Elsevier

Psychoneuroendocrinology

Volume 28, Issue 7, October 2003, Pages 916-931
Psychoneuroendocrinology

Two formulas for computation of the area under the curve represent measures of total hormone concentration versus time-dependent change

https://doi.org/10.1016/S0306-4530(02)00108-7Get rights and content

Abstract

Study protocols in endocrinological research and the neurosciences often employ repeated measurements over time to record changes in physiological or endocrinological variables. While it is desirable to acquire repeated measurements for finding individual and group differences with regard to response time and duration, the amount of data gathered often represents a problem for the statistical analysis. When trying to detect possible associations between repeated measures and other variables, the area under the curve (AUC) is routinely used to incorporate multiple time points. However, formulas for computation of the AUC are not standardized across laboratories, and existing differences are usually not presented when discussing results, thus causing possible variability, or incompatibility of findings between research groups. In this paper, two formulas for calculation of the area under the curve are presented, which are derived from the trapezoid formula. These formulas are termed ‘Area under the curve with respect to increase’ (AUCI) and ‘Area under the curve with respect to ground’ (AUCG). The different information that can be derived from repeated measurements with these two formulas is exemplified using artificial and real data from recent studies of the authors. It is shown that depending on which formula is used, different associations with other variables may emerge. Consequently, it is recommended to employ both formulas when analyzing data sets with repeated measures.

Introduction

The computation of the area under the curve (AUC) is a frequently used method in endocrinological research and the neurosciences to comprise information that is contained in repeated measurements over time. Depending on the nature of the study, it serves a variety of different purposes. In clinical trials, the AUC can be employed to monitor the effects of a specific medication over the trial period. In endocrinological studies, the AUC is used to estimate ultradian and circadian changes of hormones, and to assess the overall secretion over a specific time period. In pharmacological studies, the AUC is useful to evaluate dose/response relationships (Ghizzoni et al., 1994, Maes et al., 1994, O’Brien et al., 1996). The computation of the AUC allows the researcher to simplify the statistical analysis and increase the power of the testing without sacrificing the information contained in multiple measurements.

However, despite the proven usefulness of this method, its application across laboratories is limited, for a number of reasons. First, different formulas are used by different laboratories to derive the AUC from any given dataset, thus compromising the comparability of their findings. Second, the different formulas used are usually not explicitly elaborated or listed in the papers, thus making it impossible to compare the computation of the AUC itself. It is maybe because of these reasons that despite the usefulness of the method, some research groups do not refer to AUC but instead refer to some key time points for evaluation of their data (Gormley et al., 1992, Tucci et al., 1996). It can be assumed that researchers might be more easily convinced to employ AUC in their statistical analysis if a standardized, easy to apply formula was available.

Deriving formulas for computation of the AUC also depends on the information the researcher is interested in. A data set comprised of repeated measurements over time contains at least two different sorts of information. First, it contains the information whether any changes occurred over time (was there a change in the events being quantified in the dependent variable during the observation period?). Second, each data set also allows assessing the overall intensity at which the recorded events occurred. It is easy to imagine studies with repeated measures where either one, or both parameters are of main interest for the researcher (e.g.: Did the medication have an overall effect compared to the control group? Did a habituation of the effect of the medication occur during the trial period?)

These two different sets of information can best be described with two different formulas, which are outlined in this manuscript. One is tentatively called ‘Area under the curve with respect to ground’ (AUCG), whereas the second formula is termed ‘Area under the curve with respect to increase’ (AUCI). By using an artificial dataset, together with a recently acquired endocrinological dataset, the benefits of employing both formulas for statistical analysis are demonstrated.

Section snippets

Theory and Methods

The different formulas for the area under the curve can be derived from the trapezoid formula (Reinhardt and Soeder, 2001). A typical trapezoid separated into triangles and rectangles is illustrated in Fig. 1.

The information needed in order to calculate the formula consists of (a) the measurements themselves and (b) the time distance between the measurements. In this example with a group containing six repeated measures, the measurements have been named m1 through m6, and the time distances

Artificial data creation

A computer program (HyperCard®, Apple Computer, Cupertino, CA, USA) was employed to generate random numbers varying around a fixed value. The program was designed so that these numbers follow a normal distribution around the fixed value. Datasets can be created this way representing possible measurements over time. In order to highlight the different information provided by the two formulas presented in this study, four different groups of data were created. Each group was assigned six

Subjects and study design

Endocrinological data from a recent study with 69 teachers were chosen to validate the usefulness of the AUC formulas. In this study, 69 teachers from elementary and high schools in the region of Trier, Germany, sampled saliva for cortisol analysis at three separate days at the time of awakening and 15, 30 and 60 minutes thereafter. At the night before day three, all teachers took 0.5 mg dexamethasone (PO) to test suppression of the hypothalamic–pituitary–adrenal (HPA) axis the next morning by

Discussion

This paper describes two formulas for computation of the AUC derived from the trapezoid formula. Transformation yielded formulas universally applicable to any number of repeated measurements in any experimental design. In addition, in cases where the time distance between measurements is identical throughout the experiment, a further simplification of the formulas for computation of AUC was presented.

Applying the formulas to a set of artificially created data (Study 1) with four groups and six

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