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University of Houston • University of Houston-Clear Lake • ISSO Annual Report Y2002—pp. 70-75
Bayesian Analysis and Statistical Modeling of Venous Gas Emboli (VGE) Grade IV Onset in Hypobaric Environment
Raj S. Chhikara (UHCL) and Laura Thompson (UHCL)
Abstract
The presence of gas bubbles in venous blood (venous gas emboli, or VGE) is associated with an increased risk of decompression sickness (DCS) in hypobaric environments. A high grade level of VGE can be a precursor to serious DCS. We model time to Grade IV VGE when a subset of individuals in the population are assumed to be immune from experiencing the event. The data set we use contains both interval-censored and right-censored observations as well as repeated observations on some of the individuals. We present a Bayesian cure rate model applicable to intervalcensoring and repeated measurements. A Bayesian approach to the modeling of a limited failure population gives more flexibility in assessments of goodness-of-fit of a model and provides direct assessment of uncertainty of any estimate of survival probability or combination of survival probabilities.
HUMANS EXPOSED TO HYPOBARIC ENVIRONMENTS TYPICALLY experience the formation of gas bubbles in venous blood as a result of the decompression. The reduction in pressure can cause nitrogen gas normally dissolved in body fluids and tissue to escape too rapidly from solution, causing bubbles to form in the tissue and blood. The movement of gas bubbles into venous blood, called venous gas emboli (VGE), can contribute to serious hypobaric decompression sickness (DCS). Grade IV VGE is considered serious and a precursor to serious DCS.
Conkin et al.1 state that under certain circumstances some individuals will never experience Grade IVVGE, no matter how long they remain at high altitude. Thus, it may be appropriate to model the time to onset of Grade IVVGE by limiting the occurrence to a subset of the population. Limited failure population (LFP) models in survival analysis have frequently been used where it is known or assumed that some fraction of the population (the "cured" fraction) will never experience the event under study. Maller and Zhou2 provide a complete history of these models.
In situations where the failure event is attributed to only one known cause or where competing causes of failure are not observable, the resulting mixture population of cured and noncured individuals can involve a degenerate cured component. Thus, the survival function for the entire population, originally proposed by Berkson and Gage,3 is
| Spop(t) = pS(t) + 1 - p, | (1) |
where S(t) is the survival function for individuals who will experience the event, and p represents the probability of eventually experiencing the event, given enough time. Since S(¥) = 0 for a proper survival function, Spop(¥) = 1 - p, and is called the cure rate of the population.
In Thompson et al. (2002),4 we fit a series of LFPmodels using the model to a subset of interval- and right-censored data from the hypobaric decompression sickness databank of NASA.5 The subset we used contained measurements on the time-to-onset of Grade IV VGE for volunteer subjects who participated in denitrogenation pre-breathe procedures prior to being exposed to low pressure. The analysis of these data indicated that an LFP model predicted the time of onset of Grade IV VGE better than did a model without a cured component. However, assessment of goodness-of-fit of the model to the data was limited to graphical analysis along with an accompanying numerical description. The graphical analysis required additional bootstrapping after the estimation of the model parameters and, thus, was relatively time consuming. In addition, although we estimated the probability of onset of Grade IV VGE after certain exposure times, a measurement of uncertainty of these estimates was only available under large-sample assumptions.
A Bayesian Approach
Classical statistical inference considers the parameters of a model as fixed, unknown
quantities that are estimated given observed data. Typically, estimation is accomplished
by optimizing an objective function (e.g., maximum likelihood estimation). Another
approach to statistical inference is Bayesian which uses probability to measure
uncertainty regarding unknown parameters. As such, in Bayesian inference, parameters
may be thought of as random variables with probability distributions. The probability
distribution describes uncertainty about the values of the parameters in the actual
process being modeled. Thus, the main difference between classical and Bayesian inference
is that classical inference fails to take into account the inherent uncertainty in the
actual values of the parameters. With Bayesian inference, there is no true value, only a
plausible distribution of values.6
A Bayesian approach gives more flexibility in assessments of goodness-of-fit of a model and provides direct assessment of uncertainty of any estimate of survival probability or combination of survival probabilities. However, Bayesian estimation can be more involved than some classical estimation procedures, often requiring simulation. In spite of this, once the initial estimation is finished, additional procedures such as goodness-of-fit of the model, construction of a predictive distribution for future onset times, and assessments of uncertainty of estimates can be available almost immediately using the results from the simulation.
In taking a Bayesian approach to modeling the time to onset of Grade IV VGE, we modify a model developed by Chen, Ibrahim, and Sinha7 and Asselain et al.8 These authors propose an unknown number of latent, independent competing causes of the event under study for each individual, where the competing causes have a common lifetime distribution, which may depend on covariates. Then, the minimum of the event times of the competing causes is the observed event time for the individual. The number of competing causes is Poisson-distributed, with mean a function of observed covariates. The cure rate for an individual is equivalent to the probability of zero latent causes for that individual. To account for multiple observed failure times per individual, a random effect term is included, which multiplies the Poisson mean. This random effect term is called the "frailty" of the individual toward the event.
We apply a model developed by Chen et al. (2002) to the data on onset time of Grade IV VGE. Advantages of the Chen et al. model include simplified estimation and computational ease. Chen and his colleagues used a positive stable distribution for the frailty, as this allowed a marginal proportional hazards structure (integrating over the frailty distribution) and also gave nice mathematical results. However, a positive stable frailty distribution implies strong initial dependence between failure times for a given individual.9 Accordingly, the ratio of the joint marginal lifetime densities for a pair of observations on the same individual to the product of the two marginal densities is high or unbounded at early times. On the other hand, a gamma frailty distribution implies strong late dependence. We use a gamma frailty distribution for reasons explained later.
Neither Chen et al. (2002) nor Asselain et al. (1996) deal specifically with interval-censored data. Thus, in this paper we formulate a Bayesian cure rate model to be used with general intervalcensoring. Our model follows the concepts in Chen et al., but like Asselain et al., we allow covariates to influence both the cure rate and the lifetime distribution for the noncured individuals. Thus, we do not propose a model with marginal proportional hazards. As such, the frailty distribution we use is not restricted to be of a form that achieves proportional hazards unconditionally. We also modify the Chen et al. model for repeated measurements. Thus, we incorporate in our model the possibility of repeated measurements on each individual.
Model Formulation and Likelihood
The model formulation of Chen et al. (2002) involves the introduction of multiple
independent, latent causes of Grade IV VGE. However, latent causes do not have to be
explicitly specified (as they are latent). The formulation can be applied to situations
where multiple latent causes of an event are not physically obvious. Thus, we use the
model formulation only for convenience in sampling from the joint posterior distribution
of the resulting unknown parameters.
Chen et al. introduce individual-specific, unobserved positive frailty terms into the distribution of the number of latent causes, such that each individual has one frailty term. The frailties serve two purposes. First, they induce a correlation between pairs of event times for a single subject. Second, they account for overdispersion in the distribution of the number of latent causes for an individual. Thus, two individuals with the same covariate pattern may have different distributions of the number of latent causes of the event because the frailty term is different. In the sequel, we will refer to subsequent measurements on the same individual as different tests on that individual.
Using this terminology, suppose there are n subjects with mi tests on the ith subject. For the jth test, the ith subject provides a set of p covariates denoted by the vector xij = (x1ij,...,xpij). Let Nij be the number of latent competing causes for Grade IV VGE for the ith person on the jth test. Conditional on a positive frailty term, wi for the ith subject, we let Nij ~ Poisson(qijwi), independently of one another, where qij º q(xij) = exp (xTij). The wi are independent and identically distributed with density function fw(w|g), for some unknown parameters, g.
When Nij = 0, the ith subject on the jth test will not experience Grade IV VGE. Nij is an auxiliary latent random variable introduced to make subsequent sampling from the posterior distribution easier. It does not necessarily have a direct physical interpretation. Given Nij, let zij = (zij1,zij2,...,zijNij) be the unobserved, independent times of the Nij competing causes for the ith subject on the jth test. The cumulative distribution function of zijk is given by F2(t|y,xij) = 1 - S(t|y,xij), dependent on a set of parameters in, which do not depend on Nij. Also, let fz(t|y,xij) be its density. Then, the time to Grade IV VGE for the ith subject on the jth test is defined by the random variable Tij = min(zijk,k = 1,...,Nij). Also, Cij is the censoring time for the ijth observation. If exact onset times were recorded, then the recorded failure time would yij = min(Tij,Cij). However, we only record censoring intervals of the form Lij,Rij], 0 < Lij <Rij < inf. Denote ij as a binary indicator for the ijth observation which equals 1 if the observation is interval-censored and 0 otherwise. The observed data consist of the censoring intervals as well as d = (dij,j = 1,...,mi; i = 1,...n) and the set of covariate vectors x = (xij,j = 1,...,mi; i = 1,...,n). We denote the observed data as DObs = {n, m, x, d, L, R} where L = (Lij,j = 1,...,mi; i = 1,...n) and and R = (Rij,j = 1,...,mi; i = 1,...,n) are the censoring intervals, and m = (m1,...,mn) are the number of observations per subject.
Because yij is not observed exactly for interval-censored data, and Nij and wi are also not observed, the complete-data are D = {n, m, x, d, L, R, N, w, y}, where y = (yij,j = 1,...,mi; i = 1,...,n), w = (w1,...,wn), and N = (Nij,j = 1,...,mi; i = 1,...n) . We will assume that for interval-censored data, yij falls between Lij and Rij and that yij < Cij. Furthermore, we assume that no other zijk, k = 1,...,Nij, falls between Lij and Rij. That is, all other zijk greater than the minimum of the zijk fall at times exceeding Rij. For right-censored data, we assume that yij = Cij = Lij. In addition, we assume that all right-censoring is Type I right-censoring.10
For the frailty distribution, we use a gamma distribution, with equal scale and shape parameters. That is, the set of parameters, , includes only a single parameter which we denote as g. Thus, the density of the frailty distribution is
| |
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Application of a gamma frailty distribution is made to account for the strong late dependence between pairs of onset times for a given individual from a population where only a limited number experience Grade IV VGE. For example, if a subject is immune to Grade IV VGE, his or her onset time will be recorded as right-censored at the end of the testing period, no matter how many times he or she takes the test, provided that other covariate values that influence susceptibility do not change. Other reasons for choosing the gamma frailty are that it is commonly used as a frailty distribution and is mathematically convenient for the analysis.
We use the lognormal distribution for the cumulative distribution function of zijk, Fz(t|y,xij), which depends on a set of q covariates (x1ij,...,xqij). The covariates in the cumulative distribution function are not necessarily the same as the covariates appearing in q(xy) = exp(xtijb), the expression for the cure rate, and may have a different dimension. In fact, in the case of modeling the Grade IV VGE data, the covariates will be distinct. However, we routinely refer to both sets using the symbol, x, unless it is not clear from the context. Thus,
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where m(x) = xTa = åqb = 1xbab, y = (a = a1,...,aq), s), and F(t) is the cumulative standard normal distribution evaluated at t. A lognormal distribution was used in previous analyses of the Grade IV VGE data.4,11
With the above assumptions, given N, the contributions to the likelihood by interval and right-censored observations are given by
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(4) |
where f(x) = 1/Ö2p exp(-x2/2) is the density of the standard normal distribution. Then, the likelihood of the parameters y and b, dependent on the complete-data is given by
 |
(5) |
The likelihood based on the observed data, DObs = {n, m, x, d, L, R}, is then
 |
(6) |
where
and 
.
Prior and Posterior Distributions for Bayesian Analysis
If we use the lognormal distribution for the zijk, then we need to put
priors on the parameters y = (a,
log s), log g, and b. Note that we parameterize positively-valued parameters in
terms of logarithms so that these parameters can take all values on the real line when we
perform simulations. Thus, we need to specify the joint prior p(b, a, log s,
log g). We assume a priori independence of all
parameters in (b, a ,
log s, log g). We put
independent Uniform(-M, M) priors on each of the ajs,
and has an improper uniform prior. These last two priors can be considered relatively
noninformative with respect to the likelihood, as long as |M| is large.12
We use normal priors for log s, log g,
so that p(log g) º Normal(log g | m0, s0)
and p(log s) º Normal(log s | m0, s0). We set m0 = m0 = 0 and s0 = s0 = 10. With this prior,
the posterior distribution of (b, a
, log s, log g) based
on the observed data DObs is given by
| p(b, a, log s, log g |DObs) µ L(b, a, d, g |DObs)N(log g | m0, s0)N(log s |m0, s0) | (7) |
for -M < aj, < M, j = 1,...,q, where L(b, a , d, g |DObs) is defined in (6). We have determined that the posterior distribution in (7) is proper.
Description of the Data and Measured Variables
We use the above model and likelihood to model the onset of Grade IV VGE using a subset of
data from NASA’s Hypobaric Decompression Sickness Data Bank (HDSD).5 The
HDSD has records from volunteer subjects undergoing up to 13 denitrogenation test
procedures prior to being exposed to a hypobaric environment. On each subject, the time to
onset of Grade IV VGE was measured as well as several covariates. However, the onset time
is recorded only as being contained within certain time intervals, due to the nature of
the testing and measurement. Also, some subjects on some tests never experienced Grade IV
VGE, causing right-censoring. The 548 exposure records consisted of 452 for the males and
96 for the females who participated in a series of studies from 1983 to 1998. However, the
number of individuals tested was 238 (177 of which were male) because some subjects
participated in more than one test procedure. Each test involved one decompression, where
the subject prebreathed 100% oxygen at ambient pressure prior to exposure in the altitude
chamber. Of the 548 records, 124 were interval-censored (i.e., Grade IV bubbles were
detected), leaving over 75% of the cases right-censored. More information regarding the
monitoring for VGE in these studies is found in Conkin, et al.13 and Thompson
et al.4,11
Explanatory variables measured on each individual are included, along with summary statistics in Table 1. Variables include both experimental variables and physical characteristics of the subjects. The importance of these variables in decompression sickness is well-documented.14-17 The first variable (TR360) is a measure of decompression stress. It is the ratio of the partial pressure of nitrogen at altitude to ambient pressure prior to ascent. The higher TR360, the more quickly a high grade bubble is expected to occur. The variable NOADYN is an indicator for whether the test subject was ambulatory (NOADYN = 1) or lower body adynamic (NOADYN = 0) during the test session. The variable SEX was coded as male = 1 and female = 0. Eighty-three percent of the 548 test records correspond to males, although only 74.4% of the 238 distinct individuals were male. Finally, the variable BMI is body mass index. Continuous variables have been standardized in the analysis to avoid computational problems.
Table 1. Summary Statistics for Explanatory Variables
|
Bayesian Inference Using the Posterior Distribution
To carry out the Bayesian analysis, we must calculate the posterior distribution using the
data described above. Certain summary statistics from each of the derived marginal
posterior distributions, such as posterior means, may be used as point estimates of the
parameters. Uncertainty assessments are obtained directly from the variability of the
posterior distribution. However, the posterior distribution in (7) does not have an
analytically tractable form due to the complicated form of the likelihood. When a
likelihood function is complicated, the use of Markov Chain Monte Carlo (MCMC) sampling
can facilitate in the estimation of posterior means. In MCMC sampling, we construct a
first-order Markov chain with target distribution equal to the joint posterior
distribution. Then, estimates of the means of the marginal posterior distributions are
obtained as Monte Carlo estimates using the samples from the chain.18 Thus,
with MCMC sampling, we can obtain an approximation to the posterior distribution, which we
then use for inference about any unknown parameters and functions of those unknown
parameters, such as survival curves.
To use MCMC sampling efficiently, we take advantage of the latent auxiliary variables N = (Nij,j = 1,...,mi, i = 1,...,n) and w = (wi, i = 1,...,n). In addition to these latent variables, the variables yij = min(Tij,Cij) are also unknown for observations that are interval-censored. We will refer to the set of yij for interval-censored observations by the vector, yI. Then, given DObs, the joint posterior distribution of (b, a, log s, log g, N, w, yI) is
 |
(8) |
We use (8) to facilitate the MCMC sampling. We provide full details of the sampling in Thompson and Chhikara.19
Once the sampler has converged to a process which samples repeatedly from the posterior distribution, we can get Monte Carlo point estimates of the parameters of interest, as sample averages. For the Grade IV VGE model, these parameters are a = a0, a1,...,1q) in the location component of the likelihood, b = (b0, b1,..., bp) in the cure component, exp(xtijb), s = exp(log s), (the standard deviation of the onset times), and g = exp(log g), from the frailty distribution. Also, estimates of uncertainty around the point estimates are obtained as sample standard deviations.
We are currently preparing the MCMC sampler to fit several cure rate models to the Grade IV VGE data that vary in the number of covariates included in the cure and location components. For any models that we fit, we only consider cases where physical variables relating to the individual influence the cure rate component of the model. Thus, only SEX, AGE, and BMI can enter into the cure rate portion because only individual characteristics can influence an individual’s susceptibility to Grade IV VGE. However, the individual characteristics are not necessarily constant across tests. All of the explanatory variables given in Table 1 are allowed to enter into the lognormal location component of the model. In addition, we will include an interaction term in NOADYN and TR360. The reasoning behind its inclusion is detailed in Thompson et al. (2002)4
Estimates of the Cumulative Probability of Grade IV VGE over Time
Here we describe how to estimate the cumulative probability of Grade IV VGE after a
given number of hours of exposure to hypobaric conditions. In Thompson et al. (2002), we
estimated the cumulative probability of Grade IV VGE for individuals under different test
conditions and physical characteristics, using the best fit model. For example, we
compared the cumulative probability of Grade IV VGE after six hours of exposure for
ambulatory versus adynamic individuals, with a TR360 of 1.5. For the Bayesian cure rate
model, the general form of the marginal cumulative probability of Grade IV VGE by time t
hours, given values of a set of p covariates, is
 |
(9) |
(see Thompson and Chhikara11 for the derivation). Thus, to approximate the posterior distribution of the difference between the cumulative probabilities of Grade IV VGE after six hours for adynamic (NOADYN = 0) versus ambulatory (NOADYN = 1) individuals (with other covariates fixed at specified values), we can use, say M, sampled values of (b, a , log s, log g) from the MCMC sampler to compute (9) for two different sets of covariates x that correspond to either ambulatory or adynamic individuals, with remaining covariates fixed. For example, we can compare adynamic with ambulatory males who are 32 years old with TR360 of 1.5 and a BMI of 42. Let xadynamic denote this set of covariates for adynamic individuals, and let xambulatory denote the set for ambulatory individuals. Then, the histogram of the differences, F(t | x = xadynamic, b(r), a(r), s(r), g(r)) - F(t | x = xambulatory, b(r), a(r) , s(r), g(r), for r = 1,...,M, are an approximation to its posterior distribution. The average of these differences is a point estimate of the difference, and the variability of the values represents the spread of uncertainty in its actual value.
Model Comparison
Once we fit all of the cure rate models to the Grade IV VGE data, we can compare them
using one of the many procedures developed for Bayesian survival model comparison. We will
use the Conditional Predictive Ordinate (CPO) statistic.20 This statistic is
computed at each observation. For interval-censored data, an observation is a recorded
interval. Thus, the CPO is computed at the ijth interval (Lij, Rij)
as
 |
(10) |
where D-(ij)Obs represents the data with the ijth observation removed. CPOij is the probability of the ijth onset time Tij falling within the recorded interval (Lij, Rij) (which may be infinite at the right endpoint) given the remaining observed data. Intuitively, it is the likelihood of each data point given the model and the other data points. Higher values of CPOij indicate a better fit of the data point within the model, and the sum of the logs of all CPO values is a measure of the fit of the model to the data. This statistic is called the log of the Pseudomarginal likelihood (LPML),21
 |
(11) |
It can be shown that the CPO statistic in (10) can be approximated using a Monte Carlo average over the samples from the posterior distribution, leading to
 |
(12) |
where Q(r) = (b(r), a(r) , s(r), g(r)) represents the set of sampled parameter values at the rth iteration of the MCMC sampler.22 For the Grade IV VGE data, CPOij is then approximated as
 |
(13) |
where FT(Ri1, . . . , Rimi | Q(r), xi) is defined in (9), and xi\j is the covariate matrix for the ith individual with the jth column removed.
References
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Emboli Using a Limited Failure Population Model with Random Effects," NASA Technical
Publication 2002-210781, Houston: Johnson Space Center, 2002.
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11L. Thompson and R. Chhikara. "Modeling Grade IV Venous Gas Emboli Onset Associated
with Decompression Sickness in Hypobaric Environment," (2003). (Submitted.)
12B. Carlin. and T. Louis. Bayes and Empirical Bayes Methods for Data Analysis. 2nd
edition, London: Chapman and Hall, 2000.
13J. Conkin, M. R. Powell, P. Foster, and J. M. Waligora. "Information about Venous
Gas Emboli Improves Prediction of Hypobaric Decompression Sickness," Aviation
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Scientific Meeting of the Aerospace Medical Society, 1999.
17J. Conkin and M. R. Powell. "Lower Body Adynamia as a Factor To Reduce the Risk of
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18W. R. Gilks, S. Richardson, and D. J. Spiegelhalter. Markov Chain Monte Carlo in
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19L. Thompson and R. Chhikara. "ABayesian Cure Rate Model for the Onset of Grade IV
Venous Gas Emboli," (2003). (Unpublished manuscript.)
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Publications
Thompson, L., and R. Chhikara, R. "Modeling Grade IV Venous Gas Emboli Onset
Associated with Decompression Sickness in Hypobaric Environment," J. American
Statistical Association. (Submitted.)
Thompson, L., R. Chhikara, and J. Conkin. "Cox Proportional Hazards Models for
Modeling the Time to Onset of Decompression Sickness," NASA Technical Publication
2003-210791, Houston: Johnson Space Center, 2003.
Thompson, L., J. Conkin, R. Chhikara, and M. Powel. "Modeling Grade IV Venous Gas
Emboli Using a Limited Failure Population Model with Random Effects," NASA Technical
Publication 2002-210781, Houston: Johnson Space Center, 2002.
Investigative Team UHCL PI: Raj S. Chhikara, Ph.D., Professor NASA-JSC Co-PI: Johnny Conkin, Ph.D. UHCL PDAF: Laura A. Thompson, Ph.D. |
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Table of Contents
Institute for Space Systems Operations - Y2002
Annual Report
Copyright © 2003
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