Fast Estimation of Regression Parameters in a Broken-Stick Model for Longitudinal Data

<p>Estimation of change-point locations in the broken-stick model has significant applications in modeling important biological phenomena. In this article, we present a computationally economical likelihood-based approach for estimating change-point(s) efficiently in both cross-sectional and longitudinal settings. Our method, based on local smoothing in a shrinking neighborhood of each change-point, is shown via simulations to be computationally more viable than existing methods that rely on search procedures, with dramatic gains in the multiple change-point case. The proposed estimates are shown to have <math><msqrt><mi>n</mi></msqrt></math>-consistency and asymptotic normality—in particular, they are asymptotically efficient in the cross-sectional setting—allowing us to provide meaningful statistical inference. As our primary and motivating (longitudinal) application, we study the Michigan Bone Health and Metabolism Study cohort data to describe patterns of change in log  estradiol levels, before and after the final menstrual period, for which a two change-point broken-stick model appears to be a good fit. We also illustrate our method on a plant growth dataset in the cross-sectional setting. Supplementary materials for this article are available online.</p

Two-Stage Plans for Estimating the Inverse of a Monotone Function

<div><p>This study investigates two-stage plans based on nonparametric procedures for estimating an inverse regression function at a given point. Specifically, isotonic regression is used at stage one to obtain an initial estimate followed by another round of isotonic regression in the vicinity of this estimate at stage two. It is shown that such two-stage plans accelerate the convergence rate of one-stage procedures and are superior to existing two-stage procedures that use local parametric approximations at stage two when the available budget is moderate and/or the regression function is “ill-behaved.” Both Wald- and likelihood ratio-type confidence intervals for the threshold value of interest are investigated and the latter are recommended in applications due to their simplicity and robustness. The developed plans are illustrated through a comprehensive simulation study and an application to car fuel efficiency data.</p></div

The sample selection process.

<p>The sample selection process.</p

Long chain fatty acids that differed significantly between control and UQ groups at 2-y follow-up (<i>p</i>&lt;0.05).

<p>PC, phosphatidylcholine; PE, phosphatidylethanolamine; PG, glycerophosphoglycerol; PS, phosphatidylserine.</p

Clinical data for participants during pregnancy and at follow-up in the three study groups.

<p><i>p</i> values calculated applying ANOVA or **Chi-squared tests. #Data are geometric mean and 95% confidence intervals</p><p>Clinical data for participants during pregnancy and at follow-up in the three study groups.</p

Metabolites differing between UQ and GDM groups at 2-y follow-up (<i>p</i>&lt;0.05).

<p>Metabolites have been classified according to their molecular structures or known metabolic functions/pathway participation. Within each class, data have been separated in to those with higher and lower ratios and are then presented in order from lowest to highest <i>p</i> value. The molecular weights, calculated as the monoisotopic mass, are included. Ratios with 95% confidence intervals in parentheses are shown. CE Cholesteryl ester; CEHC, 2,5,7,8-tetramethyl-2-(2'-carboxyethyl)-6-hydroxychroman; DG, diglyceride; HEPE, hydroxy-eicosapentaenoic acid; PC, phosphatidylcholine; PG, phosphatidylglycine; The values in parentheses (for example PC(34∶0)) relate to the total fatty acid carbon chain length and number of carbon double bonds (unsaturation) in each metabolite. *Identification by matching of retention time and accurate mass to authentic chemical standard.</p><p>Metabolites differing between UQ and GDM groups at 2-y follow-up (<i>p</i>&lt;0.05).</p

Metabolites differing between control and GDM groups at 2-y follow-up (<i>p</i>&lt;0.05).

<p>Metabolites have been classified according to their molecular structures or known metabolic functions/pathway participation. Within each class the data have been separated in to those with higher and lower ratios and are then presented in order from lowest to highest <i>p</i> value. The molecular weights, calculated as the monoisotopic mass, are included. Ratios with 95% confidence intervals in parentheses are shown. CE cholesteryl ester; DG, diglyceride; PC, phosphatidylcholine; PE, phosphatidylethanolamine; PG, phosphatidylglycine; PGF, prostaglandin; PI, phosphatidylinositol; PS, phosphatidylserine; The values in parentheses (for example PC(34∶0)) relate to the total fatty acid carbon chain length and number of carbon double bonds (unsaturation) in each metabolite. *Identification by matching of retention time and accurate mass to authentic chemical standard.</p><p>Metabolites differing between control and GDM groups at 2-y follow-up (<i>p</i>&lt;0.05).</p

Phospholipids that differed significantly between control and UQ groups at 2-y follow-up (<i>p</i>&lt;0.05).

<p>*Identification by matching of retention time and accurate mass to authentic chemical standard.</p