Common Regression Coefficients Research Articles

Fusion regression methods with repeated functional data

Linear regression and classification methods with repeated functional data are considered. For each statistical unit in the sample, a real-valued parameter is observed over time under different conditions related by some neighborhood structure (spatial, group, etc.). Two regression methods based on fusion penalties are proposed to consider the dependence induced by this structure. These methods aim to obtain parsimonious coefficient regression functions, by determining if close conditions are associated with common regression coefficient functions. The first method is a generalization to functional data of the variable fusion methodology based on the 1-nearest neighbor. The second one relies on the group fusion lasso penalty which assumes some grouping structure of conditions and allows for homogeneity among the regression coefficient functions within groups. Numerical simulations and an application of electroencephalography data are presented.

HIGH-DIMENSIONAL FACTOR REGRESSION FOR HETEROGENEOUS SUBPOPULATIONS.

In modern scientific research, data heterogeneity is commonly observed owing to the abundance of complex data. We propose a factor regression model for data with heterogeneous subpopulations. The proposed model can be represented as a decomposition of heterogeneous and homogeneous terms. The heterogeneous term is driven by latent factors in different subpopulations. The homogeneous term captures common variation in the covariates and shares common regression coefficients across subpopulations. Our proposed model attains a good balance between a global model and a group-specific model. The global model ignores the data heterogeneity, while the group-specific model fits each subgroup separately. We prove the estimation and prediction consistency for our proposed estimators, and show that it has better convergence rates than those of the group-specific and global models. We show that the extra cost of estimating latent factors is asymptotically negligible and the minimax rate is still attainable. We further demonstrate the robustness of our proposed method by studying its prediction error under a mis-specified group-specific model. Finally, we conduct simulation studies and analyze a data set from the Alzheimer's Disease Neuroimaging Initiative and an aggregated microarray data set to further demonstrate the competitiveness and interpretability of our proposed factor regression model.

Spatial heterogeneity automatic detection and estimation

Spatial regression is widely used for modeling the relationship between a dependent variable and explanatory covariates. Oftentimes, the linear relationships vary across space, such that some covariates have location-specific effects on the response. One fundamental question is how to detect the systematic variation in the model and identify which locations share common regression coefficients and which do not. Only a correct model structure can assure unbiased estimation of coefficients and valid inferences. A new procedure is proposed, called Spatial Heterogeneity Automatic Detection and Estimation (SHADE), for automatically and simultaneously subgrouping and estimating covariate effects for spatial regression models. The SHADE employs a class of spatially-weighted fusion type penalty on all pairs of observations, with location-specific weight constructed using spatial information, to cluster coefficients into subgroups. Under certain regularity conditions, the SHADE is shown to be able to identify the true model structure with probability approaching one and estimate regression coefficients consistently. An alternating direction method of multiplier algorithm (ADMM) is developed to compute the SHADE. In numerical studies, the empirical performance of the SHADE is demonstrated by using different choices of weights and comparing their accuracy. The results suggest that spatial information can enhance subgroup structure analysis in challenging situations when the spatial variation among regression coefficients is small or the number of repeated measures is small. Finally, the SHADE is applied to find the relationship between a natural resource survey and a land cover data layer to identify spatially interpretable groups.

Grouped generalized estimating equationsfor longitudinal data analysis.

Generalized estimating equation(GEE) is widely adopted for regression modeling for longitudinal data, taking account of potential correlations within the same subjects. Although the standard GEE assumes common regression coefficients among all the subjects, such an assumption may not be realistic when there is potential heterogeneity in regression coefficients among subjects. In this paper, we develop a flexible and interpretable approach, called grouped GEE analysis, to modeling longitudinal data with allowing heterogeneity in regression coefficients. The proposed method assumes that the subjects are divided into a finite number of groups and subjects within the same group share the same regression coefficient. We provide a simple algorithm for grouping subjects and estimating the regression coefficients simultaneously, and show the asymptotic properties of the proposed estimator. The number of groups can be determined by the cross validation with averaging method. We demonstrate the proposed method through simulation studies and an application to a real dataset.

Coefficient tree regression: fast, accurate and interpretable predictive modeling

The proliferation of data collection technologies often results in large data sets with many observations and many variables. In practice, highly relevant engineered features are often groups of predictors that share a common regression coefficient (i.e., the predictors in the group affect the response only via their collective sum), where the groups are unknown in advance and must be discovered from the data. We propose an algorithm called coefficient tree regression (CTR) to discover the group structure and fit the resulting regression model. In this regard CTR is an automated way of engineering new features, each of which is the collective sum of the predictors within each group. The algorithm can be used when the number of variables is larger than, or smaller than, the number of observations. Creating new features that affect the response in a similar manner improves predictive modeling, especially in domains where the relationships between predictors are not known a priori. CTR borrows computational strategies from both linear regression (fast model updating when adding/modifying a feature in the model) and regression trees (fast partitioning to form and split groups) to achieve outstanding computational and predictive performance. Finding features that represent hidden groups of predictors (i.e., a hidden ontology) that impact the response only via their sum also has major interpretability advantages, which we demonstrate with a real data example of predicting political affiliations with television viewing habits. In numerical comparisons over a variety of examples, we demonstrate that both computational expense and predictive performance are far superior to existing methods that create features as groups of predictors. Moreover, CTR has overall predictive performance that is comparable to or slightly better than the regular lasso method, which we include as a reference benchmark for comparison even though it is non-group-based, in addition to having substantial computational and interpretive advantages over lasso.

Modelling larval haddock Melanogrammus aeglefinus and cod Gadus morhua densities and mortality rates in hatchery mesocosms

The numbers and instantaneous mortality rate Z of haddock Melanogrammus aeglefinus and cod Gadus morhua larvae in a hatchery from 0 to 50 days post hatch (dph) were estimated directly and non-destructively. A small mesh plankton net was raised vertically in the water column from the tank base to the water surface and larvae were examined and counted. Using this direct sampling method Z was assessed through the larval period. A semi-log plot described the relationship of larval numbers on time. The Z of M. aeglefinus larvae from 0 to 50 dph was consistent between tanks, from 11.2 to 14.8% d−1 in seven rearing tanks sampled, with a common regression coefficient of 12.7%. The Z was lower for the period 10–50 dph, from 5.5 to 11.8% d−1 with a calculated common slope for all tanks giving a Z of 9.6%. Restricted data for G. morhua suggested lower mortality of Z of 7.8% d−1 from 0 to 50 dph with 4.2% solely for the period 10 to 50 dph. Avoidance of the net sampler by larvae was generally not recorded until >50 dph. The Z in both M. aeglefinus and G. morhua larvae was similar to Z calculated for these species in sea fisheries surveys. Advantages of the technique for estimating Z in hatchery mesocosms and examining causes and timing of larval mortality are discussed. Care in sampling larvae without removal from water, minimizing tank disturbance by sampling after tank cleaning, sampling across tank locations, and thorough disinfection of sampling nets are essential.

Small area estimation with subgroup analysis

ABSTRACTIn this article, a new unit level model based on a pairwise penalised regression approach is proposed for problems in small area estimation (SAE). Instead of assuming common regression coefficients for all small domains in the traditional model, the new estimator is based on a subgroup regression model which allows different regression coefficients in different groups. The alternating direction method of multipliers (ADMM) algorithm is used to find subgroups with different regression coefficients. We also consider pairwise spatial weights for spatial areal data. In the simulation study, we compare the performances of the new estimator with the traditional small area estimator. We also apply the new estimator to urban area estimation using data from the National Resources Inventory survey in Iowa.

Effects of diet forage proportion on maintenance energy requirement and the efficiency of metabolizable energy use for lactation by lactating dairy cows

The objective of the present study was to examine the effect of dietary forage proportion (FP) on metabolizable energy (ME) requirement for maintenance (MEm) and the efficiency of ME use for lactation (kl) in lactating dairy cows. Data used were derived from 32 calorimetric chamber experiments undertaken at our institute between 1992 and 2010, including data from 818 Holstein-Friesian cows (HF), 50 Norwegian Red cows, and 62 crossbred cows (Jersey × HF or Norwegian Red × HF). Animals were offered forage-only rations (n=66) or forage and concentrate rations (n=864) with FP ranging from 18 to 100% (dry matter basis). The effect of FP was evaluated by dividing the whole data set into 4 groups according to the FP ranges, categorized as FP <30%, FP=30 to 59%, FP=60 to 99%, and FP=100%. The MEm for individual cows was calculated from heat production minus energy losses from inefficiencies of ME use for lactation, energy retention and pregnancy, and kl was obtained from milk energy output adjusted to zero energy balance (El(0)) divided by ME available for production. Increasing FP significantly reduced ME intake and milk energy output, although the differences between the 2 low FP groups were not significant. However, increasing FP significantly increased the ratio of heat production over ME intake and MEm (MJ/kg0.75), with the exception that the increases did not reach significance in heat production/ME intake between FP <30% and FP=30 to 59%, or in MEm between FP=60 to 99% and FP=100%. However, the FP had no significant effect on the kl values, which were similar among the 4 groups of cows. The effect of FP was also evaluated using the linear mixed regression technique relating El(0) to ME intake. The results demonstrated that with a common regression coefficient (slope), the regression constants (intercepts) taken as net energy requirement for maintenance significantly increased with increasing FP. However, the increase between the 2 high FP groups did not research significance. It is concluded that increasing diet FP had no effects on kl but significantly increased maintenance energy requirement (MJ/kg0.75). These results indicate that using the current energy feeding systems to ration dairy cows managed under low input systems may underestimate their nutrient requirements, because the majority of feeding systems adopted globally do not differentiate the maintenance energy requirements between low and high forage input systems.

Biodegradable mulch instead of polyethylene for weed control of processing tomato production

Black polyethylene (PE) film is used for mulching in processing tomato (Lycopersicon esculentum P. Mill.) in Spain achieving a generally high weed control but causing a serious waste problem. Few herbicides are available for tomato, so that a biodegradable mulching is the desired solution which should provide high yield, high weed control and be economically available. For 3 years, the same field trial was carried out in four areas of Spain using the following mulch materials in processing tomatoes: (1) untreated control, (2) manual weeding, (3) PE, black, 15 μm thick, (4) and (5) two types of biodegradable plastic mulch, black, 15 μm, (6) oxo-degradable plastic mulch, black, 15 μm, (7) paper mulch, black, 85 g m−2, (8) kraft paper mulch, brown, 140 g m−2 and (9) barley straw, 1 kg m−2.. Weed control, yield and economic aspects were analysed. Most frequent weed species were Convolvulus arvensis, Chenopodium album and Amaranthus spp. Here, we showed that despite differences in weed species, soil type and years, weed competition caused a similar yield decrease in all locations. A common regression coefficient could be estimated indicating that weed competition was responsible for a yield decrease of 3.3 to 4.4 t ha−1 for each 10% of efficacy loss. Weed control was high for biodegradable plastics, paper mulches and PE ranging between 80% and 100% for all of them; also, yield was similar for all plastic and paper mulching treatments ranging between 72% and 108% of the yield achieved by PE. Here, we demonstrate that technically viable biodegradable alternatives exist for all four locations. From the economic point of view, PE gave highest benefits, mainly due to its low market price and to high yields despite removal costs, showing that its substitution will depend on prohibition. For locations with low-density weed infestations, manual weeding can be an alternative.

A Heterogeneous Bayesian Regression Model for Cross-sectional Data Involving a Single Observation per Response Unit

Multiple regression is frequently used across the various social sciences to analyze cross-sectional data. However, it can often times be challenging to justify the assumption of common regression coefficients across all respondents. This manuscript presents a heterogeneous Bayesian regression model that enables the estimation of individual-level-regression coefficients in cross-sectional data involving a single observation per response unit. A Gibbs sampling algorithm is developed to implement the proposed Bayesian methodology. A Monte Carlo simulation study is constructed to assess the performance of the proposed methodology across a number of experimental factors. We then apply the proposed method to analyze data collected from a consumer psychology study that examines the differential importance of price and quality in determining perceived value evaluations.

Proportional Hazards Regression for the Analysis of Clustered Survival Data from Case-Cohort Studies

Case-cohort sampling is a commonly used and efficient method for studying large cohorts. Most existing methods of analysis for case-cohort data have concerned the analysis of univariate failure time data. However, clustered failure time data are commonly encountered in public health studies. For example, patients treated at the same center are unlikely to be independent. In this article, we consider methods based on estimating equations for case-cohort designs for clustered failure time data. We assume a marginal hazards model, with a common baseline hazard and common regression coefficient across clusters. The proposed estimators of the regression parameter and cumulative baseline hazard are shown to be consistent and asymptotically normal, and consistent estimators of the asymptotic covariance matrices are derived. The regression parameter estimator is easily computed using any standard Cox regression software that allows for offset terms. The proposed estimators are investigated in simulation studies, and demonstrated empirically to have increased efficiency relative to some existing methods. The proposed methods are applied to a study of mortality among Canadian dialysis patients.

Response to Birth Factors and Retinal Vascular Caliber in a Twin Study

We thank Cheung1 for his interest in our article and, in particular, for highlighting the issues relating to the use of birth size as a sensitive marker of fetal growth restriction and its relationship with retinal vascular caliber. The prevalence of prematurity is relatively high in twins compared with singletons, and, as such, we used birth weight for gestational score under the 10th percentile as a proxy for fetal growth restriction in our analysis. Although we could not …

Developing Optimum Sample Size and Multistage Sampling Plans for &lt;I&gt;Lobesia botrana&lt;/I&gt; (Lepidoptera: Tortricidae) Larval Infestation and Injury in Northern Greece

The purpose of this research was to quantify the spatial pattern and develop a sampling program for larvae of Lobesia botrana Denis and Schiffermüller (Lepidoptera: Tortricidae), an important vineyard pest in northern Greece. Taylor's power law and Iwao's patchiness regression were used to model the relationship between the mean and the variance of larval counts. Analysis of covariance was carried out, separately for infestation and injury, with combined second and third generation data, for vine and half-vine sample units. Common regression coefficients were estimated to permit use of the sampling plan over a wide range of conditions. Optimum sample sizes for infestation and injury, at three levels of precision, were developed. An investigation of a multistage sampling plan with a nested analysis of variance showed that if the goal of sampling is focusing on larval infestation, three grape clusters should be sampled in a half-vine; if the goal of sampling is focusing on injury, then two grape clusters per half-vine are recommended.

Testing alternative methods of analysing differences in fish condition, with an example using the mosquitofish Gambusia holbrooki

In this study we have examined two different methods, the ANCOVA and a weighted procedure, in order to highlight the problems associated with the use of ANCOVA when the assumption of homogeneity of error variances among data sets is wrongly assumed. When error variances are unequal, ANCOVA procedure may lead to estimates of common regression coefficients with larger standard error values than those obtained when a weighted method is employed. Moreover, the ANCOVA test for homogeneity of regression coefficients seems to be differently affected when error variances are not all equal. Specifically, in the case of unequal sample sizes, when large error variance is combined with the small group and the larger: smaller error variance ratio was more than 2, ANCOVA tests for homogeneity of regression coefficients could be substantially affected leading to a Type I error. Although the ANCOVA approach is a simple and robust method in testing for equality of regression coefficients and related analysis, its application should be preceded by tests to determine homogeneity of error variances to ensure its correct use and avoid erroneous interpretations of the data analysed. The weighted method could be considered as a valuable alternative method to ANCOVA when homogeneity of error variances and regression coefficients cannot be assumed. Data for Gambusia holbrooki from Lake Banyoles and Alt Emporda salt marshes (Catalonia, Spain) are used to illustrate these procedures.

Double Shrinkage Estimation of Common Coefficients in Two Regression Equations with Heteroscedasticity

The problem of estimating the common regression coefficients is addressed in this paper for two regression equations with possibly different error variances. The feasible generalized least squares (FGLS) estimators have been believed to be admissible within the class of unbiased estimators. It is, nevertheless, established that the FGLS estimators are inadmissible in light of minimizing the covariance matrices if the dimension of the common regression coefficients is greater than or equal to three. Double shrinkage unbiased estimators are proposed as possible candidates of improved procedures.

Growth Curve in Akkaraman, Ivesi x Akkaraman and Malya x Akkaraman Lambs

Weekly body weights from approximately 10 weeks of age to 19-29 weeks of age were used to study growth curve in Akkaraman, Ivesi x Akkaraman and Malya x Akkaraman lambs. A linear model was fitted to means of body weights using age as independent variable. The linear model provided a good fit with R2 values of 99%, 99.3% and 98.9% for Akkaraman, Ivesi x Akkaraman and Malya x Akkaraman, respectively. Afterthat, a common regression coefficient was calculated as within lambs regression coefficient, which gave R2 values of 95.9%, 97% and 95.8% for Akkaraman, Ivesi x Akkaraman and Malya x Akkaraman lambs, repectively. However, test of homogeneity of regression coffecient verified that there was an evidence of a statistically significant diversity of regression coefficients fitted to each lamb, which means that the growth pattern of all lambs cannot be represented by a common regression equation. Moreover, the data were analyzed considering several measurement on body-weight at each time point. In this way, residual sum of squares was divided into two parts: pure error from repeated measurements and lack of fit sum of squares. This method led to a decline in R2 values, which were 79.1%, 91.7% and 88.4% for Akkaraman, Ivesi x Akkaraman and Malya x Akkaraman lambs, respectively.

ORDMKV: a computer program fitting proportional odds model for multi-state Markov process

ORDMKV is a computer program designed to fit a multi-state discrete-time Markov model for k-stages disease processes having an ordinal structure. The model consists of k transient states representing the increasing severity of the disease process, and the final state can be optionally chosen to be an absorbing state in cases such as death. The ordinal structure of the stages of the disease is modelled by using ordinal response models. Each row of the one-step transition probability matrix is modelled using a proportional odds model based on the cumulative transition probabilities. By using these ordinal response models, the number of parameters used to model the disease process can be reduced significantly not only with respect to a general discrete-time model, but also compared with a parsimonuos continuous-time model. A restricted model can be fitted by assuming that the effect of the covariables in the cumulative probability has common regression coefficients in all stages of the disease process. This assumption, if it holds, reduces the number of regression coefficients associated with each covariate to only one. The regression coefficients of this model are estimated via the method of maximum likelihood, using a quasi-Newton optimization algorithm. When the last state is considered as an absorbing state, it is possible to compute survival curves from the transient states of the process. The program was written in standard FORTRAN 77 and is illustrated using a four-state model to determine factors influencing diabetic retinopathy in young subjects with insulin-dependent diabetes mellitus.

Robust estimation of common regression coefficients under spherical symmetry

Consider the problem of estimating the common regression coefficients of two linear regression models where the two distributions of the errors may be different and unknown. Under the spherical symmetry assumption, the paper proves the superiority of a Graybill-Deal type combined estimator and the further improvement by the Stein effect which were exhibited by Shinozaki (1978, Comm. Statist. Theory Methods, 7, 1421–1432) in the normal case. This shows the robustness of the dominations since the conditions for the dominations are independent of the errors distributions.

Prediction of the commercial yield of beef from carcasses destined for the Japanese market by using measurements from the carcass and non-carcass parts

Yield of saleable beef was predicted in 4 genotypes of bullocks (Hereford, Brahman, Brahman x Hereford and Simmental x Hereford). Simple and multiple regression equations were used to determine how well 4 fat thickness measurements (12th rib and rump measurements SC3 SC5 and SC7) individually, with or without the inclusion of hot side weight, foreshanks weight and short cut tongue weight predicted the yields of saleable beef. Each of the 4 fat thickness measurements (alone) predicted percentage yield of saleable beef with low r.s.d.'s at the means. The regression coefficients for the 4 fat thickness measuring sites were negative and significant in the Hereford, Brahman and Brahman x Hereford groups. In the Simmental x Hereford cattle, the regression coefficients for the rump sites (SC3, SC5 and SC7) were positive but significant only in SC5 which had the lowest r.s.d. (1.56%) of all sites. Twelfth rib fat thickness, with a common regression coefficient for all 4 genotypes (P&lt;0.01) and an r.s.d. of 1.81%, was the next most useful fat thickness measurement. The addition of hot side weight improved the prediction of percentage saleable beef yield at the SC3 and SC5 sites, but not in the pure-bred Brahman bullocks. For populations that include British, European and Bos indicus cattle and their crosses, SC5 fat thickness (alone) was the best determinant of percentage yield of saleable beef. Neither quadratic analyses nor the addition of foreshanks weight or short-cut tongue weight to the fat thickness measurements improved the accuracy of linear regression in any of the 4 genotypes. The errors in prediction of percentage yield of saleable beef (0.28-0.67%) attributable to incorrect rump site location (of up to 4.0 cm) are unlikely to jeopardise the commercial value of yield of saleable beef predicted from measurements made on the rump. Whilst the Brahman and Brahman x Hereford cattle produced high-yielding carcasses for the Japanese chilled beef market and the Simmental x Hereford cattle, by judicious management could equal this yield, carcass yields of the Hereford cattle were about 4-5% lower at all levels of fat thickness. The ability of Hereford cattle to satisfy this market, economically, is seriously questioned.

The influence of epidermal thickness on transcutaneous oxygen pressure measurements in normal persons

Transcutaneous oxygen pressure measurements (TcPO2) were performed in ten healthy men (age 30.6 years, range 28-35) in six regions: anterolaterally 10 cm below and above the knee on both legs, 5 cm laterally to umbilicus and on the inside of the left humerus, which was subsequently biopsied for measurements of epidermal thickness from the basal lamina to the uppermost layer of stratum granulosum. Transcutaneous oxygen pressure was on average 70 mmHg (range 42-88 mmHg), and that of epidermal thickness 70 microns (range 43-120 microns). Epidermis was thinnest on the inside of the humerus (mean +/- SD) 61.3 mu +/- 11.0 and about 25% thicker (NS) in the regions above and below the knees. The relationship between TcPO2 (y) and epidermal thickness (x) could be described by the regression equation y = alpha i - 0.26x where the intercept alpha i differed between subjects, the mean value being 88 mmHg (range 77-103). The common regression coefficient of -0.26 was significantly different from zero (p less than 0.01, r2 = 0.49). Although the oxygen gradient across the total epidermis can not be estimated from skin biopsies, correction for the thickness of the living part of the skin may prove beneficial when TcPO2 measurements are used as an indicator of wound healing. The results suggests that the change of oxygen tension across the living part of epidermis is 0.26 mmHg/micron at various skin locations in different subjects.