## Abstract

Fourier-based approaches to analysis of variability of R-R intervals or blood pressure typically compute power in a given frequency band (e.g., 0.01–0.07 Hz) by aggregating the power at each constituent frequency within that band. This paper describes a new approach to the analysis of these data. We propose to partition the blood pressure variability spectrum into more narrow components by computing power in 0.01-Hz-wide bands. Therefore, instead of a single measure of variability in a specific frequency interval, we obtain several measurements. The approach generates a more complex data structure that requires a careful account of the nested repeated measures. We briefly describe a statistical methodology based on generalized estimating equations that suitably handles this more complex data structure. To illustrate the methods, we consider systolic blood pressure data collected during psychological and orthostatic challenge. We compare the results with those obtained using the conventional methods to compute blood pressure variability, and we show that our approach yields more efficient results and more powerful statistical tests. We conclude that this approach may allow a more thorough analysis of cardiovascular parameters that are measured under different experimental conditions, such as blood pressure or heart rate variability.

- blood pressure variability
- generalized estimating equations
- repeated measures

clinical studies in psychophysiology are a rich source of repeated-measures data. In these studies, individuals' psychophysiology indexes, such as heart rate (HR) or blood pressure (BP), are monitored repeatedly during relatively different experimental periods in a laboratory setting or during different activities that occur during daily living. As a result, these studies generate large amounts of data that must be reduced and then analyzed to detect within-group differences from one condition to another or differences between different groups of subjects.

Data reduction is achieved in various ways to obtain interpretable measures. One way is to submit time series of HR or BP values to Fourier-based spectral analysis to obtain frequency domain measures of BP (BPV) or HR variability. Conventionally, power within specifically defined frequency bands is calculated. For example, lower frequency BPV, i.e., BPV measured in the 0- to 0.15-Hz range, generally thought to reflect vascular sympathetic drive, is calculated as power in the low (0.01–0.07 Hz) or the mid (0.07–0.15 Hz) frequency bands (2). Thus, for each of these frequency bands, a measurement of systolic and diastolic BPV is obtained.

The statistical analysis of these transformed data generally involves testing the effect of psychological or physical challenges. Differences between responses obtained during a resting baseline and those obtained during challenging conditions are calculated (4), and group differences are assessed using well-known statistical procedures such as *t*-tests, repeated-measures ANOVA, or multivariate ANOVA.

Although these approaches to data reduction and statistical analysis are not incorrect methodologically, they either oversimplify the complex structure of the data, resulting in loss of information, or they require model assumptions that the data often do not satisfy (1). Another limitation is that subjects with missing data, either at baseline or during a challenge, are excluded from analyses because reactivity cannot be computed.

Thus several questions arise from the current approaches to data reduction and statistical analysis applied to cardiovascular data measured repeatedly. Does computing variability as a single measure for a given frequency band result in too much data reduction and, hence, a loss of information? Do the statistical methods currently used for the analysis of psychophysiology data efficiently use all the information available in the data?

In this paper, we aim to answer these questions by presenting a new approach to the analysis of cardiovascular data. Specifically, we propose to partition the frequency spectrum into 0.01-Hz-wide intervals instead of the more conventional bands. We also describe a statistical methodology for the analysis of the data obtained. To illustrate the methods, we use data on BPV from a recently conducted study. We conclude with a brief discussion.

## MATERIALS AND METHODS

### Study Description

The data were collected in a study conducted to assess the ability of the cardiac autonomic control system to buffer BPV during psychological and orthostatic challenge (Sloan SP, unpublished observations). Three groups of subjects were recruited for the clinical experiment: 31 cardiac transplant recipients, 11 renal transplant recipients receiving equivalent immunosuppressive treatment, and 24 normal subjects. Normal and renal transplant subjects were matched to cardiac transplant recipients by age and gender.

The experimental protocol included a 30-min adaptation, a 5-min baseline, two 5-min mental arithmetic task periods, a 5-min Stroop color word task, a 2-min cold pressor task, and 10 min of 70° head-up tilt. Recovery periods followed each task. All subjects signed an informed consent form before enrollment in the study. The study was approved by the Columbia University Medical Center Institutional Review Board.

### Psychological Stressors

#### Mental arithmetic.

In this task, subjects were presented with a four-digit number on the computer monitor and were instructed to subtract serially by 7 starting with this number, which disappeared after the first answer was entered. At 1-min intervals, subjects received verbal prompts, e.g., “please subtract faster.” The task was not paced by the computer, but subjects were instructed to subtract as quickly and as accurately as possible. Subjects entered the results on the numeric keypad.

#### Stroop color word task.

In this version of the Stroop task, subjects were presented with color names (blue, green, yellow, and red) in colors that were either congruent or incongruent with the names. The subjects' task was to press the key on the keypad that corresponded to the color of the letters. The task was paced by the computer, and an incorrect response or failure to respond rapidly enough resulted in a message indicating “incorrect” on the screen.

#### Tilt.

The tilt table was elevated to the 70° head-up position over the course of 1 min. BP and respiration monitors were recalibrated after equilibration to the upright position. Subjects remained in the head-up position for 10 min unless they developed symptoms of faintness.

We present the results of both the new and conventional methods of analysis of systolic BPV data from the cardiac transplant and control subjects collected during four experimental conditions: the baseline period, the first mental arithmetic task, the Stroop color word task, and the 70° head-up tilt.

### Data Reduction

Beat-to-beat BP was recorded using a Finapres model 2300 monitor. The analog BP waveform was collected at 500 samples/s. With the use of pattern recognition software, peaks and valleys of the waveform were marked, and systolic and diastolic BP time series were generated.

Spectra were calculated on 240-s epochs using an interval method for computing Fourier transforms similar to that described by DeBoer et al. (2). Before computing Fourier transforms, the mean of the systolic and diastolic BP series were subtracted from each value in the series, and the residual series then was filtered using a Hanning window (3) and the power, i.e., variance (in mmHg^{2}), within each band was summed. Estimates of spectral power were adjusted to account for attenuation produced by this filter (3).

Because the servo self-adjustment of the Finapres was enabled during the final minute of the each 5-min recording period, only data from the first 4 min of each period were analyzed. Data from the first 5 min of the tilt period were excluded from analysis to permit full equilibration to the upright position.

Using the new approach, we applied fast Fourier transforms to the series of systolic BP values to obtain systolic BPV in 0.01-Hz-wide frequency bands. In this way, we obtained 14 systolic BPV measurements on a discrete scale from 0.01 to 0.15 Hz. Thus, for each subject in each of the four experimental conditions, instead of a single measure of variability in the 0.01- to 0.15-Hz band, we obtained 14 measurements representing systolic BPV at the midpoint of each of the 0.01-Hz-wide intervals.

These data were log-transformed before analysis to correct for skewness.

Figures 1 and 2 depict the average semicontinuous systolic BPV curves for each experimental period for the cardiac transplant patients and the normal control subjects. The values on the horizontal axis are the lower limits of the 0.01-Hz-wide frequency intervals. Thus the 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-by-point average, across subjects, of the individual curves within each experimental condition for the two study groups.

Note that this approach to data reduction does not change the interpretation of BPV. For example, low-frequency SBP power is still represented by power in the 0.01- to 0.07-Hz band. However, instead of being represented by a unique number, as in the conventional analysis, it is now represented by six values.

### Statistical Analysis of the 0.01-Hz-Wide Intervals

The statistical analysis of BPV calculated in 0.01-Hz-wide frequency bands is more complicated than the analysis of these variables calculated in the conventional way. First, in a frequency band of interest, systolic BPV is represented by several values instead of a single one. For example, in our approach, six measurements represent systolic BPV in the 0.01- to 0.07-Hz frequency band. Second, a more complicated structure of repeated measures has to be taken into account. Within the same subject, the BPV measurements in a frequency interval and the measurements obtained during different experimental conditions are nested as repeated measures. Typically, these repeated measures are positively correlated with each other, and failure to account for this correlation leads to underestimation of the standard errors of the model parameters, resulting in overly liberal tests of hypothesis about these parameters. Also, collecting large numbers of repeated measures increases the risk that some will be missing. A statistical approach to data analysis that addresses both these concerns is the generalized estimating equations (GEE) approach (6, 8). Although the approach is mostly used to model categorical or binary data, it is also widely applied to normally distributed data, such as the BPV measurements in this study.

Briefly, GEE are an extension of the standard generalized linear 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 correlation structure among the repeated measures also has to be specified. This matrix describes the dependence between the repeated measures in the model. An autoregressive correlation matrix, for example, specifies that adjacent measurements would be more highly correlated than measurements taken farther apart. However, one of the advantages of the GEE approach is that in most situations, provided that the missing data are missing at random (i.e., their missing does not depend on their values), the parameter estimates of the model and their standard errors are correct irrespective of the correctness of the working correlation matrix. Thus, as long as a generic correlation structure is specified, valid inference can be made about the regression parameters.

In the special case in which data are normally distributed and the true covariance matrix is of general form, the GEE approach produces very similar results to other methods, like mixed effects models or multivariate ANOVA. Unlike these methods, however, which require a correct specification of the covariance matrix and often extensive execution times, the GEE approach is easily implemented, is not restricted by model assumptions, and has relatively rapid computing time.

Most statistical software provides subroutines to apply the GEE approach. In our application, we used the PROC GENMOD routine provided by SAS (7). (An example of a SAS program used to perform the GEE analysis can be obtained from the corresponding author.)

Our models included the measurements of systolic BPV as the dependent variables and experimental period (baseline, mental arithmetic, Stroop, and tilt) and group membership (cardiac transplant or normal control) and their interaction as the predictors. In the following section, we present the results of the analysis performed using an unstructured correlation matrix, which specifies a completely general correlation structure among the repeated measures. However, the model produced very similar results when we used different correlation structures.

In the next section, we present the result of the analysis performed using our approach and a comparison with the analysis performed using the conventional methods for calculating BPV. For the analysis of BPV into low- and mid-frequency power, we considered two separate models for systolic BPV in the 0.01- to 0.07-Hz band and in the 0.07- to 0.15-Hz band, respectively. The analysis of low-frequency BPV using our approach considers all the points in the 0.01- to 0.07-Hz band, whereas the analysis of the mid-frequency BPV considers all the points in the 0.07- to 0.15-Hz band. With the use of the conventional approach, two values of BPV were calculated, one representing power in the 0.01- to 0.07-Hz band and one representing power in the 0.07- to 0.15-Hz band.

In a separate analysis, we also considered the 14 BPV point curves from 0.01 to 0.15 Hz.

## RESULTS

Tables 1 and 2 present the means and respective standard errors of changes in systolic BPV from baseline to each of the three experimental conditions for the cardiac transplant and control groups. The first three columns in the tables show results obtained by modeling the systolic BPV measurements calculated in 0.01-Hz-wide bands (the proposed method). The means are the least squares means (i.e., means adjusted for model effects). The *P* values for the change from baseline to the tasks also are reported. The second three columns show the same statistics obtained by modeling measures of BPV calculated as a single measurement in a specific frequency band (the conventional method). Although the means obtained from the two approaches are somewhat comparable, the standard errors of the estimates of BPV calculated in 0.01-Hz-wide bands are consistently smaller across experimental conditions. Thus partitioning the frequency spectrum into narrower components resulted in more efficient estimates of the mean changes from baseline.

Table 3 presents the results of the tests comparing the cardiac transplant and control group in reactivity from baseline with each of the experimental conditions. As for the within-group analysis, the least square means of BPV measured in 0.01-Hz-wide bands have smaller standard errors compared with the standard errors of the mean of BPV measured using the conventional method. The advantage of the smaller standard errors is reflected in significant group differences for reactivity from baseline to the mental arithmetic task and to the Stroop color word task for BPV in the range of 0.01 to 0.07 Hz that were not detected by the conventional method.

Calculating power in 0.01-Hz-wide intervals also permits us to look at BPV in a different way: it allows a detailed inspection of the differences in systolic BPV reactivity between the two experimental groups. Figures 3 and 4 show the point-by-point group differences in systolic BPV reactivity to Stroop and tilt. Values above zero indicate that reactivity was greater in the normal than in the transplant group. Conversely, values below zero indicate that reactivity was greater in the transplant compared with the normal subjects.

Figure 3 reveals that reactivity to the Stroop task was no different in the two experimental groups in the range from 0.01 to 0.03 Hz. In contrast, reactivity to Stroop was consistently greater in the control than in the transplant subjects in the range from 0.03 to 0.11 Hz and behaved somewhat erratically in the 0.11- to 0.15-Hz range. Thus examination of the full spectrum explains the significant difference between the transplant and control subjects in systolic BP low-frequency power and the lack of a statistically significant difference in the mid-frequency band.

Figure 4 shows that the group difference in reactivity to tilt has no clear pattern in the 0.01- to 0.04-Hz range, is greater in the controls in the 0.04- to 0.09-Hz range, but is greater in the transplant subjects in the 0.09- to 0.14-Hz range. This explains the lack of statistical significant differences between the two groups in mid- and low-frequency systolic BPV in response to tilt.

The confidence intervals around the curve indicates the degree of variability of the difference between cardiac transplant and control subjects with respect to reactivity to the Stroop and tilt tasks. As usual, the confidence intervals may have a hypothesis-testing interpretation indicating a significant difference between the groups at the points in which the boundaries do not include zero. This happens, in our example, for the systolic BPV reactivity to Stroop corresponding to the ranges from 0.04 to 0.05 and 0.09 to 0.10 Hz, respectively. Although informative, these pointwise differences should be interpreted with caution, especially when the sample size is small. Although understandably tempting, the tendency to point out isolated, although significant, differences should be counterbalanced by serious considerations concerning overinterpreting the results, multiplicity of testing, and clinical interpretability.

## DISCUSSION

Psychophysiology experiments generate very large amounts of data. To facilitate analysis and clinical interpretation, transformations and statistical algorithms are usually applied to reduce the data to a manageable dimension. However, the data reduction process may lead to loss of information and statistical power. This is especially true for HR and BP variability data where the series of beat-to-beat HR or BP values collected during several minutes of recording often are reduced to a single value. In view of the considerable effort required on the part of the experimental subjects and investigators to collect and process these data, it seems unwise not to exploit all the information the data provide.

In this paper, we have described an approach to analyzing HR and BP variability that permits us to achieve a level of data reduction sufficient for interpretation while retaining important characteristics of the data. In contrast to the conventional approach to analysis of BPV data that aggregates power in specified frequency bands, this new approach is based on computing variability in 0.01-Hz-wide bands. Because this approach creates a significantly greater number of repeated measurements, we employed GEE, a widely used statistical methodology that accounts for the correlation among repeated measurements and provides for valid statistical tests. Of course, other statistical methodologies, e.g., mixed linear or random effects models (5), also can be used for the analysis of these data. These methods, however, require specification of a correct covariance matrix and are not as efficient in dealing with missing values.

Applying this new method to examining group differences in systolic BPV, we have demonstrated that the approach is more efficient in that it produces more precise estimates and smaller standard deviations of the parameters of interest than the conventional approach. These results are not surprising if we consider that the multiple 0.01-Hz measurements of BPV provide considerably more information (and thus greater statistical power) than the single measure obtained using the conventional method. From a practical and computational point of view, the method proposed is easy to implement and does not require additional data management. Most programs that use fast Fourier transform to calculate power in wide bands can be easily adapted to compute power in very narrow frequency bands. The GEE methodology for handling the repeated measures is available in most statistical packages.

The proposed approach also has great flexibility. Although it is possible to perform the analysis by maintaining the conventional spectral decomposition into low- and mid-frequency variability, it is also possible to examine the behavior of BP or HR variability in much greater detail by examining the point-by-point behavior of these parameters across the entire frequency spectrum. Analyses of this type permit better understanding of differences between experimental groups or responses to experimental challenges. When the sample size is large, multivariate statistical methods can be used to construct the appropriate confidence intervals that account for the multiplicity of the points. Although we have illustrated the value of this new approach using data from a study of BPV, the approach can be implemented for any time-series data, e.g., R-R interval series.

An important point is that this approach is appropriate only when an adequately long series of values is analyzed. With the use of the 0.01-Hz component, a single cycle at this frequency lasts 100 s. So, in principle, we need at least a 300-s-long series to be able to resolve power at 0.01 Hz.

In conclusion, we have demonstrated that a new method of computing spectral power of BP or R-R time series, based on creating multiple estimates of power in narrow frequency bands, has numerous advantages over the conventional method of estimating power in a single band: greater sensitivity to group or treatment differences, more efficient use of data, and a superior approach to missing data. These advantages, along with the relative computational ease, suggest that this new approach has considerable value.

## GRANTS

The data used in this paper were collected as part of a study supported by National Institute of Mental Health Grant R01 MH-43977 (R. P. Sloan).

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 2005 the American Physiological Society