Regression imputation of missing values in longitudinal data sets

A stand-alone, menu-driven PC program, written in GAUSS, which can be used to estimate missing observations in longitudinal data sets is described and made available to interested readers. The program is limited to the situation in which we have complete data on N cases at each of the planned times of measurement t₁, t₂,…, t_T; and we wish to use this information, together with the non-missing values for n additional cases, to estimate the missing values for those cases. The augmented data matrix may be saved in an ASCII file and subsequently imported into programs requiring complete data. The use of the program is illustrated. Ten percent of the observations in a data set consisting of mandibular ramus height measurements for N = 12 young male rhesus monkeys measured at T = 5 time points are randomly discarded. The augmented data matrix is used to determine the lowest degree polynomial adequate to fit the average growth curve (AGC); the regression coefficients are estimated and confidence intervals for them are determined; and confidence bands for the AGC are constructed. The results are compared with those obtained when the original complete data set is used.     

A global goodness-of-fit statistic for Cox regression models

In this paper, a global goodness-of-fit test statistic for a Cox regression model, which has an approximate chi-squared distribution when the model has been correctly specified, is proposed. Our goodness-of-fit statistic is global and has power to detect if interactions or higher order powers of covariates in the model are needed. The proposed statistic is similar to the Hosmer and Lemeshow (1980, Communications in Statistics A10, 1043-1069) goodness-of-fit statistic for binary data as well as Schoenfeld's (1980, Biometrika 67, 145-153) statistic for the Cox model. The methods are illustrated using data from a Mayo Clinic trial in primary billiary cirrhosis of the liver (Fleming and Harrington, 1991, Counting Processes and Survival Analysis), in which the outcome is the time until liver transplantation or death. The are 17 possible covariates. Two Cox proportional hazards models are fit to the data, and the proposed goodness-of-fit statistic is applied to the fitted models.     

Can the past predict the future? Experimental tests of historically based population models

A frequently advocated approach for forecasting the population‐level impacts of climate change is to project models based on historical, observational relationships between climate and demographic rates. Despite the potential pitfalls of this approach, few historically based population models have been experimentally validated. We conducted a precipitation manipulation experiment to test population models fit to observational data collected from the 1930s to the 1970s for six prairie forb species. We used the historical population models to predict experimental responses to the precipitation manipulations, and compared these predictions to ones generated by a statistical model fit directly to the experimental data. For three species, a sensitivity analysis of the effects of precipitation and grass cover on forb population growth showed consistent results for the historical population models and the contemporary statistical models. Furthermore, the historical population models predicted population growth rates in the experimental plots as well or better than the statistical models, ignoring variation explained by spatial random effects and local density‐dependence. However, for the remaining three species, the sensitivity analyses showed that the historical and statistical models predicted opposite effects of precipitation on population growth, and the historical models were very poor predictors of experimental responses. For these species, historical observations were not well replicated in space, and for two of them the historical precipitation‐demography correlations were weak. Our results highlight the strengths and weaknesses of observational and experimental approaches, and increase our confidence in extrapolating historical relationships to predict population responses to climate change, at least when the historical correlations are strong and based on well‐replicated observations.     

Kinetics of substrate inhibition of exponential yeast growth

This work concerns mathematical modeling of the rate of microbial growth on inhibitory levels of nutrients as affected by pH, concentration of the nutrients, temperature, cultivation method, and method of data analysis. Candida utilis (ATCC 9226) was grown with sodium acetate as growth-limiting carbon and energy source in mineral salts medium in shake flask and continuous cultures to study inhibition by excess acetate. Differential shake flask cultures were grown at low yeast concentrations at temperatures (T) of 25 and 30°C, pH's between 5.5 and 7.0, and acetate concentrations (S) between 0.25 and 3.0% (w/v). Growth data were exponential with correlation coefficients greater than 0.995 in 49 of 56 experiments; the lowest correlation coefficient was 0.986. Specific growth rates (μ) determined by graphical methods showed only fair correlation with those determined by regression analysis. Both sets of specific growth rate data were grouped at constant T and pH and fitted to the three parameter equation, The improvement in use of the fitted equation instead of the mean value was significant with greater than 70% confidence in all (14 groups) and 90% confidence in only half of the data groups; the correlation does not improve with the increasing acetate inhibition at lower pH. Both defects in the model and insufficient data at each pH are responsible. A modified six parameters with hydrogen ion concentration(H⁺) as follows: Specific growth rates calculated with the six parameter equation matched observed values in all groups of isothermal data better than the means with greater than 99% confidence. The six parameter model adequately represents effects of acetate and hydrogen ion concentrations under constant or slowly changing environmental conditions and balanced growth; although better models probably exist. Thus steady-stste and transient continuous culture experiments agreed with many published growth yields, but specific growth rates could only be predicted qualitatively from the model fit to the shake flask data. The data and present models could be incorporated into published models for transient growth at low nutrient concentrations to correlate and perhaps predict microbial growth kinetics over a much wider range of growth conditions than now possible.     

Modelling the combined effect of temperature,pH and aw on the growth rate of Monascus ruber,a heat-resistant fungus isolated from green table olives

AIMS: Growth modes predicting the effect of pH (3.5-5.0), NaCl (2-10%), i.e. aw (0.937-0.970) and temperature (20-40 degrees C) on the colony growth rate of Monascus ruber, a fungus isolated from thermally-processed olives of the Conservolea variety, were developed on a solid culture medium. METHODS AND RESULTS: Fungal growth was measured as colony diameter on a daily basis. The primary predictive model of Baranyi was used to fit the growth data and estimate the maximum specific growth rates. Combined secondary predictive models were developed and comparatively evaluated based on polynomial, Davey, gamma concept and Rosso equations. The data-set was fitted successfully in all models. However, models with biological interpretable parameters (gamma concept and Rosso equation) were highly rated compared with the polynomial equation and Davey model and gave realistic cardinal pHs, temperatures and aw. CONCLUSIONS: The combined effect of temperature, pH and aw on growth responses of M. ruber could be satisfactorily predicted under the current experimental conditions, and the models examined could serve as tools for this purpose. SIGNIFICANCE AND IMPACT OF THE STUDY: The results can be successfully employed by the industry to predict the extent of fungal growth on table olives.     

A General and Dynamic Species Abundance Model, Embracing the Lognormal and the Gamma Models

One aspect of community ecology that has been given particular attention is the pattern of species abundances in a community. The species may have a wide range of abundances; some are very common and others rare. When species abundance models are fitted to observations, the lognormal model and one of the gamma models (e.g., the log-series model) are usually applied. The model that gives the best fit according to some goodness-of-fit test is then chosen. By applying a diffusion approximation for each species' dynamics with density regulation of the straight theta-logistic type, we here present a general species abundance model that embraces the two most widely applied species abundance models, the lognormal and the gamma. Our general model will, therefore, provide a better fit than the two special cases, except when it corresponds to one of them. In contrast to the classical models, ours is also dynamic, making it possible to evaluate the fluctuations in species abundance over time through both biotic and abiotic factors. The model is fitted to several species abundance data sets and our results compared to previous attempts to fit a model, usually either the lognormal or the log-series.     

Growth curve models of repeated binary response

Experimental designs that include repeated measures of binary response variables over time and under different conditions are common in biology. In such settings, it is often desirable to characterize the response pattern over time. When response variables are continuous, this characterization can be made in terms of a growth model such as the Potthoff-Roy growth curve model. We illustrate how a similar growth curve modeling strategy can be implemented using weighted least squares (WLS) methods for binary response data. The growth models are constructed in terms of polynomial functions across marginal response. However, when growth models are fit to repeated binary response, the nonsignificant higher-order polynomial functions are dropped from the model, rather than used as covariates. Dropping the nonsignificant polynomials from the model will reduce the number of response functions, and help avoid small-sample problems that can occur when the number of correlated response functions is large and sample sizes are small. The reduced set of response functions are then modeled using WLS methods. We illustrate such models with an example of binary fly oviposition response (accept or reject) exhibited by two populations of flies at four ages to two types of fruit.     

Heterochrony and sexual dimorphism in the pigtailed macaque (Macaca nemestrina)

Somatic growth is not a simple linear process with a constant rate of growth. The most successful attempts to quantify growth as a function of age or size have employed nonlinear techniques. Sexual dimorphism of primate growth, weight vs. age, was examined using nonlinear models with Sirianni and Swindler's ([1985] Growth and Development of the Pigtailed Macaque, Boca Raton, FL: CRC Press) growth data on the pigtailed macaque (Macaca nemestrina). The best fit of several exponential growth models was the Gompertz curve: Different multiple phase models were also fit, where each phase represents a distinct exponential component. The two-phase models proved to be the best (R² = .0.84 for females, 0.91 for males), suggesting that there are two growth spurts, one in infancy and one at puberty. The timing of the beginning and end of the first spurt is the same in males and females, but the rate, and value of the asymptote for this phase, is greater in males. The timing of the second spurt is earlier, and the rate of growth for this spurt is smaller in females than males. The sexual dimorphism in these species is not a simple rate change, but a complex interaction of timing and rate over the entire period of growth. It would be impossible to separate these entities with a linear, polynomial, or single-phase model of the data. While these data and results complement much of the existing work on adult dimorphism, they also emphasize the vital role that ontogenetic data have in elucidating the underlying evolutionary mechanisms that generate sexual dimorphism. © 1994 Wiley-Liss, Inc.     

The application of robust calibration to radioimmunoassay.

The minute concentrations of many biochemically and clinically important substances are currently estimated by radioimmunoassay (RIA). Traditionally, the most popular approaches to the statistical analysis of RIA data have been to linearize the data through transformation and fit the calibration curve using least squares or to directly fit a nonlinear calibration curve using least squares. Estimates of the hormone concentration in patients are then obtained using this curve. Unfortunately, the transformation is frequently unsuccessful in linearizing the data. Furthermore, the least squares fit can lead to erroneous results in both approaches since the many sources of error which exist in the RIA process often result in outlier observations. In this paper, an approach to the analysis of RIA data which incorporates robust estimation methods is described. An algorithm is presented for obtaining the M-estimates of nonlinear calibration curves. The curves to be fitted are modified hyperbolae based on 12 to 16 observations. A procedure, based on the application of the Bonferroni Inequality, is presented for obtaining tolerance-like interval estimates of the concentration of the hormone of interest in the patients. Results of simulations are cited to support the method of construction of confidence bands for the fitted calibration curve. Data obtained from the Veteran's Hospital, Buffalo, New York are used to illustrate the application of the algorithm which is presented.     

On a likelihood-based goodness-of-fit test of the beta-binomial model

When faced with proportion data that exhibit extra-binomial variation, data analysts often consider the beta-binomial distribution as an alternative model to the more common binomial distribution. A typical example occurs in toxicological experiments with laboratory animals, where binary observations on fetuses within a litter are often correlated with each other. In such instances, it may be of interest to test for the goodness of fit of the beta-binomial model; this effort is complicated, however, when there is large variability among the litter sizes. We investigate a recent goodness-of-fit test proposed by Brooks et al. (1997, Biometrics 53, 1097-1115) but find that it lacks the ability to distinguish between the beta-binomial model and some severely non-beta-binomial models. Other tests and models developed in their article are quite useful and interesting but are not examined herein.     

Noninvasive determination of upper airway resistance and flow limitation.

We have shown that a polynomial equation, FP = AP3 + BP2 + CP + D, where F is flow and P is pressure, can accurately determine the presence of inspiratory flow limitation (IFL). This equation requires the invasive measurement of supraglottic pressure. We hypothesized that a modification of the equation that substitutes time for pressure would be accurate for the detection of IFL and allow for the noninvasive measurement of upper airway resistance. The modified equation is Ft = At3 + Bt2 + Ct + D, where F is flow and t is time from the onset of inspiration. To test our hypotheses, data analysis was performed as follows on 440 randomly chosen breaths from 18 subjects. First, we performed linear regression and determined that there is a linear relationship between pressure and time in the upper airway (R2 0.96 +/- 0.05, slope 0.96 +/- 0.06), indicating that time can be a surrogate for pressure. Second, we performed curve fitting and found that polynomial equation accurately predicts the relationship between flow and time in the upper airway (R2 0.93 +/- 0.12, error fit 0.02 +/- 0.08). Third, we performed a sensitivity-specificity analysis comparing the mathematical determination of IFL to manual determination using a pressure-flow loop. Mathematical determination had both high sensitivity (96%) and specificity (99%). Fourth, we calculated the upper airway resistance using the polynomial equation and compared the measurement to the manually determined upper airway resistance (also from a pressure-flow loop) using Bland-Altman analysis. Mean difference between calculated and measured upper airway resistance was 0.0 cmH2O x l(-1) x s(-1) (95% confidence interval -0.2, 0.2) with upper and lower limits of agreement of 2.8 cmH2O x l(-1) x s(-1) and -2.8 cmH2O x l(-1) x s(-1). We conclude that a polynomial equation can be used to model the flow-time relationship, allowing for the objective and accurate determination of upper airway resistance and the presence of IFL.     

Moment arms and musculotendon lengths estimation for a three-dimensional lower-limb model

This paper presents a set of polynomial expressions that can be used as regression equations to estimate length and three-dimensional moment arms of 43 lower-limb musculotendon actuators. These equations allow one to find, at a low computational cost, the musculotendon geometric parameters required for numerical simulation of large musculoskeletal models. Nominal values for these biomechanical parameters were established using a public-domain musculoskeletal model of the lower limb (IEEE Trans. Biomed. Eng. 37 (1990) 757). To fit these nominal values, regression equations with different levels of complexity were generated, based on the number of generalized coordinates of the joints spanned by each musculotendon actuator. Least squares fitting was used to identify regression equation coefficients. The goodness of the fit and confidence intervals were assessed, and the best fitting equations selected.     

The role of fiber-matrix interactions in a nonlinear fiber-reinforced strain energy model of tendon

The objective of this study was to develop a nonlinear and anisotropic three-dimensional mathematical model of tendon behavior in which the structural components of fibers, matrix, and fiber-matrix interactions are explicitly incorporated and to use this model to infer the contributions of these structures to tendon mechanical behavior. We hypothesized that this model would show that: (i) tendon mechanical behavior is not solely governed by the isotropic matrix and fiber stretch, but is also influenced by fiber-matrix interactions; and (ii) shear fiber-matrix interaction terms will better describe tendon mechanical behavior than bulk fiber-matrix interaction terms. Model versions that did and did not include fiber-matrix interaction terms were applied to experimental tendon stress-strain data in longitudinal and transverse orientations, and the R2 goodness-of-fit was evaluated. This study showed that models that included fiber-matrix interaction terms improved the fit to longitudinal data (R2(toe) = 0.88, R2(Lin) = 0.94) over models that only included isotropic matrix and fiber stretch terms (R2(Toe) = 0.36, R2(Lin) = 0.84). Shear fiber-matrix interaction terms proved to be responsible for the best fit to data and to contribute to stress-strain nonlinearity. The mathematical model of tendon behavior developed in this study showed that fiber-matrix interactions are an important contributor to tendon behavior The more complete characterization of mechanical behavior afforded by this mathematical model can lead to an improved understanding of structure-function relationships in soft tissues and, ultimately, to the development of tissue-engineered therapies for injury or degeneration.     

A deterministic model for monophasic growth of batch cultures of bacteria

Experimental observations of bacterial numbers employing high resolution electrical conductance measurements of the culture provide the basis for a proposed deterministic model of monophasic growth of populations in batch culture. The model postulates that the production and growth of each bacterium is accompanied by the generation of a constant mass of toxic end-products and that specific growth rate declines in proportion to the ratio of the accumulated mass of these substances to the dry mass of the nutrient medium when the substrate is non-limiting. The theoretical relationship is found to fit extensive data for Escherichia coli (NCIB 9132) very closely and offers an analytical basis for the logistic curve frequently observed to represent the time-dependence of growth. These data incidentally provide substantial evidence that lag time and generation time are each independent of both inoculum number and concentration of the medium.     

苹果果实生长的数学模型及各生长指标间的相关性分析

以富士苹果(Malus domestica‘Fuji’)为试材,测定其果实生长发育期间各生长指标的动态变化,选择5种理论生长方程对纵径、横径、单果重、体积、干重进行拟合,并根据拟合结果确定合适的生长方程建立各生长指标的数学模型,采用多项式拟合建立果形指数变化的数学模型,同时对果实各生长指标之间进行相关性分析。结果表明,果实纵径、横径生长适合选择Logistic方程,单果重、体积、干重适合选择Gompertz方程,果形指数的变化适合采用多项式拟合;果实纵径、横径、单果重、体积、干重两两之间均呈显著正相关,果形指数、果实干物质相对含量均与纵径、横径、单果重、体积、干重之间呈显著负相关,果实密度与纵径、横径、单果重、体积、干重之间呈显著负相关,而与果形指数、果实干物质相对含量之间呈显著正相关。     

Polynomials to model the growth of young bulls in performance tests

The use of polynomial functions to describe the average growth trajectory and covariance functions of Nellore and MA (21/32 Charolais+11/32 Nellore) young bulls in performance tests was studied. The average growth trajectories and additive genetic and permanent environmental covariance functions were fit with Legendre (linear through quintic) and quadratic B-spline (with two to four intervals) polynomials. In general, the Legendre and quadratic B-spline models that included more covariance parameters provided a better fit with the data. When comparing models with the same number of parameters, the quadratic B-spline provided a better fit than the Legendre polynomials. The quadratic B-spline with four intervals provided the best fit for the Nellore and MA groups. The fitting of random regression models with different types of polynomials (Legendre polynomials or B-spline) affected neither the genetic parameters estimates nor the ranking of the Nellore young bulls. However, fitting different type of polynomials affected the genetic parameters estimates and the ranking of the MA young bulls. Parsimonious Legendre or quadratic B-spline models could be used for genetic evaluation of body weight of Nellore young bulls in performance tests, whereas these parsimonious models were less efficient for animals of the MA genetic group owing to limited data at the extreme ages.     

The Distribution of Ocular Chlamydia Prevalence across Tanzanian Communities Where Trachoma Is Declining

BackgroundMathematical models predict an exponential distribution of infection prevalence across communities where a disease is disappearing. Trachoma control programs offer an opportunity to test this hypothesis, as the World Health Organization has targeted trachoma for elimination as a public health concern by the year 2020. Local programs may benefit if a single survey could reveal whether infection was headed towards elimination. Using data from a previously-published 2009 survey, we test the hypothesis that Chlamydia trachomatis prevalence across 75 Tanzanian communities where trachoma had been documented to be disappearing is exponentially distributed.Methods/FindingsWe fit multiple continuous distributions to the Tanzanian data and found the exponential gave the best approximation. Model selection by Akaike Information Criteria (AIC_c) suggested the exponential distribution had the most parsimonious fit to the data. Those distributions which do not include the exponential as a special or limiting case had much lower likelihoods of fitting the observed data. 95% confidence intervals for shape parameter estimates of those distributions which do include the exponential as a special or limiting case were consistent with the exponential. Lastly, goodness-of-fit testing was unable to reject the hypothesis that the prevalence data came from an exponential distribution.ConclusionsModels correctly predict that infection prevalence across communities where a disease is disappearing is best described by an exponential distribution. In Tanzanian communities where local control efforts had reduced the clinical signs of trachoma by 80% over 10 years, an exponential distribution gave the best fit to prevalence data. An exponential distribution has a relatively heavy tail, thus occasional high-prevalence communities are to be expected even when infection is disappearing. A single cross-sectional survey may be able to reveal whether elimination efforts are on-track.     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.