First-order Autoregressive Structure Research Articles

Dynamic height growth models for highly productive pedunculate oak (Quercus robur L.) stands: explicit mapping of site index classification in Serbia

Key messageWe applied the generalized algebraic difference approach (GADA) to develop dynamic models of height growth for pedunculate oak (Quercus robur L.) in Serbia. According to the dominant heights, the studied region comprises some of Europe’s most productive sites for pedunculate oak. Therein, we have generated a map showing the current site index class of stands. Such a map could be used to enhance forest management and evaluate climate change impacts.ContextAlthough sustainable forest management requires reliable prediction of forest site productivity, such indicators are currently unavailable for pedunculate oak sites in Serbia. The site index (SI) curves represent the most commonly used indirect scale for site productivity classification. The dynamic equations derived by the Generalized Algebraic Difference Approach (GADA) are the state-of-the-art approach in growth modeling, but they have not been widely applied for studying the height dynamics of pedunculate oak.AimsThe main objectives of this study were to develop the first dynamic site index curves for pedunculate oak in Serbia and subsequently to provide stand-level maps with predicted site indices.MethodsWe have tested five flexible polymorphic equations with variable asymptotes derived by the GADA approach. Models were calibrated using artificially established growth trajectories obtained from 3636 detailed temporary sample plots. The selection of the most suitable model was accomplished according to (1) quantitative measures of goodness of fit, (2) the analysis of residual scattering, and (3) the biological plausibility of obtained height growth curves.ResultsAfter correcting the error terms with a continuous first-order autoregressive structure and conducting a three-stage performance analysis, the GADA dynamic site index model derived from the Hossfeld base equation shows the best overall properties. Insight into the oscillations of relative error suggested that 100 years is the most suitable age for site index referencing. Comparison with existing height growth models revealed greater flexibility and a considerably better representation of the height growth dynamic of pedunculate oak in the studied region. Additionally, we have produced a spatially explicit map showing the expected SI100 for 1907 stands with pedunculate oak within 22 management units.ConclusionDynamic SI-curves based on GADA will serve forest practitioners to update management plans and serve as a reference point for benchmarking the impact of climate change and for developing adaptation strategies. The utilized approach allowed unbiased estimation of SI100 across all age classes so that the results could be mapped at a broader scale. This study provides the second known application of the dynamic model for pedunculate oak in Europe but the first that includes some of the most productive sites in the species distribution range.

Variability Signatures of a Burst Process in Flaring Gamma-Ray Blazars

Blazars exhibit stochastic flux variability across the electromagnetic spectrum, often exhibiting heavy-tailed flux distributions, commonly modeled as lognormal. However, Tavecchio et al. and Adams et al. found that the high-energy gamma-ray flux distributions of several of the brightest flaring Fermi-LAT flat-spectrum radio quasars (FSRQs) are well modeled by an even heavier-tailed distribution, which we show is the inverse gamma distribution. We propose an autoregressive inverse gamma variability model in which an inverse gamma flux distribution arises as a consequence of a shot-noise process. In this model, discrete bursts are individually unresolved and averaged over within time bins, as in the analysis of Fermi-LAT data. Stochastic variability on timescales longer than the time-bin duration is modeled using first-order autoregressive structure. The flux distribution becomes approximately lognormal in the limiting case of many weak bursts. The fractional variability is predicted to decrease as the time-bin duration increases. Using simulated light curves, we show that the proposed model is consistent with the typical gamma-ray variability properties of FSRQs and BL Lac objects. The model parameters can be physically interpreted as the average burst rate, the burst fluence, and the timescale of long-term stochastic fluctuations.

The Effect of a Single Dose of Thiamine on Oxygen Consumption in Patients Requiring Mechanical Ventilation for Acute Illness: A Phase II, Randomized, Double-Blind, Placebo-Controlled Trial.

IMPORTANCE:Lower oxygen consumption is associated with worse survival in septic shock and in other forms of critical illness. No treatment that increases oxygen extraction, a key determinant of oxygen consumption, has been found. Thiamine is required for aerobic metabolism, and deficiency is common in the critically ill.OBJECTIVES:We evaluated the effect of thiamine on oxygen consumption in patients requiring mechanical ventilation for an acute illness.DESIGN:Phase II, randomized, double-blind, and placebo-controlled trial.SETTING AND PARTICIPANTS:ICUs in a tertiary care hospital in the United States. Patients admitted to the ICU and requiring mechanical ventilation were screened for enrollment.INTERVENTIONS:After enrollment, baseline measurement of oxygen consumption and baseline laboratories including lactate, central venous oxygen saturation, and pyruvate dehydrogenase, a single dose of 200 mg IV thiamine or placebo was administered. Oxygen consumption was then monitored for 6 additional hours and repeat laboratories were drawn at the end of the protocol.MAIN OUTCOMES AND MEASURES:The primary outcome was the change in oxygen consumption. Analysis was done using linear regression with a first-order autoregressive variance-covariance structure to account for repeated measures within subjects. Secondary outcomes included change in lactate, central venous oxygen saturation, and pyruvate dehydrogenase quantity and activity.RESULTS:Sixty-seven patients were enrolled. After excluding 11 patients due to inadequate quantity or quality of oxygen consumption data, 56 patients were included. There was no difference in change in oxygen consumption in the 6 hours after study drug. Results for secondary outcomes were similarly negative. In the prespecified subgroup of 18 thiamine deficient patients, there was a difference in the two oxygen consumption curves (p = 0.006), although no difference in median oxygen consumption or area under the curve.CONCLUSIONS AND RELEVANCE:A single dose of IV thiamine did not alter oxygen consumption in patients requiring mechanical ventilation for acute illness.

D-optimal population designs in linear mixed effects models for multiple longitudinal data

The main purpose of this paper is to investigate D-optimal population designs in multi-response linear mixed models for longitudinal data. Observations of each response variable within subjects are assumed to have a first-order autoregressive structure, possibly with observation error. The equivalence theorems are provided to characterise the D-optimal population designs for the estimation of fixed effects in the model. The semi-Bayesian D-optimal design which is robust against the serial correlation coefficient is also considered. Simulation studies show that the correlation between multi-response variables has tiny effects on the optimal design, while the experimental costs are important factors in the optimal designs.

The politics of urban trees: Tree planting is associated with gentrification in Portland, Oregon

This study evaluated the hypothesis that urban-tree planting increases neighborhood gentrification in Portland, Oregon. We defined gentrification as an increase in the median sales price of single-family homes in a Census tract compared to other tracts in the city after accounting for differences in the housing stock such as house size and number of bathrooms. We used tree-planting data from the non-profit Friends of Trees, who have planted 57,985 yard and street trees in Portland (1990–2019). We estimated a mixed model of gentrification (30 years and 141 tracts) including random intercepts at the tract level and a first-order auto-regressive residual structure. Tract-level house prices and tree planting may be codetermined. Therefore, to address potential endogeneity of tree planting in statistical modeling, we lagged the number of trees planted by at least one year. We found that the number of trees planted in a tract was significantly associated with a higher tract-level median sales price, although it took at least six years for this relationship to emerge. Specifically, each tree was associated with a $131 (95% CI: $53–$210; p-value = 0.001) increase in tract-level median sales price six years after planting. The magnitude of the association between the number of trees planted and median sales price generally increased as the time lag lengthened. After twelve years, each tree was associated with a $265 (95% CI: $151–$379; p-value<0.001) increase in tract-level median sales price. Tree planting was not merely a proxy for existing tree cover, as the percent of a tract covered in tree canopy was independently associated with an increase in median sales price. Specifically, each 1-percentage point increase in tree-canopy cover was associated with a $882 (95% CI: $226–$1538; p-value = 0.008) increase in median sales price. In conclusion, tree planting is associated with neighborhood-level gentrification, although the magnitude of the association is modest.

Identification of Leverage Points in Principal Component Regression and r-k Class Estimators with AR(1) Error Structure

The determination of leverage observations have been frequently investigated through ordinary least squares and some biased estimators proposed under the multicollinearity problem in the linear regression models. Recently, the identification of leverage and influential observations have been also popular on the general linear regression models with correlated error structure. This paper proposes a new projection matrix and a new quasiprojection matrix to determination of leverage observations for principal component regression and r-k class estimators, respectively, in general linear regression model with first-order autoregressive error structure. Some useful properties of these matrices are presented. Leverage observations obtained by generalized least squares and ridge regression estimators available in the literature have been compared with proposed principal component regression and r-k class estimators over a simulation study and a numerical example. In the literature, the first leverage is considered separately due to the first-order autoregressive error structure. Therefore, the behaviours of first leverages obtained by principal component regression and r-k class estimators has been also investigated according to the autocorrelation coefficient and biasing parameter through applications. The results showed that the leverage of the first observation obtained by principal component regression and r-k estimators is smaller than that obtained by generalized least squares and ridge regression estimators. In addition, as the autocorrelation coefficient goes to -1, the leverage of the first transformed observation decreases for PCR and r-k class estimators, while its increases while the autocorrelation coefficient goes to 1.

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach.

Here, we developed an individual-tree model of 5-year basal area increments based on a dataset including 21898 Picea asperata trees from 779 sample plots located in Xinjiang Province, northwest China. To prevent high correlations among observations from the same sampling unit, we developed the model using a linear mixed-effects approach with random plot effect to account for stochastic variability. Various tree- and stand-level variables, such as indices for tree size, competition, and site condition, were included as fixed effects to explain the residual variability. In addition, heteroscedasticity and autocorrelation were described by introducing variance functions and autocorrelation structures. The optimal linear mixed-effects model was determined by several fit statistics: Akaike's information criterion, Bayesian information criterion, logarithm likelihood, and a likelihood ratio test. The results indicated that significant variables of individual-tree basal area increment were the inverse transformation of diameter at breast height, the basal area of trees larger than the subject tree, the number of trees per hectare, and elevation. Furthermore, errors in variance structure were most successfully modeled by the exponential function, and the autocorrelation was significantly corrected by first-order autoregressive structure (AR(1)). The performance of the linear mixed-effects model was significantly improved relative to the model using ordinary least squares regression.

Diesel Exhaust Exposure during Farming Activities: Statistical Modeling of Continuous Black Carbon Concentrations.

Daily driving of diesel-powered tractors has been linked to increased lung cancer risk in farmers, yet few studies have quantified exposure levels to diesel exhaust during tractor driving or during other farm activities. We expanded an earlier task-based descriptive investigation of factors associated with real-time exposure levels to black carbon (BC, a surrogate of diesel exhaust) in Iowa farmers by increasing the sample size, collecting repeated measurements, and applying statistical models adapted to continuous measurements. The expanded study added 43 days of sampling, for a total of 63 sample days conducted in 2015 and 2016 on 31 Iowa farmers. Real-time, continuous monitoring (30-s intervals) of personal BC concentrations was performed using a MicroAeth AE51 microaethelometer affixed with a micro-cyclone. A field researcher recorded information on tasks, fuel type, farmer location, and proximity to burning biomass. We evaluated the influence of these variables on log-transformed BC concentrations using a linear mixed-effect model with random effects for farmer and day and a first-order autoregressive structure for within-day correlation. Proximity to diesel-powered equipment was observed for 42.5% of the overall sampling time and on 61 of the 63 sample days. Predicted geometric mean BC concentrations were highest during grain bin work, loading, and harvesting, and lower for soil preparation and planting. A 68% increase in BC concentrations was predicted for close proximity to a diesel-powered vehicle, relative to far proximity, while BC concentrations were 44% higher in diesel vehicles with open cabins compared with closed cabins. Task, farmer location, fuel type, and proximity to burning biomass explained 8% of within-day variance in BC concentrations, 2% of between-day variance, and no between-farmer variance. Our findings showed that farmers worked frequently near diesel equipment and that BC concentrations varied between tasks and by fuel type, farmer location, and proximity to burning biomass. These results could support the development of exposure models applicable to investigations of health effects in farmers associated with exposure to diesel engine exhaust.

Autoregressive repeatability model for genetic evaluation of longitudinal reproductive traits in dairy cattle.

We investigated the efficiency of the autoregressive repeatability model (AR) for genetic evaluation of longitudinal reproductive traits in Portuguese Holstein cattle and compared the results with those from the conventional repeatability model (REP). The data set comprised records taken during the first four calving orders, corresponding to a total of 416, 766, 872 and 766 thousand records for interval between calving to first service, days open, calving interval and daughter pregnancy rate, respectively. Both models included fixed (month and age classes associated to each calving order) and random (herd-year-season, animal and permanent environmental) effects. For AR model, a first-order autoregressive (co)variance structure was fitted for the herd-year-season and permanent environmental effects. The AR outperformed the REP model, with lower Akaike Information Criteria, lower Mean Square Error and Akaike Weights close to unity. Rank correlations between estimated breeding values (EBV) with AR and REP models ranged from 0.95 to 0.97 for all studied reproductive traits, when the total bulls were considered. When considering only the top-100 selected bulls, the rank correlation ranged from 0.72 to 0.88. These results indicate that the re-ranking observed at the top level will provide more opportunities for selecting the best bulls. The EBV reliabilities provided by AR model was larger for all traits, but the magnitudes of the annual genetic progress were similar between two models. Overall, the proposed AR model was suitable for genetic evaluations of longitudinal reproductive traits in dairy cattle, outperforming the REP model.

Approaches for Specifying the Level-1 Error Structure When Synthesizing Single-Case Data

ABSTRACTMultilevel modeling has been utilized for combining single-case experimental design (SCED) data assuming simple level-1 error structures. The purpose of this study is to compare various multilevel analysis approaches for handling potential complexity in the level-1 error structure within SCED data, including approaches assuming simple and complex error structures (heterogeneous, autocorrelation, and both) and those using fit indices to select between alternative error structures. A Monte Carlo study was conducted to empirically validate the suggested multilevel modeling approaches. Results indicate that each approach leads to fixed effect estimates with little to no bias and that inferences for fixed effects were frequently accurate, particularly when a simple homogeneous level-1 error structure or a first-order autoregressive structure was assumed and the inferences were based on the Kenward-Roger method. Practical implications and recommendations are discussed.

Effect of yoga on chronic non-specific neck pain: An unconditional growth model

ObjectiveChronic neck pain is a common problem that affects approximately half of the population. Conventional treatments such as medication and exercise have shown limited analgesic effects. This analysis is based on an original study that was conducted to investigate the physical and behavioral effects of a 9-week Iyengar yoga course on chronic non-specific neck pain. This secondary analysis uses linear mixed models to investigate the individual trajectories of pain intensity in participants before, during and after the Iyengar yoga course. MethodParticipants with chronic non-specific neck pain were selected for the study. The participants suffered from neck pain for at least 5 days per week for at least the preceding 3 months, with a mean neck pain intensity (NPI) of 40 mm or more on a Visual Analog Scale of 100 mm. The participants were randomized to either a yoga group (23) or to a self-directed exercise group (24). The mean age of the participants in the yoga group was 46, and ranged from 19 to 59. The participants in the yoga group participated in an Iyengar yoga program designed to treat chronic non-specific neck pain. Our current analysis only includes participants who were initially randomized into the yoga group. The average weekly neck pain intensity at baseline, during and post intervention, comprising 11 total time points, was used to construct the growth models. We performed a step-up linear mixed model analysis to investigate change in NPI during the yoga intervention. We fit nested models using restricted maximum-likelihood estimation (REML), tested fixed effects with Wald test p-values and random effects with the likelihood ratio test. We constructed 10 REML models. ResultsThe model that fit the data best was an unconditional random quadratic growth model, with a first-order auto-regressive structure specified for the residual R matrix. Participants in the yoga group showed significant variation in NPI. They demonstrated variation in their intercepts, in their linear rates of change, and most tellingly, in their quadratic rates of change. ConclusionsWhile all participants benefitted from the yoga intervention, the degree to which they benefitted varied. Additionally, they did not experience a consistent rate of reduction in NPI − their NPI fluctuated, either increasing and then decreasing, or vice-versa. We comment on the clinical and research implications of our findings.

Optimization of the time series NDVI-rainfall relationship using linear mixed-effects modeling for the anti-desertification area in the Beijing and Tianjin sandstorm source region

Degradation in drylands is a critically important global issue that threatens ecosystem and environmental in many ways. Researchers have tried to use remote sensing data and meteorological data to perform residual trend analysis and identify human-induced vegetation changes. However, complex interactions between vegetation and climate, soil units and topography have not yet been considered. Data used in the study included annual accumulated Moderate Resolution Imaging Spectroradiometer (MODIS) 250 m normalized difference vegetation index (NDVI) from 2002 to 2013, accumulated rainfall from September to August, digital elevation model (DEM) and soil units. This paper presents linear mixed-effect (LME) modeling methods for the NDVI-rainfall relationship. We developed linear mixed-effects models that considered the random effects of sample points nested in soil units for nested two-level modeling and single-level modeling of soil units and sample points, respectively. Additionally, three functions, including the exponential function (exp), the power function (power), and the constant plus power function (CPP), were tested to remove heterogeneity, and an additional three correlation structures, including the first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and the compound symmetry structure (CS), were used to address the spatiotemporal correlations. It was concluded that the nested two-level model considering both heteroscedasticity with (CPP) and spatiotemporal correlation with [ARMA(1,1)] showed the best performance (AMR = 0.1881, RMSE = 0.2576, adj-R 2 = 0.9593). Variations between soil units and sample points that may have an effect on the NDVI-rainfall relationship should be included in model structures, and linear mixed-effects modeling achieves this in an effective and accurate way.

NONLINEAR MODELS FOR DESCRIPTION OF CACAO FRUIT GROWTH WITH ASSUMPTION VIOLATIONS

ABSTRACT Cacao (Theobroma cacao L.) is an important fruit in the Brazilian economy, which is mainly cultivated in the southern State of Bahia. The optimal stage for harvesting is a major factor for fruit quality and the knowledge on its growth curves can help, especially in identifying the ideal maturation stage for harvesting. Nonlinear regression models have been widely used for description of growth curves. However, several studies in this subject do not consider the residual analysis, the existence of a possible dependence between longitudinal observations, or the sample variance heterogeneity, compromising the modeling quality. The objective of this work was to compare the fit of nonlinear regression models, considering residual analysis and assumption violations, in the description of the cacao (clone Sial-105) fruit growth. The data evaluated were extracted from Brito and Silva (1983), who conducted the experiment in the Cacao Research Center, Ilheus, State of Bahia. The variables fruit length, diameter and volume as a function of fruit age were studied. The use of weighting and incorporation of residual dependencies was efficient, since the modeling became more consistent, improving the model fit. Considering the first-order autoregressive structure, when needed, leads to significant reduction in the residual standard deviation, making the estimates more reliable. The Logistic model was the most efficient for the description of the cacao fruit growth.

Food cravings and food consumption after Roux-en-Y gastric bypass versus cholecystectomy

BackgroundFood cravings and consumption of craved foods after Roux-en-Y gastric bypass (RYGB) are poorly understood. Food cravings after bariatric surgery may explain why some patients fail to change eating behaviors after RYGB, and understanding these cravings may provide better information for nutritional counseling to either enhance weight loss or prevent weight regain. ObjectivesTo study cravings in RYGB patients and compare them with cholecystectomy (CC) control patients. SettingThis study took place in a university hospital. MethodsRYGB patients (n = 50) and CC control patients (n = 38) completed a validated food craving inventory before surgery and at 2 and 6 weeks postoperatively. In addition, RYGB patients completed the food craving inventory at 12, 24, 36, and 52 weeks postoperatively. A linear mixed-effect model with a first-order autoregressive structure for correlations was used to evaluate changes in food consumption and food cravings between visits. Correlations between food cravings and body mass index (BMI) or weight changes before and after RYGB were assessed with Spearman correlation coefficients. P<.05 was considered statistically significant. ResultsAfter RYGB, food consumption decreased significantly compared with CC control patients and was lowest at 2 weeks. Consumption progressively increased over time in the first year but remained significantly less than that from presurgery. In addition, a higher preoperative BMI was found to correlate moderately with higher preoperative cravings of the total of all 4 food groups studied (r = .3, P = .04); high-fat foods (r = .3, P = .04); and sweets (r = .3, P = .03). However, with the exception of preoperative cravings for high-fat foods, these scores were not predictive of changes in BMI after surgery. Overall, RYGB did not significantly affect food cravings after surgery compared with CC control patients. ConclusionThese findings indicate that RYGB may limit food consumption but does not affect the drive to consume certain types of food. Because food cravings are high in patients with obesity before surgery and remain high after surgery, these findings suggest a possible reason for noncompliance with dietary recommendations after RYGB.

Educational mobility across three generations of American women

We analyze the intergenerational transmission of education in a three-generation sample of women from the 20th century US. We find strong three-generation educational persistence, with the association between the education of grandmothers and their granddaughters approximately two times stronger than would be expected under the type of first-order autoregressive transmission structure that has been assumed in much of the existing two-generation mobility literature. These findings are robust to using alternative empirical specifications and sample constructions, and are successfully replicated in a second independently drawn data set. Analyses that include males in the youngest and oldest generations produce very similar estimates. A variety of potential mechanisms linking the educational outcomes of grandparents and grandchildren are discussed and where possible tested empirically.

Linear mixed-effects models to describe individual tree crown width for China-fir in Fujian Province, southeast China.

A multiple linear model was developed for individual tree crown width of Cunninghamia lanceolata (Lamb.) Hook in Fujian province, southeast China. Data were obtained from 55 sample plots of pure China-fir plantation stands. An Ordinary Linear Least Squares (OLS) regression was used to establish the crown width model. To adjust for correlations between observations from the same sample plots, we developed one level linear mixed-effects (LME) models based on the multiple linear model, which take into account the random effects of plots. The best random effects combinations for the LME models were determined by the Akaike’s information criterion, the Bayesian information criterion and the -2logarithm likelihood. Heteroscedasticity was reduced by three residual variance functions: the power function, the exponential function and the constant plus power function. The spatial correlation was modeled by three correlation structures: the first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)], and the compound symmetry structure (CS). Then, the LME model was compared to the multiple linear model using the absolute mean residual (AMR), the root mean square error (RMSE), and the adjusted coefficient of determination (adj-R 2). For individual tree crown width models, the one level LME model showed the best performance. An independent dataset was used to test the performance of the models and to demonstrate the advantage of calibrating LME models.

The impact of incentives on intrinsic and extrinsic motives for fitness-center attendance in college first-year students.

A criticism of incentives for health behaviors is that incentives undermine intrinsic motivation. The objective of this study was to determine the impact of monetary incentive provision on participation motives for exercise in first-year college students at a northeastern public university. Randomized-controlled trial. Public university in the Northeastern United States. One hundred seventeen first-year college students. Participants were randomized to one of three conditions: a control condition receiving no incentives for meeting fitness-center attendance goals; a discontinued-incentive condition receiving weekly incentives during fall semester 2011, and no incentives during spring semester 2012; or a continued-incentive condition receiving weekly incentives during fall semester, and incentives on a variable-interval schedule during spring semester. The Exercise Motivation Inventory 2 measured exercise participation motives at baseline, end of fall semester, and end of spring semester. Fitness-center attendance was monitored by using ID-card check-in/check-out records. Repeated-measures analyses using linear mixed models with first-order autoregressive covariance structures were run to compare motive changes in the three conditions. Participation motives of Enjoyment and Revitalization associated with intrinsic motivation did not decrease significantly over time in any of the conditions, F(4, 218) = 2.25, p = .065 and F(4, 220) = 1.67, p = .16, respectively. Intrinsically associated participation motives for exercise did not decrease with incentive provision. Therefore, incentives may encourage fitness-center attendance without negatively impacting participation motives for exercise.

Fitting nonlinear autoregressive models to describe coffee seed germination

Cumulative germination of coffee has a longitudinal behavior mathematically characterized by a sigmoidal model. In the seed germination evaluation, the study of the germination curve may contribute to better understanding of this process. The aim of this study was to evaluate the goodness of fit of Logistic and Gompertz models, with independent and first-order autoregressive errors structure, AR (1), in the description of coffee (Coffea arabica L.) line Catuai vermelho IAC 99 germination, at five different potential germination. The data used were from an experiment conducted in 2011 at the Seed Analysis Laboratory of the Federal University of Lavras. The Logistic and Gompertz nonlinear models were appropriately adjusted to the percentage germination data. The Gompertz model with first-order autoregressive errors structure was the best to describe the germination process

Nonlinear mixed-effects (NLME) diameter growth models for individual China-Fir (Cunninghamia lanceolata) trees in Southeast China.

An individual-tree diameter growth model was developed for Cunninghamia lanceolata in Fujian province, southeast China. Data were obtained from 72 plantation-grown China-fir trees in 24 single-species plots. Ordinary non-linear least squares regression was used to choose the best base model from among 5 theoretical growth equations; selection criteria were the smallest absolute mean residual and root mean square error and the largest adjusted coefficient of determination. To account for autocorrelation in the repeated-measures data, we developed one-level and nested two-level nonlinear mixed-effects (NLME) models, constructed on the selected base model; the NLME models incorporated random effects of the tree and plot. The best random-effects combinations for the NLME models were identified by Akaike's information criterion, Bayesian information criterion and −2 logarithm likelihood. Heteroscedasticity was reduced with two residual variance functions, a power function and an exponential function. The autocorrelation was addressed with three residual autocorrelation structures: a first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and a compound symmetry structure (CS). The one-level (tree) NLME model performed best. Independent validation data were used to test the performance of the models and to demonstrate the advantage of calibrating the NLME models.

A Comparison of Four Methods for the Analysis of N-of-1 Trials

ObjectiveTo provide a practical guidance for the analysis of N-of-1 trials by comparing four commonly used models.MethodsThe four models, paired t-test, mixed effects model of difference, mixed effects model and meta-analysis of summary data were compared using a simulation study. The assumed 3-cycles and 4-cycles N-of-1 trials were set with sample sizes of 1, 3, 5, 10, 20 and 30 respectively under normally distributed assumption. The data were generated based on variance-covariance matrix under the assumption of (i) compound symmetry structure or first-order autoregressive structure, and (ii) no carryover effect or 20% carryover effect. Type I error, power, bias (mean error), and mean square error (MSE) of effect differences between two groups were used to evaluate the performance of the four models.ResultsThe results from the 3-cycles and 4-cycles N-of-1 trials were comparable with respect to type I error, power, bias and MSE. Paired t-test yielded type I error near to the nominal level, higher power, comparable bias and small MSE, whether there was carryover effect or not. Compared with paired t-test, mixed effects model produced similar size of type I error, smaller bias, but lower power and bigger MSE. Mixed effects model of difference and meta-analysis of summary data yielded type I error far from the nominal level, low power, and large bias and MSE irrespective of the presence or absence of carryover effect.ConclusionWe recommended paired t-test to be used for normally distributed data of N-of-1 trials because of its optimal statistical performance. In the presence of carryover effects, mixed effects model could be used as an alternative.