Tuning parameter selection for nonparametric derivative estimation in random design

Estimation of a function, or its derivatives via nonparametric regression requires selection of one or more tuning parameters. In the present work, we propose a tuning parameter selection criterion called DCp for nonparametric derivative estimation in random design. Our criterion is general in that it can be applied with any nonparametric estimation method which is linear in the observed outcomes. Charnigo et al. [A generalized Cp criterion for derivative estimation. Technometrics. 2011;53(3):238–253] had proposed a GCp criterion for a similar purpose, assuming values of the covariate to be fixed and constant error variance. Here we consider the setting with random design and non-constant error variance since the covariate values will not generally be fixed and equally spaced in real data applications. We justify DCp in this setting both theoretically and by simulation. We also illustrate use of DCp with two economics data sets.</p

AT2 receptor deficiency did not change percent atherosclerotic lesion area in mice infused with AngII.

<p>Percent atherosclerotic lesion area of aortic arches was calculated as lesion area/intimal surface area (x 100%). Circles and inverted triangles represent values from individual mice, diamonds are mean values of each group, and bars are SEM. * P<0.0001 between AngII-infused groups and saline-infused groups as analyzed by two way ANOVA. There were no significant effects of genotype (AT2 +/y versus AT2 −/y) on lesion areas.</p

PD123319 did not change atherosclerotic lesion size in mice infused with AngII.

<p>Percent atherosclerotic lesion area of aortic arches was calculated as lesion area/intimal surface area (x 100%). Circles and inverted triangles represent values from individual mice, diamonds are mean values of each group, and bars are SEM. There were no significant effects of genotype (AT2 +/y versus AT2 −/y) or PD123319 on lesion areas, as demonstrated by two way ANOVA.</p

Characteristics of Mice Infused with either Saline or AngII.

<p>Body weights and plasma cholesterol concentrations were determined at termination. Systolic blood pressures were recorded during the last week of the study. Values are represented as mean±SEM. Two way ANOVA was performed for body weights and plasma cholesterol concentrations, and two way repeated measures ANOVA was used to analyze systolic blood pressures. * P = 0.03 between AT2 +/y and −/y genotypes. ^# P = 0.001 between AT2 +/y and −/y mice infused with AngII. There were no significant differences between AT2 receptor genotypes (+/y versus −/y) and between manipulations (saline versus AngII) for systolic blood pressures.</p

AT2 receptor deficiency did not change suprarenal aortic expansion in mice infused with AngII.

<p>Maximal aortic width of suprarenal aortas was measured ex vivo. Circles and inverted triangles represent values from individual mice, diamonds are mean values of each group, and bars are SEM. * P = 0.002 between AngII-infused groups and saline-infused groups as analyzed by two way ANOVA. There were no significant effects of genotype (AT2 +/y versus AT2 −/y) on aortic width.</p

PD123319 Augments Angiotensin II-Induced Abdominal Aortic Aneurysms through an AT2 Receptor-Independent Mechanism

<div><p>Background</p><p>AT2 receptors have an unclear function on development of abdominal aortic aneurysms (AAAs), although a pharmacological approach using the AT2 receptor antagonist PD123319 has implicated a role. The purpose of the present study was to determine the role of AT2 receptors in AngII-induced AAAs using a combination of genetic and pharmacological approaches. We also defined effects of AT2 receptors in AngII-induced atherosclerosis and thoracic aortic aneurysms.</p><p>Methods and Results</p><p>Male AT2 receptor wild type (AT2 +/y) and deficient (AT2 -/y) mice in an LDL receptor −/− background were fed a saturated-fat enriched diet, and infused with either saline or AngII (500 ng/kg/min). AT2 receptor deficiency had no significant effect on systolic blood pressure during AngII-infusion. While AngII infusion induced AAAs, AT2 receptor deficiency did not significantly affect either maximal width of the suprarenal aorta or incidence of AAAs. The AT2 receptor antagonist PD123319 (3 mg/kg/day) and AngII were co-infused into male LDL receptor −/− mice that were either AT2 +/y or −/y. PD123319 had no significant effect on systolic blood pressure in either wild type or AT2 receptor deficient mice. Consistent with our previous findings, PD123319 increased AngII-induced AAAs. However, this effect of PD123319 occurred irrespective of AT2 receptor genotype. Neither AT2 receptor deficiency nor PD123319 had any significant effect on AngII-induced thoracic aortic aneurysms or atherosclerosis.</p><p>Conclusions</p><p>AT2 receptor deficiency does not affect AngII-induced AAAs, thoracic aortic aneurysms and atherosclerosis. PD123319 augments AngII-induced AAAs through an AT2 receptor-independent mechanism.</p></div

PD123319 augmented AngII-induced suprarenal aortic expansion and incidence of AAAs independent of AT2 receptor genotype.

<p>A. Maximal aortic width of suprarenal aortas was measured ex vivo. Circles and inverted triangles represent values from individual mice, diamonds are mean values of each group, and bars are SEM. * P = 0.01 between AngII-infused and PD123319-infused groups as analyzed by two way ANOVA. There were no significant effects on aortic width for genotypes (AT2 +/y versus AT2 −/y). B. Incidence of AAAs. * P = 0.006 between AngII-infused and PD123319-infused groups as analyzed by Fisher's Exact test. There were no significant effects of genotype (AT2 +/y versus AT2 −/y) on incidence of AAAs.</p

AT2 receptor deficiency did not change expansion of the aortic arch region in mice infused with AngII.

<p>Intimal area of aortic arches was calculated from en face measurements. Circles and inverted triangles represent values from individual mice, diamonds are mean values of each group, and bars are SEM. * P = 0.0003 between AngII-infused and saline-infused groups as analyzed by two way ANOVA. There were no significant effects of genotype (AT2 +/y versus AT2 −/y) on intimal areas of aortic arches.</p

Characteristics of Mice Infused with AngII alone or both AngII and PD123319.

<p>Body weights and plasma cholesterol concentrations were determined at termination. Systolic blood pressures were recorded during the last week of the study. Aortic rupture was confirmed by necropsy with visible blood clot in the abdominal cavity and verification of disrupted aortic structure of abdominal aorta. Values are represented as mean±SEM. Two way ANOVA was performed for body weights and plasma cholesterol concentrations, two way repeated measures ANOVA was used to analyze systolic BP data, and Fisher's exact test was done for aortic rupture rate. There were no significant differences between AT2 receptor genotypes (+/y versus −/y) and between manipulations (AngII versus AngII+PD123319) for all parameters tested in this table.</p

PD123319 did not change AngII-induced expansion of the aortic arch region.

<p>Intimal area of aortic arches was measured using an en face method after termination. Circles and inverted triangles represent values from individual mice, diamonds are mean values of each group, and bars are SEM. There were no significant effects of genotype (AT2 +/y versus AT2 -/y) or PD123319 on intimal areas of the aortic arch region, as demonstrated by two way ANOVA.</p