Vol. 83, Issue 6, 2146-2157, December 1997
MODELING IN PHYSIOLOGY
An improved statistical methodology to estimate and analyze
impedances and transfer functions
Douglas
Curran-Everett,
Yiming
Zhang,
M. Douglas
Jones Jr., and
Richard H.
Jones
Departments of Pediatrics and of Preventive Medicine and Biometrics,
School of Medicine, University of Colorado Health Sciences Center,
Denver, Colorado 80262
ABSTRACT
- INTRODUCTION
- APPLIED TRANSFER FUNCTION ANALYSIS
- ORIGINAL EXPERIMENTAL PROBLEM
- PRELIMINARY DATA ANALYSIS
- TIME SERIES REGRESSION
- RANDOM SUBJECT EFFECT
- TRANSFER FUNCTION ANALYSIS
- DISCUSSION
- APPENDIX
- ACKNOWLEDGEMENTS
- FOOTNOTES
- REFERENCES
ABSTRACT 
Curran-Everett, Douglas, Yiming Zhang, M. Douglas Jones,
Jr., and Richard H. Jones. An improved statistical methodology to
estimate and analyze impedances and transfer functions. J. Appl.
Physiol. 83: 2146-2157, 1997.
Estimating the mathematical relationship between pulsatile time series (e.g., pressure and flow) is
an effective technique for studying dynamic systems. The
frequency-domain relationship between time series, often calculated as
an impedance (pressure/flow), is known more generally as a frequency-response or transfer function (output/input). Current statistical methods for transfer function analysis 1) assume
erroneously that repeated observations on a subject are independent,
2) have limited statistical value and power, or 3) are
restricted to use in single subjects rather than in an entire sample.
This paper develops a regression model for transfer function analysis
that corrects each of these deficiencies. Spectral densities of the input and output time series and the cross-spectral density between them are first estimated from discrete Fourier transforms and then used
to obtain regression estimates of the transfer function. Statistical
comparisons of the transfer function estimates use a test statistic
that is distributed as 
2. Confidence intervals for
amplitude and phase can also be calculated. By correctly modeling
repeated observations on each subject, this improved statistical
approach to transfer function estimation and analysis permits the
simultaneous analysis of data from all subjects in a sample, improves
the power of the transfer function model, and has broad relevance to
the study of dynamic physiological systems.
discrete Fourier transform; frequency-domain regression; frequency-response function; mixed-effects model; spectral analysis
INTRODUCTION
ESTIMATION OF THE MATHEMATICAL RELATIONSHIP between
pulsatile time series is a time-honored approach to the study of
dynamic systems. Using this strategy, physiologists have explored
respiratory mechanics (11, 15, 23, 30, 39), pulmonary ventilation (29,
33), cardiovascular function (3, 7, 24, 27, 32), and cardiorespiratory
regulation (4, 25, 34, 35). The relationship between time series is
calculated most often in the frequency domain (5, 24): this entails
first transforming the time series, originally functions of time, into
their equivalent Fourier coefficients, written as functions of
frequency. When two time series represent the input and output of a
dynamic system, the mathematical relationship between them is often
designated the transfer function (h), defined in terms of
frequency ( f ) as h( f ) = output( f )/input( f ), where the
output and input functions are the Fourier transforms of the original
time series. When input is flow and output is pressure, the transfer
function is called impedance. A transfer function is a complex
expression, with real and imaginary parts, that can be described by
amplitudes (output amplitude divided by input amplitude) and phase
angles (timing of output with respect to input).
In experimental situations, transfer function analysis is complicated
by between-subject variability as well as by random measurement error.
Current approaches to transfer function analysis 1) assume
erroneously that repeated observations on a subject are independent,
2) have limited statistical value and power, or 3)
because they are unable to account for between-subject variability, are
restricted to use in single subjects rather than in an entire sample.
This paper develops a regression model for transfer function analysis
that corrects each of these deficiencies and has broad relevance to the
analysis of dynamic physiological systems. Before we derive and
illustrate this improved methodology, we review previous approaches to
applied transfer function analysis.
APPLIED TRANSFER FUNCTION ANALYSIS
Description.
Many studies (3, 7, 11, 15, 23, 26, 28, 34, 42) simply describe the
effect of an experimental intervention on a transfer function: they
report, for example, changes in the frequencies at which maxima in
impedance amplitude occur. But to draw conclusions about an underlying
population, one must use more than rudimentary description: one must
employ the inferential techniques of confidence intervals and
significance tests (38).
Usual indexes of between-subject variability.
In most physiological studies (see Ref. 24), between-subject
variability is estimated, first by deriving the amplitude and phase
angle of a transfer function for all subjects and then by calculating
SDs. This standard approach of handling between-subject variability
fails to account for the fact that repeated observations on a subject
(e.g., observations made during baseline and then during an
experimental intervention) are correlated (21). Because of this
correlation, the true error variability is underestimated, and the
reported values for the SDs of amplitude and phase underestimate the
true variabilities (21). The APPENDIX shows how correlation decreases variability.
Separate analyses of amplitude and phase.
Some researchers (4, 35, 40) analyze the amplitude and phase of a
transfer function as if they were unrelated variables. But the real and
imaginary parts of a transfer function, from which amplitude and phase
are derived, are determined simultaneously. Therefore, the components
of a transfer function, either its real and imaginary parts or its
amplitude and phase, must also be analyzed simultaneously. There is a
quantitative benefit: the simultaneous analysis of jointly derived data
improves statistical power (10).
Derivation and analysis of analog parameters.
Because transfer function estimation simply reduces a relationship
between pulsatile time series to its mathematical form, the
correspondence of a transfer function to physical characteristics of a
system is unclear. To circumvent this limitation, some investigators (15, 39, 42) first obtain an estimate of the transfer function and
then, from that estimate, derive analog parameters that represent general characteristics (e.g., total compliance) or specific components (e.g., lung tissue elastance) of the system. It is the estimates of the
analog parameters, rather than the estimate of the transfer function,
that are subjected to statistical analysis. Although this approach
gives physiological meaning to a transfer function, it does have a
statistical drawback: the estimates of the analog parameters are
correlated (38). This means that the analysis of only one (preselected)
parameter is valid: the statistical outcomes of the remaining analyses
will be related to the outcome of the initial analysis.
Time-domain analysis.
The time-domain technique of bivariate autoregression has been used
also to estimate a transfer function (22). Bivariate autoregressions
(20) can be fit to the original time series by using the Yule-Walker
equations, which define the relationships between the covariance
functions, the cross-covariance function, and the autoregression
matrices (18, 41). The resulting bivariate autoregressive estimates are
related to the autocorrelation and cross-correlation functions, often
used in physiology to estimate a transfer function (24, 25, 32). From
the estimated autoregressive matrices, estimates of the spectral
densities of an input and output and the cross-spectral density between
them can be calculated; it is from these autoregressive spectra that a
transfer function can be computed (see PRELIMINARY DATA
ANALYSIS). The advantage of autoregression is that it can produce
smooth spectral estimates of a transfer function. The disadvantage is
that the autoregressive estimates are correlated across frequency (20,
37): therefore, the statistical properties required to compare transfer
functions at specific frequencies are unknown.
Regression techniques.
Since the 1940s, regression analysis in the frequency domain has been
used in transfer function estimation (see Ref. 6). In this ap-proach,
the real and imaginary parts of the Fourier transforms of two time
series are used to obtain a regression estimate of the transfer
function (6, 16, 31, 37). The classic regression model (see Eq. 3) assumes, however, that estimates of the transfer function during
a given condition are virtually identical for all subjects. This is
unrealistic: physiological transfer function estimates vary
considerably among subjects (7, 8, 28, 32, 34, 42). Indeed, it was
explicitly because of between-subject differences that O'Rourke and
Taylor (Ref. 28, p. 368-369) presented only
individual impedance spectra: "Although statistical analysis of a
large number of experimental results is desirable, it is obvious that
features of the impedance curves can be lost if data are pooled from
[many subjects] ..."
Because it does not allow for between-subject variability, the classic
regression model can estimate a transfer function in single subjects
only (37); this confines statistical and biological inferences to
single individuals rather than to an underlying population. The
methodology we develop here extends this classic regression approach in
such a way that all subjects in a sample can be analyzed
simultaneously. This statistical enhancement has important theoretical
and applied advantages. In the next section, we review the experimental
problem that motivated the development of this methodology.
ORIGINAL EXPERIMENTAL PROBLEM
The development of this methodology was driven by our interest in the
effects of systemic circulatory perturbations on dynamic properties of
the cerebral circulation (see Ref. 8). In this context, we considered
systemic arterial blood pressure to be the input and cerebral cortical
blood flow to be the output of a linear vascular system. In 10 rabbits,
arterial blood pressure, measured by high-fidelity transducer catheter,
and cortical blood flow, measured by laser-Doppler flowmeter, were
sampled for 1 min during two conditions: baseline and combined
hypoxia-hypercapnia. Our experimental objective was to estimate the
transfer function between arterial blood pressure and cortical blood
flow during each condition by using data from the entire sample. To do
this, we selected a 20-s segment from each time series, giving 800 pairs of observations (sampling interval 0.025 s) for each subject
during each condition (Fig. 1). The
individual transfer functions derived from each of these bivariate time
series could be estimated reliably at the frequencies of heart rate and
its first harmonic.1 For the
frequency of heart rate, Fig. 2 depicts the between-subject variability
inherent to transfer function estimation.
The methodology that follows was devised to incorporate between-subject
variability into the regression model for the transfer function between
arterial blood pressure and cortical blood flow.
Fig. 1.
Systemic arterial blood pressure and cerebral cortical blood flow
during baseline (A) and treatment (combined
hypoxia-hypercapnia; B) for 1 subject (from Ref. 8). These
2-s time series begin at an arbitrary time t and illustrate the
20-s time series used in data analysis. Analysis of changes in the
relationship between pulsatile pressure and flow is complicated by 4 sources of between-subject variability: 1) distance between
measurement sites for pressure and flow, 2) mean baseline
arterial pressure, 3) anatomic and functional properties of
cerebral circulation, and 4) systemic pressor and cerebral
circulatory responses to physiological perturbations.
[View Larger Version of this Image (0K GIF file)]
Fig. 2.
Estimated transfer function at subject-specific heart rate during
baseline (A) and treatment (combined hypoxia-hypercapnia; B) for 10 subjects (adapted from Ref. 8). Individual and
group (solid black line) estimates are portrayed in the complex plane; dashed line at Real = 0 is for reference. Amplitude
Â, designated by length of the solid line,
and phase 
of the group transfer function are labeled for
baseline (A). Imag, imaginary.
[View Larger Version of this Image (0K GIF file)]
PRELIMINARY DATA ANALYSIS
Before we derived the regression model for the transfer function,
however, we verified an important statistical assumption: that the
error term of the model, i.e., the spectral density of the residual
series, was nearly constant around the frequencies of heart rate and
its first harmonic. We did this using bivariate autoregression (20),
identifying the order of the model by Akaike's information criterion
(1, 2). The estimated autoregressive spectra of pressure and flow
(
xx and
yy, respectively) showed striking
peaks at the frequencies of heart rate and its harmonics (Fig.
3). The estimated spectral density of the
residual series was then derived in a two-step procedure analogous to
that used in regression analysis. First, the transfer function estimate
was calculated from the bivariate autoregressive spectral estimates as
where yx is the estimated
cross-spectral density between the flow and pressure series. Then, the
estimated spectral density of the residual series
(
) was computed as
where xy is the complex
conjugate of yx. The absence of
strong peaks in the residual spectrum around the frequencies of heart
rate and its first harmonic (Fig. 4) verified the requisite assumption of the regression model and enabled
smoothing across bands centered at these frequencies (19). Assuming
that the quantity being estimated is nearly constant within the
frequency band, smoothing increases the precision of the estimate: the
wider the band, the greater the precision. This strategy is valid
because discrete Fourier transforms at different frequencies are
virtually independent.
Fig. 3.
Estimated spectral densities of arterial blood pressure (A),
cortical blood flow (B), and their squared coherence
(C) during 20-s baseline period for 1 subject (see Fig. 1).
These spectral estimates were calculated from estimated autoregressive
matrices and their associated error covariance matrix (see Ref. 20). Prominent peaks in the spectra are at the frequencies of heart rate
(3.7 Hz) and its harmonics; for these frequencies, actual values of
squared coherence, rounded to 2 decimal places, are shown.
[View Larger Version of this Image (0K GIF file)]
Fig. 4.
Estimated spectral density of the residuals from autoregression
analysis shown in Fig. 3. Note absence of pronounced peaks in the
spectrum at frequencies of heart rate and its harmonics: they were
removed by the autoregression.
[View Larger Version of this Image (0K GIF file)]
Next, we review time series regression in the frequency domain. As we
develop the full regression model for the transfer function, we
illustrate the process for the transfer function (evaluated at heart
rate) between arterial blood pressure and cortical blood flow using
data from Ref. 8.
TIME SERIES REGRESSION
Hannan (16) pioneered time series regression in the frequency
domain (see also Refs. 6, 31, and 37). If
xj and yj
represent the input and output time series, respectively, each
with n observations at time intervals of 
, then the discrete Fourier transforms (z
) of these series are
|
(1)
|
where the index 
is associated with frequency f
(Hz) by f = /(n) and i = 
Each transform z can be written with real and imaginary parts
as z = 
+ i
,
where
|
(2)
|
and 
have
similar equations.
The example.
Because we study dynamic vascular properties at discrete frequencies,
the first task is to identify the frequency of heart rate; we do this
by using the spectral density of arterial pressure. Then, for each of
15 0.05-Hz frequency bands2
centered at heart rate, we calculate the real and imaginary components (Eq. 2) of the discrete Fourier transforms of both the input
and output time series (Table 1).
|
Table 1.
Discrete Fourier transforms for 1 subject
|
| fn
|
Input Series
|
Output Series
|
|
|
|
|
|
| 1
|
0.1808 |
 0.2650 |
0.0074 |
0.0156 |
| 2 |
0.2238
|
0.2936 |
0.0419 |
0.0017 |
| 3 |
0.1865 |
0.3274
|
0.0860 |
0.0989 |
| 4 |
0.3387 |
0.4377 |
0.0615
|
0.0181 |
| 5 |
0.4153 |
0.3691 |
0.0072 |
0.0174
|
| 6 |
0.4695 |
0.8715 |
0.0094 |
0.0596 |
| 7 |
0.7049
|
3.3014 |
0.3662 |
0.0492 |
| 8 |
1.6351 |
5.5360
|
0.5847 |
0.1328 |
| 9 |
0.8957 |
1.6643
|
0.1680 |
0.0681 |
| 10 |
0.6005 |
0.9256
|
0.0986 |
0.1138 |
| 11 |
0.2534 |
0.4682
|
0.0722 |
0.0360 |
| 12 |
0.0885 |
0.4068
|
0.0884 |
0.0243 |
| 13 |
0.1043 |
0.2856
|
0.0310 |
0.0347 |
| 14 |
0.0778 |
0.2603 |
0.0380
|
0.0024 |
| 15 |
0.0283 |
0.2232 |
0.0053
|
0.0218 |
|
Values are in mmHg for the real and imaginary parts of input
series, and
 respectively, and
in ml · min 1 · 100 g1 for the real and imaginary parts of output series,
and
respectively; these values were derived from discrete Fourier
transforms (Eq. 1) of input (pressure) and output (flow) time
series for 1 subject (subject 3) during baseline (from Ref. 8).
fn, Frequency number. This 15-interval
frequency band is centered at the subject's heart rate of 4.25 Hz
( f8); limits of the band are 3.90 Hz
( f1) and 4.60 Hz
( f15).
|
|
Transfer function estimation.
Using the real and imaginary parts (Eq. 2) of the Fourier
transforms, the transfer function h can be approximated in the
frequency domain as
|
(3)
|
where
is the
unobservable discrete Fourier transform of an additive noise series. An
estimate of the transfer function can be obtained by treating Eq. 3 as a complex regression equation. First, the spectral densities
of the input and output series, xx
and yy, and the cross-spectral
density yx between them are estimated within the frequency band centered at a relevant Fourier frequency f (in the worked example, the frequency of heart
rate)
where * denotes the complex conjugate. These estimates are
smoothed periodogram estimates with uniform weights, a smoothing span
of 2w + 1, and 4w + 2 degrees of freedom
(19).3 Next, the real and
imaginary parts of the cross-spectral density, the cospectrum
yx and the quadrature spectrum
yx, are estimated as
and
Last, if the residual spectral density is nearly constant
within the frequency band, an assumption verified by the preliminary analysis (see Fig. 4), then the complex regression estimate
of the transfer function h can be
calculated as
|
(4)
|
The real and imaginary parts of Eq. 4
can be used to represent the estimate
with an amplitude  and a phase angle
The computational sign of can be reversed if
the sign of the exponent in the Fourier transform (Eq. 1) is
reversed; in some software, the form of the transform is unclear. In
transfer function applications, the timing of the output time series
with respect to the input time series may be predictable: the output will lag behind the input. In impedance applications, however, the sign
of the phase angle typically reverses as frequency increases. Regardless of specific application, the safest strategy is to confirm
the correct computational sign of the estimated phase angle
by using known input and output time series.
The example.
From the real and imaginary components of the discrete Fourier
transforms within the 15-interval frequency band (see Table 1), we next
obtained a single smoothed estimate of the transfer function (Table
2; see Transfer function estimation
above). The phase estimate is meaningful only if there is high
coherence between the input and output: that is, if changes in the
output correspond closely to changes in the input. Based on the
estimated autoregressive spectrum of squared coherence (see Fig. 3), we
expect to obtain a reliable estimate of the transfer function at the
frequency of heart rate. Although smoothing increases the precision of
the transfer function estimate, it can also decrease squared coherence. Therefore, after we get the single transfer function estimate, we
verify that squared coherence across the 15-interval frequency band
remains high
where yx,
yx,
xx, and
yy are pooled over all experimental
conditions.
|
Table 2.
Transfer function estimates for 10 subjects
|
| k |
Baseline
|
Treatment
|
 h
|
 h |
Â
|
|
h
|
h |
Â
|
|
|
| 1 |
0.0256 |
0.1076 |
0.1106
|
1.804 |
0.0281 |
0.1067 |
0.1104 |
1.828 |
| 2
|
0.0554 |
0.1357 |
0.1466 |
1.183 |
0.0832
|
0.1706 |
0.1898 |
1.117 |
| 3 |
0.0053 |
0.1041
|
0.1042 |
1.520 |
0.0040 |
0.1200 |
0.1201
|
1.537 |
| 4 |
0.0060 |
0.1457 |
0.1458 |
1.530
|
0.0016 |
0.1394 |
0.1394 |
1.560 |
| 5 |
0.0153
|
0.1400 |
0.1408 |
1.462 |
0.0280 |
0.1646
|
0.1670 |
1.403 |
| 6 |
0.0022 |
0.1218 |
0.1218
|
1.589 |
0.0086 |
0.1213 |
0.1216 |
1.500 |
| 7
|
0.0034 |
0.1750 |
0.1750 |
1.551 |
0.0151
|
0.1902 |
0.1908 |
1.650 |
| 8 |
0.0476 |
0.1015
|
0.1121 |
1.132 |
0.0528 |
0.1313 |
0.1415
|
1.188 |
| 9 |
0.0139 |
0.1565 |
0.1572 |
1.660
|
0.0154 |
0.1569 |
0.1576 |
1.669 |
| 10
|
0.0162 |
0.2094 |
0.2100 |
1.648 |
0.0005
|
0.2111 |
0.2111 |
1.569 |
|
|
Values are in
ml · min1 · 100 g1 · mmHg1 for the real and
imaginary parts of transfer function estimates
h and
h, respectively; in
ml · min1 · 100 g1 · mmHg1 for amplitude
Â; and in radians for phase . For each
subject, transfer function was estimated over a band of 15 frequencies; this band was centered at subject-specific heart rate. k,
Subject no. Treatment is combined hypoxia-hypercapnia.
|
|
RANDOM SUBJECT EFFECT
In the preceding section, we reviewed the classic regression model
(Eq. 3) that assumes, for some conditions, that estimates of a
transfer function are virtually identical for all subjects. In this
section, we first add a random parameter 
to the transfer function
model, thereby transforming the classic fixed-effects model (Eq. 3) into a mixed-effects model that can handle inherent between-subject variability in the transfer function estimates. Next,
we use this enhanced statistical model to estimate the transfer function during b experimental conditions for an entire sample of N subjects. Finally, we use a likelihood ratio test to
evaluate the improvement gained by adding the random subject effect to the transfer function model.
Development of the mixed-effects model.
When a transfer function is estimated in a group of subjects, the
estimates will differ because of random between-subject variation. This
variation can be included in the transfer function model, thereby
improving its power, by using a random subject effect (see Ref. 21).
With this revision, the classic regression model (Eq. 3)
becomes, for subject k
|
(5)
|
where k is the random shift in the
transfer function about the group mean h. It is
by virtue of the random parameter k that
repeated observations on subject k are correlated. It is also
by virtue of k that the transfer function can be
estimated for an entire group of subjects. This revised statistical
model (Eq. 5), a mixed-effects model because it includes fixed
(h) and random (k) parameters, has
three assumptions: 1) each random effect
k has a complex Gaussian
distribution4 with mean 0 and
variance 
2) each 
has a complex Gaussian distribution with mean 0 and variance
2, where 2 is proportional to the error
spectral density within the frequency band; 3) the random error
effects k and
are independent. These are established assumptions for mixed-effects models in the frequency domain because, regardless of the properties of
the original time series, Fourier transforms are nearly Gaussian.
When the transfer function is estimated over a band of m
frequencies, the Fourier transforms
and
in the mixed-effects model (Eq. 5) can be arranged in column
vectors of length m. Thus, when b transfer functions,
corresponding to b experimental conditions, are estimated for
subject k, the transfer function model in matrix form is
|
(6)
|
where
z(y)k
is the mb × 1 vector of the Fourier transforms of the output
time series, z(x)k
is the block diagonal mb × b design matrix
h is the b × 1 vector that contains the
transfer functions, and
z(z)k
is the mb × 1 design matrix for the random effect
k,
Furthermore, if c2 denotes the ratio
of the error variances,
then the
complex covariance matrix 
(Hermitian) for subject k is
here, * denotes the complex conjugate transposed vector.
The covariance k is the covariance among the
transfer function estimates for subject k. The matrix
Vk is made up of an intrasubject
component I and a between-subject (error) component
c2z(z)kz(z)k*. [See Ref. 21 (section 2.2) for more detailed discussion of
k and Vk.]
For a sample of N subjects, the estimate of
the transfer functions is
|
(7)
|
Because
in the single transfer function case [i.e., when the
vector is a scalar:
z(z)k = z(x)k],
the estimate reduces to
|
(8)
|
This estimate minimizes the weighted residual sum of
squares (RSS)
These results are complex generalizations of those
presented by Jones (Ref. 21, section 1.5).
Analysis of the random effect.
Statistical evaluation of the random subject effect is based on a
likelihood ratio test, an essential component of which is the maximum
likelihood estimate of 2. [A maximum likelihood
estimate is the value of a population parameter that most likely would
have produced the sample observations (see APPENDIX).] The
maximum likelihood estimate of 2 is obtained from a
multivariate complex Gaussian distribution (12)
For N subjects, l, the 2 ln likelihood,
is
![+ <FR><NU>1</NU><DE>&sfgr;<SUP>2</SUP></DE></FR> [<B>z</B><SUP>(<IT>y</IT>)</SUP><SUB><IT>k</IT></SUB> − <B>z</B><SUP>(<IT>x</IT>)</SUP><SUB><IT>k</IT></SUB><B>h</B>]*<B>V</B><SUP>−1</SUP><SUB><IT>k</IT></SUB>[<B>z</B><SUP>(<IT>y</IT>)</SUP><SUB><IT>k</IT></SUB> − <B>z</B><SUP>(<IT>x</IT>)</SUP><SUB><IT>k</IT></SUB><B>h</B>]}](/content/vol83/issue6/fulltext/2146/img047.gif)
|
(9)
|
For a given value of c2, the transfer
function vector h can be eliminated from the likelihood
calculation by substituting Eq. 8 (to simplify, we set
h = ) into Eq. 9
Because
the likelihood
becomes
|
(10)
|
Finally, differentiating Eq. 10 with respect to
2 and setting the derivative equal to zero gives the
maximum likelihood estimate of 2
|
(11)
|
By substituting Eq. 11 into Eq. 10, the likelihood can
now be written as a function of the single variable
c2, the ratio of the error variances
The determinant
|Vk|, obtained
by Cholesky factorization (13) of Vk, is
Therefore, the likelihood is
![+ 2 <LIM><OP>∑</OP><LL><IT>k</IT>=1</LL><UL><IT>N</IT></UL></LIM> ln [1 + <IT>c</IT><SUP>2</SUP><B>z</B><SUP>(<IT>z</IT>)</SUP><SUB><IT>k</IT></SUB>*<B>z</B><SUP>(<IT>z</IT>)</SUP><SUB><IT>k</IT></SUB>] + 2<IT>mN</IT>](/content/vol83/issue6/fulltext/2146/img057.gif)
|
(12)
|
the value of c that minimizes Eq. 12 gives
2 (Eq. 11) and the maximum likelihood
estimates of the transfer functions (Eq. 7).
The example.
Before we compare the b estimates of the transfer function, we
test whether the random effect for between-subject variability, 
in Eq. 6, improves the transfer function model. To do this, we
first fit the fixed-effect (random effect absent) and mixed-effects (random effect present) models to all Fourier transforms from all
subjects and obtain, for each model, values for l and Akaike's information criterion (Table 3): the
smaller Akaike's information criterion, the better the model. Then,
using a likelihood ratio test, we evaluate the change in
2 ln likelihood, 
l, that occurs from adding the random
subject effect. (APPENDIX discusses the test statistic used
to evaluate l.) The large value of l
(P < 0.0001) signifies that the random subject effect
does improve the transfer function model. The biological interpretation
is that, for a given condition, transfer function estimates vary
considerably among subjects.
|
Table 3.
Transfer function estimates and contrasts
|
|
Transfer Function Model
|
| Without random effect |
With random effect |
|
| 2 ln likelihood, l |
972 |
1,233
|
| Akaike's information criterion |
962 |
1,221
|
| Transfer function :
h + ih
|
| Baseline, 0
|
0.0018 i0.1443
|
0.0047 i0.1446 |
| Treatment,
1 |
0.0060 i0.1534
|
0.0084 i0.1533 |
| Test statistic (Eq. 14)* |
5.44 (P = 0.07)
|
7.88 (P = 0.02) |
|
|
Transfer function model is improved by including random effect for between-subject variability: l = 972 (1,233) = 261, where l is distributed as 2 with 1 degree of freedom, P < 0.0001.
*
Test statistic for
contrast between treatment (hypoxia-hypercapnia) and baseline transfer
functions; under the null hypothesis that the contrast is 0, i.e., that
the 2 transfer functions are identical, this test statistic is
distributed as 2 with 2 degrees of freedom.
|
|
TRANSFER FUNCTION ANALYSIS
In this section, we first review statistical contrasts and contrast
coefficients; contrast coefficients are part of the test statistic used
to compare transfer function estimates. Next, we detail the test
statistic itself. Last, we calculate 100(1 )% confidence
intervals for the amplitude and phase of the transfer function.
Contrasts.
Comparisons of the transfer function estimates require contrast
coefficients, which are used to compute statistical contrasts: a
contrast represents the size of a comparison among treatment means (see Ref. 38). The sample contrast 
estimates the population contrast D and is any linear
combination
|
(13)
|
of the sample means,
for which the fixed coefficients, Ci,
satisfy
The Ci are known as contrast
coefficients. (Statistical contrasts and their coefficients are
illustrated in APPENDIX.)
The example.
The transfer function estimates, obtained by using the mixed-effects
model, are
![<IT><A><AC>s</AC><AC>ˆ</AC></A></IT><SUB><IT>xx</IT></SUB>(<IT>f</IT>) = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> [<IT>z</IT><SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>]*[<IT>z</IT><SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>] = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> {[&agr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>]<SUP>2</SUP> + [&bgr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>]<SUP>2</SUP>}](/content/vol83/issue6/fulltext/2146/img017.gif)
![<IT><A><AC>s</AC><AC>ˆ</AC></A></IT><SUB><IT>yy</IT></SUB>(<IT>f</IT> ) = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> [<IT>z</IT><SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>]*[<IT>z</IT><SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>] = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> {[&agr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>]<SUP>2</SUP> + [&bgr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>]<SUP>2</SUP>}](/content/vol83/issue6/fulltext/2146/img018.gif)
![<IT><A><AC>s</AC><AC>ˆ</AC></A></IT><SUB><IT>yx</IT></SUB>(<IT>f</IT> ) = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> [<IT>z</IT><SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>]*[<IT>z</IT><SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>] = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> {[&agr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB> − <IT>i</IT>&bgr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>][&agr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB> + <IT>i</IT>&bgr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>]}](/content/vol83/issue6/fulltext/2146/img019.gif)
![<IT><A><AC>c</AC><AC>ˆ</AC></A></IT><SUB><IT>yx</IT></SUB>(<IT>f</IT> ) = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> [&agr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>&agr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB> + &bgr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>&bgr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>]](/content/vol83/issue6/fulltext/2146/img022.gif)
![<IT><A><AC>q</AC><AC>ˆ</AC></A></IT><SUB><IT>yx</IT></SUB>(<IT>f</IT> ) = <LIM><OP>∑</OP><LL>&ngr;=<IT>k</IT>−<IT>w</IT></LL><UL><IT>k</IT>+<IT>w</IT></UL></LIM> [&agr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>&bgr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB> − &bgr;<SUP>(<IT>x</IT>)</SUP><SUB>&ngr;</SUB>&agr;<SUP>(<IT>y</IT>)</SUP><SUB>&ngr;</SUB>]](/content/vol83/issue6/fulltext/2146/img023.gif)
![RSS(<IT>c</IT><SUP>2</SUP>) = <LIM><OP>∑</OP><LL><IT>k</IT>=1</LL><UL><IT>N</IT></UL></LIM> [<B>z</B><SUP>(<IT>y</IT>)</SUP><SUB><IT>k</IT></SUB> − <B>z</B><SUP>(<IT>x</IT>)</SUP><SUB><IT>k</IT></SUB><IT><A><AC>h</AC><AC>ˆ</AC></A></IT>]*<B>V</B><SUP>−1</SUP><SUB><IT>k</IT></SUB>[<B>z</B><SUP>(<IT>y</IT>)</SUP><SUB><IT>k</IT></SUB> − <B>z</B><SUP>(<IT>x</IT>)</SUP><SUB><IT>k</IT></SUB><IT><A><AC>h</AC><AC>ˆ</AC></A></IT>]](/content/vol83/issue6/fulltext/2146/img043.gif)
![<IT>f</IT>[<B>z</B><SUP>(<IT>y</IT>)</SUP>] = <FR><NU>1</NU><DE>&pgr;<SUP><IT>m</IT></SUP>‖&sfgr;<SUP>2</SUP><B>V</B>‖</DE></FR>](/content/vol83/issue6/fulltext/2146/img044.gif)
![⋅ exp <FENCE>− <FR><NU>1</NU><DE>&sfgr;<SUP>2</SUP></DE></FR> [<B>z</B><SUP>(<IT>y</IT>)</SUP> − <B>z</B><SUP>(<IT>x</IT>)</SUP><B>h</B>]*<B>V</B><SUP>−1</SUP>[ <B>z</B><SUP>(<IT>y</IT>)</SUP> − <B>z</B><SUP>(<IT>x</IT>)</SUP><B>h</B>]</FENCE>](/content/vol83/issue6/fulltext/2146/img045.gif)
