Linear Spline Regression Research Articles

Using ‘amenable mortality’ as indicator of healthcare effectiveness in international comparisons: results of a validation study

Background and study aimsThere is widespread consensus on the need for better indicators of the effectiveness of healthcare. We carried out an analysis of the validity of amenable mortality as...

Association between plasma 25‐OH vitamin D and testosterone levels in men

A small randomized controlled trial suggested that vitamin D might increase the production of testosterone in men, which is supported by experimental studies in animals and a cross-sectional study showing positive associations between plasma 25-hydroxyvitamin D [25(OH)D] and testosterone and concordant seasonal variation of both biomarkers. We investigated the cross-sectional association of plasma 25(OH)D levels and total and free testosterone measured by immunoassay in 1362 male participants of the Health Professionals Follow-up Study who were selected for a nested case-control study on prostate cancer using multivariate-adjusted linear and restricted cubic spline regression models. 25(OH)D was positively associated with total and free testosterone levels. From the lowest to the highest 25(OH)D quintile, multivariate-adjusted means (95% confidence interval) were 18·5 (17·7; 19·4), 19·4 (18·6; 20·2), 19·6 (18·8; 20·4), 20·1 (19·3; 20·9) and 20·0 (19·1; 20·8; P-trend = 0·003) for total testosterone and 97·7 (93·9; 101·5), 98·2 (94·1; 102·2), 99·2 (95·2; 103·2), 100·7 (96·9; 104·5) and 101·5 (97·6; 105·4; P-trend = 0·03) for free testosterone. The shapes of the dose-response curves indicate that the association between 25(OH)D and total and free testosterone is linear at lower levels of 25(OH)D (below approximately 75-85 nmol/l), reaching a plateau at higher levels. Unlike for 25(OH)D, we did not observe any seasonal variation of testosterone concentrations. This study supports previously reported positive associations between vitamin D and testosterone although we did not observe parallel seasonal variation patterns. Possible causality and direction of the vitamin D-testosterone association deserve further scientific investigation.

Growth comparison of several Escherichia coli strains exposed to various concentrations of lactoferrin using linear spline regression

BackgroundWe wanted to compare growth differences between 13 Escherichia coli strains exposed to various concentrations of the growth inhibitor lactoferrin in two different types of broth (Syncase and Luria-Bertani (LB)). To carry this out, we present a simple statistical procedure that separates microbial growth curves that are due to natural random perturbations and growth curves that are more likely caused by biological differences.Bacterial growth was determined using optical density data (OD) recorded for triplicates at 620 nm for 18 hours for each strain. Each resulting growth curve was divided into three equally spaced intervals. We propose a procedure using linear spline regression with two knots to compute the slopes of each interval in the bacterial growth curves. These slopes are subsequently used to estimate a 95% confidence interval based on an appropriate statistical distribution. Slopes outside the confidence interval were considered as significantly different from slopes within. We also demonstrate the use of related, but more advanced methods known collectively as generalized additive models (GAMs) to model growth. In addition to impressive curve fitting capabilities with corresponding confidence intervals, GAM’s allow for the computation of derivatives, i.e. growth rate estimation, with respect to each time point.ResultsThe results from our proposed procedure agreed well with the observed data. The results indicated that there were substantial growth differences between the E. coli strains. Most strains exhibited improved growth in the nutrient rich LB broth compared to Syncase. The inhibiting effect of lactoferrin varied between the different strains. The atypical enteropathogenic aEPEC-2 grew, on average, faster in both broths than the other strains tested while the enteroinvasive strains, EIEC-6 and EIEC-7 grew slower. The enterotoxigenic ETEC-5 strain, exhibited exceptional growth in Syncase broth, but slower growth in LB broth.ConclusionsOur results do not indicate clear growth differences between pathogroups or pathogenic versus non-pathogenic E. coli.

PEMILIHAN PARAMETER PENGHALUS DALAM REGRESI SPLINE LINIER

This paper discusses aselection of smoothing parameters for the linier spline regression estimation on the data of electrical voltage differences in the wastewater. The selection methods are based on the mean square errorr (MSE) and generalized cross validation (GCV). The results show that in selection of smooting paranceus the mean square error (MSE) method gives smaller value , than that of the generalized cross validatio (GCV) method. It means that for our data case the errorr mean square (MSE) is the best selection method of smoothing parameter for the linear spline regression estimation.

Children's Health Insurance and Access to Care During and After the CHIP Expansion Period

We used national data from the Medical Expenditure Panel Survey and multivariate linear spline regression models to compare trends in children's health insurance coverage and access to care by income and race/ethnicity during (1998-2002) and after (2002-2006) major expansions of state insurance programs. During expansions, coverage and access for children in low-income and middle-income families improved, but these gains ceased thereafter for middle-income children, most of whom remained ineligible for public insurance. Racial and ethnic differences narrowed from 1998-2002, but persisted-and in at least one case tended to widen-from 2002-2006. Non-White children in families with incomes above most states' eligibility thresholds experienced significant declines in coverage and access to care after 2002. Gains in children's coverage and access to care during CHIP expansions have since stagnated or even reversed for some groups. Recent legislation to expand coverage for uninsured children (the PPACA of 2010) may redress these adverse changes in trends.

Circulating levels of 25-hydroxyvitamin D and human cathelicidin in healthy adults

Circulating levels of 25-hydroxyvitamin D and human cathelicidin in healthy adults

Comparison of nonlinear and spline regression models for describing mule duck growth curves

This study compared models for growth (BW) before overfeeding period for male mule duck data from 7 families of a QTL experimental design. Four nonlinear models (Gompertz, logistic, Richards, and Weibull) and a spline linear regression model were used. This study compared fixed and mixed effects models to analyze growth. The Akaike information criterion was used to evaluate these alternative models. Among the nonlinear models, the mixed effects Weibull model had the best overall fit. Two parameters, the asymptotic weight and the inflexion point age, were considered random variables associated with individuals in the mixed models. In our study, asymptotic weight had a greater effect in Akaike's information criterion reduction than inflexion point age. In this data set, the between-ducks variability was mostly explained by asymptotic BW. Comparing fixed with mixed effects models, the residual SD was reduced in about 55% in the latter, pointing out the improvement in the accuracy of estimated parameters. The mixed effects spline regression model was the second best model. Given the piecewise nature of growth, this model is able to capture different growth patterns, even with data collected beyond the asymptotic BW.

Bayesian change point analysis of abundance trends for pelagic fishes in the upper San Francisco Estuary

We examined trends in abundance of four pelagic fish species (delta smelt, longfin smelt, striped bass, and threadfin shad) in the upper San Francisco Estuary, California, USA, over 40 years using Bayesian change point models. Change point models identify times of abrupt or unusual changes in absolute abundance (step changes) or in rates of change in abundance (trend changes). We coupled Bayesian model selection with linear regression splines to identify biotic or abiotic covariates with the strongest associations with abundances of each species. We then refitted change point models conditional on the selected covariates to explore whether those covariates could explain statistical trends or change points in species abundances. We also fitted a multispecies change point model that identified change points common to all species. All models included hierarchical structures to model data uncertainties, including observation errors and missing covariate values. There were step declines in abundances of all four species in the early 2000s, with a likely common decline in 2002. Abiotic variables, including water clarity, position of the 2 per thousand isohaline (X2), and the volume of freshwater exported from the estuary, explained some variation in species' abundances over the time series, but no selected covariates could explain statistically the post-2000 change points for any species.

The Impact of Obesity on the Left Ventricle: The Multi-Ethnic Study of Atherosclerosis (MESA)

The purpose of this study was to evaluate the relationship of left ventricular (LV) remodeling assessed by cardiac magnetic resonance to various measures of obesity in a large population-based study. Obesity is a well-known risk factor for cardiovascular disease, yet its relationship with LV size and function is poorly understood. A total of 5,098 participants (age 45 to 84 years; 48% men) in the Multi-Ethnic Study of Atherosclerosis who were free of clinically apparent cardiovascular disease underwent cardiac magnetic resonance to assess LV size and function as well as measures of obesity, including body mass index, waist-to-hip ratio and waist circumference, and cardiovascular risk factors. Fat mass (FM) was estimated based on height-weight models derived from bioelectrical impedance studies. The associations of obesity measures with LV size and function were evaluated using linear spline regression models for body mass index and multivariable regression models for other measures of obesity; they were displayed graphically using generalized additive models. LV mass and end-diastolic volume were positively associated with measures of obesity in both sexes after adjustment for risk factors (e.g., 5.7-g and 6.9-g increase in LV mass per 10-kg increase in FM in women and men, respectively [p < 0.001]). LV mass-to-volume ratio was positively associated with increased body mass index, waist-to-hip ratio, waist circumference, and estimated FM (e.g., 0.02-g/ml and 0.06-g/ml increase in mass-to-volume ratio per 10-kg increase in FM in women and men, respectively [p < 0.001]). The increased mass-to-volume ratio was due to a greater increase in LV mass relative to LV end-diastolic volume. All associations were stronger for men than for women. Ejection fraction showed no significant association with measures of obesity. Obesity was associated with concentric LV remodeling without change in ejection fraction in a large, multiethnic cohort study.

Longitudinal Data Analysis in Medical Device Clinical Studies

Several statistical approaches are discussed for analysis of longitudinal clinical data from a two independent groups, randomized, prospective, multicenter study of patients with benign prostate hyperplasia (BPH) who are treated with either a new transurethral microwave thermotherapy device (TUMT, n = 147) or sham control (n = 73) at baseline and at 1, 3, and 6 months follow up. The primary clinical endpoint, based on the American Urological Association (AUA) symptom score from 7 questions (each scored 0–5) with 35 as the worst total score, is illustrated in this paper. The primary effectiveness null hypothesis is no treatment effect at any follow-up time against the alternative hypothesis of a favorable treatment effect at one or more follow-up times. Based on the prespecified clinical input of TUMT improvement in AUA mean score as a possible early effect, the expected change of slope between baseline to month 1 (baseline slope) and from month 1 to the end of month 6 (longitudinal slope), with knot at one month, were analyzed. The primary effectiveness endpoint was analyzed by a piecewise linear spline regression model with knot at month 1. Statistically significant lower mean AUA score was found in the TUMT group compared to the sham group at month 3 (P = 0.01) and at month 6 (P < 0.0001) follow-up; no significant difference in mean AUA score was found at baseline or at month 1 follow-up, after an overall statistically significant difference in mean AUA score (P = 0.004) was found between the TUMT and sham groups.

Adaptive control variates for pricing multi-dimensional American options

We explore a class of control variates for the American option pricing problem. We construct the control variates by using multivariate adaptive linear regression splines to approximate the option’s value function at each time step; the resulting approximate value functions are then combined to construct a martingale that approximates a “perfect” control variate. We demonstrate that significant variance reduction is possible even in a highdimensional setting. Moreover, the technique is applicable to a wide range of both option payoff structures and assumptions about the underlying risk-neutral market dynamics. The only restriction is that one must be able to compute certain one-step conditional expectations of the individual underlying random variables.

Corin I555(P568) Allele Is Associated With Enhanced Cardiac Hypertrophic Response to Increased Systemic Afterload

Corin activates pro-A-type naturetic peptide and pro-B-type naturetic peptide into biologically active molecules. We recently identified a minor allele in the corin gene defined by 2 highly linked single nucleotide polymorphisms (T555I and Q568P), which was associated with hypertension in blacks. Because of the direct antihypertrophic effects of the natriuretic peptide system, we hypothesized that the minor corin I555(P568) allele would be associated with an enhanced hypertrophic response to pressure overload. The relationship between systolic blood pressure and indexed left ventricular mass, derived from cardiac MRI, was analyzed in the Dallas Heart Study as a function of corin allele status. The Multi-Ethnic Study of Atherosclerosis was used as a validation cohort. All of the analyses were limited to self-identified blacks without treatment for hypertension. In addition, we genotyped 2114 markers highly informative for African ancestry in the Dallas Heart Study and derived a covariate representing African ancestry for multivariate models. In adjusted analysis, the corin I555(P568) allele was an independent predictor of left-ventricular mass in subjects with elevated systolic blood pressure. Linear spline regression analysis confirmed a significant interaction (P=0.002) between the corin I555(P568) allele and systolic blood pressure as a predictor of left ventricular mass in subjects with systolic blood pressure >120 mm Hg, and this nonlinear interaction was replicated in the Multi-Ethnic Study of Atherosclerosis. In the Dallas Heart Study, the corin I555(P568) allele was also associated with an increased odds for prevalent left ventricular hypertrophy in the presence of untreated hypertension. These data suggest that the corin I555(P568) allele represents a cardiac hypertrophy-sensitizing genetic locus in systemic hypertension.

Controlling for continuous confounding factors: non- and semiparametric approaches

Confounding is one of the major types of bias encountered in observational epidemiologic surveys designed to study the relation between an exposure factor and a health event. A common way to remove confounding bias during the statistical analysis phase is to adjust for the confounders in a regression model. If a confounding factor is assessed as a continuous variable, it is necessary to define how the variable is entered into the regression model. In the case of logistic regression, we illustrate through simulation that coding by a binary variable or a categorical variable with broad categories may lead to substantial residual confounding. Specific approaches can be used to define a coding method that limits residual confounding. Among these, we briefly present nonparametric approaches and describe in detail several semiparametric approaches (generalised partial linear models, spline regression and fractional polynomials). These can be used to estimate the relation between a continuous factor and the health event of interest by a smooth non pre-specified function. In semiparametric models, the effect of certain covariates is coded by a parametric function, whereas the coding of one or two continuous variables is represented by a nonparametric function. These models can be used in exploratory analyses to describe dose-effect relations between the confounder and the health event, and thus help to define a relevant coding for the confounder.

Stratification analysis of morphometric indices of populations of cancer cells with the phenotype of drug resistance

A new method of stratification analysis of general populations is proposed that is based on the approximation of inverse empirical distribution functions. With the help of the methods developed, it is shown that the number of modal classes in a heterogeneous population equals the number of segments of a linear regression spline. The proposed method is used for the investigation of populations of cancer cells that are sensitive or resistant to cisplatin.

Comparison of three nonlinear and spline regression models for describing chicken growth curves

This study compared three non-linear growth models (Richards, Gompertz, and logistic) and the spline linear regression model using BW measurements from an unselected, randombred chicken population. Based on the goodness of fit criteria, the nonlinear models (NLM) fitted the data better than the spline regression model. The four-parameter Richards model was expected to have the best overall fit; however, because the shape parameter predicted was close to 1.0, which corresponded with the Gompertz curve, there were no differences between the Richards and Gompertz models for the data used. The growth parameters predicted with the logistic model were different from those predicted by the Richards and Gompertz models. It was concluded that growth parameters predicted with different models with fixed inflection points are not directly comparable. The spline linear regression model is a compound function consisting of a series of linear models. It can be used to compartmentalize growth into segments and also substitute as an alternative to asymptotic models when the data are truncated before the asymptote is attained. The relationship between the growth trajectory (m) and carcass composition measurements were also investigated. Carcass yields among the different growth trajectories (m) were similar within sexes. However, the proportions of the major breast muscle and abdominal fat differed according to the m a bird followed.

Movement of teeth adjacent to posterior bounded edentulous spaces.

Bounded edentulous spaces (BES)-a missing posterior tooth with intact adjacent teeth-are thought to lead to arch collapse resulting from the movement of adjacent teeth. To determine the rate of change in distance between teeth adjacent to a BES, we examined three successive measurable radiographs of 116 untreated posterior BES cases. The distance between the teeth (DBT) adjacent to the space was measured, and change in DBT (delta DBT) between pre-extraction and follow-up radiographs was calculated. We used linear spline regression to construct models for tooth movement and to identify factors associated with delta DBT. The mean delta DBT was < 1 mm during the first year post-extraction, and the DBT continued to decrease at a successively slower rate each following year. Overall and for each tooth type, the greatest rates of decrease in DBT were seen in the zero to two-year period. In a multivariable model, time since extraction and tooth type were significantly associated with delta DBT. These findings suggest that movement of teeth adjacent to a posterior BES after the first two years is usually gradual and minor within the time frame of this study.

Service use over time and achievement of stable housing in a mentally ill homeless population.

Service use among clients of a multiservice agency serving homeless persons with severe mental illness was examined to determine whether patterns of service use reflected two stages in an adaptation of the transtheoretical model of change. In the adapted model, change occurs in five stages-pre-engagement, contemplation, engagement, the strategic moment, and consolidation. It was hypothesized that rates of service use would be highest immediately after clients obtained housing (the strategic moment) and would decrease in the months afterward (consolidation stage), with the greatest decreases occurring immediately after housing was obtained. Service use data were collected for two groups: a housed group of 58 clients who had obtained and sustained stable housing for at least 24 consecutive months at the time of sampling and an unhoused group of 55 clients who were matched with the housed clients on month of service entry. Total service use and use of three service types-a drop-in center, counseling, and health services-were examined to test the hypotheses. It was hypothesized that use of services by the unhoused group would show a consistent linear decline rather than a two-stage decline. Linear spline regression using bootstrap sampling methods was used to fit service use data for both groups. The two-stage solution significantly modeled the patterns of service use by the housed but not the unhoused clients, supporting the hypotheses. For the housed group, use of the drop-in center and counseling fit the model, and use of health services did not. The results provide limited support for the hypothesized five-stage model for achieving change.

Rainfall composition in Minnesota: integrating the chemistry, synoptic meteorology and numerical modelling

Clear quantitative differences (particularly with NH 4 + and NO 3 −) were observed in the chemical composition of rain collected by an in situ, refrigerated event sampler versus using a non-refrigerated, composite, weekly sampling protocol, at a site in the upper Midwestern USA. There was a non-linear, parabolic relationship between pH and SO 4 2− or NO 3 − concentrations. Such relationships could be stratified by the occurrences of three different classes of rainfall chemistry, governed by three different types of branching air trajectories preceding them. These overall processes were influenced by source-receptor issues. A Fourier modulated, three-way parabolic spline regression model was used to explain the non-linear relationships between H + and SO 4 2− or NO 3 −. This model performed much better ( R 2: 0.48–0.61) than the linear ( R 2: 0.008–0.31) and non-modulated three-way parabolic spline ( R 2: 0.22–0.36) regression analyses. Overall, the study allowed an integration of rainfall chemistry with synoptic meteorology and numerical modelling. ©

Controlling for Continuous Confounders in Epidemiologic Research

Multiple regression models are commonly used to control for confounding in epidemiologic research. Parametric regression models, such as multiple logistic regression, are powerful tools to control for multiple covariates provided that the covariate-risk associations are correctly specified. Residual confounding may result, however, from inappropriate specification of the confounder-risk association. In this paper, we illustrate the order of magnitude of residual confounding that may occur with traditional approaches to control for continuous confounders in multiple logistic regression, such as inclusion of a single linear term or categorization of the confounder, under a variety of assumptions on the confounder-risk association. We show that inclusion of the confounder as a single linear term often provides satisfactory control for confounding even in situations in which the model assumptions are clearly violated. In contrast, categorization of the confounder may often lead to serious residual confounding if the number of categories is small. Alternative strategies to control for confounding, such as polynomial regression or linear spline regression, are a useful supplement to the more traditional approaches.

Smooth piecewise linear regression splines with hyperbolic covariates

We-propose the use of hyperbolas as covariates in piecewise linear regression splines to fit data exhibiting a multi-phase linear response with smooth transitions between phases. The hyperbolic regression spline model, fitted by non-linear regression, provides an intuitive and easy way to extend to multiple phases the two-phase hyperbolic response model previously proposed by others. The small additional effort required to fit non-linear, as opposed to linear, regression models is particularly worthwhile when investigators are unwilling to assume that the slope of the response changes abruptly at the join points. Furthermore, undue influence on the join point and slope estimates, resulting from points in the transition region, may be avoided by using the hyperbolic regression spline. Two examples illustrate the use of this method.