Time series and compartmental analysis techniques for biological application

Smith, W. B.

English

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FINAL TECHNICAL REPORT
June 1, 1969 to May 31, 1971
COMPARTMENTAL ANALYSIS AND MISSING DATA
TECHK14MS IN STATIONARY TIME SERIES
National Aeronautics and Space Administration
Manned Spacecraft Center
Grant &GR 44-001-097
W. B. Smith
Associate Professor
Institute of Statistics
Texas A&M University
College Station, Texas 77843
FINAL TECHNICAL REPORT
COMPARTMENTAL ANALYSIS AND MISSING DATA
TECHNIQUES TN STATIONARY TIME SERIES
1. Introduction.
Hocking and Smith [196$3 have presented a technique for estimating
the parameters of a multivariate normal distribution from data vectors,
some of which have missing elements. A purpose of this grant was to
extend this technique to estimation problems of biological interest.
The procedure uses all available data, both full and partial vectors,
and can be outlined as follows: (1) group the data according to which
4 
variates are missing, (2) estimate within each group the available
parameters, (3) combine in a sequential fashion (optimally) the estimates
of part (2).
The procedure has been shown to possess some optimality problems
and, in fact, the combination of parameter estimates was done in such a
way as to minimize its variance and to retain unbiased properties if
possible.
Section 2.1 outlines in some detail the procedure and its extensions,
both theoretical and to biological problems, carried out under this grant.
It should be noted that in addition to the biological problems of interest
other applications are immediately evident. For an example missing and/or
partial data records can occur due to random machine or telemetry failure
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developed under this grant can be used to salvage those partial data
records. That is, rather than discarding partial data records the
information available within them is utilized without resorting to
estimating the missing data elements.
Another purpose of this grant was to develop and extend techniques
for biological applications of compartmental analysis. Many biological
and physical systems can be modelled mathematically by a series of
compartments. As examples, the digestive system of mammals or the flow
of interaction of documents in a research lab can be thought to be
sequences of compartments. The mathematical analysis of such models have
been limited to a deterministic situation, that is, one in which the
particles within the compartments flow at specific rates. Research on
this grant considers the sequence of compartments and the flow between
compartments in a stochastic (random) fashion. That is, within each
compartment a particle will possess a probability distribution which relates
the chances of its exiting the compartment at a particular time. Thus,
the flow rates (time dependent) between compartments are of interest.
As an example one could think of a mathematical model of an epidemic.
That is, the organism would be within a single individual, each individual
having a susceptible period and a period after which he contracts the
disease. The time interval in which the organism resides within the
individual can itself be broken down into a latent period (a period in
which no symptoms are observable), an infective period (a period in which
it may or may not exhibit symptoms but in which the individual can pass,
the organism to another individual), and a recovery period. Each of these
?"I-
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periods obviously vary in some fashion from individual to individual
depending upon the health of the individual, the severity of the case
involved, and other extraneous conditions. Thus it would be obvious
that a deterministic model would not fit such a situation whereas
a stochastic model could. We describe in Section 2.3 solutions found
to specific types of stochastic compartmental models both for the estimation
and the model itself.
In Section 2, brief descriptions of each technical advance achieved
under this grant will be given. The research carried out under the
auspices of this grant has received good reviews by colleagues, as is
evidenced by their publication in the respected journals. The lag in
publication time prevents us from reporting the final dispensation of
two of the technical reports. In total there were six technical reports
issued under the grant, four of which have been published and the other
two submitted for publication.
2. Review of Results.
2.1. Missing Data.
Technical Report #1 entitled "Selection Index Estimation from
Partial Multivariate Normal Data", represents an extension of the
Hocking-Smith technique to the estimation of the well known selection
index. The index is based upon a multivariate time dependent data
vector and constitutes a legitimate extension of the partial data
technique developed by Hocking and Staith. The procedure is spelled
out in considerable detail in the technical report but can be summarized:
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(1) The procedure is applied to the estimation of the selection index
(linear) when incomplete multivariate normal data vectors are available.
(2)The procedure utilizes all data available (both full and partial data
vectors) and represents an improvement in the precision of the estimator
over that found by using the full data vectors only. (3) Estimates of the
phenotypic means and covariance matrix are also found as a byproduct.
(4) Individuals with partial data vectors are indexed by an extension
of the technique and this index is compared to that proposed by Henderson
[1962]. (5) The results of Monte Carlo computer simulations axe also
tabulated.
That is, defining I  = Y XV where Ii is a composite index value
associated with the jth member of the population, b is an n x 1 vector
of unknown coefficients, and Xi is n x vector of phenotypic observations
on the 3th member of the population, Smith [1936] shows that b = P 1Ga.
In this context P is the n x n variance-covariance matrix of phenotypic
values, G is the n x n variance-covariance matrox of genotypic values,
and a is an n x 1 vector of weights associated with the n traits. The
problem of estimating b is attacked by estimating P, the matrix of
phenotypic values. This extension to both full and incomplete data
vectors represents the first use of an index on partial data.
Further extensions are given in Technical Report #5 in which
the parent distribution of the phenotypic observations vector is
multinomial. Again both full and partial data vectors are used, all
information being gleaned from them. The phenotypic mean vector p
and the phenotypic variance -covariance matrix P are also estimated by
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the process. Individuals with partial data vectors are indexed. For
further details of both the multivariate normal selection index and
the multinomial selection index see Technical Reports #1 and #j,
respectively.
The selection index has been quite useful in selecting on animals
in a population. Further applications are to the selection of pilots
for test purposes by Aerospace Research Labs at Wright-Patterson AFB,
Dayton, Ohio. In addition genetic applications are obvious.
2.2. Time Series.
A time series application is contained in Technical Report #4.
Consider the classical minimum variance unbiased estimator of a
population mean in a discrete parameter, covariance stationary, stochastic
process when the data is sampled by rotation sampling, as derived by
Yates [19493 and Patterson 119503. This estimator is invariant under
alterations in the rotation scheme. In Technical Report #4 we derive
by a constrained optimization procedure similar simultaneous equations.
Patterson has considered the technique but rejected it because of the
'lengthy estimator expressions, however by introducing matrix expressions
and assuaging a special rotation scheme this tectuiIque is now to be
tractable. That	 we give an alternate way of deriving the estimator.
The merits of the procedure are as follows: !1) Exact expressions
from  population mesas estimator can be found on each occasion. (2) The
f
	
matrix notation leads to easier computer programming. (3) The derivation
of the exact expression of the variance-covariance matrix leads to ease
in investigating the large sample properties of the estimator and their
relation to mwdmum likelihood.
T..
following assumption: (1) The correlation coefficient between observations
in the same unit i occasions apart is p i
 and is known, 1 < i < h. (2) The
variances are the same on each occasion and are known (they are noted by
0
a
)- (3) Sample units on each occasion are equal to n. (4) no units,
0 < 0 < l,are replaced by newly chosen units on each occasion and
(5) sampling on each unit is done mutually independently from an infinite
population, so that the correlation coefficient between two observations
on different units is zero. The resulting estimator is given by
Yh = %Yy + (1 - ^h )
 ^.yh + p(Yh-1 Xh-1 ) 1
	
{i)
where 
;-h-i is the mean of observations on occasion h-1 associated with the
n(1
	 units common to occasion h, yh is the sample mean on occasion h
associated with the same common units, yh is the mean on occasion h
associated with the newly chosen uncommon units, and Yh-i is the estimator
of µ(h-1) (the mean at occasion h-1) based on the observation up to
occasion h-l. Now % satisfies
p2(1 - 
O)CPh-1 + (1 - p2)¢	 (2)
p2(1 - 
0)%-1 + (1 - p2)0 + (1 - 0)
and
var (Yh) - %(Pfnfi given that V1 = 1 .
	
{3)
-7-
Patterson further derived the minimum variance unbiased linear
estimate of µ(h-k) when all observations up to the h occasion are
available. We derived a minimum variance unbiased linear estimate of
each of the above population means by constrained minimization of the
variance of the lire ar estimate with respect to its coefficient. That
is, we derive these estimates by minimizing the variance while guaranteeing
unbiasedness. The result of this procedure uses Aitken's generalized
least square estimator. A simplified specification of the estimator is
given and the variance-covariance matrix is spelled out. For further
details into the technical properties of the procedure, as well as, a
specific formulation of the estimator itself, see Technical Report #4.
A revised version of Technical Report #4 has appeared in the
Report an Statistical Applications published by the Japanese Union of
Scientists and Engineers, March, 1971.
2.3. Compartmental Analysis.
The introduction of compartmental analysis has stirred considerable
interest among bio-mathematicians in modelling biological clearance, that
is, the passage of a given material through a biological system.. Virtually
all current investigations are designed such that the system requires a
tracer material to enter either by a large initial pulsing or by small
continuous dosing. The initial pulse method is typically easier to
administer physically, While the continuous dosing has an advantage of
containing some checks on the underlying compartmental assumption (Hearon,
C19G8]). Both have been used to estimate turnover rates in the deterministic
system.
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Technical Report #2 "Stochas,:ic Compartmental Analysis: Model and
Least Squares Estimation from Time Series Data", by J. H. Matis and H. 0.
Hartley introduced a stochastic behavior to compartmental systcras which
are pulse labelled. A homogeneous partial differential equation which
characterized the stochastic behavior is solved for distribution theory,
and the distribution theory in turn provides a basis for the recommended
estimation procedure.
Other modelling of biological cleare»ee has been based primarily on
the idea of compartmental analysis assuming linear kinetics; the data
has been fitted to sums of negative: exponentially distributed random
variables with subsequent interpretations relative to the number of
compartments turnover rates, etc. Recently, Wise, et.ai. [1968] have
derived an alternate model based on powers of time, and they have
exhibited a great manyfits of the data to their model. Our research
contributes a generalization to the standard compartmental model by
incorporating age dependency into the system. The generalization not
only includes the standard compartmental model and -the power of times
model, in special cases, but also strengthens the power of time model
by interpreting it within the campartmental framework. Following the lines
of Technical Report #2, Technical Reports #3 and #6 present research
Introducing a gamma sojourn time distribution in the place of the previous
negative exponential distribution. The resulting data dependency of
turnover rates then induce the powers of time. A biological example
included in this research demonstrates the analytical recognition of
W dependency phenos na and the estimation procedure of the age dependency
model.
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Fo. further details on -vhe specific mathematical formulations of
the procedur,:s outlined above see Technical Reports #2, #3 and #6
issued under this grant and the papers published as results of modifications
of the above technical reports.
3. summer.
In summary this grant has supported important research in biological
statistics. Each of the three main areas outlined in Section 2 yielded
some interesting and important results. Immediate applications of each
of the resulting techniques to biological and physical situations have
resulted. Widespread interest in the results is evidenced by the
significant number of reprint requests received by the authors.
Selection index estimation utilizing the distributional properties
of the input variates is of interest to geneticists, in particular,
and other scientists involved in many variet{es of selection problems.
It was with some mee ed interest that the principal investigator
discovered the'' research on the selection index and its application to
pilot selection was being carried on at the Aerospace Research Laboratorj,
Wright-Patterson Air Force Base, Dayton, Ohio by Dr. David I.. Harville.
The compartmental analrais applications to many problems are currently
being carried out. One such problem only being investigated at its very
beginning stages is an application of stochastic compartmental analysis
through the estimation of the flow rate of certain biological micro-organiams,
which would yield information concerning tim degree of contagious infection
resulting from each. During deep space probes the extended confinement
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by a small group of space travellers would brim each in contact with the
other disease potentials. Thus if such an estimation procedure were
available before such a deep probe one would be able to simulate on a
computer the possibilities of certain contagious diseases occurring within
the crew. Further research on this problem will hopefully be carried out
under alternate funding.
4. Technical Reports.
Report #1, "Selection Index Estimation from Partial Multivariate Normal
Data", by W. B. Smith and R. C. Pfaffenberger (appeared in
Biometrics, 26, 625-640).
Report #2, "Stochastic Compartmental Analysis: Model and Least Squares
Estimation from Time Series Data", by J. H. Matis and H. 0.
Hartley (appeared in Biometrics, 27, 77-102).
Report #3, "Stochastic Compartmental Analysis: Some Applications and
Examples in Pulse Labelled Systems% by J. H. Matis and H. 0.
Hartley and W. B. Ellis (appeared in part in Biometrics
with Report 12).
Report A. "Estimation of the Mean in a Discrete Parameter, Covariance
Stationary, Stochastic Process in Rotation Sampling", by
Kin-Ya Nishikawa and W. B. Smith (appeared in Report on
Statistics Applications Research, Japanese Union of Scientists
and Engineers, 1$, No. 1, 9-21).
Report #5, "Multinamial Selection Index", by W. B. Smith and D. M. Scott.,
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Report #6, "An Example of Age Dependency in Compartmental Analysis",
by J. H. Matis (submitted for publication to Biometrics).
5. References.
Hearon, J. Z. [1968], "A Washout Curve in Tracer Kinetics", Mathematical
Biosciences, 3, 31-40.
Henderson, C. R. [1963], "Selection Index and Expected Genetic Advance",
NAS-NRC Publication 982, 141-163.
Hocking, R. R. and W. B. Smith [19683, "Parameter Estimation in the
Multivariate Normal Distribution with Missing Observations",
Journal of the American Statistical Association, 63, 159-173•
Matis, J. H. and H. 0. Hartley, "Stochastic Compartmental Analysis: Model
and Least Squares Estimation from Time Series Data% Biometrics, 27,
77-102.
Nishikwaa, K., and W. B. Smith, "Estimation of the Mean of a Discrete
Parameter, Covariance Stationary, Stochastic Process in Rotation
Sampling", Report of Statistics Applications Research, Japanese
Union of Scientists and Engineers, UP, No. 1, 9-21.
Patterson, H. D. [19503, "Sling on Successive Occasions with Partial
Replacement of Units", Journal of Royal Statistical Society, Series
B, 12, 241
-255.
Smith, H. F. [19363, "A Discriminant Function for Plant Selection",
Ann, Eugen. London 7, 240-250.
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Smith, W. B. and R. C. Pfaffenberger [1970], "Selection Index Estimation
from Partial Multivariate Normal Data", Biometrics, 26, 625-640.
Wise, M. R., S. B. Osborn, J. Anderson and R. W. S. Tomlinson [1968],
"A Stochastic Model for Turnover of Radio Calcium Based on the
Observed Power Laws", Mathematical Biosciences, 2, 199-224.
Yates, r. [1949], Sampling Methods for Censuses &iid Surveys, London,
Charles Griffin.
Ift


Compartmental analysis and missing data techniques in stationary time series

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