On the use of time constants as estimates of mean lifetimes in single channel data analysis

For Markov models of single channel kinetics, a sojourn time in a class of states has a density function which is usually a linear combination of exponential densities. There are many instances in the single channel literature where the time constants of exponentials fitted to sojourn time data have been used as estimated mean sojourn times in individual states, though the two may be very different. In the present study the nature and magnitude of this difference in the case of a two state class is illustrated analytically and numerically. The time constants should be viewed at best as approximations, possibly poor, to the estimated mean sojourn times. Estimates of kinetic parameters cannot in general be obtained explicitly from the fitted parameters of the density alone. However, this is shown to be possible in some special cases and enables direct estimation of, for example, the channel opening rate constant (or an upper limit to the estimate of in the case of multiple channels) in standard sequential three or four state models of nicotinic receptor kinetics, using only the fitted parameters of the closed-time density. Offprint requests to: R. O. Edeson     

On the robustness of Hotelling's T2

The influence of outliers in one of the samples on Hotelling's generalized T¹₀ is investigated by considering the directional derivate of the expected value with respect to the proportion of outliers in the disturbed population at the proportion zero. To make the problem tractable, the estimate of the covariance matrix thereby is replaced by its expected value under the altered distributional assumptions. For the two sample case the influence of outliers on the variance of the test criterion is also investigated. The analytical findings are confirmed by simulation results.     

The mean and standard deviation of development time

The variability in a population of the onset of a phenophase may be significantly great compared to the mean time or to a practical interest. This is illustrated by flowering of gladiolus and germination of tobacco seed andAlternaria spores. Mathematically, development is conceived to pass through a series of f stages at a rate P, the sum of P to the half-time equals f, and the variance of the times for individual completions is f divided by P² near the half-time.Presented at the Seventh International Biometeorological Congress, 17–23 August 1975, College Park, Maryland, USA.     

Effects of sample size on fish parasite prevalence, mean abundance and mean intensity estimates

This study considers the effects of sample size on estimates of three parasitological indices (prevalence, mean abundance and mean intensity) in four different host–parasite systems, each showing a different pattern of infection. Monte Carlo simulation procedures were used in order to obtain an estimation of the parasitological indices, as well as their variance and bias, based on samples of different size. Although results showed that mean values of all indices were similar irrespective of sample size, estimates of prevalence were not significantly affected by sample size whereas mean abundance and mean intensity were affected in at least one sample. Underestimation of values was more perceptible in small (<40) sample sizes. Distribution of the estimated values revealed a different arrangement according to the host–parasite system and to the parasitological parameter. Monte Carlo simulation procedures are, therefore, suggested to be included in studies concerning estimation of parasitological parameters.     

On a property of the exponential distribution

If t is an independent exponentially distributed random variable, the distribution p = [t - x] is a modified geometric distribution, similar to the result of HAWKINS and KOTZ (1976), x is uniform.     

On the interpretation and application of mean times to extinction

As a metric of population viability, conservation biologists routinely predict the mean time to extinction (MTE). Interpretation of MTE depends on the underlying distribution of times to extinction (DTE). Despite claims to the contrary, all information regarding extinction risk can be obtained from this single statistic, the MTE, provided the DTE is exponential. We discuss the proper interpretation of MTE and illustrate how to calculate any population viability statistic when only the MTE is known and the DTE is assumed to be exponential. We also discuss the restrictive assumptions underlying the exponential DTE and the conditions under which alternative models for the DTE are preferable to the conventional (exponential) model. Despite superficial similarities between the exponential and alternative DTEs, several key differences can lead to substantially different interpretations of the MTE.     

On the robustness of regular systems of inbreeding.

Half-sib, first cousin, half nth cousin, and nth cousin mating systems are robust in that small deviations from the mating structure will not significantly alter the levels of genetic identity or effect qualitative distinctions between the models. Substitution of nearest kin in matings may either increase or decrease the level of homozygosity depending on the mating structure; the effect of a single error in the mating structure is not an accurate indicator of the equilibrium resulting from recurrent errors. Models of mixed half nth cousin or nth cousin mating show that the relative frequency of the lowest order inbred mating essentially determines the level of homozygosity. Any positive relative frequency of more distant matings will reduce the probability of identity by descent under half-sib or first cousin mating to less than 1.     

Estimating the probability that the sample mean is within a desired fraction of the standard deviation of the true mean

The use of small sample sizes in human and primate evolutionary research is commonplace. Estimating how well small samples represent the underlying population, however, is not commonplace. Because the accuracy of determinations of taxonomy, phylogeny, and evolutionary process are dependant upon how well the study sample represents the population of interest, characterizing the uncertainty, or potential error, associated with analyses of small sample sizes is essential. We present a method for estimating the probability that the sample mean is within a desired fraction of the standard deviation of the true mean using small (n < 10) or very small (n ≤ 5) sample sizes. This method can be used by researchers to determine post hoc the probability that their sample is a meaningful approximation of the population parameter. We tested the method using a large craniometric data set commonly used by researchers in the field. Given our results, we suggest that sample estimates of the population mean can be reasonable and meaningful even when based on small, and perhaps even very small, sample sizes.     

On the robustness of update schedules in Boolean networks

Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems.     

On the exponential model for energy with respect to number of electrons

Using an exponential model for the variation in energy with respect to the number of electrons it is shown that, within the model, the hardness, softness, electrophilicity and other global parameters connected to higher order derivatives follow an equalization principle after a molecule is formed from two separated species. Two generalizations of the model are also discussed, one of which presents discontinuity of the chemical potential at integer values of N.     

On the robustness of complex heterogeneous gene expression networks

We analyze a continuous gene expression model on the underlying topology of a complex heterogeneous network. Numerical simulations aimed at studying the chaotic and periodic dynamics of the model are performed. The results clearly indicate that there is a region in which the dynamical and structural complexity of the system avoid chaotic attractors. However, contrary to what has been reported for Random Boolean Networks, the chaotic phase cannot be completely suppressed, which has important bearings on network robustness and gene expression modeling.     

Precision of estimates of mean and peak spinal loads in lifting

A bootstrap procedure was used to determine the statistical precision of estimates of mean and peak spinal loads during lifting as function of the numbers of subjects and measurements per subject included in a biomechanical study. Data were derived from an experiment in which 10 subjects performed 360 lifting trials each. The maximum values per lift of the lumbar flexion angle, L5S1 sagittal plane moment, and L5S1 compression force were determined. From the data set thus compiled, 3000 samples were randomly drawn for each combination of number of subjects and number of measurements considered. The coefficients of variation of mean and peak (defined as mean plus 2 standard deviations) spinal loads across these samples were calculated. The coefficients of variation of the means of the three parameters of spinal load decreased as a linear function of the number of subjects to a power of about -0.48 and number of measurements to a power of about -0.06, while the corresponding powers for peak loads were about -0.44 and -0.11.