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PubMed1997

A stochastic model for the analysis of bivariate longitudinal AIDS data.

We present a model for multivariate repeated measures that incorporates random effects, correlated stochastic processes, and measurement errors. The model is a multivariate generalization of the model for univariate longitudinal data given by Taylor, Cumberland, and Sy (1994, Journal of the American Statistical Association 89, 727-736). The stochastic process used in this paper is the multivariate integrated Ornstein-Uhlenbeck (OU) process, which includes Brownian motion and a random effects model as special limiting cases. This process is an underlying continuous-time autoregressive order [AR(1)] process for the derivatives of the multivariate observations. The model allows unequally spaced observations and missing values for some of the variables. We analyze CD4 T-cell and beta-2-microglobulin measurements of the seroconverters at multiple time points from the Los Angeles section of the Multicenter AIDS Cohort Study. The model allows us to investigate the relationship between CD4 and beta-2-microglobulin through the correlations between their random effects and their serial correlation. The data suggest that CD4 and beta-2-microglobulin follow a bivariate Brownian motion process. The fit of the model implies that an increase in beta-2-microglobulin is associated with a decrease in future CD4 but not vice versa, agreeing with immunologic postulates about the relationship between these two variables.

Acquired Immunodeficiency Syndrome↗
PubMed1992

Graphical representation of survival curves associated with a binary non-reversible time dependent covariate.

The use of time dependent covariates has allowed for incorporation into analysis of survival data intervening events that are binary and non-reversible (for example, heart transplant, initial response to chemotherapy). We can represent this type of intervening event as a three-state stochastic process with a starting state (S), an intervening state (I), and an absorbing state (D), which usually represents death. In this paper we present three procedures for calculating survivorship functions which attempt to display the prognostic significance of the time dependent covariate. The first method compares survival from baseline for the two possible paths through the stochastic process; the second method compares overall survival to survival with state I removed from the process; and, the third method compares survival for those already in state I at a landmark time x to those in state S at time x who will never enter state I. We develop discrete hazard estimates for the survival curves associated with the three methods. Two examples illustrate how these methods can yield different results and in which situations one might employ each of the three methods. Extensions to applications with reversible binary time dependent covariates and models with both baseline and time dependent covariates are suggested.

Data Interpretation, Statistical↗
PubMed1998

Does the covariance structure matter in longitudinal modelling for the prediction of future CD4 counts?

We investigate the importance of the assumed covariance structure for longitudinal modelling of CD4 counts. We examine how individual predictions of future CD4 counts are affected by the covariance structure. We consider four covariance structures: one based on an integrated Ornstein-Uhlenbeck stochastic process; one based on Brownian motion, and two derived from standard linear and quadratic random-effects models. Using data from the Multicenter AIDS Cohort Study and from a simulation study, we show that there is a noticeable deterioration in the coverage rate of confidence intervals if we assume the wrong covariance. There is also a loss in efficiency. The quadratic random-effects model is found to be the best in terms of correctly calibrated prediction intervals, but is substantially less efficient than the others. Incorrectly specifying the covariance structure as linear random effects gives too narrow prediction intervals with poor coverage rates. Fitting using the model based on the integrated Ornstein-Uhlenbeck stochastic process is the preferred one of the four considered because of its efficiency and robustness properties. We also use the difference between the future predicted and observed CD4 counts to assess an appropriate transformation of CD4 counts; a fourth root, cube root and square root all appear reasonable choices.

Acquired Immunodeficiency Syndrome↗
PubMed1993

Optimal radiation beam profiles considering uncertainties in beam patient alignment.

The often large uncertainties that exist in beam patient alignment during radiation therapy may require modification of the incident beams to ensure an optimal delivered dose distribution to the target volume. This problem becomes increasingly severe when the required dose distribution of the incident beams becomes more heterogeneous. A simple analytical formula is derived for the case when the fraction number is high, and the desired relative dose variations are small. This formula adjusts the fluence distribution of the incident beam so that the resultant dose distribution will be as close as possible to the desired one considering the uncertainties in beam patient alignment. When sharp dose gradients are important, for instance at the border of the target volume, the problem is much more difficult. It is shown here that, if the tumor is surrounded by organs at risk, it is generally best to open up the field by about one standard deviation of the positional uncertainty--that is sigma/2 on each side of the target volume. In principle it is simultaneously desirable to increase the prescribed dose by a few per cent compared to the case where the positional uncertainty is negligible, in order to compensate for the rounded shoulders of the delivered dose distribution. When the tissues surrounding the tumor no longer are dose limiting even larger increases in field size may be advantageous. For more critical clinical situations the positional uncertainty may even limit the success of radiotherapy. In such cases one generally wants to create a steeper dose distribution than the underlying random Gaussian displacement process allows. The problem is then best handled by quantifying the treatment outcome under the influence of the stochastic process of patient misalignment. Either the coincidence with the desired dose distribution, or the expectation value of the probability of achieving complication-free tumor control is maximized under the influence of this stochastic process. It is shown that the most advantageous treatment is to apply beams that are either considerably widened or slightly widened and over flattened near the field edges for small and large fraction numbers respectively.

Dose-Response Relationship, Radiation↗
PubMed1987

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation.

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.

Genetics, Population↗
PubMed1967

Neuronal spike trains and stochastic point processes. II. Simultaneous spike trains.

The statistical analysis of two simultaneously observed trains of neuronal spikes is described, using as a conceptual framework the theory of stochastic point processes.The first statistical question that arises is whether the observed trains are independent; statistical techniques for testing independence are developed around the notion that, under the null hypothesis, the times of spike occurrence in one train represent random instants in time with respect to the other. If the null hypothesis is rejected-if dependence is attributed to the trains-the problem then becomes that of characterizing the nature and source of the observed dependencies. Statistical signs of various classes of dependencies, including direct interaction and shared input, are discussed and illustrated through computer simulations of interacting neurons. The effects of nonstationarities on the statistical measures for simultaneous spike trains are also discussed. For two-train comparisons of irregularly discharging nerve cells, moderate nonstationarities are shown to have little effect on the detection of interactions.Combining repetitive stimulation and simultaneous recording of spike trains from two (or more) neurons yields additional clues as to possible modes of interaction among the monitored neurons; the theory presented is illustrated by an application to experimentally obtained data from auditory neurons.A companion paper covers the analysis of single spike trains.

Action Potentials↗
PubMed1998

Two mechanisms underlie processing of stochastic motion stimuli.

We have constructed "limited lifetime" stochastic motion stimuli using Gabor functions instead of dots, thereby controlling the local attributes of spatial frequency and orientation. Human psychophysical data for direction discrimination using these stimuli reveal two qualitatively distinct kinds of processing. For small displacements, direction discrimination performance as a function of displacement is scaled with spatial frequency in a manner consistent with a linear filtering motion mechanism. Motion perception for relatively large displacements is not directly related to the spatial frequency, and is consistent with a nonlinear process which signals motion of contrast envelopes.

Discrimination, Psychological↗
PubMed1996

Two-state stochastic models for memory in ion channels.

AIM: To study quantitatively the memory existing in ion channels. METHODS: Stochastic processes were used to model 2 categories of memory (short-term and long-term) by persisting in the standpoint of two-state, instead of multiple states, but with different transition mechanism. RESULTS: A two-state Markov process with constant transition intensities well fitted the short-term memory and a two-state Markov process within a kind of random environment well fitted the long-term memory. Statistical procedures for parameter estimation were proposed and demonstrated with 2 real examples on the channels of PC12 cells. CONCLUSION: The memory in ion channels can be quantitatively modelled as stochastic process with 2 states.

Adrenal Gland Neoplasms↗
PubMed1991

Stochastic differential equations, their interpretation and application.

The work is recommended to readers with some, maybe heuristic, imagine about stochastic processes that want to meet the notion stochastic differential equation and its interpretation. The notions like Brownian motion and stochastic integral with interpretations in concrete situations in areas of biology and medicine are discussed. The questions are related to mathematical modelling and they may be interpreted in connection with stochastic signal filtering and optimal queuing theory.

Stochastic Processes↗
PubMed1993

Thymocyte development in major histocompatibility complex-deficient mice: evidence for stochastic commitment to the CD4 and CD8 lineages.

The mechanism resulting in commitment of precursor cells in the thymus to either the CD4 or CD8 lineage remains poorly understood. In principle, this may reflect a stochastic process or may reflect instructional signals from host major histocompatibility complex (MHC) molecules. We have examined the role of MHC products in subset commitment by using mice deficient in class I or class II MHC products. Normal numbers of committed CD4 intermediates (CD4+ CD8lo) develop in the thymus in the absence of class II molecules. Similarly, CD8 transitional cells (CD4loCD8+) are present in the thymus of mice lacking class I products. These findings suggest that commitment of CD4+8+ precursor cells to either lineage is a stochastic process that does not depend on instructive signals from MHC molecules (i.e., expression of alternative differentiative options by uncommitted precursor cells is independent of this environmental signal). These studies also suggest that an interaction between the T-cell antigen receptor (TCR) and MHC molecules that is independent of CD4/CD8 coreceptor engagement enhances stochastic coreceptor downregulation substantially and leads to upregulation of TCR expression as a prelude to selective events that require joint coreceptor/TCR engagement. We suggest that this initial interaction molds the TCR repertoire of stochastically generated T-cell subsets toward recognition of self-MHC products.

Animals↗
PubMed1983

Statistical analysis of ongoing activity of neurones in the substantia gelatinosa and in lamina III of cat spinal cord.

Intracellular recordings from substantia gelatinosa (s.g.) neurones in chloralose-anaesthetized cats and in decerebrate preparations revealed the existence of ongoing synaptic activity. 59% of s.g. neurones showed ongoing spiking activity at rates of more than one per second. The ongoing activity of twenty s.g. neurones was subjected to statistical analysis. Stationarity was established for the activity of each neurone, the interspike interval (i.s.i.) distributions were bell-shaped, and no evidence of dependency of the length of an interval on the one preceding it was found. The mechanism of spike generation in these neurones is therefore an example of a renewal stochastic process. The pattern of ongoing discharge of twelve neurones recorded in lamina III was markedly different, and generated in each case a unimodal asymmetric i.s.i. histogram with a sharp rise to mode after a short dead time and a slow decay. The activity was stationary only in the long term, and there was strong evidence of dependency of intervals. The spike generating mechanism was therefore an example of a non-renewal stochastic process. The different patterns of activity are discussed in relation to differences in cutaneous input, and it is suggested that the pattern of activity in the s.g. neurones is the result of convergence on the neurones of a large number of small independent excitatory influences, whereas that of the neurones in lamina III is the result of excitation by powerful synchronous synaptic inputs. It is re-emphasized that statistical analysis of neural impulse sequences is a valuable technique in investigation of the function of a neurone within its network.

Anesthesia, General↗
PubMed1999

Analysis of protein aggregation kinetics.

Given a set of kinetic data, then, the preceding discussions suggest the following approach to its analysis. 1. For purposes of establishing the reaction, ignore the final stages and concentrate on the initial 10-20% of the reaction at first. A globally optimized model may be based on a faulty assumption for the initial steps. Thus, although the whole data set may look reasonably well fit, the reaction could be misrepresented, and thus the fit unhelpful if accuracy at the later stages has come at the expense of the initial phase of the reaction. 2. What is the time course of the initial reaction? (A) Is the reaction exponential? Exponential growth gives dramatic lag times (see Fig. 3), whereas nonexponential "lag times" have a visible signal from time 0 (i.e., Fig. 2). If the data set shows the abrupt appearance of signals after a period of quiescence, the chances are excellent that the time course is exponential. High sensitivity measurement of the signal at times during the lag phase should be used to confirm the exponential nature quantitatively. Exponential reactions mean a secondary pathway is operative. (a) A cascade (tn) can look similar to an exponential, but may proceed from a multistep single-path reaction. Thus the exponential needs to be ascertained with some accuracy. (b) It is possible that some or all of the lag results from a stochastic process, i.e., formation of a single nucleus being observed. This, however, is likely to be accompanied by a secondary process, as few techniques are sensitive enough to detect a single polymer at a time, and having one nucleus form many polymers is a hallmark of a secondary process. Thus, the reproducibility of the kinetics must be established to rule out stochastics. If data show wide variation, stochastic methods as described earlier may be employed. (c) Given a secondary process, one must separate the primary nucleation process from the secondary process (by stochastic means or by use of the product B2A, as described earlier). (B) If the reaction does not begin with an exponential, is it parabolic? If so, it falls in the general class of linear polymerizations. 3. What is the concentration dependence of the reaction(s)? This will separate nucleation processes from growth, and so on. 4. If the initial reaction is neither exponential nor parabolic, a reaction mechanism needs to be proposed and evaluated. Solving the resulting equations is best done by linearization, which has the best chance of giving equations whose solutions and their sensitivity to parameters are readily understood. If this proves fruitful, full numeric solutions may be useful. 5. At this point, the full reaction may be considered to completion. 6. The physical basis of the description (sizes of parameters and their dependencies) needs to be finally considered to ensure that the mathematical success of the description rests on tenable physical grounds.

Adenosine Triphosphate↗
PubMed1997

Stochastic Dispersal Processes in Plant Populations

A dispersal model for airborne pollen based on assumptions about wind directionality, gravity, and a wind threshold at which pollen is taken by the wind is developed, using a three dimensional diffusion approximation. The bivariate probability distribution of pollen receipt by flowers at the same height as the pollen source is derived. Gravity, vertical random movements, and vegetation density turn out to have similar effects on this distribution. Maximum likelihood methods for estimating the combined parameters from data with multiple point or continuous pollen sources, and one or more plant varieties, are developed. Using an example data set from the literature, it is shown that our model gives a better fit than more traditional descriptive dispersal models of the form e-ar b. We also show that estimates of important properties of the dispersal distribution, such as the variances, become considerably smaller using our model than for the more traditional models. Finally, we discuss potential extensions and evolutionary implications of these types of models. Copyright 1997 Academic Press

Journal Article↗
PubMed2024

Seasonal hydrological dynamics affected the diversity and assembly process of the antibiotic resistome in a canal network.

The significant threat of antibiotic resistance genes (ARGs) to aquatic environments health has been widely acknowledged. To date, several studies have focused on the distribution and diversity of ARGs in a single river while their profiles in complex river networks are largely known. Here, the spatiotemporal dynamics of ARG profiles in a canal network were examined using high-throughput quantitative PCR, and the underlying assembly processes and its main environmental influencing factors were elucidated using multiple statistical analyses. The results demonstrated significant seasonal dynamics with greater richness and relative abundance of ARGs observed during the dry season compared to the wet season. ARG profiles exhibited a pronounced distance-decay pattern in the dry season, whereas no such pattern was evident in the wet season. Null model analysis indicated that deterministic processes, in contrast to stochastic processes, had a significant impact on shaping the ARG profiles. Furthermore, it was found that Firmicutes and pH emerged as the foremost factors influencing these profiles. This study enhanced our comprehension of the variations in ARG profiles within canal networks, which may contribute to the design of efficient management approaches aimed at restraining the propagation of ARGs.

Seasons↗