Michele R. Norton, Richard P. Sloan and Emilia Bagiella
J Appl Physiol 98:2298-2303, 2005. First published Feb 17, 2005; d...
J Appl Physiol 98: 2298 –2303, 2005.
Innovative Methodology                                                               ...
Innovative Methodology
                                            STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA            ...
Innovative Methodology
2300                                           STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA

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Innovative Methodology
                                         STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA               ...
Innovative Methodology
2302                                          STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA




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Innovative Methodology
                                                 STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA       ...
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Michele R. Norton, Richard P. Sloan and Emilia Bagiella

  1. 1. Michele R. Norton, Richard P. Sloan and Emilia Bagiella J Appl Physiol 98:2298-2303, 2005. First published Feb 17, 2005; doi:10.1152/japplphysiol.00772.2004 You might find this additional information useful... This article cites 7 articles, 1 of which you can access free at: http://jap.physiology.org/cgi/content/full/98/6/2298#BIBL This article has been cited by 1 other HighWire hosted article: Restoration of baroreflex function in patients with end-stage renal disease after renal transplantation D. Rubinger, R. Backenroth and D. Sapoznikov Nephrol. Dial. Transplant., April 1, 2009; 24 (4): 1305-1313. [Abstract] [Full Text] [PDF] Updated information and services including high-resolution figures, can be found at: http://jap.physiology.org/cgi/content/full/98/6/2298 Additional material and information about Journal of Applied Physiology can be found at: http://www.the-aps.org/publications/jappl Downloaded from jap.physiology.org on August 6, 2010 This information is current as of August 6, 2010 . Journal of Applied Physiology publishes original papers that deal with diverse areas of research in applied physiology, especially those papers emphasizing adaptive and integrative mechanisms. It is published 12 times a year (monthly) by the American Physiological Society, 9650 Rockville Pike, Bethesda MD 20814-3991. Copyright © 2005 by the American Physiological Society. ISSN: 8750-7587, ESSN: 1522-1601. Visit our website at http://www.the-aps.org/.
  2. 2. J Appl Physiol 98: 2298 –2303, 2005. Innovative Methodology First published February 17, 2005; doi:10.1152/japplphysiol.00772.2004. New approach to the statistical analysis of cardiovascular data Michele R. Norton,1 Richard P. Sloan,2,3,4 and Emilia Bagiella1 1 Department of Biostatistics, Mailman School of Public Health, 2Department of Psychiatry, Columbia University, New York, 3Behavioral Medicine Program, Columbia-University Medical Center, New York, and 4New York State Psychiatric Institute, New York, New York Submitted 22 July 2004; accepted in final form 15 February 2005 Norton, Michele R., Richard P. Sloan, and Emilia Bagiella. bands (2). Thus, for each of these frequency bands, a measure- New approach to the statistical analysis of cardiovascular data. J Appl ment of systolic and diastolic BPV is obtained. Physiol 98: 2298 –2303, 2005. First published February 17, 2005; The statistical analysis of these transformed data generally doi:10.1152/japplphysiol.00772.2004.—Fourier-based approaches to involves testing the effect of psychological or physical chal- analysis of variability of R-R intervals or blood pressure typically lenges. Differences between responses obtained during a rest- compute power in a given frequency band (e.g., 0.01– 0.07 Hz) by ing baseline and those obtained during challenging conditions aggregating the power at each constituent frequency within that band. are calculated (4), and group differences are assessed using This paper describes a new approach to the analysis of these data. We well-known statistical procedures such as t-tests, repeated- propose to partition the blood pressure variability spectrum into more measures ANOVA, or multivariate ANOVA. narrow components by computing power in 0.01-Hz-wide bands. Although these approaches to data reduction and statistical Downloaded from jap.physiology.org on August 6, 2010 Therefore, instead of a single measure of variability in a specific frequency interval, we obtain several measurements. The approach analysis are not incorrect methodologically, they either over- generates a more complex data structure that requires a careful simplify the complex structure of the data, resulting in loss of account of the nested repeated measures. We briefly describe a information, or they require model assumptions that the data statistical methodology based on generalized estimating equations that often do not satisfy (1). Another limitation is that subjects with suitably handles this more complex data structure. To illustrate the missing data, either at baseline or during a challenge, are methods, we consider systolic blood pressure data collected during excluded from analyses because reactivity cannot be com- psychological and orthostatic challenge. We compare the results with puted. those obtained using the conventional methods to compute blood Thus several questions arise from the current approaches to pressure variability, and we show that our approach yields more data reduction and statistical analysis applied to cardiovascular efficient results and more powerful statistical tests. We conclude that data measured repeatedly. Does computing variability as a this approach may allow a more thorough analysis of cardiovascular single measure for a given frequency band result in too much parameters that are measured under different experimental conditions, data reduction and, hence, a loss of information? Do the such as blood pressure or heart rate variability. statistical methods currently used for the analysis of psycho- blood pressure variability; generalized estimating equations; repeated physiology data efficiently use all the information available in measures the data? In this paper, we aim to answer these questions by present- ing a new approach to the analysis of cardiovascular data. CLINICAL STUDIES IN PSYCHOPHYSIOLOGY are a rich source of Specifically, we propose to partition the frequency spectrum repeated-measures data. In these studies, individuals’ psycho- into 0.01-Hz-wide intervals instead of the more conventional physiology indexes, such as heart rate (HR) or blood pressure bands. We also describe a statistical methodology for the (BP), are monitored repeatedly during relatively different ex- analysis of the data obtained. To illustrate the methods, we use perimental periods in a laboratory setting or during different data on BPV from a recently conducted study. We conclude activities that occur during daily living. As a result, these with a brief discussion. studies generate large amounts of data that must be reduced MATERIALS AND METHODS and then analyzed to detect within-group differences from one condition to another or differences between different groups of Study Description subjects. The data were collected in a study conducted to assess the ability Data reduction is achieved in various ways to obtain inter- of the cardiac autonomic control system to buffer BPV during psy- pretable measures. One way is to submit time series of HR or chological and orthostatic challenge (Sloan SP, unpublished observa- BP values to Fourier-based spectral analysis to obtain fre- tions). Three groups of subjects were recruited for the clinical exper- quency domain measures of BP (BPV) or HR variability. iment: 31 cardiac transplant recipients, 11 renal transplant recipients Conventionally, power within specifically defined frequency receiving equivalent immunosuppressive treatment, and 24 normal subjects. Normal and renal transplant subjects were matched to bands is calculated. For example, lower frequency BPV, i.e., cardiac transplant recipients by age and gender. BPV measured in the 0- to 0.15-Hz range, generally thought to The experimental protocol included a 30-min adaptation, a 5-min reflect vascular sympathetic drive, is calculated as power in the baseline, two 5-min mental arithmetic task periods, a 5-min Stroop low (0.01– 0.07 Hz) or the mid (0.07– 0.15 Hz) frequency color word task, a 2-min cold pressor task, and 10 min of 70° head-up Address for reprint requests and other correspondence: E. Bagiella, Dept. of The costs of publication of this article were defrayed in part by the payment Biostatistics, Mailman School of Public Health, Columbia Univ., 722 West of page charges. The article must therefore be hereby marked “advertisement” 168th St., New York, NY 10032 (E-mail: eb51@columbia.edu). in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. 2298 8750-7587/05 $8.00 Copyright © 2005 the American Physiological Society http://www. jap.org
  3. 3. Innovative Methodology STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA 2299 Fig. 1. Mean log systolic blood pressure (BP) variability by study period in cardiac transplant subjects. Values on the x-axis are the lower limits of the 0.01-Hz-wide frequency intervals. tilt. Recovery periods followed each task. All subjects signed an the use of pattern recognition software, peaks and valleys of the informed consent form before enrollment in the study. The study was waveform were marked, and systolic and diastolic BP time series were approved by the Columbia University Medical Center Institutional generated. Review Board. Spectra were calculated on 240-s epochs using an interval method for computing Fourier transforms similar to that described by DeBoer Psychological Stressors Downloaded from jap.physiology.org on August 6, 2010 et al. (2). Before computing Fourier transforms, the mean of the Mental arithmetic. In this task, subjects were presented with a systolic and diastolic BP series were subtracted from each value in the four-digit number on the computer monitor and were instructed to series, and the residual series then was filtered using a Hanning subtract serially by 7 starting with this number, which disappeared window (3) and the power, i.e., variance (in mmHg2), within each after the first answer was entered. At 1-min intervals, subjects re- band was summed. Estimates of spectral power were adjusted to ceived verbal prompts, e.g., “please subtract faster.” The task was not account for attenuation produced by this filter (3). paced by the computer, but subjects were instructed to subtract as Because the servo self-adjustment of the Finapres was enabled quickly and as accurately as possible. Subjects entered the results on during the final minute of the each 5-min recording period, only data the numeric keypad. from the first 4 min of each period were analyzed. Data from the first Stroop color word task. In this version of the Stroop task, subjects 5 min of the tilt period were excluded from analysis to permit full were presented with color names (blue, green, yellow, and red) in equilibration to the upright position. colors that were either congruent or incongruent with the names. The Using the new approach, we applied fast Fourier transforms to the subjects’ task was to press the key on the keypad that corresponded to series of systolic BP values to obtain systolic BPV in 0.01-Hz-wide the color of the letters. The task was paced by the computer, and an frequency bands. In this way, we obtained 14 systolic BPV measure- incorrect response or failure to respond rapidly enough resulted in a ments on a discrete scale from 0.01 to 0.15 Hz. Thus, for each subject message indicating “incorrect” on the screen. in each of the four experimental conditions, instead of a single Tilt. The tilt table was elevated to the 70° head-up position over the measure of variability in the 0.01- to 0.15-Hz band, we obtained 14 course of 1 min. BP and respiration monitors were recalibrated after measurements representing systolic BPV at the midpoint of each of equilibration to the upright position. Subjects remained in the head-up the 0.01-Hz-wide intervals. position for 10 min unless they developed symptoms of faintness. These data were log-transformed before analysis to correct for We present the results of both the new and conventional methods skewness. of analysis of systolic BPV data from the cardiac transplant and Figures 1 and 2 depict the average semicontinuous systolic BPV control subjects collected during four experimental conditions: the curves for each experimental period for the cardiac transplant patients baseline period, the first mental arithmetic task, the Stroop color word and the normal control subjects. The values on the horizontal axis are task, and the 70° head-up tilt. the lower limits of the 0.01-Hz-wide frequency intervals. Thus the Data Reduction value at 0.01 Hz represents log systolic BP power in the interval 0.01– 0.02 Hz. Each average BPV curve was obtained as the point- Beat-to-beat BP was recorded using a Finapres model 2300 mon- by-point average, across subjects, of the individual curves within each itor. The analog BP waveform was collected at 500 samples/s. With experimental condition for the two study groups. Fig. 2. Mean log systolic BP variability by study period in normal control subjects. Values on the x-axis are the lower limits of the 0.01-Hz-wide frequency intervals. J Appl Physiol • VOL 98 • JUNE 2005 • www.jap.org
  4. 4. Innovative Methodology 2300 STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA Note that this approach to data reduction does not change the restricted by model assumptions, and has relatively rapid computing interpretation of BPV. For example, low-frequency SBP power is still time. represented by power in the 0.01- to 0.07-Hz band. However, instead Most statistical software provides subroutines to apply the GEE of being represented by a unique number, as in the conventional approach. In our application, we used the PROC GENMOD routine analysis, it is now represented by six values. provided by SAS (7). (An example of a SAS program used to perform the GEE analysis can be obtained from the corresponding author.) Statistical Analysis of the 0.01-Hz-Wide Intervals Our models included the measurements of systolic BPV as the dependent variables and experimental period (baseline, mental arith- The statistical analysis of BPV calculated in 0.01-Hz-wide fre- metic, Stroop, and tilt) and group membership (cardiac transplant or quency bands is more complicated than the analysis of these variables normal control) and their interaction as the predictors. In the following calculated in the conventional way. First, in a frequency band of section, we present the results of the analysis performed using an interest, systolic BPV is represented by several values instead of a unstructured correlation matrix, which specifies a completely general single one. For example, in our approach, six measurements represent correlation structure among the repeated measures. However, the systolic BPV in the 0.01- to 0.07-Hz frequency band. Second, a more model produced very similar results when we used different correla- complicated structure of repeated measures has to be taken into tion structures. account. Within the same subject, the BPV measurements in a fre- In the next section, we present the result of the analysis performed quency interval and the measurements obtained during different using our approach and a comparison with the analysis performed experimental conditions are nested as repeated measures. Typically, using the conventional methods for calculating BPV. For the analysis these repeated measures are positively correlated with each other, and of BPV into low- and mid-frequency power, we considered two failure to account for this correlation leads to underestimation of the separate models for systolic BPV in the 0.01- to 0.07-Hz band and in standard errors of the model parameters, resulting in overly liberal the 0.07- to 0.15-Hz band, respectively. The analysis of low-fre- tests of hypothesis about these parameters. Also, collecting large quency BPV using our approach considers all the points in the 0.01- Downloaded from jap.physiology.org on August 6, 2010 numbers of repeated measures increases the risk that some will be to 0.07-Hz band, whereas the analysis of the mid-frequency BPV missing. A statistical approach to data analysis that addresses both considers all the points in the 0.07- to 0.15-Hz band. With the use of these concerns is the generalized estimating equations (GEE) ap- the conventional approach, two values of BPV were calculated, one proach (6, 8). Although the approach is mostly used to model representing power in the 0.01- to 0.07-Hz band and one representing categorical or binary data, it is also widely applied to normally power in the 0.07- to 0.15-Hz band. distributed data, such as the BPV measurements in this study. In a separate analysis, we also considered the 14 BPV point curves Briefly, GEE are an extension of the standard generalized linear from 0.01 to 0.15 Hz. models that allow modeling of correlated data. As with conventional regression models, the dependent and independent variables need to be specified. A working correlation matrix that describes the correla- RESULTS tion structure among the repeated measures also has to be specified. This matrix describes the dependence between the repeated measures Tables 1 and 2 present the means and respective standard in the model. An autoregressive correlation matrix, for example, errors of changes in systolic BPV from baseline to each of the specifies that adjacent measurements would be more highly correlated three experimental conditions for the cardiac transplant and than measurements taken farther apart. However, one of the advan- control groups. The first three columns in the tables show tages of the GEE approach is that in most situations, provided that the results obtained by modeling the systolic BPV measurements missing data are missing at random (i.e., their missing does not calculated in 0.01-Hz-wide bands (the proposed method). The depend on their values), the parameter estimates of the model and means are the least squares means (i.e., means adjusted for their standard errors are correct irrespective of the correctness of the model effects). The P values for the change from baseline to working correlation matrix. Thus, as long as a generic correlation the tasks also are reported. The second three columns show the structure is specified, valid inference can be made about the regression same statistics obtained by modeling measures of BPV calcu- parameters. In the special case in which data are normally distributed and the lated as a single measurement in a specific frequency band (the true covariance matrix is of general form, the GEE approach produces conventional method). Although the means obtained from the very similar results to other methods, like mixed effects models or two approaches are somewhat comparable, the standard errors multivariate ANOVA. Unlike these methods, however, which require of the estimates of BPV calculated in 0.01-Hz-wide bands are a correct specification of the covariance matrix and often extensive consistently smaller across experimental conditions. Thus par- execution times, the GEE approach is easily implemented, is not titioning the frequency spectrum into narrower components Table 1. Mean changes from baseline in systolic BPV for cardiac transplant subjects New Method Conventional Method Contrast LS Mean Standard error P value LS mean Standard error P value Low-frequency power (0.01–0.07 Hz) MA - Bs 0.0667 0.1135 0.5567 0.2255 0.2075 0.2772 Stroop - Bs 0.5541 0.1212 0.0001 0.6689 0.2152 0.0019 Tilt - Bs 0.2687 0.1216 0.0271 0.0189 0.2028 0.9259 Mid-frequency power (0.07–0.15 Hz) MA - Bs 0.1143 0.1033 0.2684 0.0134 0.2109 0.9495 Stroop - Bs 0.2629 0.1207 0.0293 0.2027 0.2448 0.3978 Tilt - Bs 0.7101 0.1208 0.0001 0.7730 0.2557 0.0025 MA, mental arithmetic task; Stroop, Stroop color word task; Bs, baseline; Tilt, 70° head-up tilt. LS Mean, least squares mean; BPV, blood pressure variability. J Appl Physiol • VOL 98 • JUNE 2005 • www.jap.org
  5. 5. Innovative Methodology STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA 2301 Table 2. Mean changes from baseline in systolic BPV for control subjects New Method Conventional Method Contrast LS mean Standard error P value LS mean Standard error P value Low-frequency power (0.01–0.07 Hz) MA - Bs 0.5021 0.1399 0.0003 0.4841 0.2685 0.0715 Stroop - Bs 1.0156 0.1368 0.0001 0.8193 0.2549 0.0013 Tilt - Bs 0.5876 0.1288 0.0001 0.3984 0.2344 0.0743 Mid-frequency power (0.07–0.15 Hz) MA - Bs 0.0993 0.1003 0.3220 0.0977 0.1338 0.4655 Stroop - Bs 0.4098 0.1162 0.0004 0.4339 0.1903 0.0226 Tilt - Bs 0.6988 0.1142 0.0001 0.7383 0.1915 0.0001 resulted in more efficient estimates of the mean changes from the controls in the 0.04- to 0.09-Hz range, but is greater in the baseline. transplant subjects in the 0.09- to 0.14-Hz range. This explains Table 3 presents the results of the tests comparing the the lack of statistical significant differences between the two cardiac transplant and control group in reactivity from baseline groups in mid- and low-frequency systolic BPV in response to with each of the experimental conditions. As for the within- tilt. group analysis, the least square means of BPV measured in The confidence intervals around the curve indicates the 0.01-Hz-wide bands have smaller standard errors compared degree of variability of the difference between cardiac trans- Downloaded from jap.physiology.org on August 6, 2010 with the standard errors of the mean of BPV measured using plant and control subjects with respect to reactivity to the the conventional method. The advantage of the smaller stan- Stroop and tilt tasks. As usual, the confidence intervals may dard errors is reflected in significant group differences for have a hypothesis-testing interpretation indicating a significant reactivity from baseline to the mental arithmetic task and to the difference between the groups at the points in which the Stroop color word task for BPV in the range of 0.01 to 0.07 Hz boundaries do not include zero. This happens, in our example, that were not detected by the conventional method. for the systolic BPV reactivity to Stroop corresponding to the Calculating power in 0.01-Hz-wide intervals also permits us ranges from 0.04 to 0.05 and 0.09 to 0.10 Hz, respectively. to look at BPV in a different way: it allows a detailed inspec- Although informative, these pointwise differences should be tion of the differences in systolic BPV reactivity between the interpreted with caution, especially when the sample size is two experimental groups. Figures 3 and 4 show the point-by- small. Although understandably tempting, the tendency to point group differences in systolic BPV reactivity to Stroop point out isolated, although significant, differences should be and tilt. Values above zero indicate that reactivity was greater counterbalanced by serious considerations concerning over- in the normal than in the transplant group. Conversely, values interpreting the results, multiplicity of testing, and clinical below zero indicate that reactivity was greater in the transplant interpretability. compared with the normal subjects. Figure 3 reveals that reactivity to the Stroop task was no DISCUSSION different in the two experimental groups in the range from 0.01 to 0.03 Hz. In contrast, reactivity to Stroop was consistently Psychophysiology experiments generate very large amounts greater in the control than in the transplant subjects in the range of data. To facilitate analysis and clinical interpretation, trans- from 0.03 to 0.11 Hz and behaved somewhat erratically in the formations and statistical algorithms are usually applied to 0.11- to 0.15-Hz range. Thus examination of the full spectrum reduce the data to a manageable dimension. However, the data explains the significant difference between the transplant and reduction process may lead to loss of information and statisti- control subjects in systolic BP low-frequency power and the cal power. This is especially true for HR and BP variability lack of a statistically significant difference in the mid-fre- data where the series of beat-to-beat HR or BP values collected quency band. during several minutes of recording often are reduced to a Figure 4 shows that the group difference in reactivity to tilt single value. In view of the considerable effort required on the has no clear pattern in the 0.01- to 0.04-Hz range, is greater in part of the experimental subjects and investigators to collect Table 3. Analysis of group differences: cardiac transplant vs. control subjects New Method Conventional Method Contrast LS mean Standard error P value LS mean Standard error P value Low-frequency power (0.01–0.07 Hz) MA - Bs 0.4354 0.1803 0.0157 0.2586 0.3394 0.4462 Stroop - Bs 0.4615 0.1838 0.0116 0.1504 0.3336 0.6521 Tilt - Bs 0.3189 0.1771 0.0718 0.3796 0.3100 0.2208 Mid-frequency power (0.07–0.15 Hz) MA - Bs 0.2136 0.1440 0.1378 0.1110 0.2497 0.6566 Stroop - Bs 0.1470 0.1675 0.3803 0.2269 0.3101 0.4644 Tilt - Bs 0.0113 0.1662 0.9457 0.0441 0.3196 0.8903 J Appl Physiol • VOL 98 • JUNE 2005 • www.jap.org
  6. 6. Innovative Methodology 2302 STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA Fig. 3. Mean group differences and 95% confidence intervals in reactivity to Stroop in the range from 0.01 to 0.15 Hz. and process these data, it seems unwise not to exploit all the frequency bands. The GEE methodology for handling the information the data provide. repeated measures is available in most statistical packages. In this paper, we have described an approach to analyzing The proposed approach also has great flexibility. Although it HR and BP variability that permits us to achieve a level of data is possible to perform the analysis by maintaining the conven- reduction sufficient for interpretation while retaining important tional spectral decomposition into low- and mid-frequency Downloaded from jap.physiology.org on August 6, 2010 characteristics of the data. In contrast to the conventional variability, it is also possible to examine the behavior of BP or approach to analysis of BPV data that aggregates power in HR variability in much greater detail by examining the point- specified frequency bands, this new approach is based on by-point behavior of these parameters across the entire fre- computing variability in 0.01-Hz-wide bands. Because this quency spectrum. Analyses of this type permit better under- approach creates a significantly greater number of repeated standing of differences between experimental groups or re- measurements, we employed GEE, a widely used statistical sponses to experimental challenges. When the sample size is methodology that accounts for the correlation among repeated large, multivariate statistical methods can be used to construct measurements and provides for valid statistical tests. Of the appropriate confidence intervals that account for the mul- course, other statistical methodologies, e.g., mixed linear or tiplicity of the points. Although we have illustrated the value of random effects models (5), also can be used for the analysis of this new approach using data from a study of BPV, the these data. These methods, however, require specification of a approach can be implemented for any time-series data, e.g., correct covariance matrix and are not as efficient in dealing R-R interval series. with missing values. An important point is that this approach is appropriate only Applying this new method to examining group differences in when an adequately long series of values is analyzed. With the systolic BPV, we have demonstrated that the approach is more use of the 0.01-Hz component, a single cycle at this frequency efficient in that it produces more precise estimates and smaller lasts 100 s. So, in principle, we need at least a 300-s-long series standard deviations of the parameters of interest than the to be able to resolve power at 0.01 Hz. conventional approach. These results are not surprising if we In conclusion, we have demonstrated that a new method of consider that the multiple 0.01-Hz measurements of BPV computing spectral power of BP or R-R time series, based on provide considerably more information (and thus greater sta- creating multiple estimates of power in narrow frequency tistical power) than the single measure obtained using the bands, has numerous advantages over the conventional method conventional method. From a practical and computational of estimating power in a single band: greater sensitivity to point of view, the method proposed is easy to implement and group or treatment differences, more efficient use of data, and does not require additional data management. Most programs a superior approach to missing data. These advantages, along that use fast Fourier transform to calculate power in wide bands with the relative computational ease, suggest that this new can be easily adapted to compute power in very narrow approach has considerable value. Fig. 4. Mean group differences and 95% confidence intervals in reactivity to tilt in the range from 0.01 to 0.15 Hz. J Appl Physiol • VOL 98 • JUNE 2005 • www.jap.org
  7. 7. Innovative Methodology STATISTICAL ANALYSIS OF CARDIOVASCULAR DATA 2303 GRANTS 4. Kamarck TW, Jennings JR, Debski TT, Glickman-Weiss E, Johnson The data used in this paper were collected as part of a study supported by PS, Eddy JJ, and Manuck SB. Reliable measures of behaviorally-evoked National Institute of Mental Health Grant R01 MH-43977 (R. P. Sloan). cardiovascular reactivity from a PC-based test battery: results from student and community samples. Psychophysiology 29: 17–28, 1992. REFERENCES 5. Laird NM and Ware JH. Random-effects models for longitudinal data. 1. Bagiella E, Sloan RP, and Heitjan DF. Mixed effects models in pyscho- Biometrics 38: 963–974, 1982. physiology. Psychophysiology 37: 13–20, 2000. 6. Liang KY and Zeger SL. Longitudinal data analysis using generalized 2. deBoer RW, Karemaker JM, and Strackee J. Comparing spectra of a linear models. Biometrika 73: 13–22, 1986. series of point events, particularly for heart rate variability spectra. IEEE 7. SAS Institute. SAS/STAT User’s Guide, Version 8. Cary, NC: SAS Insti- Trans Biomed Eng 31: 384 –387, 1984. tute, 1999. 3. Harris FJ. On the use of windows for harmonic analysis with the discrete 8. Zeger SL and Liang KY. Longitudinal data analysis for discrete and Fourier transform. Proc IEEE 66: 51– 83, 1978. continuous outcomes. Biometrics 42: 121–130, 1986. Downloaded from jap.physiology.org on August 6, 2010 J Appl Physiol • VOL 98 • JUNE 2005 • www.jap.org

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