Application of logistic regression equation analysis using derivatives for optimal cutoff discriminative criterion estimation

Author(s): Andrey Bokov* and Svitlana Antonenko Background: Sigmoid curve function is frequently applied for modeling in clinical studies. The main task of scientific research relevant to medicine is to find rational cutoff criterion for decision making rather than finding just equation for probability calculation. The objective of this study is to analyze the specific features of logistic regression curves in ord ... Abstract View Full Article View DOI: 10.17352/amp.000016

Citation: Bokov A, Antonenko S (2020) Application of logistic regression equation analysis using derivatives for optimal cutoff discriminative criterion estimation. research relevant to medicine is to fi nd rational cutoff criterion for decision making rather than fi nding just equation for probability calculation.
It has been estimated that sigmoid curve that is used for population growth modeling has several phases: lag phase with minimal growth responding to changes in predictor values, phase of initial acceleration, phase of exponential growth with maximal changes of dependent variable per unit change of predictor value, phase of negative acceleration and stationary phase with minimal changes of dependent variable per unit change of predictor [5]. Taking into account that sigmoid curve is used for logistic regression modeling it is assumed that sigmoid curve estimated for logistic regression modeling have the same segments that has different patterns in dependent variable growth. Knowing the borders between those segments may help to convert continuous predictor into dichotomized achieving optimal receiver operative characteristics providing optimal discrimination of cases.
An analysis of function using derivatives is used to fi nd critical points of function and changes in patterns in dependent variable growth [6,7]. Zero values of fi rst order derivative correspond to maximal and minimal values of function while second derivative zero value corresponds to infl ection point on the graph [6,8]. The meaning of third derivative in geometry is aberrance -the torsion of curve, in mechanics it relates to jerk defi nition in other words it represents a rate at which acceleration changes [6,9]. Critical points that can be estimated using derivative analysis can defi ne an infl ection point of graph and points with maximal torsion that may delineate segments of graph providing optimal values for continuous data dichotomization.
The objective of this study is to analyze the specifi c features of logistic regression curves in order to evaluate critical points and to assess their implication for continuous predictor variable dichotomization in order to provide optimal cutoff criterion for decision making.

Methods
Three logistic regression models were used for this study    provides an opportunity of a threshold selection with optimal discriminative function for cases.
The better ability of discriminative function has the greater area under curve [12,13]. Finally logistic regression function can be characterized by accuracy of classifi cation, sensitivity and specifi city [14]. The problem of sensitivity and specifi city balance is akin to the problem of type I error and type 2 errors Analysis using derivatives of function is frequently used in physics and differential geometry for graph analysis [6][7][8]. Using

Limitation
Feasibility of suggested analysis can be limited by goodness-of-fi t of initial equation with continuous predictor.

Discussion
Estimation of discriminative function for classifi cation of cases is one of the most frequent goals of medical research [4]. Logistic regression is frequently used for those purposes if dependent variable is registered in dichotomized scale. The result of the analysis is an equation for probability calculation that can be used to forecast likelihood of complication or probability of unsatisfactory results of particular treatment modality application [4,10]. In certain cases practitioners need also optimal thresholds for decision making, however equations with continuous predictors do not provide this information consequently optimal dichotomization of continuous data is required [11].
The effectiveness of discrimination system worked out using logistic regression analysis is assessed using ROC-curves analysis that refl ects the ability to classify cases correctly.
Citation: Bokov A, Antonenko S (2020) Application of logistic regression equation analysis using derivatives for optimal cutoff discriminative criterion estimation.

Conclusion
Analysis of logistic regression equation with continuous predictor applying derivatives help to choose optimal thresholds that provide maximally effective discriminative functions with priority sensitivity or specifi city. Using this dichotomization discriminative function can be adjusted to the needs of particular task or study depending which characteristic is in priority -sensitivity or specifi city.