Statistical Analysis of the Three-Dimensional Data of Software Metrics RFC, CBO, and WMC that are not Normally Distributed

Abstract

Empirical data of RFC (response for a class), CBO (coupling between object classes), and WMC (weighted methods per class) software metrics, that can be used for estimation of software quality, deviate from normality. These metrics unveil multivariate skewness and kurtosis that do not conform to a multivariate Gaussian distribution. At the same time, well-known statistical methods that assume data normality may not be appropriate for the analysis of non-Gaussian data. To detect the outliers in the three-dimensional data of RFC, CBO, and WMC metrics and to estimate the confidence and prediction intervals of nonlinear regressions for these metrics, we need to use three-variate normalizing transformations. For statistical analysis of RFC, CBO, and WMC metrics, their normalization using the three-variate Box-Cox transformation was applied. Mardia’s test for the transformed data after applying the multivariate Box-Cox transformation points that the transformed dataset is Gaussian. A technique for detecting outliers in multivariate non-Gaussian data based on the squared Mahalanobis distance for normalized data was applied to ensure the removal of outliers. Three nonlinear regression models for each of the RFC, CBO, and WMC metrics were constructed. The confidence and prediction intervals of nonlinear regressions for each of the RFC, CBO, and WMC metrics were built. Well-known statistical characteristics PRED(0.25) and MMRE for both the primary and the test datasets show that the model quality is satisfactory. The confidence and prediction intervals of nonlinear regressions for these metrics can be used for estimation of the quality of the object-oriented design of the software.