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ORIGINAL ARTICLES |
From the Biostatistics and Epidemiology Branch (D.B.F., M.E.A., C.M.B., T.A.H., L.E.C.), Toxicology and Molecular Biology Branch (D.B.M.), Health Effects Laboratory Division, National Institute for Occupational Safety and Health, Centers for Disease Control and Prevention, Morgantown, West Virginia; Department of Social and Preventive Medicine (J.M.V.), School of Public Health and Health Professions, State University of New York at Buffalo, Buffalo, New York.
Address correspondence and reprint requests to Desta B. Fekedulegn, Biostatistics and Epidemiology Branch, National Institute for Occupational Safety and Health, HELD/BEB, MS 4050, 1095 Willowdale Rd., Morgantown, WV 26505. E-mail: djf7{at}cdc.gov
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONNOTESREFERENCES |
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Methods: In this study, interpretation and generic definition of the area under the curve was provided through graphical analyses and examination of its association with other summary measures using data from the Buffalo Cardio-Metabolic Occupational Police Stress (BCOPS) Pilot Study. In generic form, AUCI is derived as the area under the curve above the baseline value minus the area above the curve below the baseline value.
Results: The sign and magnitude of AUCI are related to the profile and the rate of change of the measurements over time. The parameter showed significant associations with other summary indicators that measure pattern or rate of change of the measurements over time.
Conclusion: Principal components analyses revealed that summary parameters derived from repeated cortisol measurements can be grouped into two meaningful general categories: measures of the magnitude of response and measures of the pattern of response over time.
Key Words: repeated cortisol measurements • area under the curve • principal component analysis • total hormonal secretion • time course of salivary cortisol
Abbreviations: AUCG = area under the curve with respect to ground; AUCI = area under the curve with respect to increase; AUCB = area under the curve with respect to baseline; HPA = hypothalamic-pituitary-adrenal; PCA = principal component analyses.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONNOTESREFERENCES |
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In repeated data, each measurement is comprised of two types of information (Figure 1A); its distance from the ground (i.e., the intensity or magnitude of the response) and its distance from its neighbor (i.e., changes over time). Unlike other summary measures (e.g., baseline value, average, maximum), both pieces of information can be captured through the use of AUC. The use of AUC simplifies the statistical analyses by transforming the multivariate data into univariate space, especially when the numbers of repeated measurements are high and there is a need to summarize the information (7). This approach also reduces the number of statistical comparisons between groups, which minimizes the need for adjustment of the significance level. With AUC, the number of statistical comparisons only depends on the number of groups to be compared, as opposed to the original repeated data. In addition, when the time interval between repeated measurements is not identical, the use of AUC provides an alternative because repeated measures analysis of variance, using the original data, has no proven method to adjust for these differences (10).
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A recent study by Pruessner et al. (10) provides a simple formula for the computation of two types of AUC that reveal different information embodied in the repeated multivariate measurements (Figure 1B): area under the curve with respect to ground (AUCG) and area under the curve with respect to increase (AUCI). The formulas are basically simple additions of areas consisting of triangles and rectangles. AUCG is the total area under the curve of all measurements. It takes into account both sensitivity (the difference between the single measurements from each other) and intensity (the distance of these measures from ground). AUCI is calculated with reference to the baseline measurement and it ignores the distance from zero for all measurements and emphasizes the changes over time. With endocrinological data, AUCG is assumed to be a measure more related to total hormonal output, whereas AUCI is a parameter that emphasizes the changes over time and is more related to sensitivity of the system.
Despite the simple computational formula for AUCI, the parameter has had limited application in research compared with AUCG. This could be attributed to the lack of a generic formula and clear interpretation of the parameter and the limited literature on its association with other summary indicators. In addition, graphical representation of the region accounted for by AUCI is not well documented for situations where one or more of the measurements have values lower than the baseline measurement. The main objectives of this study, therefore, are to provide more insight into the interpretation of AUCI by defining a generic formula for AUCI through graphical analyses, to study its association with other well defined cortisol parameters known to measure different aspects of the multivariate data, and to select optimal parameters that capture meaningful information about the original data using dimension reduction analytical techniques.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONNOTESREFERENCES |
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Average cortisol is the average of the four measurements. Peak cortisol is the maximum value of the measurements regardless of when it occurred. Time from baseline, where baseline is the measurement at awakening, to peak is the number of minutes from the baseline to the peak value. Slope from baseline to peak is the slope of the line connecting the baseline to the peak value. The reactivity was defined as the change in salivary cortisol during the observation period and was calculated as the difference between the first and the last sample (sample 4 – sample 1). Slope of the line through the baseline and last measurement is a scaled version of reactivity; it is reactivity divided by time in minutes from baseline to last measurement.
Previous studies (3,11) used the slope from a fitted regression model and the time-weighted average to describe the pattern and the amount of cortisol secreted, respectively. Hence, the last three parameters (INT, SP2, and AUR) are derived after fitting a simple linear regression model to the waking data. The area under the regression line (AUR) was computed by using the estimated equation and integrating the resulting function as follows:
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where 
is the time interval in minutes from the baseline measurement to the last measurement, x is time from baseline (predictor variable), a is the intercept, and b is the slope of the fitted regression line.
Data From BCOPS Study
Endocrinological data from a recent study with 100 police officers were chosen to study the association of AUCI with other cortisol parameters. Participants are 100 police officers from Buffalo, New York, who were randomly selected in the Buffalo Cardio-Metabolic Occupational Police Stress (BCOPS) Pilot Study and were examined between 2001 and 2003. The study design, methods, and participant characteristics are described by Violanti and colleagues (12). Cortisol was measured in saliva samples and was analyzed using a competitive chemiluminescence immunoassay technique. The participants collected 13 saliva samples at various times during a 3-day period. On the second day, awakening cortisol measures were obtained from four salivary samples taken at 15-minute intervals on first awakening (at awakening, 15, 30, and 45 minutes after waking). Morning cortisol response is widely studied because it serves as a useful index of adrenocortical activity. Hence, the waking data were used for statistical analyses in this study. From the 100 participants, 68 subjects with complete (nonmissing) data on all four waking cortisol samples and times of measurement were used in this study. Summary statistics of the waking measurements and the derived parameters for the BCOPS waking data are shown in Table 2.
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We examined the association between the area measures (AUCI and AUCG) and the other cortisol parameters by using correlation and cluster analyses. We used cluster analysis (k-means clustering) to classify participants into two clusters, using AUCI as the clustering variable and Student's t test to assess differences in other cortisol parameters between these two clusters. Principal component analysis (PCA) was employed to understand the correlation structure among the derived cortisol parameters and reduce the dimension of the summary measures to a few new meaningful components. The Eigenvector loadings associated with the new components were examined graphically to assess what aspect of the multivariate data the components measure. All statistical analyses were performed using the SAS/STAT software, version 9.0 for Windows.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONNOTESREFERENCES |
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Scenario I
Four repeated measurements taken at 15-inute intervals (sample 1 (baseline) measurement = 9.6 at 0720 AM; sample 2 = 13.4 at 0735 AM; sample 3 = 16.5 at 0750 AM; and sample 4 = 14.2 at 0805 AM). The pattern of the measurements is shown in Figure 2. The area under the curve with respect to the baseline (AUCB), area under the curve with respect to the ground (AUCG), area under the curve with respect to increase (AUCI), and area under the curve above the baseline (AUCAB) are computed as:
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Scenario II
Four repeated measurements taken at 15-minute intervals (sample 1 (baseline) measurement = 9.6 at 0720 AM; sample 2 = 13.4 at 0735 AM; sample 3 = 11.1 at 0750 AM; and sample 4 = 7.2 at 05 AM). The pattern of the measurements is shown in Figure 3. As in scenario I, AUCB, AUCG and AUCI are calculated using the computational formula as:
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However, to compute AUCAB and to illustrate the generic definition of AUCI, the region between two sampling points is partitioned into a series of triangles (T) and rectangles (R). The area (A) of each triangle and rectangle was calculated and given as: AT1 = 28.5, AT2 = 17.25, AT3 = 4.33, AT4 = 11.08, AT5 = 11.08, AR1 = 144, AR2 = 22.5, AR3 = 144, AR4 = 13.85, and AR5 = 108. The generic definition entails that AUCI is the difference of two areas. The first component, area under the curve above the baseline value (AUCAB), is the sum of areas of T1, T2, T3 and R2 (28.5 + 17.25 + 4.33 + 22.5 = 72.58). The second component, area above the curve below the baseline, is the area of T5 (11.08). AUCI is then 72.58 – 11.08 = 61.5; the same result as in the computational formula.
Note that to compute the area of the triangles named T3, T4, T5 and the rectangle named R4, the coordinates of the point labeled "a" (Figure 3) need to be known. The y coordinate that represents the cortisol level is 9.6 but the x coordinate that represents time of measurement (in minutes) from baseline is unknown. The x coordinate was then computed by solving the simultaneous equation of the two intersecting lines: the horizontal line through the baseline measurement (y = 9.6) and the line connecting the 3rd and 4th measurements. The equation of the line passing through the third and fourth measurements is computed by solving the equation:
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where (x1, y1) and (x2, y2) are the coordinates of the 3rd and 4th sample points, respectively. After algebraic manipulation, the equation of the line is
Y = 18.9 – 0.26X.
The x coordinate of the intersection point of the two lines is obtained by solving the simultaneous equations: y = 9.6 and y = 18.9 – 0.26X. This yields a value of x = 35.76923 minutes from baseline.
Sign and Magnitude of AUCI
Previous studies (13–15) have shown that the response magnitude and time course of salivary cortisol levels after awakening are significantly related to various psychological and physical conditions (e.g., pain, stress). Therefore, the change in the sign and magnitude of AUCI associated with various cortisol patterns is important in understanding the definition of the parameter. In this section, the variation of AUCI associated with various cortisol patterns is examined numerically and graphically. Artificial data on eight waking cortisol patterns (Table 1) was used for illustration and the corresponding graphical depiction of these patterns is shown in Figure 4 (A–H). Although simulated for analyses here, the patterns in Figure 4 can also be observed in real life. In each figure, the area represented by AUCI is shown. There are four distinct patterns in Figure 4: a steadily increasing pattern (Figure A); a pattern where cortisol peaks 15 minutes after waking and decreases thereafter (Figures B–D); a steadily decreasing pattern (Figures E–G); and a pattern where cortisol drops 15 minutes after waking and increases thereafter (Figure H).
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For the first example (A), AUCI is positive and clearly defined; it is the area of the region under the curve above the reference baseline measurement. The examples in (B), (C), and (D) depict similar patterns but the AUCI values differ substantially. Note that, unlike the first example, AUCI is not clearly defined in a single region but it is obtained through subtraction of area from two regions: area above the baseline and below the curve minus area below the baseline and above the curve. This illustrates that AUCI balances the amount of "increase" versus "decrease" and can be considered as an index of change. AUCI therefore is the difference in area under and above the curve with reference to the baseline. In (B) and (C), the amount of "increase" (area under the curve above the baseline) is larger than the amount of "decrease" (area above the curve below the baseline). In (D), the amount of "increase" is smaller than the amount of "decrease"; hence, AUCI takes a negative value.
The examples in (E), (F), and (G) also present similar patterns but the AUCI values show large differences in magnitude. Note that, for these cases, the amount of increase (area above the baseline = AUCAB) is zero; hence, AUCI is the amount of decrease and takes a negative value. The slope of the regression line fitted to the patterns in (E), (F), and (G) are –0.11, –0.15, and –0.26, respectively (Table 1). The rate of decrease is highest for (G) because the cortisol value decreases by 0.26 nmol/l for 1- minute increase in time. AUCI has the largest magnitude for the curve that has the steepest slope. This illustrates that AUCI also measures or indicates the rate of decrease relative to the baseline. The large differences in magnitude of AUCI for these three examples are related to the differences in the rate of decrease. The pattern depicted in (H) is a good example illustrating that large and negative AUCI does not necessarily imply that the pattern of the measurements is strictly decreasing. In this example, cortisol decreased 15 minutes after waking and then picked up. There is a stronger decrease than increase and, hence, a negative AUCI. In summary, AUCI is an index of change that takes negative or positive values and translating the magnitude of AUCI into patterns over time in reference to the baseline requires caution.
Association of AUCI with Other Cortisol Parameters
A correlation analysis of AUCI and AUCG with each of the original measurements as well as with other derived parameters was performed to gain further insight in interpreting the parameter (Table 3). The correlations with the raw measurements indicate that AUCI is negatively and significantly correlated to the baseline measurement. This can be seen by examining the computational formula for AUCI (AUCI = AUCG – AUCB). AUCB is an area solely determined by the baseline value and larger baseline measurements yield larger AUCB and, hence, smaller AUCI. It also shows that the association between AUCI and the raw measurements is weaker compared with AUCG. This is because AUCI does not take into account the distance of the measurements from the ground; rather, it only considers distance of the measurements from the reference baseline. A similar result was reported by Pruessner et al. (10). Note that area under the curve above the baseline (AUCAB) shows stronger correlation to AUCI than AUCG because it is the first component in the generic formula for AUCI.
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Five parameters largely measure the pattern of the measurements over time: area under the curve above the baseline (AUCAB); time from baseline to peak; reactivity; slope from baseline to last measurement; and slope of the regression line fitted to the data. The slope of the regression line fitted to the data can be thought of as a more informative measure of pattern than the other four because it utilizes all the information in the multivariate data. Large positive values for all four parameters generally indicate an increasing trend in the measurements over time and, hence, a positive or large AUCI.
Two observations can be ascertained based on the correlation of the two area measures with the other derived parameters (1). AUCI showed strong positive correlations with the four parameters (reactivity, slope from baseline to last measurement, slope of the regression line fitted to the data, and time from baseline to peak) although the corresponding correlations with AUCG were weak or negligible (2). AUCG showed significant positive correlations with the intercept of the regression line fitted to the data (r = .85) and the corresponding area under the regression line (AUR) (r = .99) although the associations of AUCI with these two parameters were weaker. The intercept from the regression line is a model estimate of the baseline value; hence, the two parameters (INT and baseline cortisol) are strongly correlated (r = .97, p < .001).
The two clusters of participants created using AUCI as a clustering variable were compared with respect to the measurements and the derived parameters (Table 4). The first cluster contained 38 participants and the average AUCI for the subjects in the first cluster was –109.5 ± 167. The second cluster had 30 subjects and an average AUCI of 329.6 ± 178. The largest differences between the two clusters of participants, as shown by the test statistic, were observed for the following parameters: area under the curve above the baseline; time from baseline to peak; reactivity; slope between the 1st and 4th sample; and slope of the regression line. The result is in agreement with the data in Table 3 that these parameters are strongly correlated to AUCI and they all measure a common aspect of the repeated measurements: time course of salivary cortisol (profile). The baseline measurement, the area under the curve with respect to the baseline (AUCB), and the intercept from the fitted regression model (INT) showed the least differences between the two clusters (Table 4).
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Principal Component Analyses
Based on the data in Tables 3 and 4, there seem to be two broad categories of derived parameters: parameters related to total hormonal secretion and parameters indicating the time course of the measurements. A formal analytical tool, principal component analysis (PCA), was used to uncover if there are two underlying sets of parameters. The Eigenvalues and the Eigenvectors of the correlation matrix of the 13 derived cortisol parameters were computed. Only the first two principal components had Eigenvalues >1.0 and the majority of the variability (nearly 90%) among the 13 parameters was accounted by these two principal components. The magnitude and sign of the Eigenvector loadings were examined to assign meaningful interpretation to the two components. Several observations can be made based on the results of the PCAs. The first observation is that the first principal component has large loadings for the average of the waking measurements, AUCB, AUCG, peak cortisol, intercept of the regression line, and AUR. Thus, the first principal component, which accounts for 49.3% of the variability, seems to dominantly measure total hormonal secretion. Second, the second principal component has large loadings for AUCI, reactivity, SP1, SP2, and AUCAB. The second component that accounts for 39.5% of the variability in the original data seems to dominantly measure time course of salivary cortisol. Finally, the results from PCAs demonstrate that the large number of parameters can be reduced to two components: the first principal component measures intensity or total hormonal secretion and a second component measures sensitivity or profile of cortisol over time (Table 5).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONNOTESREFERENCES |
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The effect of both measurement error and biological variability are important to consider. As in any other scientific measurements, each awakening cortisol observation is composed of the true value plus some random error value (X = T + e). Random error is caused by any factor that affects the measurement of the variable across the sample (subjects) and is always part of any measurement. The accuracy of a cortisol measurement may be as important as the measurement itself. In addition, the inherent (biological) variability associated with each measurement is another important source of variation. For example, the baseline waking cortisol measurement may have the largest variability of all cortisol measurements within a day as it is obtained during the time of rapid rise from the late night nadir to the morning peak. Multiple-day awakening cortisol sampling would be ideal to estimate and understand the biological variation associated with each measurement. Such extensive observations involving multiple day sampling of cortisol may not be economically feasible and contribute to participant burden. The analyses in this study were based on 1 day of awakening cortisol sampling (no retest reliability); hence, we cannot provide an estimate of the variability associated with each of the waking measurements. However, we believe that the variability in the baseline waking measurement has the largest influence in the magnitude of AUCI. This is because AUCI is the difference between AUCG and AUCB and the baseline measurement solely determines AUCB. For example, if we assume a random error of 1.5 nmol/l for the baseline value of the data shown in Figure 3, the value of AUCI could change by ±56.3 area units.
In analyses of multivariate data, PCA can serve a dual role: 1) it is often used to reduce the dimension of a correlated set of variables; and 2) PCA is also used to partition experimental units into subgroups so that units that are similar are in the same group. In the latter case, the principal components can be used as input into clustering programs (16). In this study, the technique significantly reduced the number of summary measures and the new components explained the majority of the variability in the original measures and had meaningful biological interpretation. According to PCA, the most important measures of total hormonal secretion are area under the curve with respect to the ground (AUCG), peak cortisol, average of the measurements, and intercept of the fitted regression line (INT), whereas the corresponding measures of time course of salivary cortisol are area under the curve with respect to increase (AUCI), reactivity (RT), SP1 and SP2. This result is consistent with previous studies (3,11). Note that the important measures of total hormonal secretion (AUCG, PK, AVE, and INT) are strongly correlated with the baseline measurement (r = .81 for AUCG; r = .83 for PK; r = .84 for AVE; and r = .97 for INT; p < .0001 for all cases). This suggests that the baseline measurement is a natural measure of total waking hormonal concentration. PCA was rerun including the baseline measurement in the list of cortisol parameters and the result was consistent with the previously mentioned conclusion.
The use of AUC is advantageous from both statistical and biological stand points. Statistically, it simplifies the analyses by creating a single summary response from multiple measurements and also increases the power of testing without sacrificing the information contained in the multiple measurements. Biologically, it provides a means to incorporate both pieces of information (intensity and sensitivity) contained in repeated measurements into the statistical analyses. The results of this study (generic definition and interpretation) regarding AUCI are applicable in other designs; for example, when studying cortisol secretion over the course of a day (diurnal rhythm) or when studying cortisol response after a specific challenge. One limitation of the study is that it is based on a small sample size of 68 participants.
Given the high level of intercorrelation among derived summary measures of cortisol data, it seems prudent to reduce to a manageable number the set of parameters to be considered in analyses. This has implications for ease in presentation and interpretation of analysis results as well as for issues of multiple testing. We have provided an approach that allows the number of parameters to be reduced by choosing a smaller set from those most highly loaded on the first two principal components.
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NOTES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONNOTESREFERENCES |
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This work was supported by Contract No. 200 to 2003 to 01580 from the National Institute for Occupational Safety and Health. The findings and conclusions in this report are those of the authors and do not necessarily represent the views of the National Institute for Occupational Safety and Health.
DOI:10.1097/PSY.0b013e31814c405c
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