Statistics in Cancer: Diagnosis, Disease Progression, Treatment Efficacy, and Patient Survival Studies

Abstract

This article proposes a simple nonparametric measure for diagnosis of cancer and assessing cancer intensity for an individual, without resorting to group data or reduction of dimensionality or scaling or finding weights. The measure also identifies the critical areas/variables requiring attention, can be applied for all non-nominal data, can be used to find mean, variance, and confidence interval for group data, and facilitates statistical tests of hypothesis.

The cancer intensity facilitates ranking/classifying a group of patients along with quantifying progress of treatment at individual and group level. Using suitably designed group data, attempt can be made to find a small interval of values of cancer intensity for each type of cancer, which may be associated with Stage IV cancer or metastatic cancer. The proposed measure of cancer intensity offers an alternative approach for estimation of survival function of cancer patients. This study leads to a number of new areas of statistical analysis in cancer treatment. An empirical study will be of vital interest based on this theoretical study.

Appendix

1.1 Statistical Notes

• 1. Chi-square test is a nonparametric test that makes comparisons (usually of cross tabulated data) between two or more samples on the observed frequency of values with expected frequency of values and also, used as test of goodness of fit of log linear models.

• 2. Regression analysis establishes relationship of the dependent variable with one or more of the independent variables.

• 3. Logistic regression analysis deals with dependent variable in binary and one or more independent variables in nominal, ordinal, interval, or ratio level.

• 4. Factor Analysis (FA)/Principal Component Analysis (PCA) are both multivariate statistical techniques for reduction of data/variables. PCA considers linear combination of weighted observed variables to minimize the variance of the observed variables, while FA explains the covariance between the variables.

• 5. Wilks’ Lambda is the ratio of the within group sum of squares to the total sum of squares. When observed group means are nearly equal, Wilks’ Lambda will be high and a small lambda occurs when group means differ.

• 6. A Bayesian network consists of a set of nodes (random variables) and a set of directed edges (direct dependencies between the variables). Major difficulties are specifying prior probability (a priori probability) and computing a posterior probabilities (a posteriori probability).

• 7. The k-nearest neighbors (KNN) algorithm is used primarily in classification problems.

• 8. Neural network starts with a set of variables X_i and associated weights W_i for all i = 1, 2, …, n. A function f is determined whose domain is the sums of the weights and range is an output Y. Neural Networks, which have multiple solutions associated with local minima, may not be robust over different samples.

• 9. Nearest shrunken centroid classification calculates a standardized centroid for each class in terms of ratio of average gene expression for each gene and the within-class standard deviation for that gene.

• 10. Random forest or random decision forest is a machine learning algorithm used for classification, regression, and other tasks by constructing multitude of ensemble of decision trees and merging them together for more accurate and stable prediction.

• 11. Support vector machine (SVM) is a nonparametric method for analysis consisting of both classification of tissue samples and explorations of the data for mislabeled or questionable tissue results. Here, the marginal contribution of each component ratio to the score is variable. Moreover, the choice of the input variables has a decisive influence on the performance results.

• 12. Cluster analysis groups a set of objects in such a way that objects in the same group are more similar to each other than to those in the other groups.

• 13. When the dependent variable is categorical and the independent variables are in interval scale or in ratio scale, Discriminant analysis develops Discriminant functions (df) that discriminates between the categories of the dependent variables.

• 14. Log-rank test compares the survival distributions of two samples when the data are right skewed and censored.