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# Older males secrete luteinizing hormone and testosterone more irregularly, and jointly more asynchronously, than youngermales

^{†}

^{‡}Thomas Mulligan,

^{§}Ali Iranmanesh,

^{¶}Sylvia Gheorghiu,

^{§}Michael Godschalk,

^{§}and JohannesD. Veldhuis

^{‖}

^{†}990 Moose Hill Road, Guilford, CT 06437;

^{§}Hunter Holmes McGuire Veterans Affairs Medical Center and Medical College of Virginia, Richmond, VA 23249;

^{¶}Endocrine Section, Veterans Affairs Medical Center, Salem, VA 24153; and

^{‖}Department of Internal Medicine, and National Science Foundation Center for Biological Timing, University of Virginia Health Sciences Center, Charlottesville, VA 22908

^{‡}To whom reprint requests should be addressed.

## Abstract

New statistical perspectives on the secretory patterns of both
luteinizing hormone (LH) and testosterone (T) may prove useful in
further understanding the aging process, and possibly ultimately in
improving the diagnosis and treatment of spermatogenetic failure and
loss of sexual interest. We examined serum concentration time-series
for LH and T in 14 young (21–34 years of age) and 11 aged (62–74
years of age) healthy men. For each subject, blood samples were
obtained at 2.5-min intervals during a sleep period, with an average
sampling duration of 7 hr. For each of LH and T, we used the
model-independent statistic approximate entropy (ApEn) to quantify the
irregularity of the serum concentration time-series; to quantify joint
LH–T secretory asynchrony, we employed the recently introduced
cross-ApEn. Although mean (and SD) LH and T concentrations were
indistinguishable in the two age groups (*P* >
0.25), for LH, aged subjects had greater ApEn values (1.525 ±
0.221) than younger individuals (1.207 ± 0.252),
*P* < 0.003, indicating more irregular secretion in
the older cohort. For T, aged subjects also had greater ApEn values
(1.622 ± 0.120) than younger counterparts (1.384 ± 0.228),
*P* < 0.004. In young, but not older men, ApEn(T)
significantly exceeded ApEn(LH), *P* < 0.02. Aged
subjects had greater cross-ApEn values (1.961 ± 0.121) than
younger subjects (1.574 ± 0.249), *P* <
10^{−4}, with nearly 100% sensitivity and specificity,
indicating greater LH–T asynchrony in the older group. In conjunction
with previous findings of greater irregularity of growth hormone
release with increasing age, we propose that increased secretory
irregularity with advancing age may be a widespread hormonal
phenomenon. Finally, theoretically, we clarify the need for
quantifications such as ApEn and cross-ApEn via a study of a
“variable lag” pulsatile process, and empirically note the
potential wide applicability of cross-ApEn to quantify asynchrony in
interconnected (hormonal) networks.

**Keywords:**approximate entropy, aging, pulsatility, reproduction, variable lag process

Greater understanding of the evolution of the
hypothalamo–pituitary–testicular axis with aging is of vital
importance both scientifically, in elucidating the physiology of
reproductive capacity, and clinically, in assessing, e.g., a loss of
libido or decreased reproductive performance. In recent years, there
has been considerable study of luteinizing hormone (LH) and
testosterone (T) serum concentration time-series in both younger and
older males to develop such understanding, and to determine whether a
hypothesized male climacteric (or so-called andropause) at least
partially analogous to menopause in the female exists, and if so, in
what precise sense. Such studies have evaluated changes in
(*i*) mean concentrations of total and free T, and LH and the
ratio of biological to immunological (B/I) LH activity (1, 2, 3, 4, 5, 6);
(*ii*) “near-term” (circhoral) pulsatility
characteristics of LH and T, via changes in mean frequencies and
amplitudes (7, 8, 9, 10); (*iii*) “longer term,” i.e.,
nyctohemeral characteristics of LH and T release (1). While
considerable insight has already been gained, there remain nontrivial
controversies, e.g., primary determinations of whether overall mean
levels of LH and T decrease with increasing age, as discussed below.
Furthermore, the precise neuroendocrine mechanisms that underlie such
age-related changes remain largely unresolved.

Here, we consider possible reproductive aging changes from two perspectives entirely different from those mentioned above, namely by directly evaluating the degree of irregularity of each of the LH and T time-series, via approximate entropy (ApEn) (11, 12) and by quantifying the degree of asynchrony in the joint LH–T series, via cross-ApEn (12). In this study, we reanalyzed data collected by frequent venous sampling (every 2.5 min) overnight (8) to delineate the nature of changes in the secretion of these two hormones in healthy older men. Methodologically, first, it is imperative to note the importance of matching corresponding parts of the circadian epoch in the older and younger age cohorts, because aged men lose circadian fluctuations in serum T concentrations (1).

In addition to the biological relevance of assessing LH and T
release from a distinct statistical perspective, we particularly note
the potential broad statistical utility of the recently introduced
cross-ApEn to quantify asynchrony or conditional irregularity in
interconnected (hormonal) networks. In the Appendix we
consider: (*i*) the manner in which ApEn and cross-ApEn have a
quite different and complementary primary orientation from both linear
correlation and the power spectrum and (*ii*) why this
separate perspective affords the biologist distinct tools from which
changes in the extent of synchrony in interconnected hormonal systems
can be clearly determined. This quantification strategy is relevant to
many feedback and/or control systems and models for which
cross-correlation and cross-spectral methods fail to fully highlight
markedly changing features of the data sets under consideration.

## MATERIALS AND METHODS

The study group comprised 14 young (21–34 years of
age) and 11 aged (62–74 years of age) healthy nonsmoking men within
20% of ideal body weight. For each subject, blood samples were
obtained during a sleep period on a second night of study in the
General Clinical Research Center at the University of Virginia, at
2.5-min intervals commencing at 2300 hr, with sampling terminated when
the subject spontaneously awakened, for an average sampling duration of
7 hr. Serum LH concentrations were measured in duplicate by using a
two-site monoclonal immunoradiometric assay (Nichols Institute, San
Juan Capistrano, CA). Assay sensitivity was 0.2 unit/liter according
to the First International Reference Preparation. Serum total T
concentrations were quantified in duplicate for each sample by using a
solid-phase RIA (Diagnostic Products, Los Angeles). Assay sensitivity
was 20 ng/dl. For both LH and T assays, intra- and interassay
imprecision was less than 10%. Further subject and assay descriptions
are given by Mulligan *et al.* (8).

#### ApEn.

To quantify irregularity, we use ApEn, defined in ref.
11, further mathematical properties, and biological applications as
given in refs. 12, 13, 14, 15, 16, 17, 18. ApEn is complementary to pulse-detection
algorithms widely employed to evaluate hormone secretion time-series
(19). ApEn evaluates both dominant and subordinant patterns in data;
notably, it will detect changes in underlying episodic behavior not
reflected in peak occurrences or amplitudes (17). Additionally, ApEn
provides an explicit barometer of feedback system change in many
coupled systems (17, 20). Within various endocrine contexts, ApEn has
unveiled vivid distinctions (*P* < 10^{−10})
between normal and tumor-bearing subjects for GH (21) and aldosterone
release (22), a pronounced and consistent gender difference in GH
time-series irregularity in both human and rat (23), and a positive
correlation between greater irregularity of GH secretion and advancing
age (24).

ApEn assigns a nonnegative number to a time-series, with larger values
corresponding to greater apparent process randomness (serial
irregularity), and smaller values corresponding to more instances of
recognizable patterns in the data. Two input parameters, *m*
and *r*, must be specified to compute ApEn. Briefly, ApEn
measures the logarithmic likelihood that runs of patterns that are
close (within *r*) for *m* contiguous observations
remain close (within the same tolerance width *r*) on next
incremental comparisons; the precise mathematical definition is given
in ref. 11.

In this study, we calculated
ApEn(*m*,*r*) values for all data sets,
*m* = 1 and *r* = 20% of the SD of the
individual subject’s hormone time-series.^{**}
Normalizing *r* to each time-series SD gives ApEn a
translation- and scale-invariance to absolute serum concentration
levels (14). ApEn is a relative measure of process regularity, and can
show significant variation with changing background noise
characteristics. Because ApEn generally increases with increasing
process noise, it is appropriate to compare data sets with similar
assay coefficients of variation, as we do here.

Previous studies that included both theoretical analysis (16, 17, 25)
and clinical applications (13, 14, 15, 18, 21, 22, 23, 24) have demonstrated that
the input parameters indicated above produce good statistical validity
(reproducibility) for ApEn applied to time-series of the lengths
considered here. The ApEn application with *m* = 1
estimates the rate of entropy for a first-order (*m* = 1)
approximating Markov Chain to the underlying true process (26). Further
technical discussion of mathematical and statistical properties of
ApEn, including robustness to noise and artifacts, mesh interplay,
relative consistency of (*m*,*r*) pair choices,
asymptotic normality under general assumptions, statistical bias, and
error estimation for general processes can be found elsewhere (16, 25).

#### Cross-ApEn.

To quantify asynchrony (conditional irregularity), we use cross-ApEn, as introduced in ref. 12, definition 5. As noted there, cross-ApEn can be employed to compare sequences from two distinct yet intertwined variables in a network, herein applied to the joint LH–T time-series. The precise definition is thematically similar to that for ApEn:

Let *u* = (*u*(1), *u*(2), …
*u*(*N*)) and *v* = (*v*(1),
*v*(2), … *v*(*N*)) be two
length-*N* sequences. Fix input parameters *m* and
*r*. Form vector sequences *x*(*i*) =
(*u*(*i*), *u*(*i* + 1), …
*u*(*i* + *m* − 1)) and
*y*(*j*) =
(*v*(*j*), *v*(*j*
+ 1), … *v*(*j* + *m* −
1)) from *u* and *v*, respectively. For each
*i* ≤ *N* − *m* + 1, set
*C*_{i}^{m}(*r*)(*v*
*u*) = (number of *j* ≤ *N* −
*m* + 1 such that d[*x*(*i*),
*y*(*j*)] ≤
*r*)/(*N* − *m* + 1), where
*d*[*x*(*i*),
*y*(*j*)] = max_{k = 1,
2,… , m}
(|*u*(*i* + *k* − 1) −
*v*(*j* + *k* − 1)|), i.e.,
the maximum difference in their respective scalar components. The
C_{i}^{m}(*r*)’s measure *within a
tolerance r* the regularity, or frequency, of (*v*-)
patterns similar to a given (*u*-) pattern *of window
length m*. Then define Φ^{m}(*r*) (*v*
*u*) as the average value of ln
*C*_{i}^{m}(*r*) (*v*
*u*), and finally, define
cross-ApEn(*m*,*r*,*N*)(*v*
*u*) = Φ^{m}(*r*) (*v*
*u*) − Φ^{m+1}(*r*) (*v*
*u*).

For this study, we applied cross-ApEn with *m* = 1 and
*r* = 0.2 to standarized LH (= *u*) and
testosterone (= *v*) time-series data, i.e., for each subject,
we applied cross-ApEn(1, 0.2) to the {*u**(*i*),
*v**(*i*)} series, where
*u**(*i*) = (*u*(*i*) − mean
*u*)/SD *u* and *v**(*i*) =
(*v*(*i*) − mean *v*)/SD *v*.
This standardization, in conjunction with the choice of *m*
and *r*, ensures good replicability properties for cross-ApEn
for the data lengths studied.

## RESULTS

All statistical comparisons below employ the two-sided
*t* test, except for the ApEn(LH) vs. ApEn(T) comparisons
within each of the younger and older cohorts, for which we employed the
paired *t* test. Results are given as mean ± SD.

Inspection of serum hormone concentration profiles suggests that clear
pulse identification is a nontrivial endeavor, especially for the aged
subjects’ T series (see Fig. 4, ref. 8). For LH, aged subjects had
greater ApEn values (1.525 ± 0.221) than younger subjects
(1.207 ± 0.252), *P* < 0.003. For T, aged subjects
also had greater ApEn values (1.622 ± 0.120) than younger
subjects (1.384 ± 0.228), *P* < 0.004. In Fig.
Fig.1,1, scatterplots of mean LH level vs. ApEn(LH),
and of mean T level vs. ApEn(T) visually confirm this statistical
distinction. The decision rule that associates ApEn(LH) values greater
than 1.445 with aged subjects has a specificity of 93% and a
sensitivity of 82%, whereas the decision rule that associates ApEn(T)
values greater than 1.60 with aged subjects has a specificity of 100%
and a sensitivity of 64%.

*Upper*) Individual subject ApEn

_{LH}(

*m*= 1,

*r*= 20% SD) values vs. mean serum LH concentrations. (

*Lower*) Individual subject ApEn

_{TESTO}(

*m*= 1,

*r*= 20% SD) values

**...**

Notably, there was no difference in *mean* serum LH levels
between the younger (2.409 ± 0.658 units/liter) and aged
subjects (2.830 ± 1.064 units/liter) levels, *P*
= 0.26; and there was no difference in mean T levels between the
younger (459 ± 148 ng/dl) and aged subjects (415 ± 115
ng/dl) levels, *P* = 0.41.

Aged subjects had greater cross-ApEn values (1.961 ± 0.121) than
younger subjects (1.574 ± 0.249), *P* <
10^{−4}. Importantly, there was nearly complete separation of
younger and older subject cross-ApEn values, as observed in Fig.
Fig.2,2, with all younger subjects’ cross-ApEn values
smaller than all but a single older subject’s value. The decision rule
that associates cross-ApEn values greater than 1.85 with aged subjects
has a specificity of 100% and a sensitivity of 91%. In counterpoint,
cross-correlation (Pearson “R”), reveals no significant
differences, either in the Pearson R values directly, older subjects
(0.078 ± 0.210) vs. younger subjects (0.030 ± 0.284),
*P* = 0.629; or in the magnitude of the correlation,
assessed by |Pearson R|, older subjects (0.150 ± 0.162) vs.
younger subjects (0.231 ± 0.155), *P* = 0.220.

As another perspective on changes with aging in the joint LH–T
variable system, we ascertained that for younger subjects serum T
concentration time-series were more irregular than the corresponding LH
concentration series, *P* < 0.02, whereas this per
subject distinction vanished in the aged group, with no significance in
the pairwise ApEn(LH) and ApEn(T) values, *P* >
0.28.^{‡‡}

## DISCUSSION

#### Summary and Resultant Biological Questions.

Summarizing the primary statistical results, for each of LH and T, older males have consistently and significantly more irregular serum reproductive-hormone concentrations than younger males. The distinction between ApEn(T) and ApEn(LH) indicating greater irregularity of the former in young men was lost in older men. Furthermore, cross-ApEn quantitatively supports a mechanistic hypothesis, a loss of synchrony with aging in the coupled LH–T system. The cross-ApEn finding reinforces the utility of studying network aspects, in addition to single-variable or nodal aspects, of hormone systems, both in statistical analysis and in modeling, and ultimately, in evaluating therapies. The determinations that mean serum LH and T concentrations in the young and older males were not significantly different, nor were linear cross-correlations, further suggest the need for the distinct perspectives assessed by quantification of irregularity and (a)synchrony.

Our inferences in the aging male reproductive axis (above) are in agreement with findings of greater growth hormone irregularity of release with increased aging (24). Thus, we hypothesize that greater secretory irregularity, and possibly greater asynchrony, with increased aging may be a more general paradigm for many hormones, potentially indicating a diminution of subsystem integrity (20) and/or of (synchronous) control.

It seems worthwhile to compare the results for the male to
corresponding findings for the female, although sex-steroid levels
decline more markedly in postmenopausal individuals than in aging men.
Any comparisons between male and female evolution of
“reproductive” hormone secretion as a function of increasing age
are at best partial, given the cessation of female reproductive
capacity in the aged, in contrast to continued, albeit diminished male
fertility in advanced age. However, the above findings clearly indicate
distinct quantitative shifts in male hormonal secretory dynamics with
aging. Thus, the question arises as to how mechanistically greater
individual signal irregularity (in LH or T release) or joint signal
asynchrony are linked directly or causally to clinical changes in,
e.g., spermatogenetic function or libido, as commonly seen in elderly
males. Moreover, does the increased LH–T irregularity and asynchrony
in older males occur gradually, at a relatively constant rate
throughout life, or instead develop rather abruptly during a relatively
shorter time-frame of months or years, the latter analogous to estrogen
transitions in females across the menopause? For this last point, we
hypothesize a more gradual evolution based on the somewhat analogous
determination of a modest, slow continuous decrease in mean total T
serum concentrations with increasing age seen, e.g. by Zumoff *et
al.* (6) in a study of normal men 21–85 years of age.

In principle, there are several possibilities for the source of the
erosion of LH–T synchrony quantified above. These include:
(*i*) decreased multi-synaptic modulation and/or synchrony
of the hypothalamic gonadotropin-releasing hormone (GnRH) neuronal
network that produces the GnRH drive to pituitary LH synthesis and
secretion; (*ii*) altered feedback control of individual
and/or coupled GnRH–LH secretory activity by gonadal hormones, via a
disrupted feedback signal, e.g., of T itself, or deficient
responsiveness to the feedback signal; (*iii*) decreased GnRH
and non-GnRH-dependent paracrine or autocrine coordination of LH
secretion by gonadotroph cells; and/or (*iv*) disruption of
effective (LH–T) stimulus-secretion coupling at the level of the
Leydig cell in the testis. Further physiological studies will be
required to clarify the precise basis of this change. Nonetheless,
because there is increased ApEn of LH release after short-term
ketoconazole treatment in young men when T secretion
falls,^{††} and increased ApEn of GH
release with fasting as IGF-1 falls (21), we favor decreased feedback
signal strength, or diminished GnRH–LH system responsiveness to
feedback signal intensity, as a unifying hypothesis.

#### Complementarity of Present Findings to Previous LH and T Age-Related Changes.

A number of age-related changes have been
established earlier for both LH and T secretion. Our findings provide
an entirely distinct and complementary perspective to previously
identified differences in means or amplitudes of suitable physiological
variables, so that secretory typicality can be assessed quantitatively
both on the basis of mean and amplitude level of output and on the
basis of orderliness of serial output. Mathematically, we observe a
primary difference between regularity measures, such as ApEn, and
moment statistics (e.g., means, standard deviations); namely, moment
statistics and their nonparametric counterparts are computed without
regard to the *order* of the series to which they are applied.
For ApEn, the serial data order is the crucial factor. Additionally, to
the best of our knowledge, a direct statistical assessment of joint
LH–T network characteristics of either younger or aged men has not
previously been accomplished, which cross-ApEn now addresses.

The relative clarity of the young/old separation by ApEn and
cross-ApEn takes on enhanced importance in light of reassessment of
age-related changes in mean reproductive hormone levels, especially in
the case of T, for which there is no clear consensus. Touitou (27)
elucidates the controversy for T, in part due to diurnal variations,
and time-of-day sampling; e.g., Vermeulen *et al*.
reports a decrease in plasma concentrations in the aged (4), based on
morning sampling, whereas Harman and Tsitouras (2) show unchanged T
levels in the aged group, based on early afternoon studies. Bremner
*et al.* (1) showed an overall 24-hr decrease in T
concentrations (*P* < 0.05) in the aged, although none
between mid-afternoon and late evening. Zumoff *et al*.
(6) concluded that total T levels decrease in the aged, though there
was considerable overlap between younger and older mean levels (Fig. (Fig.1,
1,
ref. 6). Nankin and Calkins (3) reported that mean serum total and free
T were similar in young and older groups, whereas the mean absolute
non-sex hormone-binding globulin-bound T level, as an index of
bioavailable T, was significantly lower in the older group. And above,
we deduced that for nocturnal observations, mean serum T concentrations
are not necessarily distinguishable between very healthy young and
older males. For immunoreactive LH, the inference of no overall mean
level changes between young and older subjects has been verified by
several studies (3, 5, 6), as well as by our analysis. Even so, Warner
*et al.* (5) and Urban *et al.* (28) suggested that
whereas mean serum immunoreactive and bioactive LH concentrations are
age-invariant, the ratio of biological to immunological LH activity
decreases in the aged basally or after stimulation, respectively.

Our findings also augment, from a distinct perspective, previous analyses of mean frequency and amplitude characteristics of LH and T release episodes, where elderly men exhibit more frequent (low amplitude) LH secretory bursts, and amplitude-attenuation of T secretory bursts (8). Indeed, the present appraisal by ApEn and cross-ApEn offers a clearer young/old group separation than the aforementioned significant frequency and amplitude differences, especially for T (compare figures 5 and 6 of ref. 8 to figures 1 and 2 herein). In addition, we observe that for parameters such as pulse characteristics and irregularity, a rapid sampling protocol is crucial in some settings (e.g., the 2.5-min sampling paradigm employed herein) to obtain a fine probabilistic description of the contiguous measurement series, whereas for mean level analysis such as those described in the previous paragraph, the 20-min sampling protocol used by many researchers is generally sufficient to provide an accurate estimate (of the mean). In instances in which mean secretory burst frequency and amplitude differences are relatively subtle (e.g., young vs. old LH and T characterizations), inferences from the finer sampling can actually differ qualitatively from those based on significantly coarser sampling (e.g., compare results from ref. 8 to those from coarser sampling protocols employed in refs. 7 and 9).

It is crucial to note the counterpoint between the perspectives of
irregularity and that of diurnal variation. Bremner *et al.*
(1) observed a clear loss of nyctohemeral T variation in older men,
which, as indicated above, was critically noted in establishing the
experimental protocol in this study. From a broader perspective, as
summarized by Copinschi and van Cauter (29), changes in circadian
rhythms toward lower amplitude and/or phase advance with increasing
age have been established for the peripheral levels of many other
hormones (30, 31, 32, 33). However, this change in circadian variation is a
very different notion of rhythm change from that of a change in
irregularity, as quantified here. Statistically, the extent of 24-hr
variation [e.g., quantified by Bremner *et al.* (1) as the
highest point minus the lowest point] is basically a measure of
overall day-night amplitude. The nyctohemeral changes with age reflect
an evolution from a decidedly nonstationary time-series in younger
subjects, with pronounced day-night secretory differences, toward a
more stationary output, with blunted overall variation. Thus, the very
real attenuation of circadian variation, while linguistically often
labeled as a “loss of rhythmicity,” is more precisely an
appropriately quantified change in a notion of overall amplitude or
variation, typically applied as a measure of the extent of time-series
non-stationarity over a relatively long time period (24 hr). This is
juxtaposed with the quantification of changes in
*irregularity* or *disorderliness* of serial data
seen above and elsewhere (21, 23), inasmuch as irregularity and
amplitude measure epistemologically distinct concepts. Nonetheless, the
generality of the finding of consistently blunted overall daily
circadian variation with increasing aging is most convincing, and the
interpretation that this is at least partially due to changes in
central nervous system control (29) is thematically consistent with our
hypothesis of an age-related increased asynchrony/network
dissociation in broad classes of hormonal networks.

#### Potential Applications.

More generally, quantification of signal regularity of both LH and T release, as well as of their mutual relationship and synchrony, could be employed to evaluate a variety of clinical disorders and the efficacy of medical interventions. Furthermore, if a disorder is most prominently characterized by diminution of synchrony, means to restore synchrony may require putatively novel therapeutic strategies. From an experimental perspective, studies will be required ultimately to specify the source(s) determining synchrony, e.g., from the possibilities indicated above, and to perturb this source directly. However, even prior to this identification, one could attempt to restore synchrony obliquely, by providing dual, synchronous administration of agents that respectively induce LH and T production. The point is that if a disorder is biologically determined by an overall system decoupling, a recoupling or reestablishment of temporal concordance may be required to restore physiological function, rather than any means of perturbing a single target node.

## Acknowledgments

This work was completed at the University of Virginia, the Salem Veterans Affairs Medical Center, and the McGuire Veterans Affairs Medical Center. We thank the staff at the University of Virginia General Clinical Research Center for their assistance with frequent venous sampling. This work was supported in part by the U.S. Department of Veterans Affairs, National Institutes of Health Grant RCDA-1-KO4-HD00634 from National Institute of Child Health and Human Development, National Science Foundation Science and Technology Center for Biological Timing, Baxter Health Care Corporation (Round Lake, IL), General Clinical Research Center Grant RR-008477, and Diabetes and Endocrine Research Center National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-38942.

## Mathematical and Statistical Considerations for ApEn and Cross-ApEn

#### Replicability of Cross-ApEn.

To establish a theoretical
statistical validity of cross-ApEn as employed here, we studied a range
of two-variable vector AR(2) processes, and several types of coupled
two-variable analogs of the “variable lag” process described
below, for each of which we applied cross-ApEn(1, 0.2) to standardized
time-series (*x*–*y* pair) outputs, 50 replicates of
*n* = 150 point data lengths per process. For each
process studied, SD (cross-ApEn) was ≤0.06, the SD calculated
from the cross-ApEn values from the 50 replicates; this imparts
reasonable replicability properties similar to that given elsewhere for
ApEn (16, 25). This degree of reproducibility is not unexpected,
because cross-ApEn is a parameter that aggregates *low*-order,
two variable joint distributions at a moderately coarse resolution
(determined by *r*).

#### Complementarity of ApEn and Cross-ApEn to Correlation and Spectral Analyses.

Mathematically, the need for ApEn, and particularly for cross-ApEn, is clarified by considering alternative parameters that might address similar concepts. In comparing two distinct signals or variables (e.g., to assess a degree of synchrony), primary parameters that one might employ include the cross-correlation function, and the cross-spectrum (34), with single variable counterparts the auto-correlation function and the power spectrum. Evaluation of these parameters often is insightful, but with relatively small length data sets statistical estimation issues are nontrivial and, moreover, interpretation of the sample cross-correlation function is highly problematic, unless one employs a model-based prefiltering procedure (ref. 34, p. 139). Furthermore, “standard” spectral estimation methods such as the fast Fourier transform can be shown to be inconsistent and/or so badly biased that findings may be qualitatively incorrect, especially in the presence of outliers and nonstationarities. This is vividly demonstrated by Thomson (35), who recently developed a superior multiple-data-window technique with major advantages compared with other spectral estimation techniques (35, 36). These difficulties are mirrored in the cross-spectrum, in addition to an often serious bias in estimation of coherency in short series.

Most importantly, the autocorrelation function and power spectrum, and
their bivariate counterparts, are most illuminating in linear systems,
e.g., SARIMA models, for which a rich theoretical development exists
(37). For many other classes of processes, these parameters often are
much less effective at highlighting certain model characteristics, even
apart from statistical considerations. To illustrate this point
consider the following simple model, which we denote as a “variable
lag” process: this consists of a series of quiescent periods of
variable length duration, interspersed with identical positive pulses
of a fixed amplitude and frequency. Formally, we recursively define an
integer time-valued process denoted VarLag whose
*i*^{th} epoch consists of (a quiescent period of)
values = 0 at times *t*_{i-1} + 1,
*t*_{i-1} + 2, … ,
*t*_{i-1} + lag_{i},
immediately followed by the successive values sin (π/6), sin
(2π/6), sin (3π/6), sin (4π/6), sin (5π/6),
sin(6π/6) = 0 at the next 6 time units, where
lag_{i} is a random variable uniformly distributed
on (randomly chosen between) the integers between 0 and 60, and
*t*_{i-1} denotes the last time-value of
the (*i*-1)^{st} sine pulse. Fig.
Fig.33*A* displays representative process
output, with Fig. Fig.33*B* a closer view near time
*t* = 400. The power spectrum and autocorrelation
function calculations shown in Fig. Fig.33 *C* and *E*
were calculated from a realization of length *n* =
100,000 points. (The coarse pulse sampling in the above definition was
chosen to approximate typical sampling resolution in clinical studies.)

### Figure 3

Processes consisting of alternatingly quiescent and active
periods seem reasonable for biologists to consider, as they appear to
model a wide variety of phenomena. However, within mathematics, such
processes with a variable quiescent period are not commonly studied. To
the endocrinologist, output from the above model would be considered
smoothly pulsatile, especially with the identical pulses; the variable
lag process would be most readily distinguished from its constant lag
counterpart (for which lag_{i} = 30 time units for
all *i*) via a decidedly positive SD for the interpulse
duration time-series, in the variable lag setting, as opposed to
SD = 0 (constant interpulse duration) in the constant lag setting.
The essential point here, however, is that for VarLag, the power
spectrum and autocorrelation function somewhat confound, as seen in
Fig. Fig.33 *C* and *E*. Based on these figures alone, the
pulsatile nature of the time-series realizations is hardly evident, and
for all *k* ≥ 6, the autocorrelation coefficient
*r*_{k} at lag *k* is
insignificantly different from 0. In contrast, the power spectrum and
autocorrelation function confirm the periodicity of the constant lag
analogue, shown in Fig. Fig.33 *D* and *F*, as expected.
Significantly, the issues here are in the parameters, rather than
statistical inadequacies based on an insufficiently long output, or on
artifacts (outliers), since Fig. Fig.33 *C*–*F* were
derived from calculations based on 100,000 points from a purely
theoretical model.

Similar limitations of the spectra and autocorrelation function are inherent to wide classes of mathematical processes. We can construct large classes of variable lag processes simply by considering point processes (38), in which we replace the “point” occurrence by a pulse occurrence, the pulse itself of either a fixed or variable form. The associated counting process could be of any character, and need not be so special as Poisson or renewal (as in the above example). Also, variable lags between events to be compared are the normative case in nonlinear differential equations, in Poisson clumping models (39), and in output variables in typical (adaptive) control theory models and queueing network models. Notably, for many two-dimensional analogs of variable lag processes, and for many two-dimensional systems in which no small set of dominant frequencies encapsulates most of the total power, the cross-spectrum and the cross-correlation function often will similarly fail to highlight episodicities in the underlying model and data, and thus fail to highlight concomitant changes to such episodic components.

In contrast to the autocorrelation function and spectral differences between the above variable lag and constant lag processes, respective ApEn(1, 20% SD) values for the two processes are in close agreement: mean ApEn = 0.195 for the variable lag process, while ApEn = 0.199 for the constant lag setting. This agreement in ApEn values manifests the primary requirement of matching (sub)patterns within data, while relaxing the requirement of a dominant set of frequencies at which these subpatterns occur. The two-variable analogue of ApEn, given by cross-ApEn, similarly enables one to assess synchrony in many classes of models. It thus should not be surprising that in this study cross-correlation (Pearson R) does not show significant group differences, whereas cross-ApEn does.

We emphasize, nonetheless, that Fig. Fig.33 *C*–*F*
neither invalidate spectral power and (lagged) autocorrelation
calculations, nor do they violate a properly oriented intuition. The
broad-banded spectrum in Fig. Fig.33*C*, and the negligible lagged
autocorrelation in Fig. Fig.33*E* for lag ≥ 6 time-units,
primarily reflect the independent, identically distributed, relatively
broad distribution of the variable lag_{i}.
Visually, this conforms to viewing Fig. Fig.33*A* from afar, in
effect (nearly) ignoring the nature of each pulse, instead de facto
primarily focusing on the “random” timing of the peaks as the
process of interest. The viewpoint taken by ApEn is thus complementary
to the spectrum and correlogram, more de facto focusing on (close-up)
similarities between active pulses, e.g., from the perspective given in
Fig. Fig.33*B*, while in effect nearly ignoring the nature of the
quiescent epoch aspect of the process. The putative utility of ApEn and
cross-ApEn to endocrinologists is based on the recognition that in many
settings, changes in the episodic character of the *active*
periods within pulsatile secretory time-series appear to mark
physiologic and pathophysiologic changes; thus, there is a concomitant
need for quantitative methods that primarily address this perspective,
e.g., ApEn and cross-ApEn.

Simple variable lag processes similar to VarLag would not confound pulse-identification statistics widely used within endocrinology (19). In many settings, such pulse-identification methods are quite sufficient to characterize and distinguish distinct physiologic states. However, in other settings, clear pulse identification appears to be a challenging endeavor, e.g., clinically, for GH time-series in acromegalics (21) and for healthy female rats (23), as well as for the T data studied here and, theoretically, for several distinct classes of mathematical processes (17, 20). Crucially, as well, there has been no quantification of two-variable synchrony from a pulse-identification perspective, except to determine whether or not co-pulsatility of discrete events within concomitant hormone series is nonrandom (40).

## Footnotes

The publication costs of this
article were defrayed in part by page
charge payment. This article
must therefore be hereby marked
“*advertisement*”
in accordance with 18 U.S.C.
§1734 solely to indicate this fact.

Abbreviations: LH, luteinizing hormone; T, testosterone; ApEn, approximate entropy; GnRH, gonadotropin-releasing hormone; GH, growth hormone.

^{**}As a consequence of the sleep protocol, the measured
time-series show a modest variation in length, about a
*N* = 168 point mean. In comparing ApEn of different
data set lengths, we neglect a small statistical bias in the estimator
ApEn(*m*,*r*,*N*), as a function
of *N*. As indicated in ref. 16, the expected value of
ApEn(*m*,*r*,*N*) increases
asymptotically with *N* to a limit
ApEn(*m*,*r*) for virtually all processes. If
we were to incorporate a bias or length correction, we would first
observe that the younger group data set lengths were, on average, 7%
larger than those of the aged subjects. Accordingly, the ApEn values of
younger subjects should be *reduced* by a correction term
that is a function of an unknown process, to accommodate an average 7%
reduction in data set length for an unbiased comparison of group ApEn
values between younger and aged subjects. Because ApEn values for the
younger subjects are seen below to be significantly smaller than those
for aged subjects, a reduction in ApEn values for the younger subject
values would enhance the degree of younger-aged group separation. Thus,
the reported findings are slightly conservative as to the extent of
younger vs. aged group distinctions shown via ApEn. Furthermore, we
established an upper bound of 0.06 for such a bias correction, for the
range of data set lengths encountered here, so that any correction
would have minimal qualitative effect on our results. A similar
observation applies to the cross-ApEn analysis.

^{‡‡}It is appropriate to apply the statistic
ApEn(T) − ApEn(LH), based on the serum concentration time-series, only
to establish that there are significant differences between two
subgroups, here younger vs. aged, based on the joint [LH–T]
dynamics. Given that LH and T were measured via different assays, and
have distinct clearance rates, we do not compare LH secretory
irregularity to T secretory irregularity for either young or old.

^{††}Zwart, A. D., Iranmanesh, A. & Veldhuis,
J. D., 77th Annual Endocrine Society Meeting, Washington, DC, June
14–17, 1995, abstract OR29-4.

## References

**National Academy of Sciences**

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