Bent Line Quantile Regression with Application to an Allometric Study of Land Mammals' Speed and Mass

Summary Quantile regression, which models the conditional quantiles of the response variable given covariates, usually assumes a linear model. However, this kind of linearity is often unrealistic in real life. One situation where linear quantile regression is not appropriate is when the response variable is piecewise linear but still continuous in covariates. To analyze such data, we propose a bent line quantile regression model. We derive its parameter estimates, prove that they are asymptotically valid given the existence of a change‐point, and discuss several methods for testing the existence of a change‐point in bent line quantile regression together with a power comparison by simulation. An example of land mammal maximal running speeds is given to illustrate an application of bent line quantile regression in which this model is theoretically justified and its parameters are of direct biological interests.     

A mathematical framework for examining whether a minimum number of chiasmata is required per metacentric chromosome or chromosome arm in human

We introduce a piecewise linear regression called "hockey stick regression" to model the relationship between genetic and physical lengths of chromosomes in a genome. This piecewise linear regression is an extension of the two-parameter linear regression we proposed earlier [W. Li and J. Freudenberg, Two-parameter characterization of chromosome-scale recombination rate, Genome Res., 19 (2009) 2300-2307]. We use this, as well as the one-piece regression with a fixed y-intercept, to compare the two competing hypotheses concerning the minimum number of required chiasmata for meiosis: minimum one chiasma per chromosome (PC) and per chromosome arm (PA). Using statistical model selection and testing, we show that for human genome data, one-piece PC (PC1) is often in a statistical tie with two-piece PA model (PA2). If an upper bound for the segmentation point in two-piece regression is imposed, PC is usually the preferred model. This indicates that a presence of more than one chiasmata is rather caused by the relationship between chromosome size and chiasma formation than by cytogenetic constraints.     

Iwao’s patchiness regression through the origin: biological importance and efficiency of sampling applications

Iwao’s mean crowding-mean density relation can be treated both as a linear function describing the biological characteristics of a species at a population level, or a regression model fitted to empirical data (Iwao’s patchiness regression). In this latter form its parameters are commonly used to construct sampling plans for insect pests, which are characteristically patchily distributed or overdispersed. It is shown in this paper that modifying both the linear function and statistical model to force the intercept or lower functional limit through the origin results in more intuitive biological interpretation of parameters and better sampling economy. Firstly, forcing the function through the origin has the effect of ensuring that zero crowding occurs when zero individuals occupy a patch. Secondly, it ensures that negative values of the intercept, which do not yield an intuitive biological interpretation, will not arise. It is shown analytically that sequential sampling plans based on regression through the origin should be more efficient compared to plans based on conventional regression. For two overdispersed data sets, through-origin based plans collected a significantly lower sample size during validation than plans based on conventional regression, but the improvement in sampling efficiency was not large enough to be of practical benefit. No difference in sample size was observed when through-origin and conventional regression based plans were validated using underdispersed data. A field researcher wishing to adopt a through-origin form of Iwao’s regression for the biological reasons outlined above can therefore be confident that their sampling strategies will not be affected by doing so.     

The Application of Linear Piecewise Regression to Basal Body Temperature Data

In many situations one wishes to fit a piecewige regression which enables one to obtain estimates of the join points as well as the slopes and intercepts of the fitted submodels. This study developes a technique for fitting piecewise models to data which contain measurement error in an independent variable. The technique developed here combines the HUDSON (1966) procedure for estimating parameters in piecewise regression and the WALD (1940) Grouping Technique which obviates the problem of measurement error. If one assumes some knowledge of the position of the join point in relation to the data, methodology has been developed to estimate the parameters and study the asymptotic properties of the means and variances of the parameter estimates. However, in the more realistic case, when additional knowledge is limited, it is only possible to obtain the parameter estimates using an iterative technique (TEETER, 1982). The general technique for obtaining the join point estimate in the presence of measurement error is presented here and an example is given using data on women's basal body temperature during menstrual cycles.     

Survival analysis with time-varying regression effects using a tree-based approach

Nonproportional hazards often arise in survival analysis, as is evident in the data from the International Non-Hodgkin's Lymphoma Prognostic Factors Project. A tree-based method to handle such survival data is developed for the assessment and estimation of time-dependent regression effects under a Cox-type model. The tree method approximates the time-varying regression effects as piecewise constants and is designed to estimate change points in the regression parameters. A fast algorithm that relies on maximized score statistics is used in recursive segmentation of the time axis. Following the segmentation, a pruning algorithm with optimal properties similar to those of classification and regression trees (CART) is used to determine a sparse segmentation. Bootstrap resampling is used in correcting for overoptimism due to split point optimization. The piecewise constant model is often more suitable for clinical interpretation of the regression parameters than the more flexible spline models. The utility of the algorithm is shown on the lymphoma data, where we further develop the published International Risk Index into a time-varying risk index for non-Hodgkin's lymphoma.     

Respiration of midges (Diptera; Chironomidae) in British Columbian lakes: oxy-regulation, temperature and their role as palaeo-indicators

1. The specific respiration rate of 13 chironomid taxa and Chaoborus were measured to test the hypothesis of the relation between a species' ability to regulate their oxygen uptake and their distributional patterns among nine study lakes in British Columbia, Canada.
2. Respiration patterns of individual taxa were modelled using piecewise linear regression with break point and simple hyperbolic functions. Three types of respiration curves were identified: (i) classical oxy-conformers (e.g. littoral Cricotopus ) which cannot sustain a sufficient oxygen uptake with decreasing oxygen availability; (ii) oxy-regulators (e.g. profundal Chironomus ) which can regulate and maintain a constant respiration until a certain critical point and (iii) oxy-stressors ( Micropsectra ) which increase their respiration rate with decreasing oxygen availability until a critical point.
3. Respiration was measured at two different temperatures (10 and 20 °C), and over the range of oxygen saturation conditions studied here (0–90%) mean Q ₁₀ values varied from 1.3 to 2.5.
4. The results show that different chironomid taxa have varying sensitivity to low oxygen concentrations and different respiratory responses to increased temperature. The critical point increased to higher oxygen saturation for six taxa, decreased for one taxon and was unchanged for two taxa.
5. The results illustrate one of the possible biological mechanisms behind the use of chironomids as temperature and climate indicators in palaeoecological studies by exploring the link between temperature and respiration physiology.     

Statistical analysis of functional response experiments

The aim of this paper is to highlight the benefits of a full data analysis of a functional response data set, compared to the usual one‐pass regression analysis. In a biological control setting where the choice of organism is often based on comparative studies of the functional responses, it is imperative to have both reliable estimates and a feeling of the degree of confidence one is willing to put on the figures. We analyzed a data set involving the freshwater predator Notonecta glauca (Hemiptera) preying on Asellus aquaticus during 24 h. The specific aim of the analysis was to test whether the functional response is of type II or type III. The different stages of a complete analysis are (1) a preliminary inspection of the data, (2) model building, (3) a model check and (4) a combination of the results with independent information. We argue that the analysis is best done with the predation rate as response and define a test for the location of its maximum. The existence of a maximum is typical for type III functional response. We explain why the binomial distribution is a natural error distribution, and how to implement the regression analysis within the family of generalized linear models using two competing link functions, the logit and the reciprocal. There is marked overdispersion which increases with increasing prey numbers. We use prior weights to take account of it. Using all available data, a type III functional response is warranted with the reciprocal link, but not with the logit link Model checks using Pearson residuals and regression diagnostics based on point deletions show that three points have a particularly strong influence on the parameter estimates. If these are deleted, the functional response type III is then warranted for both link functions. The complete analysis enables us to determine the various degrees of uncertainty and to draw biological conclusions with corresponding confidence. We are convinced that the data set shows a type III functional response, but we are less sure about which link function to choose. Furthermore, the marked overdispersion at high density, the regression diagnostics, as well as independent information on a change in the behaviour of the prey at high density, indicate that the experimental conditions may have changed as a function of the prey density.     

Vestibular integrator neurons have quadratic functions due to voltage dependent conductances

The nonlinear properties of the dendrites of the prepositus hypoglossi nucleus (PHN) neurons are essential for the operation of the vestibular neural integrator that converts a head velocity signal to one that controls eye position. A novel system of frequency probing, namely quadratic sinusoidal analysis (QSA), was used to decode the intrinsic nonlinear behavior of these neurons under voltage clamp conditions. Voltage clamp currents were measured at harmonic and interactive frequencies using specific nonoverlapping stimulation frequencies. Eigenanalysis of the QSA matrix reduces it to a remarkably compact processing unit, composed of just one or two dominant components (eigenvalues). The QSA matrix of rat PHN neurons provides signatures of the voltage dependent conductances for their particular dendritic and somatic distributions. An important part of the nonlinear response is due to the persistent sodium conductance (g_NaP), which is likely to be essential for sustained effects needed for a neural integrator. It was found that responses in the range of 10 mV peak to peak could be well described by quadratic nonlinearities suggesting that effects of higher degree nonlinearities would add only marginal improvement. Therefore, the quadratic response is likely to sufficiently capture most of the nonlinear behavior of neuronal systems except for extremely large synaptic inputs. Thus, neurons have two distinct linear and quadratic functions, which shows that piecewise linear?+?quadratic analysis is much more complete than just piecewise linear analysis; in addition quadratic analysis can be done at a single holding potential. Furthermore, the nonlinear neuronal responses contain more frequencies over a wider frequency band than the input signal. As a consequence, they convert limited amplitude and bandwidth input signals to wider bandwidth and more complex output responses. Finally, simulations at subthreshold membrane potentials with realistic PHN neuron models suggest that the quadratic functions are fundamentally dominated by active dendritic structures and persistent sodium conductances.     

Bayesian isotonic regression and trend analysis

In many applications, the mean of a response variable can be assumed to be a nondecreasing function of a continuous predictor, controlling for covariates. In such cases, interest often focuses on estimating the regression function, while also assessing evidence of an association. This article proposes a new framework for Bayesian isotonic regression and order-restricted inference. Approximating the regression function with a high-dimensional piecewise linear model, the nondecreasing constraint is incorporated through a prior distribution for the slopes consisting of a product mixture of point masses (accounting for flat regions) and truncated normal densities. To borrow information across the intervals and smooth the curve, the prior is formulated as a latent autoregressive normal process. This structure facilitates efficient posterior computation, since the full conditional distributions of the parameters have simple conjugate forms. Point and interval estimates of the regression function and posterior probabilities of an association for different regions of the predictor can be estimated from a single MCMC run. Generalizations to categorical outcomes and multiple predictors are described, and the approach is applied to an epidemiology application.     

Modeling convergent ON and OFF pathways in the early visual system

For understanding the computation and function of single neurons in sensory systems, one needs to investigate how sensory stimuli are related to a neuron’s response and which biological mechanisms underlie this relationship. Mathematical models of the stimulus–response relationship have proved very useful in approaching these issues in a systematic, quantitative way. A starting point for many such analyses has been provided by phenomenological “linear–nonlinear” (LN) models, which comprise a linear filter followed by a static nonlinear transformation. The linear filter is often associated with the neuron’s receptive field. However, the structure of the receptive field is generally a result of inputs from many presynaptic neurons, which may form parallel signal processing pathways. In the retina, for example, certain ganglion cells receive excitatory inputs from ON-type as well as OFF-type bipolar cells. Recent experiments have shown that the convergence of these pathways leads to intriguing response characteristics that cannot be captured by a single linear filter. One approach to adjust the LN model to the biological circuit structure is to use multiple parallel filters that capture ON and OFF bipolar inputs. Here, we review these new developments in modeling neuronal responses in the early visual system and provide details about one particular technique for obtaining the required sets of parallel filters from experimental data.     

Reliability of pattern separation by the cerebellar mossy fiber — granule cell system

As a simple example of a neuronal network in which synaptic connectivity among neurons is probabilistic, Marr's model for the granular layer of cat cerebellar cortex is examined. The mean and variance are computed for the fraction of granule cells activated, and for the extent of pattern separation by granule cells, for various mossy fiber inputs and various values of connectivity and electrical parameters of the network structure. Results suggest different functions for the network, and different optimal ranges for its parameters, depending on whether Golgi cells are present or absent. The model network does not perform the functions originally prescribed for it with high reliability.     

Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210-219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine "interior" and "exterior" species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210–219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine “interior” and “exterior” species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Lag‐phases in alien plant invasions: separating the facts from the artefacts

Temporal trends in biological invasions are often described by a lag‐phase of little or no increase in species occurrence followed by an increase‐phase in which species occurrence rises rapidly. While several biological and environmental mechanisms may underlie lag‐phases, they may also represent statistical artefacts or temporal changes in sampling effort. To date, distinguishing the facts from these artefacts has not been possible. Here we describe a method for estimating the lag‐phase in cumulative records of species occurrence, using a piecewise regression model that explicitly differentiates the lag and increase phases. We used the von Bertalanffy, logistic, linear and exponential functions to model the increase phase, and identified the best‐fitting function using model selection techniques. We confirmed the accuracy of our method using simulated data and then estimated the length of the lag‐phase (t_lag), the maximum collection rate (r) and the projected asymptotic number of records (K) using herbarium records for 105 weed species in New Zealand, while accounting for changes in sampling effort. Nearly all the New Zealand weed species had a lag‐phase, which averaged around 20–30 years, with 4% of species having a lag‐phase greater than 40 years. In more than two thirds of the cases, the accumulation of records was best modelled with the decelerating von Bertalanffy function, despite the tendency for temporal variation in sampling effort to force cumulative herbarium records to follow the sigmoidal shape of a logistic curve. A positive correlation between r and K is consistent with the assumption that the final distribution of an alien plant species reflects its rate of spread. Seemingly rare but fast‐spreading aliens may thus become tomorrow's noxious weeds. A positive correlation between inflection year and r warns that the weeds that have only begun to spread relatively recently may spread faster than previously known invaders.