Prediction Intervals in Nonlinear Regression

By treating the nonlinear model as if it were linear in the parameterization θ in the neighbourhood of the least squares estimate θ, we construct two-sided nominally-q-prediction intervals by applying the usual linear model theory. The derivation of the truncated series expansion of the expected coverage of the prediction intervals at a feasible value of the parameter vector is described. The quadratic approximation of the expected coverage is then obtained for a two-parameter nonlinear model. Finally we show how we may construct the prediction intervals when a certain type of nonlinear transformation of the parameter vector has been applied.     

Approximate Confidence Intervals for Contrasts in the Balanced Three Factor Mixed Model

When conducting a statistical analysis of data from a designed experiment, an investigator is often interested in confidence intervals for contrasts of the fixed effects. If the analysis involves a mixed linear model, exact confidence intervals for contrasts of the fixed effects are not always available. In such cases, confidence intervals with approximate coverage probabilities must be used. As will be shown, this problem may be generalized to that of constructing a confidence interval for the parameter μ, where X is a normal random variable with mean μ and variance ∑ a_qθ_q, where a₁…,a_Q are known constants, U_q = n_qS/θ_q is a chi-squared random variable with n_q degrees of freedom, for each q = 1,…, Q, and X,U₁,…, U_Q are mutually independent. In this paper, we consider the case where Q = 3 and a₃ ≤0.     

不同生态修复措施和环境因素对亚热带红壤区针叶纯林坡面水土保持功能的影响

使用"地理探测器（GeoDetector）"对亚热带红壤区水土流失影响因素的定量分析结果可为当地森林生态修复和侵蚀模型完善提供科学依据。基于福建省龙岩市新罗区龙门溪小流域森林调查数据和径流小区监测数据,利用地理探测器探测不同生态修复措施和环境因素对针叶纯林坡面水土保持功能的影响及交互作用,结果表明：（1）对比中幼龄针叶纯林,补植阔叶树使针阔混交比例为7 ： 2可减少46%的径流量和76%的泥沙量,生态修复效果较好。对重侵蚀区"老头树"少量施肥难以产生效果。（2）影响坡面径流的因素由强到弱依次是：降雨因子（0.53）,土壤容重、林分密度、灌草层盖度、树高和针阔比（均在0.08左右）;影响泥沙流失的因素依次是：地表径流量（0.84）,降雨因子（0.2）,林分密度、土壤容重、灌草层盖度、土壤含水率、灌草层生物量（均在0.12左右）。（3）各影响因素交互后主要呈增强作用;林分密度、灌草层盖度和土壤容重还可与其他因子产生强烈非线性增强作用（交互后影响力>0.9）,是在森林修复和模型参数优化时需重点关注的对象。     

Variability of the electric organ discharge interval duration in resting Gymnotus carapo

 We recorded the electric organ discharges of resting Gymnotus carapo specimens. We analyzed the time series formed by the sequence of interdischarge intervals. Nonlinear prediction, false nearest neighbor analyses, and comparison between the performance of nonlinear and linear autoregressive models fitted to the data indicated that nonlinear correlations between intervals were absent, or were present to a minor extent only. Following these analyses, we showed that linear autoregressive models with combined Gaussian and shot noise reproduced the variability and correlations of the resting discharge pattern. We discuss the implications of our findings for the mechanisms underlying the timing of electric organ discharge generation. We also argue that autoregressive models can be used to evaluate the changes arising during a wide variety of behaviors, such as the modification in the discharge intervals during interaction between fish pairs. Received: 14 March 2000 / Accepted in revised form: 9 October 2000     

UNDESIRABLE CONSEQUENCES OF NEGLECTING NONLINEARITY: RESPONSE TO COMMENTS BY LIEFTING ET AL. (2013) ON ROCHA & KLACZKO (2012)

In response to our previous study, Liefting et al. argue, in defense of their work on latitudinal variation of developmental‐rate reaction norms (RNs), that (1) developmental rate (the reciprocal of development time: rate = time^?1) is a more biologically relevant variable than development time; (2) the linear RN model is a valid approximation; and (3) three experimental points suffice to estimate RN parameters. Here, we reply to their comments. First, we give evidence that the complexity of actual development challenges the appealing simplicity of developmental rate. Using the same analysis as Liefting et al. to test their hypothesis with development time, instead of rate, reveals a pattern that is the opposite of their conclusion. Second, we show that a quadratic model is consistent with the whole development‐time RNs and explains this contradiction. Third, with the quadratic model, we introduce two parameters to study plasticity: the RN shape (the quadratic coefficient) and RN local plasticity (the derivative of the RN function). The first showed a statistically significant correlation with latitude; and the second showed a continuous variation pattern where all localized patterns can be found (positive, negative, or nonsignificant correlations with latitude) but certainly cannot be generalized.     

Correlation of antibody production rate with glucose and lactate metabolism in Chinese hamster ovary cells

A linear relationship was found between the antibody production rate (q _mAb) and the glucose and lactate consumption rate (q _GL) in Chinese hamster ovary cells. Under a series of q _mAb-perturbing conditions, q _GL was determined and a linear relationship between q _mAb and q _GL was further established (R ²  = 0.914). Mitochondrial dehydrogenase activity was monitored in all the q _mAb-perturbing conditions and showed a linear correlation with q _GL (R ² = 0.874) as well as with q _mAb. Taken collectively, our results establish that the metabolic parameter, q _GL, is linearly correlated with q _mAb; this finding strengthens our current understanding of process optimization for antibody production.     

Dynamic gene expression regulation model for growth and penicillin production in Penicillium chrysogenum

As is often the case for microbial product formation, the penicillin production rate of Penicillium chrysogenum has been observed to be a function of the growth rate of the organism. The relation between the biomass specific rate of penicillin formation (q_p) and growth rate (µ) has been measured under steady state conditions in carbon limited chemostats resulting in a steady state q_p(µ) relation. Direct application of such a relation to predict the rate of product formation during dynamic conditions, as they occur, for example, in fed‐batch experiments, leads to errors in the prediction, because q_p is not an instantaneous function of the growth rate but rather lags behind because of adaptational and regulatory processes. In this paper a dynamic gene regulation model is presented, in which the specific rate of penicillin production is assumed to be a linear function of the amount of a rate‐limiting enzyme in the penicillin production pathway. Enzyme activity assays were performed and strongly indicated that isopenicillin‐N synthase (IPNS) was the main rate‐limiting enzyme for penicillin‐G biosynthesis in our strain. The developed gene regulation model predicts the expression of this rate limiting enzyme based on glucose repression, fast decay of the mRNA encoding for the enzyme as well as the decay of the enzyme itself. The gene regulation model was combined with a stoichiometric model and appeared to accurately describe the biomass and penicillin concentrations for both chemostat steady‐state as well as the dynamics during chemostat start‐up and fed‐batch cultivation. Biotechnol. Bioeng. 2010;106: 608–618. © 2010 Wiley Periodicals, Inc.     

Systematic designs for experiments with mixtures

In this paper systematic designs for experiments with mixtures are developed. The plan of analysis of the experiment is discussed for a quadratic model of SCHEFF É (1958) for q-component mixture with orthogonal polynomials of third degree describing the time trends.     

Sharp simultaneous confidence intervals for the means of selected populations with application to microarray data analysis

Summary Simultaneous inference for a large number, N, of parameters is a challenge. In some situations, such as microarray experiments, researchers are only interested in making inference for the K parameters corresponding to the K most extreme estimates. Hence it seems important to construct simultaneous confidence intervals for these K parameters. The naïve simultaneous confidence intervals for the K means (applied directly without taking into account the selection) have low coverage probabilities. We take an empirical Bayes approach (or an approach based on the random effect model) to construct simultaneous confidence intervals with good coverage probabilities. For N= 10,000 and K= 100, typical for microarray data, our confidence intervals could be 77% shorter than the naïve K‐dimensional simultaneous intervals.     

On Confidence Interval Construction for Establishing Equivalence of Two Binary-Outcome Treatments in Matched-Pair Studies in the Presence of Incomplete Data

Matched-pair design is often adopted in equivalence or non-inferiority trials to increase the efficiency of binary-outcome treatment comparison. Briefly, subjects are required to participate in two binary-outcome treatments (e.g., old and new treatments via crossover design) under study. To establish the equivalence between the two treatments at the α significance level, a (1−α)100% confidence interval for the correlated proportion difference is constructed and determined if it is entirely lying in the interval (−δ ₀,δ ₀) for some clinically acceptable threshold δ ₀ (e.g., 0.05). Nonetheless, some subjects may not be able to go through both treatments in practice and incomplete data thus arise. In this article, a hybrid method for confidence interval construction for correlated rate difference is proposed to establish equivalence between two treatments in matched-pair studies in the presence of incomplete data. The basic idea is to recover variance estimates from readily available confidence limits for single parameters. We compare the hybrid Agresti–Coull, Wilson score and Jeffreys confidence intervals with the asymptotic Wald and score confidence intervals with respect to their empirical coverage probabilities, expected confidence widths, ratios of left non-coverage probability, and total non-coverage probability. Our simulation studies suggest that the Agresti–Coull hybrid confidence intervals is better than the score-test-based and likelihood-ratio-based confidence interval in small to moderate sample sizes in the sense that the hybrid confidence interval controls its true coverage probabilities around the pre-assigned coverage level well and yield shorter expected confidence widths. A real medical equivalence trial with incomplete data is used to illustrate the proposed methodologies.     

Predicting the light attenuation coefficient through Secchi disk depth and beam attenuation coefficient in a large, shallow, freshwater lake

The diffuse attenuation coefficient of photosynthetically active radiation (PAR) (400–700 nm) (K _d(PAR)) is one of the most important optical properties of water. Our purpose was to create K _d(PAR) prediction models from the Secchi disk depth (SDD) and beam attenuation coefficient of particulate and dissolved organic matter (C _t−w(PAR), excluding pure water) in the PAR range. We compare their performance and prediction precision by using the determination coefficient (r ²), relative root mean square error (RRMSE), and mean relative error (MRE). Our dataset comprised 1,067 measurements, including K _d(PAR), SDD, and C _t−w(PAR) taken in shallow, eutrophic, Lake Taihu, China, from 2005 to 2010. The prediction models of K _d(PAR) were based on the linear model with an intercept of zero, using the inverse SDD, and the nonlinear model using SDD. The linear model generated a slope of 1.369, which was not significantly different from 1.7, the index used worldwide, but significantly lower than the value of 2.26. The nonlinear model gave a slightly more reliable prediction of K _d(PAR) with a r ² of 0.804. Compared to the SDD, C _t−w(PAR) was more significantly correlated to K _d(PAR) based on the linear model, with a significantly higher r ² and lower RMSE and RE. Considering the measurement simplicity of C _t−w(PAR) and data acquisition feasibility from high-frequency autonomous buoys and satellites, our results demonstrated that this prediction model reliably estimates K _d(PAR), and could be used to significantly expand optical observations in an environment where the conditions for underwater PAR measurement are limited.     

Generalized prediction intervals for treatment effects in random‐effects models

This article derives generalized prediction intervals for random effects in linear random‐effects models. For balanced and unbalanced data in two‐way layouts, models are considered with and without interaction. Coverage of the proposed generalized prediction intervals was estimated in a simulation study based on an agricultural field experiment. Generalized prediction intervals were compared with prediction intervals based on the restricted maximum likelihood (REML) procedure and the approximate methods of Satterthwaite and Kenward and Roger. The simulation study showed that coverage of generalized prediction intervals was closer to the nominal level 0.95 than coverage of prediction intervals based on the REML procedure.     

Selective prediction-set models with coverage rate guarantees

The current approach to using machine learning (ML) algorithms in healthcare is to either require clinician oversight for every use case or use their predictions without any human oversight. We explore a middle ground that lets ML algorithms abstain from making a prediction to simultaneously improve their reliability and reduce the burden placed on human experts. To this end, we present a general penalized loss minimization framework for training selective prediction-set (SPS) models, which choose to either output a prediction set or abstain. The resulting models abstain when the outcome is difficult to predict accurately, such as on subjects who are too different from the training data, and achieve higher accuracy on those they do give predictions for. We then introduce a model-agnostic, statistical inference procedure for the coverage rate of an SPS model that ensembles individual models trained using K-fold cross-validation. We find that SPS ensembles attain prediction-set coverage rates closer to the nominal level and have narrower confidence intervals for its marginal coverage rate. We apply our method to train neural networks that abstain more for out-of-sample images on the MNIST digit prediction task and achieve higher predictive accuracy for ICU patients compared to existing approaches.     

Five Confidence Intervals of the Closed Population Size in the Capture‐recapture Problem under Inverse Sampling with Replacement

In the capture‐recapture problem for two independent samples, the traditional estimator, calculated as the product of the two sample sizes divided by the number of sampled subjects appearing commonly in both samples, is well known to be a biased estimator of the population size and have no finite variance under direct or binomial sampling. To alleviate these theoretical limitations, the inverse sampling, in which we continue sampling subjects in the second sample until we obtain a desired number of marked subjects who appeared in the first sample, has been proposed elsewhere. In this paper, we consider five interval estimators of the population size, including the most commonly‐used interval estimator using Wald's statistic, the interval estimator using the logarithmic transformation, the interval estimator derived from a quadratic equation developed here, the interval estimator using the χ²‐approximation, and the interval estimator based on the exact negative binomial distribution. To evaluate and compare the finite sample performance of these estimators, we employ Monte Carlo simulation to calculate the coverage probability and the standardized average length of the resulting confidence intervals in a variety of situations. To study the location of these interval estimators, we calculate the non‐coverage probability in the two tails of the confidence intervals. Finally, we briefly discuss the optimal sample size determination for a given precision to minimize the expected total cost. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Threshold and other departures from linear-quadratic curvature in the non-cancer mortality dose-response curve in the Japanese atomic bomb survivors

Recently released data on non-cancer mortality in Japanese atomic bomb survivors are analysed using a variety of generalised relative risk models that take account of errors in estimates of dose to assess the dose-response at low doses. If linear-threshold, quadratic-threshold or linear-quadratic-threshold relative risk models (the dose-response is assumed to be linear, quadratic or linear-quadratic above the threshold, respectively) are fitted to the non-cancer data there are no statistically significant (p>0.10) indications of threshold departures from linearity, quadratic curvature or linear-quadratic curvature. These findings are true irrespective of the assumed magnitude of dosimetric error, between 25%–45% geometric standard deviations. In general, increasing the assumed magnitude of dosimetric error had little effect on the central estimates of the threshold, but somewhat widened the associated confidence intervals. If a power of dose model is fitted, there is little evidence (p>0.10) that the power of dose in the dose-response is statistically significantly different from 1, again irrespective of the assumed magnitude of dosimetric errors in the range 25%–45%. Again, increasing the size of the errors resulted in wider confidence intervals on the power of dose, without marked effect on the central estimates. In general these findings remain true for various non-cancer disease subtypes.     

Prediction of Higher Heating Value of Solid Biomass Fuels Using Artificial Intelligence Formalisms

The higher heating value (HHV) is an important property defining the energy content of biomass fuels. A number of proximate and/or ultimate analysis based predominantly linear correlations have been proposed for predicting the HHV of biomass fuels. A scrutiny of the relationships between the constituents of the proximate and ultimate analyses and the corresponding HHVs suggests that all relationships are not linear and thus nonlinear models may be more appropriate. Accordingly, a novel artificial intelligence (AI) formalism, namely genetic programming (GP) has been employed for the first time for developing two biomass HHV prediction models, respectively using the constituents of the proximate and ultimate analyses as the model inputs. The prediction and generalization performance of these models was compared rigorously with the corresponding multilayer perceptron (MLP) neural network based as also currently available high-performing linear and nonlinear HHV models. This comparison reveals that the HHV prediction performance of the GP and MLP models is consistently better than that of their existing linear and/or nonlinear counterparts. Specifically, the GP- and MLP-based models exhibit an excellent overall prediction accuracy and generalization performance with high (>0.95) magnitudes of the coefficient of correlation and low (<4.5 %) magnitudes of mean absolute percentage error in respect of the experimental and model-predicted HHVs. It is also found that the proximate analysis-based GP model has outperformed all the existing high-performing linear biomass HHV prediction models. In the case of ultimate analysis-based HHV models, the MLP model has exhibited best prediction accuracy and generalization performance when compared with the existing linear and nonlinear models. The AI-based models introduced in this paper due to their excellent performance have the potential to replace the existing biomass HHV prediction models.     

Likelihood-Weighted Confidence Intervals for the Difference of Two Binomial Proportions

Two new methods for computing confidence intervals for the difference δ = p₁ — p₂ between two binomial proportions (p₁, p₂) are proposed. Both the Mid-P and Max-P likelihood weighted intervals are constructed by mapping the tail probabilities from the two-dimensional (p₁, p₂)-space into a one-dimensional function of δ based on the likelihood weights. This procedure may be regarded as a natural extension of the CLOPPER-PEARSON (1934) interval to the two-sample case where the weighted tail probability is α/2 at each end on the δ scale. The probability computation is based on the exact distribution rather than a large sample approximation. Extensive computation was carried out to evaluate the coverage probability and expected width of the likelihood-weighted intervals, and of several other methods. The likelihood-weighted intervals compare very favorably with the standard asymptotic interval and with intervals proposed by HAUCK and ANDERSON (1986), COX and SNELL (1989), SANTNER and SNELL (1980), SANTNER and YAMAGAMI (1993), and PESKUN (1993). In particular, the Mid-P likelihood-weighted interval provides a good balance between accurate coverage probability and short interval width in both small and large samples. The Mid-P interval is also comparable to COE and TAMHANE'S (1993) interval, which has the best performance in small samples.