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Surrogate Gaussian First Derivative Curves for Determination of Decision Levels and Confidence Intervals by Binary Logistic Regression

Address correspondence to Vincent A. DeBari, Ph.D., School of Health and Medical Sciences, Seton Hall University, 400 South Orange Avenue, South Orange, NJ 07079, USA; tel 973 877 2813; fax 973 877 5767; email: debarivi{at}shu.edu.

It has been demonstrated that decision levels (DL) and their confidence intervals (CI) can be estimated from the second derivative, f ’’ (P), of the logistic regression probability curve (LRPC). Although this method generally provides smooth curves from which DL and CI can be obtained, there are datasets that generate "noisy" curves making these measurements difficult. The purpose of this study was to develop a procedure to obviate this noise, thus allowing the more facile estimation of DL and CI. Data from two clinical studies were examined. Logistic regression analysis was performed and the first derivatives, f ’ (P), were fitted to Gaussian models. The derivatives of these surrogate f ’ (P) were generated to provide f ’’ (P) and were compared with data from receiver operating characteristic (ROC) curves. For both sets of data, the surrogate curves demonstrated strong fits to the natural f ’ (P) with r² = 0.986 for one study and 0.832 for the second. The f ’’ (P) generated from the surrogate curves demonstrated single maxima (M) and minima (m), compared with the f ’’ (P) generated from the natural f ’ (P) in which multiple M and m were observed. Easily discernible DL and CI were observed for both datasets with differences from ROC-estimated DL of 1.7% for the first study and 4.8% for the second. The use of a surrogate Gaussian simulation of f ’ (P) may be a useful alternative to natural f ’ (P) when using the f ’’ (P) of the LRPC to determine DL and CI.

Keywords: logistic regression, curve-fitting, medical decision making, receiver-operator characteristic curve, epidemiology, biostatistics