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Threshold models first appeared in the literature nearly half a century ago. Threshold segments have been added to many commonly used forms of models from linear models and generalized linear models through mixed models for the analysis of cross-sectional data. Nonlinear models with thresholds for cross-sectional data are less prevalent in the literature. Nonlinear models with thresholds for longitudinal data are new. The historical developments leading to this point are reviewed as a means of introducing terms necessary for discussing features of these newer models. Nonlinear models for longitudinal data with thresholds are presented and discussed.
Trauma excepted, muscle forces cause the largest loads on bones and the largest bone strains. In children, steadily increasing muscle strength increases bone loads and strains above a modeling threshold, which allows modeling to increase bone strength and "mass" and conservation-mode remodeling to retain existing bone. As a result, bone strength and "mass" both increase. In young adults, muscle strength plateaus, so bone strength can increase enough to reduce strains below the modeling threshold, turn modeling off, and also plateau. Those strains still exceed the lower remodeling threshold, so conservation-mode remodeling retains existing bone. Most aging adults lose momentary muscle strength, so their bone strains fall toward the remodeling threshold. That drop leaves modeling off and switches remodeling to its disuse mode to begin removing bone next to marrow, contributing to the well-known age-related loss of bone. Although "vigorous" exercise by aging adults can raise strains above the remodeling threshold to turn conservation-mode remodeling back on and reduce or stop further bone losses, causing the much larger strains needed to reach or exceed the modeling threshold would require larger increases in momentary muscle strength and muscle mass than most such adults could achieve. Thus, exercises that can readily increase bone strength and "mass" in children and adolescents (in whom modeling is already turned on) only seem to reduce bone loss in aging adults. This difference makes their bones seem partly unresponsive to physical exercise. This effect would occur in addition to possible nonbiomechanical explanations that others have suggested for the phenomenon.
An important goal of microarray studies is the detection of genes that show significant changes in expression when two classes of biological samples are being compared. We present an ANOVA-style mixed model with parameters for array normalization, overall level of gene expression, and change of expression between the classes. For the latter we assume a mixing distribution with a probability mass concentrated at zero, representing genes with no changes, and a normal distribution representing the level of change for the other genes. We estimate the parameters by optimizing the marginal likelihood. To make this practical, Laplace approximations and a backfitting algorithm are used. The performance of the model is studied by simulation and by application to publicly available data sets.
We implemented statistical models of Bayesian inference that included direct and maternal genetic effects for genetic parameter estimation of categorical traits by Gibbs sampling. The estimation errors and variances of estimates of animal versus sire and maternal grandsire models, of linear versus threshold models, of single-trait versus multiple-trait models, and of treating herd-year-season as fixed versus random effects in the model were compared. The results indicated that linear models yielded biased estimates of genetic parameters for categorical traits. The animal model was improper for analysis of categorical traits using a threshold model and the Gibbs sampler. Moreover, linear versus threshold models and animal versus sire-maternal grandsire models resulted in larger Monte Carlo errors and increased auto-correlations among posterior samples. Treating herd-year-seasons as random effects in the threshold models decreased the Monte Carlo error, auto-correlations, and the variances of estimates. Efficiency of the single-trait threshold sire model, as measured by the variance of the estimates, was lower than for a multiple-trait model that included a correlated continuous trait, but both estimates were unbiased. Therefore, the threshold single-trait sire and maternal grandsire model is a feasible alternative to the multiple-trait model for analysis of variance components of categorical traits affected by direct and maternal genetic factors.
In Swedish Holstein dairy cattle, genetic effects on stillbirth and calving difficulty were studied in 411,409 first- and 281,193 second-calvers. A linear single-trait sire-maternal grandsire model and a threshold model using a Gibbs sampling technique were used to analyse calving data from 1985 to 1996. In first calving when using the linear model, the heritability of stillbirth on the visible scale was 4% for the direct effect and 3% for the maternal effect. For calving difficulty it was 6% and 5% for direct and maternal effects, respectively. In second calving the corresponding heritabilities for the two traits were considerably lower, less than 1%. Adjusting for calving difficulty in linear analysis of stillbirth halved the heritabilities for the direct and maternal effects in first calving. When using a threshold model, heritabilities for stillbirth in first-calvers were 12% and 8% for direct and maternal effects, respectively, and for calving difficulty they were 17% and 12%. At second calving corresponding heritabilities were 2 to 4% for stillbirth and 4 to 7% for calving difficulty. The correlation between direct and maternal effects was around -0.1, irrespective of whether the linear or the threshold model was used for first-calvers. The genetic correlations between bulls' EBV from first and second calving were 0.4 to 0.5 for direct and maternal effects in stillbirth, whereas they were 0.6 to 0.7 for calving difficulty. In first-calvers there was a substantial genetic variation in both traits, expressed by differences between breeding values of bulls, despite fairly low heritability. The results obtained in this study suggest that first-parity records should preferably be used for genetic evaluation of bulls for calving performance. In such routine evaluations both stillbirth and calving difficulty, and both direct and maternal effects, should be included.
A bivariate censored threshold-linear model was used to study genetic parameters of number of services to conception (STC) and days open (DO) in first-lactation Norwegian Red (NRF) cows. Records of 1,454,916 NRF cows, with a first insemination from 1980 to 2004, were analyzed. It was assumed that every cow had at least a first insemination. The number of inseminations was recorded until a cow conceived or was culled, whichever occurred first. If a cow was culled before conception, it was considered censored at the number of services until culling was recorded. Twenty-one percent of cows were censored for both STC and DO. Using an univariate probit link function for STC, unobserved liabilities to STC and DO were modeled jointly as a linear function of age at first calving, month-year at first calving, herd-5-yr period, sire of cow and residual effects. Heritability of liability to STC and DO was 4% for each trait. The genetic and residual correlations between STC and DO were 0.77 and 0.68, respectively. There has been little or no genetic change for DO, whereas STC had favorable genetic and phenotypic trends.
Several approaches for analysis of survival in the first three lactations were compared using data from approximately 700,000 Canadian Holsteins. Two approaches (linear model and threshold model) were used to analyze a binary measure of survival. Other approaches were survival analyses to evaluate two measures of the number of days that cows were in milk during their first three lactations. One measure restricted days per lactation to < or = 305; the other was based on the actual number of days in milk without an upper limit on days per lactation. Variance components and breeding values (EBV) were estimated. Sire models were used almost exclusively, but one set of EBV was obtained using a linear animal model. Effects in the models were herd-year of calving, age at first calving, interaction of several factors related to herd, and production. Thus, all EBV were for functional herd life. Heritabilities were approximately 0.04, 0.07, and 0.10 from linear, threshold, and survival analyses, respectively. Correlations among sire EBV from all analyses using sire models were high, particularly for linear and threshold models (0.98). In contrast, correlations of EBV from sire models with EBV from the linear animal model were less than 0.90, regardless of the approach taken. In Canada, the current linear animal model remains in use for sire evaluation of herd life, but research with survival analyses will continue.
Randomized clinical trials often are planned to study a specific intervention. However, the collection of data on treatment actually received often reveals variable levels of treatment exposure (or 'dose') across subjects, due to non-compliance or other reasons. This paper presents a new method, using such 'dose' data as well as control group responses, to assess a causal dose-response relationship. The specific model utilizes a threshold function and incorporates a random effect term to allow for heterogeneous treatment responses among subjects. Further modelling of the random effects allows for reduction of error variance and control for potential confounders. The threshold dose is estimated using a residual variance criterion based on a transformed model. Estimates of standard errors and confidence intervals are obtained using a bootstrap procedure. The method is applied to data from an AIDS clinical trial. A simulation study demonstrates the adequacy of the threshold estimates for particular sample sizes and error variances. The limitations of this essentially exploratory method, as well as some possible extensions, are discussed. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government Work and is in the public domain in the United States.
1. A "threshold-crossing" motoneuron model was developed to relate recently described biophysical features of cat alpha-motoneurons to motoneuron discharge behavior. This model incorporated three features not included in precedent models: 1) a low-threshold, persistent calcium current; 2) realistic voltage dependencies of the major ionic conductances; and 3) a variable spike threshold. The effects of these additional biophysical features on model behavior were investigated by successively adding them to a fixed threshold model with a single potassium conductance. 2. Fixed-threshold models with either one or two potassium conductances could not produce appropriate discharge behavior. Steady-state frequency-current (F-I) relations were characterized by a continuously increasing slope, unlike the piecewise linear relations observed in real motoneurons. These models also produced unrealistically high discharge rates at the highest levels of "injected" current. 3. The addition of a variable spike threshold, which was made to increase linearly with the magnitude of injected current, could limit maximum discharge rates to more realistic levels. However, steady-state F-I relations still did not exhibit the appropriate shape. 4. The incorporation of a low-threshold calcium current led to a good quantitative agreement between the steady-state F-I relations produced by the model and those obtained in real motoneurons. In addition, the steady-state relation between total membrane current and membrane voltage (I-V relation) of the model was very similar to those measured in real motoneurons. The model's I-V and F-I relations were both very sensitive to the exact form of the steady-state relation between the magnitude of the calcium conductance and membrane voltage. 5. Additional modifications, which included a second calcium conductance and a factor relating spike threshold to membrane voltage, helped to produce more realistic afterhyperpolarizations and first-interval F-I relations. 6. Bistable discharge behavior could be produced by reducing the slow potassium conductance and increasing the time constants governing the activation and deactivation of the low-threshold calcium conductance. 7. The final model thus reproduces a wide range of motoneuron behaviors including subthreshold rectification, piecewise linear first interval and steady-state F-I relations, and, with appropriate modifications, bistable discharge behavior. Nonetheless, by simplifying the representation of fast spike conductances as well as the kinetics of the other ionic conductances, the model remains simple enough to be incorporated into a larger neural network.
In this paper, we consider selection based on the best predictor of animal additive genetic values in Gaussian linear mixed models, threshold models, Poisson mixed models, and log normal frailty models for survival data (including models with time-dependent covariates with associated fixed or random effects). In the different models, expressions are given (when these can be found - otherwise unbiased estimates are given) for prediction error variance, accuracy of selection and expected response to selection on the additive genetic scale and on the observed scale. The expressions given for non Gaussian traits are generalisations of the well-known formulas for Gaussian traits - and reflect, for Poisson mixed models and frailty models for survival data, the hierarchal structure of the models. In general the ratio of the additive genetic variance to the total variance in the Gaussian part of the model (heritability on the normally distributed level of the model) or a generalised version of heritability plays a central role in these formulas.
Formaldehyde inhalation at 6 ppm and above causes nasal squamous cell carcinoma (SCC) in F344 rats. The human health implications of this effect are of significant interest since human exposure to environmental formaldehyde is widespread, though at lower concentrations than those that cause cancer in rats. In this article, which is part of a larger effort to predict the human cancer risks of inhaled formaldehyde, we describe biologically motivated quantitative modeling of the exposure-tumor response continuum in the rat. An anatomically realistic, three-dimensional fluid dynamics model of the F344 rat nasal airways was used to predict site-specific flux of formaldehyde from inhaled air into tissue, since both SCC and preneoplastic lesions develop in a characteristic site-specific pattern. Flux into tissue was used as a dose metric for two modes of action, direct mutagenicity and cytolethality-regenerative cellular proliferation (CRCP), which in turn were linked to key parameters of a two-stage clonal growth model. The direct mutagenicity mode of action was represented by a low dose linear dose-response model of DNA-protein cross-link (DPX) formation. An empirical J-shaped dose-response model and a threshold model fit to the empirical data were used for CRCP. In the clonal growth model, the probability of mutation per cell generation was a function of the tissue concentration of DPX while the rate of cell division was calculated from the CRCP data. Maximum likelihood methods were used to estimate parameter values. Survivor (a nontumor outcome) and tumor data for controls from the National Toxicology Program database and from two formaldehyde inhalation bioassays were used for likelihood calculations. The J-shaped dose-response for CRCP provided a better description of the SCC data than did the threshold model. Sensitivity analyses indicated that the rodent tumor response is due to the CRCP mode of action, with the directly mutagenic pathway having little, if any, influence. When evaluated in light of modeling and database uncertainties, particularly the specification of the clonal growth model and the dose-response data for CRCP, this work provides suggestive though not definitive evidence for a J-shaped dose-response for formaldehyde-mediated nasal SCC in the F344 rat.
Traditional factorial designs for evaluating interactions among chemicals in a mixture may be prohibitive when the number of chemicals is large. Using a mixture of chemicals with a fixed ratio (mixture ray) results in an economical design that allows estimation of additivity or nonadditive interaction for a mixture of interest. This methodology is extended easily to a mixture with a large number of chemicals. Optimal experimental conditions can be chosen that result in increased power to detect departures from additivity. Although these designs are used widely for linear models, optimal designs for nonlinear threshold models are less well known. In the present work, the use of D-optimal designs is demonstrated for nonlinear threshold models applied to a fixed-ratio mixture ray. For a fixed sample size, this design criterion selects the experimental doses and number of subjects per dose level that result in minimum variance of the model parameters and thus increased power to detect departures from additivity. An optimal design is illustrated for a 2:1 ratio (chlorpyrifos:carbaryl) mixture experiment. For this example, and in general, the optimal designs for the nonlinear threshold model depend on prior specification of the slope and dose threshold parameters. Use of a D-optimal criterion produces experimental designs with increased power, whereas standard nonoptimal designs with equally spaced dose groups may result in low power if the active range or threshold is missed.
The search for efficient and powerful statistical methods and optimal mapping strategies for categorical traits under various experimental designs continues to be one of the main tasks in genetic mapping studies. Methodologies for genetic mapping of categorical traits can generally be classified into two groups, linear and non-linear models. We develop a method based on a threshold model, termed mixture threshold model to handle ordinal (or binary) data from multiple families. Monte Carlo simulations are done to compare its statistical efficiencies and properties of the proposed non-linear model with a linear model for genetic mapping of categorical traits using multiple families. The mixture threshold model has notably higher statistical power than linear models. There may be an optimal sampling strategy (family size vs number of families) in which genetic mapping reaches its maximal power and minimal estimation errors. A single large-sibship family does not necessarily produce the maximal power for detection of quantitative trait loci (QTL) due to genetic sampling of QTL alleles. The QTL allelic model has a marked impact on efficiency of genetic mapping of categorical traits in terms of statistical power and QTL parameter estimation. Compared with a fixed number of QTL alleles (two or four), the model with an infinite number of QTL alleles and normally distributed allelic effects results in loss of statistical power. The results imply that inbred designs (e.g. F2 or four-way crosses) with a few QTL alleles segregating or reducing number of QTL alleles (e.g. by selection) in outbred populations are desirable in genetic mapping of categorical traits using data from multiple families.
BACKGROUND: Hematologic toxicity after cancer chemotherapy and other drug effects that occur late compared to the exposure are usually modeled with use of some summary exposure variable such as the area under the concentration-time curve (AUC model) or the time of exposure above a threshold concentration (threshold model). An underlying assumption for both of these models is that the drug exerts a direct effect while present in the body and that it is the time integral of this direct effect that is related to the ultimate observed effect, either linearly (AUC model) or by a step function (threshold model). We propose a more general model that allows this relationship to be characterized by a nonlinear continuous function. METHODS: Data on survival fraction of neutrophiles and time course of leukopenia from 92 courses of paclitaxel therapy in 21 patients with breast or ovarian cancer was related to paclitaxel concentration-time profiles with the AUC, threshold, and general models. The properties of the general model were also investigated with use of simulations. RESULTS: For both pharmacodynamic end points, the general model described the data significantly better than the AUC or threshold models. CONCLUSION: The general model is an extension to the present way of relating concentration-time profiles to late-effect measures, and it may provide an improved description of the concentration-response relationship and more accurate predictions of the ultimate effect when doses and schedules are varied. It can explain complex relationships between concentration-time profiles and the observed effect, and predictions from it lack some of the counterintuitive properties that the AUC or threshold model have when extrapolations are made.
United States regulatory agencies use no-threshold models for estimating carcinogenic risks. Other countries use no-threshold models for carcinogens that are genotoxic and threshold models for carcinogens that are not genotoxic, such as 2, 3, 7, 8-tetrachlorodibenzo-p-dioxin (TCDD or "dioxin"). The U.S. Environmental Protection Agency has proposed a revision of the carcinogenic potency estimate for TCDD that is based on neither a threshold nor a no-threshold model; instead, it is a compromise between risk numbers generated by the two irreconcilably different models. This paper discusses the revision and its implications.
In commercial pig production, many important traits are recorded as binary phenotypes. For such traits, threshold models offer an appropriate framework but are computationally intensive. Thus, linear models are widely used to obtain genomic estimated breeding values (GEBV); however, these are on the observed scale (phenotypic). This creates the need for a robust method to approximate GEBV from linear models to the liability scale. A recently proposed approximation showed good concordance for low-prevalence traits (<5%) but has not yet been tested for a wider range of prevalence values and for models with more than one random effect. We aimed to evaluate the performance of this approximation for pig binary traits with prevalences ranging from <5% to >86%, in both animal and maternal animal models. Data were available for five fitness traits (FT1-FT5), with up to 233k animals with phenotypes, of which 204k animals were genotyped with a 25k SNP array. Variance component estimates were obtained using threshold models. Classical animal models were used for FT1-FT3, and maternal animal models for FT4 and FT5. Variance components on the observed scale were then obtained by multiplying estimates from a threshold model by the square of the height of the standard normal density evaluated at the threshold. GEBV were predicted using single-step genomic best linear unbiased prediction under both linear and threshold models. The approximation tested involved scaling the GEBV using the height of the ordinate of the standard normal distribution evaluated at the threshold as a scaling factor. The agreement between GEBV from the scaled linear model and the threshold model on the probability scale was evaluated using Pearson and Spearman correlations, mean squared error (MSE), regression parameters, overlapping coefficient (OVL), distribution overlap, and classification accuracy (CACC). Correlations between linear and threshold GEBV ranged from 0.94 (low-prevalence traits) to 0.99 (high-prevalence traits) for the direct GEBV and were 0.99 for the maternal GEBV. MSE were close to zero. The OVL exceeded 0.83 for all traits. CACC ranged from 95.10% to 98.33% for the direct GEBV and from 92.54% to 97.42% for the maternal GEBV. Regardless of model and trait prevalence, this approximation yielded GEBV that are highly consistent with threshold model GEBV, providing a reliable, practical approach for large-scale pig genetic evaluations for binary traits using linear models.
Threshold measures have been made as a function of the repetition rate and pulse duration of biphasic electrical pulses applied to the cochlea through a cochlear implant (Shannon, 1985). Nonmonotonicities in those data suggest that at least two separate processes are involved in the translation of an electrical stimulus into a threshold perception. This paper presents a phenomenological model which accounts for the key features of the threshold data. The model consists of two parallel processes which are each power-law functions of the instantaneous current amplitude. The output of each process is then integrated with a short time constant (approximately 1-2 ms). The maximum of these two outputs represents the sensory magnitude of that electrical stimulus. Threshold data from 14 patients implanted with three different devices are compared to model predictions over a wide range of pulse durations and pulse rates. Since the model accurately predicts thresholds over such a wide range of stimuli, it is possible that it can predict the threshold of an arbitrary electrical stimulus. This model could be used to construct a speech processor that would convert any acoustic waveform into an equivalent electrical waveform that would preserve threshold relationships.