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PubMed1997

Multiple linear regression as an analytical tool in cephalometric studies.

When the effect is studied of a factor like 'orthodontic therapy' on linear craniofacial growth, the concomitant consequence of age and gender on size cannot be ignored. The methodologically correct solution is division of the study group into smaller units, each of which is homogeneous with respect to age, gender, and therapy, and to compare these with matched controls. Yet, apart from matched controls being hard to find, this method of subdivision has the serious drawback that smaller groups decrease statistical power. A solution without the need to create sub-groups lies in the application of multiple linear regression analysis. It has been applied to biological data in other studies, but verification of the outcome has not been reported so far. Indeed, testing the mathematical assumptions underlying the regression model created unresolvable obstacles and, therefore, it was decided to perform verification by means of practical examples. Two separate tests for the applicability of the multiple linear regression method, on different data, with differing predictor sets, and with different control samples have been performed.

Adolescent↗

PubMed2003

Linear regression with spatial constraint to generate parametric images of ligand-receptor dynamic PET studies with a simplified reference tissue model.

For the quantitative analysis of ligand-receptor dynamic positron emission tomography (PET) studies, it is often desirable to apply reference tissue methods that eliminate the need for arterial blood sampling. A common technique is to apply a simplified reference tissue model (SRTM). Applications of this method are generally based on an analytical solution of the SRTM equation with parameters estimated by nonlinear regression. In this study, we derive, based on the same assumptions used to derive the SRTM, a new set of operational equations of integral form with parameters directly estimated by conventional weighted linear regression (WLR). In addition, a linear regression with spatial constraint (LRSC) algorithm is developed for parametric imaging to reduce the effects of high noise levels in pixel time activity curves that are typical of PET dynamic data. For comparison, conventional weighted nonlinear regression with the Marquardt algorithm (WNLRM) and nonlinear ridge regression with spatial constraint (NLRRSC) were also implemented using the nonlinear analytical solution of the SRTM equation. In contrast to the other three methods, LRSC reduces the percent root mean square error of the estimated parameters, especially at higher noise levels. For estimation of binding potential (BP), WLR and LRSC show similar variance even at high noise levels, but LRSC yields a smaller bias. Results from human studies demonstrate that LRSC produces high-quality parametric images. The variance of R(1) and k(2) images generated by WLR, WNLRM, and NLRRSC can be decreased 30%-60% by using LRSC. The quality of the BP images generated by WLR and LRSC is visually comparable, and the variance of BP images generated by WNLRM can be reduced 10%-40% by WLR or LRSC. The BP estimates obtained using WLR are 3%-5% lower than those estimated by LRSC. We conclude that the new linear equations yield a reliable, computationally efficient, and robust LRSC algorithm to generate parametric images of ligand-receptor dynamic PET studies.

Adult↗

PubMed1999

Gait analysis before and after unilateral total knee arthroplasty. Study using a linear regression model of normal controls -- women without arthropathy.

Stepwise multiple regression analysis (forward method) was performed with 22 gait variables obtained from the free and slow gait of 35 normal controls (women without knee arthropathy). These 22 variables were target variables, and velocity, age, body height, and body weight were explanatory variables. Velocity showed the greatest effect on the gait variables, followed by weight, age, and height. Of the 22 target variables, 16 could be explained by a significant level of difference of P < 0.01. A linear regression model of normal gait was then established, based on the judgment that these 16 gait variables were greatly affected by four variables -- velocity, age, height, and weight. Since this model does not perfectly represent the observed values, we compared the observed value/predicted value ratios in different groups to compensate for their differences. The free gait velocity in 20 osteoarthritic patients, 1 year or more after unilateral total knee arthroplasty (TKA), and who had no pain in the contralateral knee was lower than in the normal controls. In comparisons using linear regression models, step length was shorter, step width was longer, and gait cycle was shorter than in controls. Single support time was shorter and double support time was longer. Of the ground reaction forces, the first peak of the vertical component and the peak of the driving force of the fore-aft component were smaller than in controls. The total range of motion (TRM) in the stance phase was less than in controls. These results show quantitatively not only that the velocity of gait after TKA is lower than in normal controls, but also that gait patterns are different. In eight osteoarthritic patients assessed before and after TKA, and who had no pain in the contralateral knee, free gait velocity increased 6 months post-operation, but showed no further changes at 1 year. A linear regression study comparing the gait before TKA and 6 months post-TKA revealed that step time, single support time, double support time, and step width, and -- with regard to ground reaction forces -- the peaks of the driving force and the braking force of the fore-aft component, and TRM in the stance phase all approached the levels in the normal controls. No further changes were observed 1 year postoperatively. Although our models were not perfect, we were able to clarify the differences in gait variables between a TKA group and normal controls and the improvements from pre- to post TKA by normalizing the influence of the four independent variables, (velocity, age, weight, and height) with particular emphasis on gait velocity.

Aged↗

PubMed1992

Use of a spreadsheet program for Deming's linear regression analysis.

A spreadsheet program for Deming's linear regression analysis suitable for use with popular software packages such as Lotus 1-2-3 or Quattro is described. The program is controlled by an autoexecuting simple menu of operations allowing use by those with only limited experience of spreadsheet operation. Regression coefficients, identification of suspect outlying data values and confidence limits and graphical displays of fitted regression lines are provided.

Confidence Intervals↗

PubMed1992

Interpretation of linear regression models that include transformations or interaction terms.

In linear regression analyses, we must often transform the dependent variable to meet the statistical assumptions of normality, variance stability, or linearity. Transformations, however, can complicate the interpretation of results because they change the scale on which the dependent variable is measured. In this setting, the inclusion of product terms or the transformation of some independent (or predictor) variables may further complicate interpretation. In this article, we present some interpretations of linear models that include transformations or product terms. We illustrate these interpretations using regression analyses designed to study determinants of serum testosterone levels. These examples show how one can present results using simple measures, such as medians, and interpret regression parameters.

Epidemiologic Methods↗

PubMed2005

Pointwise linear regression for evaluation of visual field outcomes and comparison with the advanced glaucoma intervention study methods.

OBJECTIVE: To investigate pointwise linear regression (PLR) for longitudinal evaluation of visual fields and to compare results with those of the Advanced Glaucoma Intervention Study (AGIS) criteria. METHODS: We selected 509 eyes (401 patients) from the AGIS with 3 or more years of follow-up, 7 or more visual field examinations, and an AGIS reference score of 16 or lower. Visual field change at test locations was defined as a change of threshold sensitivity of 1 dB/y or higher and P<or=.01. Several sets of criteria were investigated for defining change of visual field series with PLR. MAIN OUTCOME MEASURES: Progression or improvement of visual field series with PLR and AGIS criteria. RESULTS: Mean (SD) follow-up time and baseline AGIS score were 7.4 (1.7) years and 7.7 (4.4), respectively. Pairwise agreement between AGIS and various PLR criteria ranged from 52% to 64% with the kappa statistic varying between 0.22 (95% confidence interval, 0.15-0.29) and 0.30 (95% confidence interval, 0.22-0.38). One hundred thirty-eight (27%) and 151 (30%) eyes progressed (85 eyes or 17% detected by both methods) while 72 (14%) and 11 (2%) eyes improved (5 eyes or 1% detected by both methods) based on AGIS and the most rigorous PLR criteria, respectively. CONCLUSIONS: Based on rigorous, clinically relevant criteria, PLR detects progression in a similar proportion of eyes compared with AGIS criteria. Pointwise linear regression may be superior to AGIS methods since it identifies fewer visual field series as improving.

Adult↗

PubMed1984

Some simple multiple linear regression equations for estimation of maximal aerobic power in healthy Indian males.

An attempt has been made to evolve some simple multiple linear regression equations for the prediction of VO2 max from body weight, time for 3.2 km run and exercise dyspnoeic index ( DIstd Ex%). The predictor variables have been selected by examining the product moment correlations of body weight, relative body weight indices, time for 3.2 km run, chest expansion, height, and DIstd Ex% with VO2 max, based on data collected on 320 healthy Indian males (17-22 years). It has been observed that body weight, time for 3.2 km run and DIstd Ex% attained maximum correlations with VO2 max. Thus, two regression equations with two and three predictor variables have been established in this paper to predict VO2 max. The first regression equation yielded a multiple correlation of 0.608 (P less than 0.001) with a standard error of 0.214 1 X min-1. In this equation, body weight and time for 3.2 km run were considered as significant predictors. To increase the precision of this equation, another multiple linear regression equation based on body weight, time for 3.2 km run and DIstd Ex% as predictors has been developed. This equation yielded a multiple correlation of 0.658 (P less than 0.001) with a standard error of 0.204 1 X min-1. Applications of these regression equations will be of practical importance to biomedical scientists engaged in the development of a simple procedure for indirect assessment of VO2 max, and may serve well as preliminary screening procedures for personnel selection.

Adolescent↗

PubMed1999

Linear regression for bivariate censored data via multiple imputation.

Bivariate survival data arise, for example, in twin studies and studies of both eyes or ears of the same individual. Often it is of interest to regress the survival times on a set of predictors. In this paper we extend Wei and Tanner's multiple imputation approach for linear regression with univariate censored data to bivariate censored data. We formulate a class of censored bivariate linear regression methods by iterating between the following two steps: 1. the data is augmented by imputing survival times for censored observations; 2. a linear model is fit to the imputed complete data. We consider three different methods to implement these two steps. In particular, the marginal (independence) approach ignores the possible correlation between two survival times when estimating the regression coefficient. To improve the efficiency, we propose two methods that account for the correlation between the survival times. First, we improve the efficiency by using generalized least squares regression in step 2. Second, instead of generating data from an estimate of the marginal distribution we generate data from a bivariate log-spline density estimate in step 1. Through simulation studies we find that the performance of the two methods that take the dependence into account is close and that they are both more efficient than the marginal approach. The methods are applied to a data set from an otitis media clinical trial.

Anti-Bacterial Agents↗

PubMed1998

Comparative study of different weighting methods in non-linear regression analysis: implications in the parametrization of carebastine after intravenous administration in healthy volunteers.

The influence of different weighting methods in non-linear regression analysis was evaluated in the pharmacokinetics of carebastine after a single intravenous dose of 10 mg in 8 healthy volunteers. Plasma concentrations were measured by HPLC using an on-line solid-phase extraction method and automated injection. The analytical method was fully validated and the function of the analytical error subsequently determined. The parametric approach was performed using different weighting methods, including the homoscedastic method (W = 1) and heteroscedastic methods using weights of 1/C, 1/C2, and the inverse of the concentration variance calculated through the analytical error function (1/V), and the results were statistically evaluated according to the normal distribution. Statistically significant differences were observed in the representative parameters of the disposition kinetics of carebastine. The use of a multiple comparison test for statistical analysis of all differences among group means indicated that differences were generated between the homoscedastic method (W = 1) and the heteroscedastic methods (1/C, 1/C2, and 1/V). The results obtained in the present study confirmed the utility of the analytical error function as a weighting method in non-linear regression analysis and reinforced the importance of the correct choice of weights to avoid the estimation of imprecise or erroneous pharmacokinetic parameters.

Adult↗

PubMed2006

Pointwise linear regression analysis for detection of visual field progression with absolute versus corrected threshold sensitivities.

PURPOSE: To compare the performance of point-wise linear regression analysis (PLR) with total deviation (TD) versus corrected or pattern deviation (PD) threshold sensitivities for detection of visual field progression. METHODS: Four hundred two eyes (402 patients) enrolled in the Advanced Glaucoma Intervention Study (AGIS) were selected. Criteria for progression according to PLR were a slope or=2 worsening points within the same Glaucoma Hemifield Test cluster. PLR was performed on TD and PD threshold sensitivities and compared to clinical evaluation. Eyes were classified into three groups based on mean deviation (MD): mild (MD>or=-6 dB), moderately advanced (-6 dB>MD>or=-12 dB), and advanced (MD<-12 dB) glaucoma. RESULTS: Visual field progression was observed in 154 (38%), 85 (21%), and 175 (44%) eyes, according to PLR(TD) and PLR(PD), and clinical evaluation. The pair-wise agreement between clinicians and PLR(TD) was significantly greater than that of clinicians and PLR(PD) (kappa=0.48, 95% CI: 0.44-0.52 vs. kappa=0.31, 95% CI: 0.27-0.35). Agreement between PLR(TD) and PLR(PD) decreased with increasing glaucoma severity: kappa (95% CI)=0.60 (0.52-0.67), 0.41 (0.35-0.47), and 0.33 (0.27-0.40) for mild, moderately advanced, and advanced glaucoma, respectively. CONCLUSIONS: Point-wise linear regression analysis on TD threshold sensitivities performed better than the same analysis on PD when clinical evaluation was used as a reference. Agreement between the two methods was less in moderately advanced and advanced glaucoma.

Adult↗

PubMed2006

Linear regression modeling to compare fluoride release profiles of various restorative materials.

OBJECTIVES: The aim of this study was to compare the released fluoride profiles of various restorative materials by using linear regression analysis. METHODS: Specimens were prepared using a cylindrical Teflon mold with a height of 2 mm and a radius of 8 mm. After being prepared, specimens were immediately placed into artificial saliva which was replaced at various times during 6 weeks. These released intrinsic fluoride amounts were measured by using an ion selective electrode. Then, data obtained cumulatively were statistically analyzed, and the released profiles were compared. RESULTS: It was observed that the materials released fluoride at different levels of concentration and the largest fluoride release was obtained from the conventional glass ionomer cement. This was followed by resin modified glass ionomer cement, polyacid modified composite resin, and fluoride releasing composite resin, respectively. Although the released fluoride amounts of the materials were different, their release profiles were found to be similar in that the release was initially fast and then it became steady as time passed. SIGNIFICANCE: The statistical modeling of the release profiles helps to compare the fluoride release behavior of materials and also to predict fluoride release amounts for the future. In literature, for these purposes, separate nonlinear statistical models have extensively been utilized. However, the single linear statistical modeling approach has numerous advantages such as providing estimators having good statistical properties, exact results, precise inference and simplicity in calculation. Therefore, this study was conducted to introduce the use of single linear regression modeling to compare release profiles statistically.

Compomers↗

PubMed1988

Confidence intervals for estimates based on linear regression in forensic anthropology.

Forensic anthropologists commonly use simple linear regression to estimate the value of a dependent variable, such as stature, for a single specimen where the value of the independent variable, such as humerus length, is known. Published studies providing regression equations for such use almost invariably include the standard error of estimate. Unfortunately, it is exceptional for forensic anthropologists to use the standard error to calculate correctly the confidence limits for their single predicted value. We attempt to show why this may be and provide explicit guidelines for the proper construction of confidence interval in such circumstances.

Anthropology, Physical↗

PubMed2005

Differential gene expression detection and sample classification using penalized linear regression models.

Differential gene expression detection and sample classification using microarray data have received much research interest recently. Owing to the large number of genes p and small number of samples n (p >> n), microarray data analysis poses big challenges for statistical analysis. An obvious problem owing to the 'large p small n' is over-fitting. Just by chance, we are likely to find some non-differentially expressed genes that can classify the samples very well. The idea of shrinkage is to regularize the model parameters to reduce the effects of noise and produce reliable inferences. Shrinkage has been successfully applied in the microarray data analysis. The SAM statistics proposed by Tusher et al. and the 'nearest shrunken centroid' proposed by Tibshirani et al. are ad hoc shrinkage methods. Both methods are simple, intuitive and prove to be useful in empirical studies. Recently Wu proposed the penalized t/F-statistics with shrinkage by formally using the (1) penalized linear regression models for two-class microarray data, showing good performance. In this paper we systematically discussed the use of penalized regression models for analyzing microarray data. We generalize the two-class penalized t/F-statistics proposed by Wu to multi-class microarray data. We formally derive the ad hoc shrunken centroid used by Tibshirani et al. using the (1) penalized regression models. And we show that the penalized linear regression models provide a rigorous and unified statistical framework for sample classification and differential gene expression detection.

Algorithms↗

PubMed1978

Panel studies of acute health effects of air pollution. II. A methodologic study of linear regression analysis of asthma panel data.

Panel studies relating illness, such as asthma attacks, cardiopulmonary symptoms, and acute respiratory symptoms, to daily air pollution and weather are important in environmental epidemiology. A study of the practical robustness of multiple linear regression procedures, which have been the preferred statistical models in analyses, is presented. The study is based on data from three asthma panels in Chattanooga, Tennessee, collected in 1972-1973. Linear regression models, commonly used, which incorporate only minimum temperature and an air pollutant were found to be potentially misleading; such models are highly sensitive to reporting trends in the data and do not correct adequately for weather variables. Temporal and spatial control strategies were employed and proved to be useful in detecting problems in the data due to undiscovered intervening variables. True day-to-day relationships estimated by a pair-day analysis were frequently inconsistent with "daily" effects estimated by the usual regression models and suggested that, in fact, the asthma panel data contained no useful information concerning day-to-day relationships.

Adolescent↗

PubMed2002

Linear regression calibration: theoretical framework and empirical results in EPIC, Germany.

Large scale dietary assessment instruments are usually based on the food frequency technique and have therefore to be tailored to the involved populations with respect to mode of application and inquired food items. In multicenter studies with different populations, the direct comparability of dietary data is therefore a challenge because each local dietary assessment tool might have its specific measurement error. Thus, for risk analysis the direct use of dietary measurements across centers requires a common reference. For example, in the European prospective cohort study EPIC (European Prospective Investigation into Cancer and Nutrition) a 24-hour recall was chosen to serve as such a reference instrument which was based on a highly standardized computer-assisted interview (EPIC-SOFT). The 24-hour recall was applied to a representative subset of EPIC participants in all centers. The theoretical framework of combining multicenter dietary information was previously published in several papers and is called linear regression calibration. It is based on a linear regression of the food frequency questionnaire to the reference. The regression coefficients describe the absolute and proportional scaling bias of the questionnaire with the 24-hour recall taken as reference. This article describes the statistical basis of the calibration approach and presents first empirical results of its application to fruit, cereals and meat consumption in EPIC Germany represented by the two EPIC centers, Heidelberg and Potsdam. It was found that fruit could be measured well by the questionnaire in both centers (lambdacirc; = 0.98 (males) and lambdacirc; = 0.95 (females) in Heidelberg, and lambdacirc; = 0.86 (males) and lambdacirc; = 0.7 (females) in Potsdam), cereals less (lambdacirc; = 0.53 (males) and lambdacirc; = 0.4 (females) in Heidelberg, and lambdacirc; = 0.53 (males) and lambdacirc; = 0.44 (females) in Potsdam), and that the assessment of meat (lambdacirc; = 0.72 (males) and lambdacirc; = 0.65 (females) in Heidelberg, and lambdacirc; = 0.49 (males) and lambdacirc; = 0.42 (females) in Potsdam) has a center-specific bias. The application of the calibration approach to the questionnaire data will change the ranking of the two centers following the data of the reference instrument, and not well-measured food items will exhibit considerably less variation compared to the original data. We conclude that calibration is a necessary step in multicenter studies. However, this exercise shows that the current statistical framework is not yet sufficiently developed for a broad application.

Adult↗

PubMed1991

Effect of airway inertance on linear regression estimates of resistance and compliance in mechanically ventilated infants: a computer model study.

Respiratory inertance (I) is usually ignored when resistance (R) and compliance (C) of mechanically ventilated infants are estimated by least squares linear regression. Values of I that have been reported for these patients can cause impedances whose magnitudes approximate respiratory resistance. We show theoretically that if inertance is neglected no error is expected in resistance estimates, but a positive bias in compliance can be, proportional to the inertance, the compliance, and the sinusoidal frequency at which the measurements are made. To determine the errors in parameter estimates when the pressure waveform is non-sinusoidal, we simulated linear regression based on non-inertive and inertive models. R, C, and I of the simulated lung were varied over the range expected in an infant intensive care unit. The ventilator was simulated as a critically damped second order system with a square pulse input. The rise time (TR) of the pressure pulse was varied over the range reported in infant ICUs. Simulated measurements confirmed that resistance is correctly estimated if inertance is neglected. Maximum error in compliance estimates (13%) occurred when TR and R were low, and C and I were high. The variation in the error in estimated compliance was consistent with the theory. Coefficients of variation of the parameters, the standard errors, and R2 of the regressions tended to deteriorate with increasing compliance error, but the relationships were not single valued. These statistics may alert investigators to possible bias in compliance caused by neglected inertance, but cannot be used to correct any bias.

Airway Resistance↗

PubMed2005

Generalized linear regression analysis of association of universal helmet laws with motorcyclist fatality rates.

This study evaluates the association of universal helmet laws with U.S. motorcyclist fatality rates from 1993 through 2002 using climate measures as statistical controls for motorcycling activity via quasi-maximum likelihood generalized linear regression analyses. Results revealed that motorcyclist fatalities and injuries are strongly associated with normalized heating degree days and precipitation inches, and that universal helmet laws are associated with lower motorcyclist fatality rates when these climate measures, and their interaction, are statistically controlled. This study shows that climate measures have considerable promise as indirect measures (proxies) of motorcycling activity in generalized linear regression studies.

Accidents, Traffic↗

PubMed2004

Differential gene expression detection using penalized linear regression models: the improved SAM statistics.

UNLABELLED: Differential gene expression detection using microarrays has received lots of research interests recently. Many methods have been proposed, including variants of F-statistics, non-parametric approaches and empirical Bayesian methods etc. The SAM statistics has been shown to have good performance in empirical studies. SAM is more like an ad hoc shrinkage method. The idea is that for small sample microarray data, it is often useful to pool information across genes to improve efficiency. Under Bayesian framework Smyth formally derived the test statistics with shrinkage using the hierarchical models. In this paper we cast differential gene expression detection in the familiar framework of linear regression model. Commonly used test statistics correspond to using least squares to estimate the regression parameters. Based on the vast literature of research on linear models, we can naturally consider other alternatives. Here we explore the penalized linear regression. We propose the penalized t-/F-statistics for two-class microarray data based on [Formula: see text] penalty. We will show that the penalized test statistics intuitively makes sense and through applications we illustrate its good performance. AVAILABILITY: Supplementary information including program codes, more detailed analysis results and R functions for the proposed methods can be found at http://www.biostat.umn.edu/~baolin/research CONTACT: baolin@biostat.umn.edu SUPPLEMENTARY INFORMATION: http://www.biostat.umn.edu/~baolin/research.

Cell Line, Tumor↗