Multiple linear regression with compositional response and covariates

<p>The standard regression model designed for real space is not suitable for compositional variables; it should be considered, whether the response and/or covariates are of compositional nature. There are usually three types of multiple regression model with compositional variables: Type 1 refers to the case where all the covariates are compositional data and the response is real; Type 2 is the opposite of Type 1; Type 3 relates to the model with compositional response and covariates. There have been some models for the three types. In this paper, we focus on Type 3 and propose multiple linear regression models including model in the simplex and model in isometric log-ratio (ilr) coordinates. The model in the simplex is based on matrix product, which can project a <math><msub><mi>D</mi><mrow><mn>1</mn></mrow></msub></math>-part composition to another <math><msub><mi>D</mi><mrow><mn>2</mn></mrow></msub></math>-part composition, and can deal with different number of parts of compositional variables. Some theorems are given to point out the relationship of parameters between the proposed models. Moreover, the inference for parameters in proposed models is also given. Real example is studied to verify the validity and usefulness of proposed models.</p

Intrathecal administration of U0126 attenuates CFA-induced heat hyperalgesia.

<p>a. PWLs to thermal stimuli significantly decreases after intraplantar injection of CFA; **p<0.01. b. Compared to the PWL of the control group treated with DMSO, PWL to thermal stimuli significantly increases following intrathecal injection of U0126 30 min before and once per day for 1 week after CFA injection. The symbols represent mean and vertical lines represented SD. Data were analyzed by two-way ANOVA with repeated measures followed by Bonferroni's multiple-comparison tests, *p<0.05, **p<0.01. c. 2 µg U0126 does not affect rat movement as detected by the inclined plane test before CFA administration. N = 6.</p

CFA injection increases the levels of p-ERK and p-Cdk5^S159 expression in SCDH.

<p>a. Time course of ERK1/2 activity (panel 1) and p-Cdk5^S159expression (panel 2) from 2 h to 1 week after CFA injection in rats as compared to naïve condition (lane1) in the Western blot analysis. b. Quantification of ERK1 activity in the dorsal horn. Columns represent means ± SEM. Data were normalized to naïve control and ANOVA followed by Dunnett's Multiple Comparison Test. *P<0.05, **P<0.01, as compared with naive rats. c. Quantification of ERK2 activity in the dorsal horn. d. Quantification of the levels in p-Cdk5^S159 in the dorsal horn. e. CFA injection does not affect total ERK1/2 and total Cdk5 expression in SCDH compared to naïve control (lane 1) in the Western blot analysis. N = 3.</p

Roscovitine does not affect ERK1/2 activity in SCDH induced by CFA injection.

<p>a. Effects of roscovitine on the heat hyperalgesia compare to the DMSO treatment 2-way ANOVA with repeated measures followed by Bonferroni's multiple-comparison tests. *p<0.05, **p<0.01, n = 6. b. 100 µg roscovitine does not affect rat movement as detected by the inclined plane test before CFA administration, n = 6. c. Effects of roscovitine administration on ERK1/2 activity and p-Cdk5^S159 expression after CFA in the Western blot analysis. d and e. Quantification of the Western blot shows that roscovitine treatment does not affect ERK1/2 activity by CFA, **p<0.01, n = 3. f. Quantification of the Western blot shows that the increase of p-Cdk5^S159 by CFA is attenuated after roscovitine treatment, *p<0.05, **p<0.01, n = 3. g. Roscovitine treatment does not affect total-Cdk5 expression in spinal cord 1 d after CFA injection.</p

Intra-specific genetic distance of 8 species of <i>Actinidia</i>.

Intra-specific genetic distance of 8 species of Actinidia.</p

U0126 attenuates p-Cdk5^S159 increase and Cdk5 activity in SCDH induced by CFA administration.

<p>a. U0126 mitigates CFA-induced p-ERK1/2 and p-Cdk5 increases in spinal cord in the Western blot analysis. b and c. Quantification of the Western blot shows that ERK1/2 activity by CFA is attenuated after U0126 treatment, *p<0.05, **p<0.01. d. Quantification of the Western blot shows that the increase of p-Cdk5^S159 by CFA is attenuated after U0126 treatment, **p<0.01, ***p<0.001. e. U0126 treatment does not affect total-ERK1/2 and total-Cdk5 expression in spinal cord 1 d after CFA injection. f. U0126 mitigates CFA-induced Cdk5 kinase activity in spinal cord detected by an in vitro phosphorylation assay. Histone H1 was used as a substrate. g. Quantification of Cdk5 activity (density of phosphorylated H1 band) in the SCDH. Columns represent means ± SEM for three separate experiments. h. Double-immunofluorescence staining for p-ERK (red), p-Cdk5 (green), Hoechst [a cell nuclear marker (blue)] in ipsilateral spinal enlargement at 1 day after CFA injection. P-Cdk5^S159 colocalizes with p-ERK (white arrow head).</p

Leaf image of 8 species of <i>Actinidia</i>.

Note: All pictures from Plant Photo Bank of China.</p

Genetic distances among 8 species of <i>Actinidia</i>.

Genetic distances among 8 species of Actinidia.</p

Phylogenetic tree of 8 species of <i>Actinidia</i>.

(a) Analysis of ITS sequences fragments; (b) Analysis of ITS1 sequences fragments; (c) Analysis of ITS2 sequences fragments.</p

Levels of p-ERK and p-Cdk5^S159 increase in SCDH after formalin injection.

<p>a. Time courses of p-ERK1/2 (panel 1) and t-ERK1/2 expression (panel 3) from 5 min to 1 d after formalin injection in rats as compared to naïve condition (lane1) in the Western blot analysis. b. Quantification of ERK1 activity in the dorsal horn. Columns represent means ± SEM. Data were normalized to naïve control and ANOVA followed by Dunnett's Multiple Comparison Test. *P<0.05, **P<0.01, as compared with naive rats. c. Quantification of ERK2 activity in the dorsal horn. d. Time courses of p-Cdk5^S159 (panel 1) and t-Cdk5 (panel 3) expression from 5 min to 1 d after formalin injection in rats as compared to naïve condition (lane1) in the Western blot analysis. e. Quantification of levels in p-Cdk5^S159 in the dorsal horn.</p