter name Category 1. Clump Thickness: 1,2,3,4,5,6,7,8,9,10. 2. Uniformity of Cell Size: 1,2,3,4,5,6,7,8,9,10. 3. Uniformity of Cell Shape: 1,2,3,4,5,6,7,8,9,10. 4. Marginal Adhesion: 1,2,3,4,5,6,7,8,9,10. 5. Single epithelial cell size 1,2,3,4,5,6,7,8,9,10. 6. Bare Nuclei: 1,2,3,4,5,6,7,8,9,10. 7. Bland Chromatin: 1,2,3,4,5,6,7,8,9,10. 8. Normal Nucleoli: 1,2,3,4,5,6,7,8,9,10. 9. Mitose: 1,2,3,4,5,6,7,8,9,10. 10. Class: 2-benign, 4-malignant 2. Calculate the uncertainty coefficients ij c 1 , 10 ≤ ≤ i j for the parameters of the Wisconsin Breast Cancer Database and construct the [ ]ij c matrix of uncertainty coefficients (Table 1). Table 1. Uncertainty coefficient matrix [ ]ij c . No Parameter name 1 2 3 4 5 6 7 8 9 10 1 Clump thickness 1 .222 .194 .144 .170 .205 .136 .180 .171 .439 2 Uniformity of cell size .181 1 .406 .263 .305 .319 .243 .301 .251 .676 3 Uniformity of cell shape .165 .419 1 .238 .250 .282 .211 .266 .209 .556 4 Marginal adhesion .105 .232 .205 1 .211 .239 .198 .196 .175 .382 5 Single epithelial cell size .135 .295 .235 .231 1 .240 .191 .256 .255 .503 6 Bare nuclei .137 .259 .223 .219 .202 1 .197 .222 .154 .491 7 Bland chromatin .123 .267 .226 .247 .218 .268 1 .245 .155 .499 8 Normal nucleoli .121 .245 .211 .181 .216 .223 .181 1 .180 .386 9 Mitoses .071 .125 .103 .100 .133 .096 .071 .111 1 .226 10 Class .180 .339 .269 .215 .259 .301 .225 .236 .223 1 3. Rank the [ ]ij c matrix (Table 1) columns and obtain a rank matrix [ ]ij r (Table 2). Table 2. Rank matrix [ ]ij r . No Parameter name 1 2 3 4 5 6 7 8 9 10 Sum of ranks 1 Clump thickness 10 2 2 2 2 2 2 2 3 4 31 2 Uniformity of cell size 9 10 9 9 9 9 9 9 8 9 90 3 Uniformity of cell shape 7 9 10 7 7 7 7 8 6 8 76 4 Marginal adhesion 2 3 3 10 4 4 6 3 4 2 41 5 Single epithelial cell size 5 7 7 6 10 5 4 7 9 7 67 6 Bare nuclei 6 5 5 5 3 10 5 4 1 5 49 7 Bland chromatin 4 6 6 8 6 6 10 6 2 6 60 8 Normal nucleoli 3 4 4 3 5 3 3 10 5 3 43 9 Mitoses 1 1 1 1 1 1 1 1 10 1 19 10 Class 8 8 8 4 8 8 8 5 7 10 74 Consider Table 2 as the Friedman statistical model [15] and examine the row effect of this table. International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 4 Hypotheses H0: There is no row effect (“null hypothesis”). H1: The null hypothesis is invalid. Critical range. The sample is “large”; therefore, the critical range is the upper 1%-range of 2 9 distribution. Calculation of the 2 -criterion [15] gives 2 = 48.48. The critical range is 2 9 > 21.67. Since 48.48 > 21.67, the null hypothesis with respect to Table 2 is rejected. Thus, according to the Friedman test, the row effect exists. Hence, there is a difference between the rows under consideration. 4. For multiple comparisons, we use the Newman-Keuls test [13]. We obtain 1 8.15 j j R R + − > , where Rj and Rj+1 are the j-th and (j+1)-th elements of “Sum of ranks” column of Table 2. Using the multiple comparisons method, we construct the parameter clustering shown in Table 3. Table 3. Parameter Clustering. No Cluster Parameter name Sum of ranks 1 Cluster 1 Uniformity of cell size 90 2 Cluster 2 Uniformity of cell shape 76 3 Class "benign" or "malignant" 74 4 Single epithelial cell size 67 5 Bland chromatin 60 6 Cluster 3 Bare nuclei 49 7 Normal nucleoli 43 8 Marginal adhesion 41 9 Cluster 4 Clump thickness 31 10 Cluster 5 Mitoses 19 The obtained clustering has the following properties: For two neighboring clusters of Table 3, the smallest element of one cluster and the greatest element of another cluster located nearby are significantly different ( T =0.01); Elements belonging to the same cluster do not differ from each other ( T =0.01); The differences between Cluster 1 (Uniformity of cell size) and all the characteristics are statistically significant ( T =0.01). The characteristic "Uniformity of cell size" contains the greatest amount of information about all the characteristics under study. It corresponds to the thesis that the uniformity of cell size in unicellular organisms and within tissues in multicellular organisms could be necessary to maintain the homeostasis of transcription and proliferation [16]. Thus, an adequate result has been obtained using the proposed algorithm. We believe that the processing other biomedical data using the proposed algorithm will also give adequate results. International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 5 4. Conclusions In the present paper, a method of sele